Performance Modelling of Computer Systems and Computer Networks

Performance Modelling of Computer Systems and Computer Networks

PERFORMANCE MODELLING OF COMPUTER SYSTEMS AND COMPUTER NETWORKS (part 1) Ramon Puigjaner Universitat de les Illes Balears Palma, Spain [email protected] Facultade de Informatica. A Corua. Junio 2005 OUTLINE INTRODUCTION CONCEPT OF QUEUE CONCEPT OF QUEUEING NETWORK NUMERICAL TECHNIQUES EXACT ANALYTICAL SOLUTIONS APPROXIMATE ANALYTICAL SOLUTIONS SIMULATION TECHNIQUES Facultade de Informatica. A Corua. Junio 2005 2 INTRODUCTION What is the performance of a Computer Network? Performance is how a software is using a hardware when they are serving a load. Facultade de Informatica. A Corua. Junio 2005 3 INTRODUCTION

This definition considers the three elements intervening in a system: The load that is externally defined. The hardware to be used. The basic software that controls the hardware. Facultade de Informatica. A Corua. Junio 2005 4 INTRODUCTION: Performance measures The performance of a Computer Network is not a unique value but a set of them to take into account the heterogeneous composition of such kind of systems. External performance measures o response time o throughput (flow through the system) o loss rate Facultade de Informatica. A Corua. Junio 2005 5 INTRODUCTION: Performance measures Internal performance measures o mean queue length o device utilisation (% of busy time) o overlap o overhead (operating system utilisation, paging, etc.) Facultade de Informatica. A Corua. Junio 2005 6

INTRODUCTION: Performance tools Measuring Monitors Logs Hardware probes Software probes Modelling Benchmarking Facultade de Informatica. A Corua. Junio 2005 7 INTRODUCTION: Performance tools Measuring Modelling Queuing networks Petri nets Markov chains Benchmarking Workload modelling Facultade de Informatica. A Corua. Junio 2005 8 INTRODUCTION: Measuring Measuring is the technique to be used when system is installed and running. It is used to verify whether the performance requirements are met or not. Facultade de Informatica. A Corua. Junio 2005

9 INTRODUCTION: Modelling A model is an abstract mathematical representation of the system behaviour in steady state. It is the appropriate technique when the computer network, partially or totally, does not exist. Main existing techniques are: Petri nets o Better suited to represent synchronisation mechanisms o Solving techniques may be either numerical (based on Markov chains) or simulation. Facultade de Informatica. A Corua. Junio 2005 10 INTRODUCTION: Modelling Queuing networks o Better suited to represent customer-server mechanisms o Solving techniques may be either analytical (closed form formulae) or numerical (based on Markov chains) or simulation. Markov chains o High abstraction level o Solving techniques are most frequently numerical. Facultade de Informatica. A Corua. Junio 2005 11 OUTLINE

INTRODUCTION CONCEPT OF QUEUE CONCEPT OF QUEUEING NETWORK NUMERICAL TECHNIQUES EXACT ANALYTICAL SOLUTIONS APPROXIMATE ANALYTICAL SOLUTIONS SIMULATION TECHNIQUES Facultade de Informatica. A Corua. Junio 2005 12 CONCEPT OF QUEUE Queue: A customer that arrives and finds the server busy joins the queue Service mechanism: It consists of one or more servers that give service to the customers from the queue Facultade de Informatica. A Corua. Junio 2005 13 CONCEPT OF QUEUE Customer source characteristics finite or infinite distribution of inter-arrival consecutive customer arrivals customer service request times between Service station characteristics queue number and capacity server number server capacity service discipline queue policy

Facultade de Informatica. A Corua. Junio 2005 14 CONCEPT OF QUEUE Single queue with single server Single queue with single server with state dependent capacity m(k ) Facultade de Informatica. A Corua. Junio 2005 15 CONCEPT OF QUEUE Single queue with multiple servers Facultade de Informatica. A Corua. Junio 2005 16 CONCEPT OF QUEUE Multi-server with no queue Facultade de Informatica. A Corua. Junio 2005 17 CONCEPT OF QUEUE Infinite server . . .

Facultade de Informatica. A Corua. Junio 2005 18 OUTLINE INTRODUCTION CONCEPT OF QUEUE CONCEPT OF QUEUEING NETWORK NUMERICAL TECHNIQUES EXACT ANALYTICAL SOLUTIONS APPROXIMATE ANALYTICAL SOLUTIONS SIMULATION TECHNIQUES Facultade de Informatica. A Corua. Junio 2005 19 CONCEPT OF QUEUEING NETWORK A queuing network is nothing else but a collection of single queues, which are arbitrarily interconnected. A queuing network is an oriented graph that has in each node a server of some type. The time in traversing the network is spent in the nodes and the arcs are traversed in a null time. Facultade de Informatica. A Corua. Junio 2005 20 CONCEPT OF QUEUEING NETWORK Queuing closed. networks may be either open or In an open queuing network, customers arrive from outside, circulate through the nodes, and finally

they depart from the network. In a closed queuing network, there is a fixed number of customers constantly circulating through the nodes. Neither departures from the network nor arrivals to the network are allowed. It is possible to have a queuing network which is both open and closed. Such a network is known as a mixed network. Facultade de Informatica. A Corua. Junio 2005 21 EXAMPLES OF OPEN QUEUING NETWORKS Tandem configuration Facultade de Informatica. A Corua. Junio 2005 22 EXAMPLES OF OPEN QUEUING NETWORKS Tree-like configuration Facultade de Informatica. A Corua. Junio 2005 23 EXAMPLES OF OPEN QUEUING NETWORKS Tree-like configuration Facultade de Informatica. A Corua. Junio 2005 24 EXAMPLES OF CLOSED QUEUING NETWORKS Cyclic network (closed tandem configuration)

Facultade de Informatica. A Corua. Junio 2005 25 EXAMPLES OF CLOSED QUEUING NETWORKS Arbitrary configuration Facultade de Informatica. A Corua. Junio 2005 26 EXAMPLES OF CLOSED QUEUING NETWORKS Central server model D is k 1 D is k 2 CPU A rriv a ls D is k 3 D is k 4 E x its Facultade de Informatica. A Corua. Junio 2005 27 EXAMPLES OF CLOSED QUEUING NETWORKS Central server model D is k 1 D is k 2 A rriv a ls

CPU D is k 3 D is k 4 E x its Facultade de Informatica. A Corua. Junio 2005 28 EXAMPLES OF CLOSED QUEUING NETWORKS Central server model D is k 1 T e rm in a ls D is k 2 A rriv a ls CPU D is k 3 D is k 4 E x its Facultade de Informatica. A Corua. Junio 2005 29 EXAMPLES OF MIXED QUEUING NETWORKS C o n v e rs a tio n a l ta s k s T e rm in a ls T ra n s a c tio n s

Facultade de Informatica. A Corua. Junio 2005 C e n tra l s y s te m 30 CONCEPT OF QUEUEING NETWORK Observations Each node can have any of the single-node characteristics described above. In order to specify the queuing network we need to provide information concerning the routing; that is to specify how a customer chooses the next node when it leaves the current node. This routing can be deterministic, probabilistic, function of the state, etc. Facultade de Informatica. A Corua. Junio 2005 31 CONCEPT OF QUEUEING NETWORK How to set-up a queuing network model? The notion of customer o Typically a customer may be a piece of software in a computer system, an information packet in a packetswitched environment, a phone call in a circuitswitched environment, etc. o Customer classes will be defined if there are differences in the resource consumption or in the routing across the network Facultade de Informatica. A Corua. Junio 2005 32

CONCEPT OF QUEUEING NETWORK How to set-up a queuing network model? The notion of node o A node is a service mechanism that may be a hardware component or a piece of software or a combination of both, e.g. a CPU, a disk, a memory module, a bus, a trunk, a switching node, etc. o Each service mechanism has a buffer (the queue), where customers wait until they are served. The buffer capacity is finite; that is, they can accommodate a finite number of customers. However, if a finite buffer has low probability of being full, then it can be assumed as infinite. Facultade de Informatica. A Corua. Junio 2005 33 CONCEPT OF QUEUEING NETWORK How to set-up a queuing network model? Collecting information o Once customers and server have been identified, it is necessary to characterise service time distributions at each node, routing probabilities and inter-arrival time distributions. o In many cases, this information can be compiled from raw data (technical information, measurements, etc.); in other cases it is based on an educated guess. Facultade de Informatica. A Corua. Junio 2005 34 CONCEPT OF QUEUEING NETWORK Solution techniques for queuing networks

To study the steady state behavior of a network the following techniques can be used: Analytic solutions Numerical techniques Simulation techniques Facultade de Informatica. A Corua. Junio 2005 35 OUTLINE INTRODUCTION CONCEPT OF QUEUE CONCEPT OF QUEUEING NETWORK NUMERICAL TECHNIQUES EXACT ANALYTICAL SOLUTIONS APPROXIMATE ANALYTICAL SOLUTIONS SIMULATION TECHNIQUES Facultade de Informatica. A Corua. Junio 2005 36 NUMERICAL TECHNIQUES The behaviour of a queuing network can be described in terms of linear equations (known as the steady-state Kolmogorov equations). These equations can be solved numerically to obtain the solution.

Facultade de Informatica. A Corua. Junio 2005 37 NUMERICAL TECHNIQUES To highlight this approach, let us consider the following two-node closed queuing network 1 2 Let us assume that: there are 5 customers in the system. 1 and 2 are the service rates. both services are exponentially distributed. Facultade de Informatica. A Corua. Junio 2005 38 NUMERICAL TECHNIQUES The state of the system is described by (n , n ), 1 2 that there are the number of customers in each queue. The numerical analysis approach involves the following steps: Generation of all feasible states. Setting-up the rate matrix.

Solving the steady state equations. Facultade de Informatica. A Corua. Junio 2005 39 NUMERICAL TECHNIQUES Generation of all feasible states. The states for our example are: (5,0) (4,1) (3,2) (2,3) (1,4) (0,5) Facultade de Informatica. A Corua. Junio 2005 40 NUMERICAL TECHNIQUES Setting-up the rate matrix. This matrix which contains all the transitions and their associated rates between each pair of states. (5,0) (4,1) (3,2) (2,3) (1,4) (0,5) (5,0) * 1 (4,1) 2 * 1 (3,2) 2 * 1 (2,3) 2 * 1 (1,4) 2 * 1 (0,5) 2 *

Facultade de Informatica. A Corua. Junio 2005 41 NUMERICAL TECHNIQUES Setting-up the rate matrix. Let us refer to the this matrix as Q. All blanks are assumed to be zero. Each diagonal element marked with * is equal to the negative sum of the off-diagonal elements of the same row. Facultade de Informatica. A Corua. Junio 2005 42 NUMERICAL TECHNIQUES Solving the steady state equations. Let p(n1, n2) be the steady-state probability that the system is in state (n1, n2) and P the row vector of these probabilities. To determine them we must solve the following system of equations: PxQ=0 together with the condition p n , n 1 1

2 n1 ,n2 Facultade de Informatica. A Corua. Junio 2005 43 NUMERICAL TECHNIQUES Solving the steady state equations. From the knowledge of these probabilities we can determine performance measures such as: o Server utilisation: r1 = p(5,0) + p(4,1) + p(3,2) + p(2,3) + p(1,4): r2 = p(4,1) + p(3,2) + p(2,3) + p(1,4) + p(0,5) o Throughputs: l1 = r1 x 1 l 2 = r 2 x 2 o Queue lengths: N1 = 5p(5,0) + 4p(4,1) + 3p(3,2) + 2p(2,3) + p(1,4) N2 = p(4,1) + 2p(3,2) + 3p(2,3) + 4p(1,4) + 5p(0,5) Facultade de Informatica. A Corua. Junio 2005 44 NUMERICAL TECHNIQUES Solving the steady state equations. Advantages/disadvantages o There are packages, like QNAP2, that automatically set-up the rate matrix Q, solve it to find the P vector and give the performance results. Other packages give the vector P if the user is able to create the matrix Q. o This numerical technique gives the exact solution. There are also approximated solutions in some cases in order to reduce the amount of computation. o The approach is limited to cases where the number of

states is not very large. o In queuing networks, quite often, the rate matrix is sparse. In this cases, one can analyse larger systems by using compact storage techniques. Facultade de Informatica. A Corua. Junio 2005 45 OUTLINE INTRODUCTION CONCEPT OF QUEUE CONCEPT OF QUEUEING NETWORK NUMERICAL TECHNIQUES EXACT ANALYTICAL SOLUTIONS APPROXIMATE ANALYTICAL SOLUTIONS SIMULATION TECHNIQUES Facultade de Informatica. A Corua. Junio 2005 46 EXACT ANALYTICAL SOLUTIONS An analytical solution means that we can obtain the probabilities of the steady steady by the application of a closed formula. This formula will obviously be a function of the parameters of the system. Quite often an analytic solution is so complicated that we can not evaluate it "on the back of an envelope". In fact, one might need to write a fairly sophisticated program. Facultade de Informatica. A Corua. Junio 2005 47 EXACT ANALYTICAL SOLUTIONS A

certain class of queuing networks has an analytic solution, known as a product-form solution because the steady state probability has the form of the product of the state probabilities of each node. Its solution can be easily evaluated. Facultade de Informatica. A Corua. Junio 2005 48 EXACT ANALYTICAL SOLUTIONS Product-form networks have been proved to be very useful in computer and communication systems performance modelling. Also, there are a lot of queuing networks which do not have product-form solutions. These networks are analysed approximately. Facultade de Informatica. A Corua. Junio 2005 49 EXACT ANALYTICAL SOLUTIONS The BCMP theorem It is the general theorem concerning queuing networks with product-form solutions. Let us consider a BCMP queuing network with: N nodes arbitrarily linked. Multiple classes of customers Probabilistic routing External arrivals with state-dependent rates Different service mechanisms Facultade de Informatica. A Corua. Junio 2005

50 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Classes of customers Customers are grouped in different classes. Each class has its own service characteristics at each node and its own routing probabilities. A class may be open or closed. Thus, a BCMP network, in its most general form, can be seen as consisting of several open classes and closed classes of customers. Facultade de Informatica. A Corua. Junio 2005 51 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Classes of customers It is possible that upon departure from a node a customer may change of class. A superclass or a subchain is the set of classes among those the customers can change. The use of classes of customers provides the modeller with a lot of modelling flexibility. Facultade de Informatica. A Corua. Junio 2005 52 EXACT ANALYTICAL SOLUTIONS

The BCMP theorem: Classes of customers Example 1. A queuing network model of a multiprogramming system D is k 1 D is k 2 A rriv a ls CPU D is k 3 D is k 4 E x its Facultade de Informatica. A Corua. Junio 2005 53 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Classes of customers Example 1. A queuing network model of a multiprogramming system The total number of customers constantly circulating through the system reflects the degree of multiprogramming. Implicitly it is assumed that when a job completes its execution and departs from the system, another job takes its place. That is, there is always at least one job waiting to get into the multiprogramming environment. This rather simplistic model captures the main features of a multiprogramming system. Facultade de Informatica. A Corua. Junio 2005 54

EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Classes of customers Example 1. A queuing network model of a multiprogramming system Let us see, how can make this model more useful by introducing classes. We can introduce different classes for different types of jobs, i. e.: o Class 1: Interactive jobs o Class 2: Short batch jobs o Class 3: Medium batch jobs o Class 4: Long batch jobs Also, in a multiprogramming system, a process originally classified in some class may be changed to another one. Facultade de Informatica. A Corua. Junio 2005 55 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Classes of customers Example 1. A queuing network model of a multiprogramming system Features as service requirements, visit rates to each node, class change, etc. are captured through the use of classes. However, the BCMP theorem is limited as it does not allow other features, such as priorities among classes. Facultade de Informatica. A Corua. Junio 2005 56

EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Classes of customers Example 2. Queuing network of a packetswitching system 2 1 4 3 Facultade de Informatica. A Corua. Junio 2005 57 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Classes of customers Example 2. Queuing network of a packetswitching system A packet is represented by a customer in the queuing network and each logical end-to-end connection is represented by a class. This allows us to assign a different routing to each class, and, if need be, different service times at each node. Facultade de Informatica. A Corua. Junio 2005 58 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Classes of customers Example 2. Queuing network of a packetswitching system So, we consider the following classes: o Class 1: packets arrive at node 1, go to node 2, then to node 3 and then they depart from the system. o Class 2: packets arrive at node 1, go to node 3, then to

node 4 and then they depart from the system. o Class 3: packets arrive at node 2, go to node 3 and then they depart from the system. o etc. Facultade de Informatica. A Corua. Junio 2005 59 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Arrival processes If we have open classes of customers, one needs to specify how these customers arrive from outside. In general, the rate of arrivals is allowed to be statedependent, i. e. it can be an arbitrary function of the number of customers in the system. Facultade de Informatica. A Corua. Junio 2005 60 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Arrival processes Single arrival stream o All external arrivals come from a single stream. When a customer arrives to the network, it joins node i as class r with probability pi,r. o The inter-arrival times must be exponentially distributed. The rate of arrivals may be constant or it may be dependent upon the total number of customers in the network. Facultade de Informatica. A Corua. Junio 2005 61

EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Arrival processes One arrival stream per class o Each open class has its own arrival stream. A new arrival of class r joins node i with probability pi. o The inter-arrival times must be exponentially distributed. The rate of class r arrivals may be constant or it may be dependent upon the total number of class r customers in the network. Facultade de Informatica. A Corua. Junio 2005 62 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Classes of customers Example 2. Queuing network of a packetswitching system Window-flow control allows only up to a prespecified number of packets in the system. Any additional packets are forced to wait in an input queue. In order to model a sliding-window flow-control scheme, we need to model the input queue. However, the BCMP theorem does not provide such features. Facultade de Informatica. A Corua. Junio 2005

63 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Classes of customers Example 2. Queuing network of a packetswitching system We can consider in some way the window-flow control by making the arrival process of customers state-dependent. That is, arrivals will occur as long as the total number of customers of some class is less than some threshold. When it becomes equal to this value, the arrival stream will be turned off. The arrival stream will start again when a customer of the considered class departs from the network. Facultade de Informatica. A Corua. Junio 2005 64 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Classes of customers Example 2. Queuing network of a packetswitching system In this way, we make sure that the total number of customers of each class does not exceed its threshold. However, this is done by introducing the erroneous assumption that no arrivals occur during the time the window is full. Facultade de Informatica. A Corua. Junio 2005 65 EXACT ANALYTICAL SOLUTIONS

The BCMP theorem: Service mechanisms Type 1 o State dependent exponentially distributed service times. o Class independent service time distribution o FIFO irrespective of classes. o Single server Type 2 o Class and state dependent Coxian distributed service times. o Processor sharing (PS) discipline (or RR, round robin) o Single servers Facultade de Informatica. A Corua. Junio 2005 66 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Service mechanisms Type 3 o Class-dependent Coxian distributed service times o Infinite servers Type 4 o Class and state dependent Coxian distributed service times o Pre-emptive server LIFO queue (PI) o Single server Facultade de Informatica. A Corua. Junio 2005 67 EXACT ANALYTICAL SOLUTIONS

The BCMP theorem: Classes of customers Example 1. A queuing network model of a multiprogramming system In this model it makes sense to assume that the CPU node is a type 2 node, i. e. customer are processor-shared, while the peripheral (disks 1 to 4) are type 1. If we assume an interactive system we would represent the terminals by means of a type 3 node. Facultade de Informatica. A Corua. Junio 2005 68 EXACT ANALYTICAL SOLUTIONS The BCMP theorem Assumption that the system reaches a steady state with state probabilities p(S) Balance equations: p S exit rate from S p S ' rate of going from S' to S S ' Normalising equation p S 1 S Facultade de Informatica. A Corua. Junio 2005 69

EXACT ANALYTICAL SOLUTIONS The BCMP theorem N 1 p S d S g i S i G i 1 G normalisation constant to obtain the addition the probabilities of all states is equal to 1. o If the system is closed, the number of states is finite and the problem is numerical o If the system is open, the number of states is infinite and the problem is analytical Facultade de Informatica. A Corua. Junio 2005 70 EXACT ANALYTICAL SOLUTIONS The BCMP theorem d(S) is a function that if the network is closed its value is 1 and if the network is open its value is M S 1 i i 0 if the arrival rate depends on M(S), K M S , Ek 1 i k

k 1 i 0 if the arrival rate to each subchain depends on M(S,Ek) Facultade de Informatica. A Corua. Junio 2005 71 EXACT ANALYTICAL SOLUTIONS The BCMP theorem The expressions of gi(Si) are o If the station is of type 1 C 1 mc 1 g i S i mi ! eic c 1 mic ! m i mc o If the station is of type 2 or 4 1 eic g i S i mi ! c 1 mic ! m ic C Facultade de Informatica. A Corua. Junio 2005

mc 72 EXACT ANALYTICAL SOLUTIONS The BCMP theorem The expressions of gi(Si) are o If the station is of type 3 1 eic g i S i c 1 mic ! m ic C Facultade de Informatica. A Corua. Junio 2005 mc 73 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Solving of a BCMP queuing network The expression that gives the probability that the system is in a precise state is quite complicated.

However it is still possible to write down the solution in form of product of terms, each term consisting of parameters related to the node. Facultade de Informatica. A Corua. Junio 2005 74 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Solving of a BCMP queuing network In the case of closed networks, as the number of states is finite, it is necessary to compute a normalising constant. o There are various algorithms to do that, such as the convolution algorithm and the mean value analysis. o These algorithms are available through various network analysers, such as QNAP2, BEST-1 and RESQ. The user simply specifies the network characteristics, and the package produces the solution. Facultade de Informatica. A Corua. Junio 2005 75 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Solving of a BCMP queuing network In the case of open networks, as the number of states is infinite, it is necessary to compute the result of a series. o This computation is only possible for specific combinations of node characteristics.

o As for closed networks, these algorithms are available through various network analysers, such as QNAP2, BEST-1 and RESQ. The user simply specifies the network characteristics, and the package produces the solution. Facultade de Informatica. A Corua. Junio 2005 76 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Case studies Transactional system The case proposed is a throughput input system with two types of transactions (messages arriving to the system and requiring some process) that have different arrival frequency, CPU consumption and profile (number of accesses to the disks), but the same mean service time to each disk (but different from disk to disk). Facultade de Informatica. A Corua. Junio 2005 77 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Case studies Transactional system We assume that these transactions are executed concurrently on the computer system and that the conflicts in the its execution are due to the access to the same servers (CPU and disks) but not to any kind of synchronisation or use of critical objects. Facultade de Informatica. A Corua. Junio 2005 78

EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Case studies Transactional system D is k 1 D is k 2 C PU A rriv a ls D is k 3 D is k 4 E x its Facultade de Informatica. A Corua. Junio 2005 79 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Transactional system 1 /DECLARE/ QUEUE CPU,DISC(4),ENTRADA1,ENTRADA2; 2 REAL PROF1(4)=(2,1.5,1,0.5); 3 REAL PROF2(4)=(1.5,2,3,3.5); 4 REAL TR1,TR2; 5 CLASS C1,C2; 6 INTEGER I; 7 /STATION/ NAME=CPU; 8 SCHED=PS; 9 SERVICE(C1)=CST(8.52); 10 SERVICE(C2)=CST(12.);

11 TRANSIT(C1)=DISC,PROF1,OUT,1; 12 TRANSIT(C2)=DISC,PROF2,OUT,1; Facultade de Informatica. A Corua. Junio 2005 80 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Transactional system 13 14 15 16 17 18 19 20 21 22 /STATION/ NAME=DISC; TRANSIT=CPU; /STATION/ NAME=DISC(1); SERVICE=EXP(23.); /STATION/ NAME=DISC(2); SERVICE=EXP(22.); /STATION/ NAME=DISC(3); SERVICE=EXP(21.); /STATION/ NAME=DISC(4); SERVICE=EXP(20.); Facultade de Informatica. A Corua. Junio 2005 81 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Transactional system 23 /STATION/ NAME=ENTRADA1; 24

TYPE=SOURCE; 25 SERVICE=EXP(1000./7.); 26 TRANSIT=CPU,C1; 27 /STATION/ NAME=ENTRADA2; 28 TYPE=SOURCE; 29 SERVICE=EXP(1000./3.); 30 TRANSIT=CPU,C2; 31 /CONTROL/ CLASS=ALL QUEUE; Facultade de Informatica. A Corua. Junio 2005 82 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Transactional system 32 /EXEC/ 33 34 35 36 37 38 39 40 41 42 43 44 45 46 BEGIN PRINT; SOLVE; TR1:=MCUSTNB(CPU,C1); TR2:=MCUSTNB(CPU,C2); FOR I:= 1 STEP 1 UNTIL 4 DO

BEGIN TR1:=TR1+MCUSTNB(DISC(I),C1); TR2:=TR2+MCUSTNB(DISC(I),C2); END; TR1:=TR1/MTHRUPUT(ENTRADA1); TR2:=TR2/MTHRUPUT(ENTRADA2); PRINT("RESPONSE TIME OF CLASS C1 =",TR1); PRINT("RESPONSE TIME OF CLASS C2 =",TR2); END; Facultade de Informatica. A Corua. Junio 2005 83 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Transactional system - CONVOLUTION METHOD ("CONVOL") ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * * * * * * * * CPU * 10.05 *0.7538 * 3.062 * 40.83 *0.7500E-01* *(C1 )* 8.520 *0.3578 * 1.454 * 34.61 *0.4200E-01* *(C2

)* 12.00 *0.3960 * 1.609 * 48.75 *0.3300E-01* * * * * * * * * DISC 1 * 23.00 *0.4255 *0.7406 * 40.03 *0.1850E-01* *(C1 )* 23.00 *0.3220 *0.5605 * 40.03 *0.1400E-01* *(C2 )* 23.00 *0.1035 *0.1802 * 40.03 *0.4500E-02* * * * * * * * * DISC 2 * 22.00 *0.3630 *0.5699 * 34.54 *0.1650E-01*

*(C1 )* 22.00 *0.2310 *0.3626 * 34.54 *0.1050E-01* *(C2 )* 22.00 *0.1320 *0.2072 * 34.54 *0.6000E-02* * * * * * * * Facultade de Informatica. A Corua. Junio 2005 84 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Transactional system ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * DISC 3 * 21.00 *0.3360 *0.5060 * 31.63 *0.1600E-01* *(C1 )* 21.00 *0.1470 *0.2214 * 31.63

*0.7000E-02* *(C2 )* 21.00 *0.1890 *0.2846 * 31.63 *0.9000E-02* * * * * * * * * DISC 4 * 20.00 *0.2800 *0.3889 * 27.78 *0.1400E-01* *(C1 )* 20.00 *0.7000E-01*0.9722E-01* 27.78 *0.3500E-02* *(C2 )* 20.00 *0.2100 *0.2917 * 27.78 *0.1050E-01* * * * * * * * * ENTRADA1 * 142.9 * 1.000 * 1.000 * 142.9 *0.7000E-02* *

* * * * * * * ENTRADA2 * 333.3 * 1.000 * 1.000 * 333.3 *0.3000E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 385.1 RESPONSE TIME OF CLASS C2 = 857.5 47 /END/ Facultade de Informatica. A Corua. Junio 2005 85 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Case studies Conversational system We assume that these programs are executed concurrently on the computer system and that the conflicts in the its execution are due to the access to the same servers (CPU and disks) but not to any kind of synchronisation or use of critical objects. Also we assume that the human behaviour in front of the terminal is different for each class. Facultade de Informatica. A Corua. Junio 2005 86 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Case studies Conversational system

D is k 1 T e rm in a ls D is k 2 A rriv a ls CPU D is k 3 D is k 4 E x its Facultade de Informatica. A Corua. Junio 2005 87 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Conversational system 1 /DECLARE/ QUEUE CPU,DISC(4),TERMINAL; 2 REAL PROB1(4)=(2,1.5,1,0.5); 3 REAL PROB2(4)=(1.5,2,3,3.5); 4 REAL TR1,TR2; 5 CLASS C1,C2; 6 INTEGER I,N; 7 /STATION/ NAME=CPU; 8 SCHED=PS; 9 SERVICE(C1)=CST(8.52); 10 SERVICE(C2)=CST(12.); 11 TRANSIT(C1)=DISC,PROB1,TERMINAL,C1,0.6,TERMINAL,C2,0 ==> .4;

12 TRANSIT(C2)=DISC,PROB2,TERMINAL,C1,0.6,TERMINAL,C2,0 ==> .4; Facultade de Informatica. A Corua. Junio 2005 88 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Conversational system 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 /STATION/ NAME=DISC; TRANSIT=CPU; /STATION/ NAME=DISC(1); SERVICE=EXP(23.); /STATION/ NAME=DISC(2); SERVICE=EXP(22.); /STATION/ NAME=DISC(3); SERVICE=EXP(21.); /STATION/ NAME=DISC(4); SERVICE=EXP(20.); /STATION/ NAME=TERMINAL; TYPE=INFINITE; INIT(C1)=N; SERVICE(C1)=EXP(30000.); SERVICE(C2)=EXP(60000.); TRANSIT=CPU;

/CONTROL/ CLASS=ALL QUEUE; Facultade de Informatica. A Corua. Junio 2005 89 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Conversational system 30 /EXEC/ 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 FOR N:=150 STEP 150 UNTIL 750 DO BEGIN PRINT; PRINT("NOMBRE DUSUARIS =",N); SOLVE; TR1:=MCUSTNB(CPU,C1); TR2:=MCUSTNB(CPU,C2); FOR I:= 1 STEP 1 UNTIL 4 DO BEGIN TR1:=TR1+MCUSTNB(DISC(I),C1); TR2:=TR2+MCUSTNB(DISC(I),C2); END; TR1:=TR1/MTHRUPUT(TERMINAL,C1); TR2:=TR2/MTHRUPUT(TERMINAL,C2);; PRINT("RESPONSE TIME OF CLASS C1 =",TR1); PRINT("RESPONSE TIME OF CLASS C2 =",TR2);

END; Facultade de Informatica. A Corua. Junio 2005 90 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Conversational system NOMBRE DUSUARIS = 150 - MEAN VALUE ANALYSIS ("MVA") ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * * * * * * * * CPU * 10.43 *0.2960 *0.4189 * 14.76 *0.2837E-01* *(C1 )* 8.520 *0.1088 *0.1539 * 12.06 *0.1277E-01* *(C2 )* 12.00 *0.1873 *0.2650 * 16.98 *0.1560E-01* * * *

* * * * * DISC 1 * 23.00 *0.1468 *0.1719 * 26.92 *0.6384E-02* *(C1 )* 23.00 *0.9790E-01*0.1146 * 26.92 *0.4257E-02* *(C2 )* 23.00 *0.4894E-01*0.5729E-01* 26.92 *0.2128E-02* * * * * * * * * DISC 2 * 22.00 *0.1326 *0.1528 * 25.33 *0.6029E-02* *(C1 )* 22.00 *0.7023E-01*0.8088E-01* 25.33 *0.3192E-02* *(C2 )* 22.00 *0.6242E-01*0.7188E-01* 25.33 *0.2837E-02* Facultade de Informatica. A Corua. Junio 2005

91 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Conversational system ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * DISC 3 * 21.00 *0.1341 *0.1546 * 24.22 *0.6384E-02* *(C1 )* 21.00 *0.4469E-01*0.5155E-01* 24.22 *0.2128E-02* *(C2 )* 21.00 *0.8937E-01*0.1031 * 24.22 *0.4256E-02* * * * * * * * * DISC 4 * 20.00 *0.1206 *0.1370 * 22.72 *0.6029E-02* *(C1 )* 20.00 *0.2128E-01*0.2418E-01* 22.72 *0.1064E-02* *(C2

)* 20.00 *0.9930E-01*0.1128 * 22.72 *0.4965E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 149.0 *0.4200E+05*0.3547E-02* *(C1 )*0.3000E+05*0.0000E+00* 63.84 *0.3000E+05*0.2128E-02* *(C2 )*0.6000E+05*0.0000E+00* 85.12 *0.6000E+05*0.1419E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 199.8 RESPONSE TIME OF CLASS C2 = 430.0 Facultade de Informatica. A Corua. Junio 2005 92 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Conversational system NOMBRE DUSUARIS = 300 - MEAN VALUE ANALYSIS ("MVA") ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * * * *

* * * * CPU * 10.43 *0.5905 * 1.426 * 25.19 *0.5659E-01* *(C1 )* 8.520 *0.2170 *0.5239 * 20.57 *0.2547E-01* *(C2 )* 12.00 *0.3735 *0.9017 * 28.97 *0.3112E-01* * * * * * * * * DISC 1 * 23.00 *0.2929 *0.4134 * 32.46 *0.1273E-01* *(C1 )* 23.00 *0.1953 *0.2756 * 32.46 *0.8490E-02* *(C2 )* 23.00 *0.9761E-01*0.1378

* 32.46 *0.4244E-02* * * * * * * * * DISC 2 * 22.00 *0.2646 *0.3592 * 29.87 *0.1203E-01* *(C1 )* 22.00 *0.1401 *0.1902 * 29.87 *0.6367E-02* *(C2 )* 22.00 *0.1245 *0.1690 * 29.87 *0.5659E-02* Facultade de Informatica. A Corua. Junio 2005 93 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Conversational system ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * DISC 3 * 21.00 *0.2674

*0.3644 * 28.62 *0.1273E-01* *(C1 )* 21.00 *0.8914E-01*0.1215 * 28.62 *0.4245E-02* *(C2 )* 21.00 *0.1783 *0.2429 * 28.62 *0.8488E-02* * * * * * * * * DISC 4 * 20.00 *0.2405 *0.3162 * 26.30 *0.1203E-01* *(C1 )* 20.00 *0.4245E-01*0.5582E-01* 26.30 *0.2122E-02* *(C2 )* 20.00 *0.1981 *0.2604 * 26.30 *0.9903E-02* * * * * * *

* * TERMINAL *0.4200E+05*0.0000E+00* 297.1 *0.4200E+05*0.7074E-02* *(C1 )*0.3000E+05*0.0000E+00* 127.3 *0.3000E+05*0.4245E-02* *(C2 )*0.6000E+05*0.0000E+00* 169.8 *0.6000E+05*0.2830E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 274.9 RESPONSE TIME OF CLASS C2 = 605.0 Facultade de Informatica. A Corua. Junio 2005 94 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Conversational system NOMBRE DUSUARIS = 450 - MEAN VALUE ANALYSIS ("MVA") ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * * * * * * * * CPU * 10.43 *0.8763 * 6.438 * 76.66 *0.8399E-01* *(C1

)* 8.520 *0.3220 * 2.366 * 62.60 *0.3780E-01* *(C2 )* 12.00 *0.5543 * 4.072 * 88.16 *0.4619E-01* * * * * * * * * DISC 1 * 23.00 *0.4347 *0.7667 * 40.57 *0.1890E-01* *(C1 )* 23.00 *0.2898 *0.5112 * 40.57 *0.1260E-01* *(C2 )* 23.00 *0.1449 *0.2555 * 40.57 *0.6298E-02* * * * * * * *

* DISC 2 * 22.00 *0.3926 *0.6451 * 36.14 *0.1785E-01* *(C1 )* 22.00 *0.2079 *0.3415 * 36.14 *0.9449E-02* *(C2 )* 22.00 *0.1848 *0.3035 Facultade de Informatica. A Corua. Junio 2005 * 36.14 *0.8398E-02* 95 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Conversational system ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * DISC 3 * 21.00 *0.3968 *0.6564 * 34.74 *0.1890E-01* *(C1 )* 21.00

*0.1323 *0.2188 * 34.74 *0.6300E-02* *(C2 )* 21.00 *0.2645 *0.4376 * 34.74 *0.1260E-01* * * * * * * * * DISC 4 * 20.00 *0.3569 *0.5541 * 31.05 *0.1785E-01* *(C1 )* 20.00 *0.6300E-01*0.9779E-01* 31.05 *0.3150E-02* *(C2 )* 20.00 *0.2939 *0.4563 * 31.05 *0.1470E-01* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 440.9 *0.4200E+05*0.1050E-01* *(C1

)*0.3000E+05*0.0000E+00* 189.0 *0.3000E+05*0.6299E-02* *(C2 )*0.6000E+05*0.0000E+00* 252.0 *0.6000E+05*0.4199E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 561.2 RESPONSE TIME OF CLASS C2 = 1316. Facultade de Informatica. A Corua. Junio 2005 96 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Conversational system NOMBRE DUSUARIS = 600 - MEAN VALUE ANALYSIS ("MVA") ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * * * * * * * * CPU * 10.43 * 1.000 * 93.51 * 975.7 *0.9584E-01* *(C1 )* 8.520 *0.3675 * 34.37 * 796.7

*0.4313E-01* *(C2 )* 12.00 *0.6325 * 59.15 * 1122. *0.5271E-01* * * * * * * * * DISC 1 * 23.00 *0.4960 *0.9841 * 45.63 *0.2157E-01* *(C1 )* 23.00 *0.3307 *0.6561 * 45.63 *0.1438E-01* *(C2 )* 23.00 *0.1653 *0.3280 * 45.63 *0.7187E-02* * * * * * * * * DISC 2 * 22.00 *0.4481 *0.8118

* 39.86 *0.2037E-01* *(C1 )* 22.00 *0.2372 *0.4298 * 39.86 *0.1078E-01* *(C2 )* 22.00 *0.2108 *0.3820 Facultade de Informatica. A Corua. Junio 2005 * 39.86 *0.9583E-02* 97 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Conversational system ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * DISC 3 * 21.00 *0.4528 *0.8276 * 38.38 *0.2156E-01* *(C1 )* 21.00 *0.1510 *0.2759 * 38.38 *0.7189E-02*

*(C2 )* 21.00 *0.3019 *0.5517 * 38.38 *0.1437E-01* * * * * * * * * DISC 4 * 20.00 *0.4073 *0.6872 * 33.74 *0.2037E-01* *(C1 )* 20.00 *0.7189E-01*0.1213 * 33.74 *0.3594E-02* *(C2 )* 20.00 *0.3354 *0.5659 * 33.74 *0.1677E-01* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 503.2 *0.4200E+05*0.1198E-01* *(C1 )*0.3000E+05*0.0000E+00* 215.7 *0.3000E+05*0.7188E-02* *(C2

)*0.6000E+05*0.0000E+00* 287.5 *0.6000E+05*0.4792E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 4987. RESPONSE TIME OF CLASS C2 = 0.1272E+05 Facultade de Informatica. A Corua. Junio 2005 98 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Conversational system NOMBRE DUSUARIS = 750 - MEAN VALUE ANALYSIS ("MVA") ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * * * * * * * * CPU * 10.43 * 1.000 * 243.5 * 2541. *0.9584E-01* *(C1 )* 8.520 *0.3675 * 89.49 * 2075. *0.4313E-01* *(C2 )* 12.00 *0.6325

* 154.0 * 2922. *0.5271E-01* * * * * * * * * DISC 1 * 23.00 *0.4960 *0.9841 * 45.64 *0.2157E-01* *(C1 )* 23.00 *0.3307 *0.6561 * 45.64 *0.1438E-01* *(C2 )* 23.00 *0.1653 *0.3280 * 45.64 *0.7187E-02* * * * * * * * * DISC 2 * 22.00 *0.4481 *0.8118 * 39.86 *0.2037E-01* *(C1 )* 22.00

*0.2372 *0.4298 * 39.86 *0.1078E-01* *(C2 )* 22.00 *0.2108 *0.3820 Facultade de Informatica. A Corua. Junio 2005 * 39.86 *0.9583E-02* 99 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Conversational system ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * DISC 3 * 21.00 *0.4528 *0.8276 * 38.38 *0.2156E-01* *(C1 )* 21.00 *0.1510 *0.2759 * 38.38 *0.7189E-02* *(C2 )* 21.00 *0.3019 *0.5517

* 38.38 *0.1437E-01* * * * * * * * * DISC 4 * 20.00 *0.4073 *0.6872 * 33.74 *0.2037E-01* *(C1 )* 20.00 *0.7189E-01*0.1213 * 33.74 *0.3594E-02* *(C2 )* 20.00 *0.3354 *0.5659 * 33.74 *0.1677E-01* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 503.2 *0.4200E+05*0.1198E-01* *(C1 )*0.3000E+05*0.0000E+00* 215.7 *0.3000E+05*0.7188E-02* *(C2 )*0.6000E+05*0.0000E+00* 287.5 *0.6000E+05*0.4792E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 0.1266E+05

RESPONSE TIME OF CLASS C2 = 0.3252E+05 47 /END/ Facultade de Informatica. A Corua. Junio 2005 100 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Case studies Communication network There are four sources of variable length messages. Routing across the net is fixed. All the lines have the same capacity and the transmission is full duplex. We consider negligible the time spent in each node for protocol verifications, routing, etc. The end-to-end traffic is known and we want to determine the response time also end-to-end. Facultade de Informatica. A Corua. Junio 2005 101 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 7 D 6 8 A 5 1 C 4

2 B Facultade de Informatica. A Corua. Junio 2005 3 102 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network to from A B C D A 75 80 100 B 60 120 150 Facultade de Informatica. A Corua. Junio 2005 C 80 50 D 100 25 40 50

103 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network A-1-2-4-3-B A-1-2-4-6-5-C A-1-2-8-7-D B-3-4-2-1-A B-3-4-6-5-C B-3-4-8-7-D C-5-6-8-2-1-A C-5-6-4-3-B C-5-6-8-7-D D-7-8-2-1-A D-7-8-4-3-B D-7-8-6-5-C Facultade de Informatica. A Corua. Junio 2005 104 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 1 /DECLARE/ QUEUE GEN,LINIA(8,8); 2 REAL TRAB,TRAC,TRAD,TRBA,TRBC,TRBD,TRCA,TRCB,TRCD, ==> TRDA,TRDB,TRDC; 3 INTEGER I; 4 CLASS CLAB,CLAC,CLAD,CLBA,CLBC,CLBD,CLCA,CLCB,CLCD ==> ,CLDA,CLDB,CLDC; 5 Facultade de Informatica. A Corua. Junio 2005 105 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 6 /STATION/ NAME = GEN;

7 TYPE = SOURCE; 8 SERVICE = EXP(60000./930.); 9 TRANSIT = LINIA(1,2),CLAB,60,LINIA(1,2),CLAC,80,L ==> INIA(1,2),CLAD,100, 10 LINIA(3,4),CLBA,75,LINIA(3,4),CLBC,50,L ==> INIA(3,4),CLBD,25, 11 LINIA(5,6),CLCA,80,LINIA(5,6),CLCB,120, ==> LINIA(5,6),CLCD,40, 12 LINIA(7,8),CLDA,100,LINIA(7,8),CLDB,150 ==> ,LINIA(7,8),CLDC,50; 13 Facultade de Informatica. A Corua. Junio 2005 106 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

/STATION/ NAME = LINIA; SERVICE = EXP(256.*8./64.); /STATION/ NAME = LINIA(1,2); TRANSIT(CLAB) = LINIA(2,4); TRANSIT(CLAC) = LINIA(2,4); TRANSIT(CLAD) = LINIA(2,8); /STATION/ NAME = LINIA(2,1); TRANSIT = OUT; /STATION/ NAME = LINIA(2,4); TRANSIT(CLAB) = LINIA(4,3); TRANSIT(CLAC) = LINIA(4,6); /STATION/ NAME = LINIA(2,8); TRANSIT(CLAD) = LINIA(8,7); Facultade de Informatica. A Corua. Junio 2005 107 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 /STATION/ NAME = LINIA(3,4); TRANSIT(CLBA) = LINIA(4,2); TRANSIT(CLBC) = LINIA(4,6);

TRANSIT(CLBD) = LINIA(4,8); /STATION/ NAME = LINIA(4,2); TRANSIT(CLBA) = LINIA(2,1); /STATION/ NAME = LINIA(4,3); TRANSIT = OUT; /STATION/ NAME = LINIA(4,6); TRANSIT(CLAC) = LINIA(6,5); TRANSIT(CLBC) = LINIA(6,5); /STATION/ NAME = LINIA(4,8); TRANSIT(CLBD) = LINIA(8,7); Facultade de Informatica. A Corua. Junio 2005 108 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 /STATION/ NAME = LINIA(5,6); TRANSIT(CLCA) = LINIA(6,8); TRANSIT(CLCB) = LINIA(6,4); TRANSIT(CLCD) = LINIA(6,8);

/STATION/ NAME = LINIA(6,4); TRANSIT(CLCB) = LINIA(4,3); /STATION/ NAME = LINIA(6,5); TRANSIT = OUT; /STATION/ NAME = LINIA(6,8); TRANSIT(CLCA) = LINIA(8,2); TRANSIT(CLCD) = LINIA(8,7); /STATION/ NAME = LINIA(7,8); TRANSIT(CLDA) = LINIA(8,2); TRANSIT(CLDB) = LINIA(8,4); TRANSIT(CLDC) = LINIA(8,6); Facultade de Informatica. A Corua. Junio 2005 109 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 70 71 72 73 74 75 76 77 78 79 80 81 82 /STATION/ NAME = LINIA(8,2); TRANSIT(CLCA) = LINIA(2,1); TRANSIT(CLDA) = LINIA(2,1); /STATION/ NAME = LINIA(8,4); TRANSIT(CLDB) = LINIA(4,3); /STATION/ NAME = LINIA(8,6); TRANSIT(CLDC) = LINIA(6,5); /STATION/ NAME = LINIA(8,7); TRANSIT = OUT;

Facultade de Informatica. A Corua. Junio 2005 110 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 83 /EXEC/ 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 BEGIN NETWORK(GEN, LINIA(1,2),LINIA(2,1),LINIA(2,4),LINIA(2,8), LINIA(3,4),LINIA(4,2),LINIA(4,3),LINIA(4,6), LINIA(4,8),LINIA(5,6),LINIA(6,4),LINIA(6,5), LINIA(6,8),LINIA(7,8),LINIA(8,2),LINIA(8,4), LINIA(8,6),LINIA(8,7)); PRINT; SOLVE; TRAB:=MRESPONSE(LINIA(1,2))+ MRESPONSE(LINIA(2,4))+MRESPONSE(LINIA(4,3)); TRAC:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,4))+ MRESPONSE(LINIA(4,6))+MRESPONSE(LINIA(6,5)); TRAD:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,8))+ MRESPONSE(LINIA(8,7)); TRBA:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,2))+ MRESPONSE(LINIA(2,1));

Facultade de Informatica. A Corua. Junio 2005 111 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 TRBC:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,6))+ MRESPONSE(LINIA(6,5)); TRBD:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,8))+ MRESPONSE(LINIA(8,7)); TRCA:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+ MRESPONSE(LINIA(8,2))+MRESPONSE(LINIA(2,1)); TRCB:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,4))+ MRESPONSE(LINIA(4,3)); TRCD:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+ MRESPONSE(LINIA(8,7)); TRDA:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,2))+ MRESPONSE(LINIA(2,1)); TRDB:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,4))+ MRESPONSE(LINIA(4,3)); TRDC:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,6))+ MRESPONSE(LINIA(6,5)); Facultade de Informatica. A Corua. Junio 2005

112 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 116 117 118 119 120 PRINT(TRAB,TRAC,TRAD); PRINT(TRBA,TRBC,TRBD); PRINT(TRCA,TRCB,TRCD); PRINT(TRDA,TRDB,TRDC); END; Facultade de Informatica. A Corua. Junio 2005 113 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network - CONVOLUTION METHOD ("CONVOL") ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * * * * * * * * GEN * 64.52 * 1.000 * 1.000 * 64.52 *0.1550E-01* * *

* * * * * * LINIA 2 * 32.00 *0.1280 *0.1468 * 36.70 *0.4000E-02* * * * * * * * * LINIA 9 * 32.00 *0.1360 *0.1574 * 37.04 *0.4250E-02* * * * * * * * * LINIA 12 * 32.00 *0.7467E-01*0.8069E-01* 34.58 *0.2333E-02* * * * * * * * * LINIA 16 * 32.00 *0.5333E-01*0.5634E-01* 33.80 *0.1667E-02* *

* * * * * * * LINIA 20 * 32.00 *0.8000E-01*0.8696E-01* 34.78 *0.2500E-02* * * * * * * * * LINIA 26 * 32.00 *0.4000E-01*0.4167E-01* 33.33 *0.1250E-02* * * * * * * * Facultade de Informatica. A Corua. Junio 2005 114 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * LINIA 27 * 32.00 *0.1760 *0.2136 * 38.83 *0.5500E-02*

* * * * * * * * LINIA 30 * 32.00 *0.6933E-01*0.7450E-01* 34.38 *0.2167E-02* * * * * * * * * LINIA 32 * 32.00 *0.1333E-01*0.1351E-01* 32.43 *0.4167E-03* * * * * * * * * LINIA 38 * 32.00 *0.1280 *0.1468 * 36.70 *0.4000E-02* * * * * * * * * LINIA 44 * 32.00 *0.6400E-01*0.6838E-01* 34.19 *0.2000E-02* *

* * * * * * * LINIA 45 * 32.00 *0.9600E-01*0.1062 * 35.40 *0.3000E-02* * * * * * * * * LINIA 48 * 32.00 *0.6400E-01*0.6838E-01* 34.19 *0.2000E-02* * * * * * * * * LINIA 56 * 32.00 *0.1600 *0.1905 * 38.10 *0.5000E-02* * * * * * * * * LINIA 58 * 32.00 *0.9600E-01*0.1062 * 35.40 *0.3000E-02*

* * * * * * * Facultade de Informatica. A Corua. Junio 2005 115 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * LINIA 60 * 32.00 *0.8000E-01*0.8696E-01* 34.78 *0.2500E-02* * * * * * * * * LINIA 62 * 32.00 *0.2667E-01*0.2740E-01* 32.88 *0.8333E-03* * * * * * * * * LINIA 63 * 32.00 *0.8800E-01*0.9649E-01* 35.09 *0.2750E-02* *

* * * * * * ******************************************************************* 110.1 105.2 143.3 110.5 141.1 104.6 109.7 111.7 105.6 102.3 106.0 106.4 Facultade de Informatica. A Corua. Junio 2005 116 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Case studies Communication network considering the nodes This case is identical to the previous one but considering the process at each node. Facultade de Informatica. A Corua. Junio 2005 117 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 1 /DECLARE/ QUEUE GEN,LINIA(8,8),CPU(8);

2 REAL TRAB,TRAC,TRAD,TRBA,TRBC,TRBD,TRCA,TRCB,TRCD,TRD ==> A,TRDB,TRDC; 3 INTEGER I; 4 CLASS CLAB,CLAC,CLAD,CLBA,CLBC,CLBD,CLCA,CLCB,CLCD,CL ==> DA,CLDB,CLDC; 5 6 /STATION/ NAME = GEN; 7 TYPE = SOURCE; 8 SERVICE = EXP(60000./930.); 9 TRANSIT = CPU(1),CLAB,60,CPU(1),CLAC,80,CPU(1),CLAD,1 ==> 00, 10 CPU(3),CLBA,75,CPU(3),CLBC,50,CPU(3),CLBD,2 ==> 5, 11 CPU(5),CLCA,80,CPU(5),CLCB,120,CPU(5),CLCD, ==> 40, 12 CPU(7),CLDA,100,CPU(7),CLDB,150,CPU(7),CLDC ==> ,50, 13 Facultade de Informatica. A Corua. Junio 2005 118 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 14 15 16 17 18 19 20 21

22 23 24 25 26 27 28 29 30 31 /STATION/ NAME = LINIA; SERVICE = EXP(256.*8./64.); /STATION/ NAME = LINIA(1,2); TRANSIT = CPU(2); /STATION/ NAME = LINIA(2,1); TRANSIT = CPU(1); /STATION/ NAME = LINIA(2,4); TRANSIT = CPU(4); /STATION/ NAME = LINIA(2,8); TRANSIT = CPU(8); /STATION/ NAME = LINIA(3,4); TRANSIT = CPU(4); Facultade de Informatica. A Corua. Junio 2005 119 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 32 33 34 35 36 37 38 39 40 41 42 43

44 45 46 47 48 49 /STATION/ NAME = LINIA(4,2); TRANSIT = CPU(2); /STATION/ NAME = LINIA(4,3); TRANSIT = CPU(3); /STATION/ NAME = LINIA(4,6); TRANSIT = CPU(6); /STATION/ NAME = LINIA(4,8); TRANSIT = CPU(8); /STATION/ NAME = LINIA(5,6); TRANSIT = CPU(6); /STATION/ NAME = LINIA(6,5); TRANSIT = CPU(5); Facultade de Informatica. A Corua. Junio 2005 120 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

66 67 /STATION/ NAME = LINIA(6,4); TRANSIT = CPU(4); /STATION/ NAME = LINIA(6,8); TRANSIT = CPU(8); /STATION/ NAME = LINIA(7,8); TRANSIT = CPU(8); /STATION/ NAME = LINIA(8,2); TRANSIT = CPU(2); /STATION/ NAME = LINIA(8,7); TRANSIT = CPU(7); /STATION/ NAME = LINIA(8,6); TRANSIT = CPU(6); Facultade de Informatica. A Corua. Junio 2005 121 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 /STATION/ NAME = LINIA(8,4);

TRANSIT = CPU(4); /STATION/ NAME = CPU; SERVICE = EXP(1.); /STATION/ NAME = CPU(1); TRANSIT(CLAB,CLAC,CLAD) = LINIA(1,2); TRANSIT(CLBA,CLCA,CLDA) = OUT; /STATION/ NAME = CPU(2); TRANSIT(CLAB) = TRANSIT(CLAC) = TRANSIT(CLAD) = TRANSIT(CLBA) = TRANSIT(CLCA) = TRANSIT(CLDA) = LINIA(2,4); LINIA(2,4); LINIA(2,8); LINIA(2,1); LINIA(2,1); LINIA(2,1); Facultade de Informatica. A Corua. Junio 2005 122 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 86 /STATION/ NAME = CPU(3); 87 TRANSIT(CLBA,CLBC,CLBD) = LINIA(3,4); 88 TRANSIT(CLAB,CLCB,CLDB) = OUT; 89 90 /STATION/ NAME = CPU(4); 91 TRANSIT(CLAB) = LINIA(4,3); 92 TRANSIT(CLAC) = LINIA(4,6); 93 TRANSIT(CLBA) = LINIA(4,2); 94 TRANSIT(CLBC) = LINIA(4,6);

95 TRANSIT(CLBD) = LINIA(4,8); 96 TRANSIT(CLCB) = LINIA(4,3); 97 TRANSIT(CLDB) = LINIA(4,3); 98 99 /STATION/ NAME = CPU(5); 100 TRANSIT(CLCA,CLCB,CLCD) = LINIA(5,6); 101 TRANSIT(CLAC,CLBC,CLDC) = OUT; 102 Facultade de Informatica. A Corua. Junio 2005 123 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 103 /STATION/ NAME = CPU(6); 104 TRANSIT(CLAC) = LINIA(6,5); 105 TRANSIT(CLBC) = LINIA(6,5); 106 TRANSIT(CLCA) = LINIA(6,8); 107 TRANSIT(CLCB) = LINIA(6,4); 108 TRANSIT(CLCD) = LINIA(6,8); 109 TRANSIT(CLDC) = LINIA(6,5); 110 111 /STATION/ NAME = CPU(7); 112 TRANSIT(CLDA,CLDB,CLDC) = LINIA(7,8); 113 TRANSIT(CLAD,CLBD,CLCD) = OUT; 114 115 /STATION/ NAME = CPU(8); 116

TRANSIT(CLAD) = LINIA(8,7); 117 TRANSIT(CLBD) = LINIA(8,7); 118 TRANSIT(CLCA) = LINIA(8,2); 119 TRANSIT(CLCD) = LINIA(8,7); 120 TRANSIT(CLDA) = LINIA(8,2); 121 TRANSIT(CLDB) = LINIA(8,4); 122 TRANSIT(CLDC) = LINIA(8,6); 123 Facultade de Informatica. A Corua. Junio 2005 124 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 124 /CONTROL/ CLASS = ALL QUEUE; 125 /EXEC/ BEGIN 126 NETWORK(GEN, 127 CPU(1 STEP 1 UNTIL 8), 128 LINIA(1,2),LINIA(2,1),LINIA(2,4),LINIA(2,8), 129 LINIA(3,4),LINIA(4,2),LINIA(4,3),LINIA(4,6), 130 LINIA(4,8),LINIA(5,6),LINIA(6,4),LINIA(6,5), 131 LINIA(6,8),LINIA(7,8),LINIA(8,2),LINIA(8,4), 132 LINIA(8,6),LINIA(8,7)); 133 PRINT; 134 SOLVE; 135

TRAB := MRESPONSE(CPU(1))+MRESPONSE(LINIA(1,2))+ 136 MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,4))+ 137 MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,3))+ 138 MRESPONSE(CPU(3)); 139 TRAC := MRESPONSE(CPU(1))+MRESPONSE(LINIA(1,2))+ 140 MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,4))+ 141 MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,6))+ 142 MRESPONSE(CPU(6))+MRESPONSE(LINIA(6,5))+ 143 MRESPONSE(CPU(5)); Facultade de Informatica. A Corua. Junio 2005 125 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161

162 163 164 TRAD := MRESPONSE(CPU(1))+MRESPONSE(LINIA(1,2))+ MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,8))+ MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,7))+ MRESPONSE(CPU(7)); TRBA := MRESPONSE(CPU(3))+MRESPONSE(LINIA(3,4))+ MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,2))+ MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,1))+ MRESPONSE(CPU(1)); TRBC := MRESPONSE(CPU(3))+MRESPONSE(LINIA(3,4))+ MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,6))+ MRESPONSE(CPU(6))+MRESPONSE(LINIA(6,5))+ MRESPONSE(CPU(5)); TRBD := MRESPONSE(CPU(3))+MRESPONSE(LINIA(3,4))+ MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,8))+ MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,7))+ MRESPONSE(CPU(7)); TRCA := MRESPONSE(CPU(5))+MRESPONSE(LINIA(5,6))+ MRESPONSE(CPU(6))+MRESPONSE(LINIA(6,8))+ MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,2))+ MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,1))+ MRESPONSE(CPU(1)); Facultade de Informatica. A Corua. Junio 2005 126 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 165 166 167 168 169 170 171 172 173 174

175 176 177 178 179 180 TRCB := MRESPONSE(CPU(5))+MRESPONSE(LINIA(5,6))+ MRESPONSE(CPU(6))+MRESPONSE(LINIA(6,4))+ MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,3))+ MRESPONSE(CPU(3)); TRCD := MRESPONSE(CPU(5))+MRESPONSE(LINIA(5,6))+ MRESPONSE(CPU(6))+MRESPONSE(LINIA(6,8))+ MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,7))+ MRESPONSE(CPU(7)); TRDA := MRESPONSE(CPU(7))+MRESPONSE(LINIA(7,8))+ MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,2))+ MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,1))+ MRESPONSE(CPU(1)); TRDB := MRESPONSE(CPU(7))+MRESPONSE(LINIA(7,8))+ MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,4))+ MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,3))+ MRESPONSE(CPU(3)); Facultade de Informatica. A Corua. Junio 2005 127 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network 181 TRDC := +MRESPONSE(LINIA(7,8))+ 182 +MRESPONSE(LINIA(8,6))+ 183 +MRESPONSE(LINIA(6,5))+ 184 MRESPONSE(CPU(5)); 185 PRINT(TRAB,TRAC,TRAD); 186

PRINT(TRBA,TRBC,TRBD); 187 PRINT(TRCA,TRCB,TRCD); 188 PRINT(TRDA,TRDB,TRDC); 189 END; Facultade de Informatica. A Corua. Junio 2005 MRESPONSE(CPU(7)) MRESPONSE(CPU(8)) MRESPONSE(CPU(6)) 128 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network - CONVOLUTION METHOD ("CONVOL") ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * GEN * 64.52 * 1.000 * 1.000 * 64.52 *0.1550E-01* * * * * * * * * LINIA 2 * 32.00 *0.1280 *0.1468 * 36.70 *0.4000E-02* *(CLAB )* 32.00

*0.3200E-01*0.3670E-01* 36.70 *0.1000E-02* *(CLAC )* 32.00 *0.4267E-01*0.4893E-01* 36.70 *0.1333E-02* *(CLAD )* 32.00 *0.5333E-01*0.6116E-01* 36.70 *0.1667E-02* * * * * * * * * LINIA 9 * 32.00 *0.1360 *0.1574 * 37.04 *0.4250E-02* *(CLBA )* 32.00 *0.4000E-01*0.4630E-01* 37.04 *0.1250E-02* *(CLCA )* 32.00 *0.4267E-01*0.4938E-01* 37.04 *0.1333E-02* *(CLDA )* 32.00 *0.5333E-01*0.6173E-01* 37.04 *0.1667E-02* * * * * * * * * LINIA 12 * 32.00 *0.7467E-01*0.8069E-01* 34.58

*0.2333E-02* *(CLAB )* 32.00 *0.3200E-01*0.3458E-01* 34.58 *0.1000E-02* *(CLAC )* 32.00 *0.4267E-01*0.4611E-01* 34.58 *0.1333E-02* * * * * * * * Facultade de Informatica. A Corua. Junio 2005 129 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * LINIA 16 * 32.00 *0.5333E-01*0.5634E-01* 33.80 *0.1667E-02* *(CLAD )* 32.00 *0.5333E-01*0.5634E-01* 33.80 *0.1667E-02* * * * * * * * * LINIA 20 * 32.00

*0.8000E-01*0.8696E-01* 34.78 *0.2500E-02* *(CLBA )* 32.00 *0.4000E-01*0.4348E-01* 34.78 *0.1250E-02* *(CLBC )* 32.00 *0.2667E-01*0.2899E-01* 34.78 *0.8333E-03* *(CLBD )* 32.00 *0.1333E-01*0.1449E-01* 34.78 *0.4167E-03* * * * * * * * * LINIA 26 * 32.00 *0.4000E-01*0.4167E-01* 33.33 *0.1250E-02* *(CLBA )* 32.00 *0.4000E-01*0.4167E-01* 33.33 *0.1250E-02* * * * * * * * * LINIA 27 * 32.00 *0.1760 *0.2136 * 38.83 *0.5500E-02* *(CLAB )* 32.00 *0.3200E-01*0.3883E-01* 38.83

*0.1000E-02* *(CLCB )* 32.00 *0.6400E-01*0.7767E-01* 38.83 *0.2000E-02* *(CLDB )* 32.00 *0.8000E-01*0.9709E-01* 38.83 *0.2500E-02* * * * * * * * Facultade de Informatica. A Corua. Junio 2005 130 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * LINIA 30 * 32.00 *0.6933E-01*0.7450E-01* 34.38 *0.2167E-02* *(CLAC )* 32.00 *0.4267E-01*0.4585E-01* 34.38 *0.1333E-02* *(CLBC )* 32.00 *0.2667E-01*0.2865E-01* 34.38 *0.8333E-03* * * * *

* * * * LINIA 32 * 32.00 *0.1333E-01*0.1351E-01* 32.43 *0.4167E-03* *(CLBD )* 32.00 *0.1333E-01*0.1351E-01* 32.43 *0.4167E-03* * * * * * * * * LINIA 38 * 32.00 *0.1280 *0.1468 * 36.70 *0.4000E-02* *(CLCA )* 32.00 *0.4267E-01*0.4893E-01* 36.70 *0.1333E-02* *(CLCB )* 32.00 *0.6400E-01*0.7339E-01* 36.70 *0.2000E-02* *(CLCD )* 32.00 *0.2133E-01*0.2446E-01* 36.70 *0.6667E-03* * * * * * * * * LINIA 44 * 32.00 *0.6400E-01*0.6838E-01* 34.19

*0.2000E-02* *(CLCB )* 32.00 *0.6400E-01*0.6838E-01* 34.19 *0.2000E-02* * * * * * * * Facultade de Informatica. A Corua. Junio 2005 131 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * LINIA 45 * 32.00 *0.9600E-01*0.1062 * 35.40 *0.3000E-02* *(CLAC )* 32.00 *0.4267E-01*0.4720E-01* 35.40 *0.1333E-02* *(CLBC )* 32.00 *0.2667E-01*0.2950E-01* 35.40 *0.8333E-03* *(CLDC )* 32.00 *0.2667E-01*0.2950E-01* 35.40 *0.8333E-03* * *

* * * * * * LINIA 48 * 32.00 *0.6400E-01*0.6838E-01* 34.19 *0.2000E-02* *(CLCA )* 32.00 *0.4267E-01*0.4558E-01* 34.19 *0.1333E-02* *(CLCD )* 32.00 *0.2133E-01*0.2279E-01* 34.19 *0.6667E-03* * * * * * * * * LINIA 56 * 32.00 *0.1600 *0.1905 * 38.10 *0.5000E-02* *(CLDA )* 32.00 *0.5333E-01*0.6349E-01* 38.10 *0.1667E-02* *(CLDB )* 32.00 *0.8000E-01*0.9524E-01* 38.10 *0.2500E-02* *(CLDC )* 32.00 *0.2667E-01*0.3175E-01* 38.10 *0.8333E-03* * * *

* * * * * LINIA 58 * 32.00 *0.9600E-01*0.1062 * 35.40 *0.3000E-02* *(CLCA )* 32.00 *0.4267E-01*0.4720E-01* 35.40 *0.1333E-02* *(CLDA )* 32.00 *0.5333E-01*0.5900E-01* 35.40 *0.1667E-02* * * * * * * * Facultade de Informatica. A Corua. Junio 2005 132 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * LINIA 60 * 32.00 *0.8000E-01*0.8696E-01* 34.78 *0.2500E-02* *(CLDB )* 32.00 *0.8000E-01*0.8696E-01* 34.78 *0.2500E-02*

* * * * * * * * LINIA 62 * 32.00 *0.2667E-01*0.2740E-01* 32.88 *0.8333E-03* *(CLDC )* 32.00 *0.2667E-01*0.2740E-01* 32.88 *0.8333E-03* * * * * * * * * LINIA 63 * 32.00 *0.8800E-01*0.9649E-01* 35.09 *0.2750E-02* *(CLAD )* 32.00 *0.5333E-01*0.5848E-01* 35.09 *0.1667E-02* *(CLBD )* 32.00 *0.1333E-01*0.1462E-01* 35.09 *0.4167E-03* *(CLCD )* 32.00 *0.2133E-01*0.2339E-01* 35.09 *0.6667E-03* * * * * * * *

* CPU 1 * 1.000 *0.8250E-02*0.8319E-02* 1.008 *0.8250E-02* *(CLAB )* 1.000 *0.1000E-02*0.1008E-02* 1.008 *0.1000E-02* *(CLAC )* 1.000 *0.1333E-02*0.1344E-02* 1.008 *0.1333E-02* *(CLAD )* 1.000 *0.1667E-02*0.1681E-02* 1.008 *0.1667E-02* *(CLBA )* 1.000 *0.1250E-02*0.1260E-02* 1.008 *0.1250E-02* *(CLCA )* 1.000 *0.1333E-02*0.1344E-02* 1.008 *0.1333E-02* *(CLDA )* 1.000 *0.1667E-02*0.1681E-02* 1.008 *0.1667E-02* Facultade de Informatica. A Corua. Junio 2005 133 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU 2 * 1.000

*0.8250E-02*0.8319E-02* 1.008 *0.8250E-02* *(CLAB )* 1.000 *0.1000E-02*0.1008E-02* 1.008 *0.1000E-02* *(CLAC )* 1.000 *0.1333E-02*0.1344E-02* 1.008 *0.1333E-02* *(CLAD )* 1.000 *0.1667E-02*0.1681E-02* 1.008 *0.1667E-02* *(CLBA )* 1.000 *0.1250E-02*0.1260E-02* 1.008 *0.1250E-02* *(CLCA )* 1.000 *0.1333E-02*0.1344E-02* 1.008 *0.1333E-02* *(CLDA )* 1.000 *0.1667E-02*0.1681E-02* 1.008 *0.1667E-02* * * * * * * * * * * * * * * * CPU 3 * 1.000 *0.8000E-02*0.8065E-02* 1.008

*0.8000E-02* *(CLAB )* 1.000 *0.1000E-02*0.1008E-02* 1.008 *0.1000E-02* *(CLBA )* 1.000 *0.1250E-02*0.1260E-02* 1.008 *0.1250E-02* *(CLBC )* 1.000 *0.8333E-03*0.8401E-03* 1.008 *0.8333E-03* *(CLBD )* 1.000 *0.4167E-03*0.4200E-03* 1.008 *0.4167E-03* *(CLCB )* 1.000 *0.2000E-02*0.2016E-02* 1.008 *0.2000E-02* *(CLDB )* 1.000 *0.2500E-02*0.2520E-02* 1.008 *0.2500E-02* Facultade de Informatica. A Corua. Junio 2005 134 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU 4 * 1.000 *0.9333E-02*0.9421E-02* 1.009 *0.9333E-02* *(CLAB )* 1.000

*0.1000E-02*0.1009E-02* 1.009 *0.1000E-02* *(CLAC )* 1.000 *0.1333E-02*0.1346E-02* 1.009 *0.1333E-02* *(CLBA )* 1.000 *0.1250E-02*0.1262E-02* 1.009 *0.1250E-02* *(CLBC )* 1.000 *0.8333E-03*0.8412E-03* 1.009 *0.8333E-03* *(CLBD )* 1.000 *0.4167E-03*0.4206E-03* 1.009 *0.4167E-03* *(CLCB )* 1.000 *0.2000E-02*0.2019E-02* 1.009 *0.2000E-02* *(CLDB )* 1.000 *0.2500E-02*0.2524E-02* 1.009 *0.2500E-02* * * * * * * * * CPU 5 * 1.000 *0.7000E-02*0.7049E-02* 1.007 *0.7000E-02* *(CLAC )* 1.000 *0.1333E-02*0.1343E-02* 1.007 *0.1333E-02* *(CLBC )* 1.000

*0.8333E-03*0.8392E-03* 1.007 *0.8333E-03* *(CLCA )* 1.000 *0.1333E-02*0.1343E-02* 1.007 *0.1333E-02* *(CLCB )* 1.000 *0.2000E-02*0.2014E-02* 1.007 *0.2000E-02* *(CLCD )* 1.000 *0.6667E-03*0.6714E-03* 1.007 *0.6667E-03* *(CLDC )* 1.000 *0.8333E-03*0.8392E-03* 1.007 *0.8333E-03* Facultade de Informatica. A Corua. Junio 2005 135 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU 6 * 1.000 *0.7000E-02*0.7049E-02* 1.007 *0.7000E-02* *(CLAC )* 1.000 *0.1333E-02*0.1343E-02* 1.007 *0.1333E-02* *(CLBC )* 1.000 *0.8333E-03*0.8392E-03* 1.007 *0.8333E-03* *(CLCA

)* 1.000 *0.1333E-02*0.1343E-02* 1.007 *0.1333E-02* *(CLCB )* 1.000 *0.2000E-02*0.2014E-02* 1.007 *0.2000E-02* *(CLCD )* 1.000 *0.6667E-03*0.6714E-03* 1.007 *0.6667E-03* *(CLDC )* 1.000 *0.8333E-03*0.8392E-03* 1.007 *0.8333E-03* * * * * * * * * CPU 7 * 1.000 *0.7750E-02*0.7811E-02* 1.008 *0.7750E-02* *(CLAD )* 1.000 *0.1667E-02*0.1680E-02* 1.008 *0.1667E-02* *(CLBD )* 1.000 *0.4167E-03*0.4199E-03* 1.008 *0.4167E-03* *(CLCD )* 1.000 *0.6667E-03*0.6719E-03* 1.008 *0.6667E-03* *(CLDA )* 1.000 *0.1667E-02*0.1680E-02* 1.008 *0.1667E-02* *(CLDB

)* 1.000 *0.2500E-02*0.2520E-02* 1.008 *0.2500E-02* *(CLDC )* 1.000 *0.8333E-03*0.8398E-03* 1.008 *0.8333E-03* Facultade de Informatica. A Corua. Junio 2005 136 EXACT ANALYTICAL SOLUTIONS The BCMP theorem: Communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU 8 * 1.000 *0.9083E-02*0.9167E-02* 1.009 *0.9083E-02* *(CLAD )* 1.000 *0.1667E-02*0.1682E-02* 1.009 *0.1667E-02* *(CLBD )* 1.000 *0.4167E-03*0.4205E-03* 1.009 *0.4167E-03* *(CLCA )* 1.000 *0.1333E-02*0.1346E-02* 1.009 *0.1333E-02* *(CLCD )* 1.000 *0.6667E-03*0.6728E-03* 1.009 *0.6667E-03* *(CLDA )* 1.000 *0.1667E-02*0.1682E-02* 1.009 *0.1667E-02*

*(CLDB )* 1.000 *0.2500E-02*0.2523E-02* 1.009 *0.2500E-02* *(CLDC )* 1.000 *0.8333E-03*0.8410E-03* 1.009 *0.8333E-03* ******************************************************************* 114.1 109.2 148.4 114.6 146.1 108.6 113.8 115.7 109.6 106.3 110.0 110.4 190 191 /END/ Facultade de Informatica. A Corua. Junio 2005 137 OUTLINE INTRODUCTION CONCEPT OF QUEUE CONCEPT OF QUEUEING NETWORK NUMERICAL TECHNIQUES EXACT ANALYTICAL SOLUTIONS APPROXIMATE ANALYTICAL SOLUTIONS SIMULATION TECHNIQUES Facultade de Informatica. A Corua. Junio 2005

138 APPROXIMATE ANALYTICAL SOLUTIONS Approximate solutions for queuing networks are useful for tackling problems not covered by the product-form solutions as, for instance, priorities class dependent or non-exponential service times at FIFO stations finite buffers simultaneous resource possession Facultade de Informatica. A Corua. Junio 2005 139 APPROXIMATE ANALYTICAL SOLUTIONS There are a lot of different types of approximations in the literature. Basically, they can be grouped in the following classes of methods: decomposition-aggregation diffusion iterative Facultade de Informatica. A Corua. Junio 2005 140 APPROXIMATE ANALYTICAL SOLUTIONS

Decomposition-aggregation methods The idea behind decomposition is to break up the queuing network into smaller subsystems, so that each subsystem can be easily analysed in isolation, and then put together these partial solutions, in order to obtain the solution of the queuing network. What is difficult to do is to find a good way to decompose the network under study into smaller more manageable subsystems and then put the solution together for the whole network. Facultade de Informatica. A Corua. Junio 2005 141 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation methods Various decomposition procedures have been developed specifically for analysing queuing networks with particular features. Criteria for decomposing can be: o Parts with different timing behaviour o Norton theorem: inspired in the Norton theorem for electric circuits Facultade de Informatica. A Corua. Junio 2005 142 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Norton theorem

BCMP network with a non-BCMP node m2 m1 m4 m3 Facultade de Informatica. A Corua. Junio 2005 143 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Norton theorem BCMP network with a non-BCMP node o Let us assume that one node violate the BCMP assumptions. In our particular case, node 3 is the culprit. o Let us assume that we know all the parameters of the network, i. e. the service times, routing probabilities, etc. Also, for simplicity, we assume a single class of customers. Let N be the total number of customers in it. Facultade de Informatica. A Corua. Junio 2005 144 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Norton theorem Decomposition step o "Short-out" node 3 and analyse the queuing network as if this node did not exist. By "short-out" we mean simply that node 3 is removed without changing the incoming and outgoing flows. o Now, the resulting network is a BCMP network and it can be easily solved. We study this network to compute the throughput along the "short".

o We do this computation assuming that there are k customers in the "shorted" network, where k = 1, 2, ..., N. Let (k) be the throughput along the "short" when there are k customers in it. Facultade de Informatica. A Corua. Junio 2005 145 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Norton theorem m2 m1 m4 (k ) Facultade de Informatica. A Corua. Junio 2005 146 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Norton theorem Aggregation step o Now, the original network can be reduced to node 3 and another node, known as the composite node, which represents approximately the shorted-out network. The service rate of the composite node is (k), where k is the number of customers in the composite node (statedependent service). The total number of customers in the network is still N. Facultade de Informatica. A Corua. Junio 2005 147 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Norton theorem (k)

m3 Facultade de Informatica. A Corua. Junio 2005 148 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Norton theorem Aggregation step o In general, this network is not a BCMP network because of node 3. So, we have to be able to analyse this aggregate system by either a numerical approach, or another approximate method or a simulation method. However, in any case as the system has only two nodes its analysis will be easier. o Let us assume that we can obtain the queue length distribution of nodes 3 and composite. Then, the queue length distribution of node 3 is an approximation to the queue length distribution of node 3 in the original network. Facultade de Informatica. A Corua. Junio 2005 149 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Norton theorem Aggregation step o What about the queue length distributions of nodes 1, 2 and 4? They can be obtained by combining the results obtained in step 1 with the queue length distribution of the composite node obtained in step 2. To show how to obtain them, let us assume that we want to obtain the queue length of node 1. Facultade de Informatica. A Corua. Junio 2005 150

APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Norton theorem Aggregation step o Let pc(k) be the probability that there are n customers in the composite node, where k = 1, 2, ..., N, and q1(n|k) be the probability that there are n customers in the node 1 when there are k customers in the shorted network. Then N q n | k p k , for n 1, 2, ..., N q1 n 1 c k 1 o It is also possible to proceed in a similar way when instead of just one non-BCMP node we have a non-BCMP sub-network Facultade de Informatica. A Corua. Junio 2005 151 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system D is k 1 D is k 2 C PU D is k 3 T e rm in a ls M e m o ry

m anagem ent Facultade de Informatica. A Corua. Junio 2005 D is k 4 152 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system D is k 1 D is k 2 C PU D is k 3 D is k 4 Facultade de Informatica. A Corua. Junio 2005 153 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system 1 /DECLARE/ QUEUE CPU,DISC(4),TERMINAL,SC; 2 REAL PROB1(4)=(2,1.5,1,0.5); 3 REAL PROB2(4)=(1.5,2,3,3.5); 4 REAL TR,CAP(20); 5 REAL TR1,TR2;

6 CLASS C1,C2; 7 INTEGER I,N,M; 8 /STATION/ NAME=CPU; 9 SCHED=PS; 10 INIT(C1)=N; 11 SERVICE(C1)=CST(8.52); 12 SERVICE(C2)=CST(12.); 13 TRANSIT(C1)=DISC,PROB1,CPU,C1,0.6,CPU,C2,0.4; 14 TRANSIT(C2)=DISC,PROB2,CPU,C1,0.6,CPU,C2,0.4; Facultade de Informatica. A Corua. Junio 2005 154 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system 15 16

17 18 19 20 21 22 23 24 /STATION/ NAME=DISC; TRANSIT=CPU; /STATION/ NAME=DISC(1); SERVICE=EXP(23.); /STATION/ NAME=DISC(2); SERVICE=EXP(22.); /STATION/ NAME=DISC(3); SERVICE=EXP(21.); /STATION/ NAME=DISC(4); SERVICE=EXP(20.); Facultade de Informatica. A Corua. Junio 2005 155 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system 25 /EXEC/ BEGIN 26 NETWORK(CPU,DISC); 27 FOR N := 1 STEP 1 UNTIL 20 DO 28 BEGIN 29 PRINT; 30 PRINT("FACTOR DE MULTIPROGRAMACIO =",N); 31 SOLVE; 32 CAP(N) := MTHRUPUT(CPU);

33 FOR I:=1 STEP 1 UNTIL 4 DO CAP(N):=CAP(N)-MTHRUP ==> UT(DISC(I)); 34 END; 35 FOR N := 1 STEP 1 UNTIL 20 DO PRINT(N,CAP(N)); 36 END; Facultade de Informatica. A Corua. Junio 2005 156 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system FACTOR DE MULTIPROGRAMACIO = 1 - MEAN VALUE ANALYSIS ("MVA") ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.3566 *0.3566 * 10.43 *0.3418E-01* * * * * * * * * DISC 1 * 23.00 *0.1769 *0.1769 * 23.00 *0.7690E-02*

* * * * * * * * DISC 2 * 22.00 *0.1598 *0.1598 * 22.00 *0.7263E-02* * * * * * * * * DISC 3 * 21.00 *0.1615 *0.1615 * 21.00 *0.7690E-02* * * * * * * * * DISC 4 * 20.00 *0.1453 *0.1453 * 20.00 *0.7263E-02* ******************************************************************* Facultade de Informatica. A Corua. Junio 2005

157 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system FACTOR DE MULTIPROGRAMACIO = 5 - MEAN VALUE ANALYSIS ("MVA") ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.8852 * 2.552 * 30.08 *0.8484E-01* * * * * * * * * DISC 1 * 23.00 *0.4390 *0.7037 * 36.87 *0.1909E-01* * * * * * * * * DISC 2 * 22.00 *0.3966 *0.6041 * 33.51

*0.1803E-01* * * * * * * * * DISC 3 * 21.00 *0.4009 *0.6137 * 32.15 *0.1909E-01* * * * * * * * * DISC 4 * 20.00 *0.3606 *0.5263 * 29.19 *0.1803E-01* ******************************************************************* Facultade de Informatica. A Corua. Junio 2005 158 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system FACTOR DE MULTIPROGRAMACIO = 10 - MEAN VALUE ANALYSIS ("MVA") ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * *******************************************************************

* CPU * 10.43 *0.9905 * 6.801 * 71.65 *0.9493E-01* * * * * * * * * DISC 1 * 23.00 *0.4913 *0.9443 * 44.21 *0.2136E-01* * * * * * * * * DISC 2 * 22.00 *0.4438 *0.7855 * 38.94 *0.2017E-01* * * * * * * * * DISC 3 * 21.00 *0.4485 *0.8003

* 37.47 *0.2136E-01* * * * * * * * * DISC 4 * 20.00 *0.4034 *0.6685 * 33.14 *0.2017E-01* ******************************************************************* Facultade de Informatica. A Corua. Junio 2005 159 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system FACTOR DE MULTIPROGRAMACIO = 15 - MEAN VALUE ANALYSIS ("MVA") ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.9995 * 11.70 * 122.1 *0.9579E-01* * * * * * *

* * DISC 1 * 23.00 *0.4957 *0.9807 * 45.50 *0.2155E-01* * * * * * * * * DISC 2 * 22.00 *0.4478 *0.8098 * 39.78 *0.2036E-01* * * * * * * * * DISC 3 * 21.00 *0.4526 *0.8256 * 38.31 *0.2155E-01* * * * * * * * * DISC 4 * 20.00 *0.4071

*0.6860 * 33.70 *0.2036E-01* ******************************************************************* Facultade de Informatica. A Corua. Junio 2005 160 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system FACTOR DE MULTIPROGRAMACIO = 20 - MEAN VALUE ANALYSIS ("MVA") ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 * 1.000 * 16.69 * 174.1 *0.9584E-01* * * * * * * * * DISC 1 * 23.00 *0.4960 *0.9838 * 45.62 *0.2156E-01* * * * * *

* * * DISC 2 * 22.00 *0.4480 *0.8117 * 39.85 *0.2037E-01* * * * * * * * * DISC 3 * 21.00 *0.4528 *0.8275 * 38.38 *0.2156E-01* * * * * * * * * DISC 4 * 20.00 *0.4073 *0.6872 * 33.74 *0.2037E-01* ******************************************************************* Facultade de Informatica. A Corua. Junio 2005 161 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system

1 0.4272E-02 11 0.1191E-01 2 0.6940E-02 12 0.1194E-01 3 0.8680E-02 13 0.1196E-01 4 0.9835E-02 14 0.1197E-01 5 0.1060E-01 15 0.1197E-01 6 0.1111E-01

16 0.1198E-01 7 0.1144E-01 17 0.1198E-01 8 0.1165E-01 18 0.1198E-01 9 0.1179E-01 19 0.1198E-01 10 0.1187E-01 20 0.1198E-01 Facultade de Informatica. A Corua. Junio 2005 162 APPROXIMATE ANALYTICAL SOLUTIONS

Decomposition-aggregation: Multiprogramming system T e rm in a ls Facultade de Informatica. A Corua. Junio 2005 163 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system 37 38 /STATION/ NAME=TERMINAL; 39 TYPE=INFINITE; 40 INIT(C1)=N; 41 SERVICE(C1)=EXP(30000.); 42 SERVICE(C2)=EXP(60000.); 43 TRANSIT=SC; 44 /STATION/ NAME=SC; 45 SERVICE=EXP(1.); 46 RATE=CAP(1 STEP 1 UNTIL M); 47 TRANSIT=TERMINAL,C1,0.6,TERMINAL,C2; 48 /CONTROL/ OPTION=NRESULT; Facultade de Informatica. A Corua. Junio 2005 164 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system 49 /EXEC/ BEGIN 50 NETWORK(TERMINAL,SC);

51 PRINT(" TERM FM PRODUCTIVITAT ==> RESPOSTA"); 52 FOR N := 150 STEP 150 UNTIL 750 DO 53 FOR M := 1 STEP 1 UNTIL 20 DO 54 BEGIN 55 SOLVE; 56 PRINT(N,M,MTHRUPUT(SC),MRESPONSE(SC)); 57 END; 58 END; Facultade de Informatica. A Corua. Junio 2005 165 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system TERM 150 150 150 150 150 150 150 150 . 150 FM PRODUCTIVITAT 1 0.3478E-02

2 0.3541E-02 3 0.3546E-02 4 0.3547E-02 5 0.3547E-02 6 0.3547E-02 7 0.3547E-02 8 0.3547E-02 20 0.3547E-02 Facultade de Informatica. A Corua. Junio 2005 RESPOSTA 1124. 358.5 304.1 294.4 292.4 292.0 291.9 291.9 291.9 166 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system TERM 300 300 300 300 300 300

300 300 300 300 300 300 . 300 FM PRODUCTIVITAT RESPOSTA 1 0.4272E-02 0.2822E+05 2 0.6735E-02 2545. 3 0.7031E-02 668.5 4 0.7062E-02 482.4 5 0.7070E-02 433.6 6 0.7073E-02 417.0 7 0.7074E-02 410.8 8 0.7074E-02 408.4 9 0.7074E-02 407.5 10 0.7074E-02 407.2 11 0.7074E-02 407.0 12

0.7074E-02 407.0 20 0.7074E-02 407.0 Facultade de Informatica. A Corua. Junio 2005 167 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system TERM 450 450 450 450 450 450 450 450 450 450 450 450 450 450 450 450 450 450 450 450 FM PRODUCTIVITAT RESPOSTA 1 0.4272E-02 0.6334E+05 2 0.6940E-02 0.2284E+05 3

0.8679E-02 9847. 4 0.9792E-02 3956. 5 0.1026E-01 1872. 6 0.1040E-01 1270. 7 0.1045E-01 1052. 8 0.1048E-01 956.3 9 0.1049E-01 910.5 10 0.1049E-01 887.4 11 0.1050E-01 875.5 12 0.1050E-01 869.4 13 0.1050E-01 866.3 14 0.1050E-01 864.7 15 0.1050E-01 864.0 16 0.1050E-01 863.6 17 0.1050E-01

863.4 18 0.1050E-01 863.3 19 0.1050E-01 863.2 20 0.1050E-01 863.2 Facultade de Informatica. A Corua. Junio 2005 168 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system TERM 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 FM PRODUCTIVITAT RESPOSTA 1 0.4272E-02 0.9845E+05

2 0.6940E-02 0.4446E+05 3 0.8679E-02 0.2713E+05 4 0.9835E-02 0.1901E+05 5 0.1060E-01 0.1458E+05 6 0.1111E-01 0.1199E+05 7 0.1144E-01 0.1043E+05 8 0.1165E-01 9486. 9 0.1179E-01 8910. 10 0.1187E-01 8565. 11 0.1191E-01 8360. 12 0.1194E-01 8241. 13 0.1196E-01 8172. 14 0.1197E-01 8133. 15 0.1197E-01 8111. 16 0.1198E-01 8099. 17 0.1198E-01 8092. 18

0.1198E-01 8088. 19 0.1198E-01 8086. 20 0.1198E-01 8085. Facultade de Informatica. A Corua. Junio 2005 169 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Multiprogramming system TERM 750 750 750 750 750 750 750 750 750 750 750 750 750 750 750 750 750 750 750 750 59 /END/ FM PRODUCTIVITAT RESPOSTA 1 0.4272E-02 0.1336E+06 2 0.6940E-02 0.6607E+05

3 0.8679E-02 0.4441E+05 4 0.9835E-02 0.3426E+05 5 0.1060E-01 0.2873E+05 6 0.1111E-01 0.2549E+05 7 0.1144E-01 0.2354E+05 8 0.1165E-01 0.2236E+05 9 0.1179E-01 0.2164E+05 10 0.1187E-01 0.2121E+05 11 0.1191E-01 0.2095E+05 12 0.1194E-01 0.2080E+05 13 0.1196E-01 0.2071E+05 14 0.1197E-01 0.2067E+05 15 0.1197E-01 0.2064E+05 16 0.1198E-01 0.2062E+05 17 0.1198E-01 0.2061E+05 18 0.1198E-01 0.2061E+05 19 0.1198E-01 0.2061E+05 20 0.1198E-01 0.2061E+05 Facultade de Informatica. A Corua. Junio 2005 170 APPROXIMATE ANALYTICAL SOLUTIONS

Decomposition-aggregation: Sliding window flow control mechanisms f(S ) tj tf N e tw o rk 1 e (S ) N e tw o rk 2 Facultade de Informatica. A Corua. Junio 2005 171 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Sliding window flow control mechanisms Step 1: Decomposition N e tw o rk 1 N e tw o rk 2 Facultade de Informatica. A Corua. Junio 2005 172 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Sliding window flow control mechanisms Step 1: Decomposition o We analyze this queuing network to obtain its throughput when there are k customers where k = 1, 2, , C. o Let (k) be the throughput we obtain when there are k customers in the

network. The final result of this step is a set of values (1), (2), , (C). Facultade de Informatica. A Corua. Junio 2005 173 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Sliding window flow control mechanisms Step 2: Aggregation tj f(S ) g( k ) e (S ) Facultade de Informatica. A Corua. Junio 2005 174 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Sliding window flow control mechanisms Step 2: Aggregation o The arrival process at queue f(S) is Poisson distributed of mean o There are C tokens. o The inter-arrival times at queue e(S) are exponentially distributed with a rate g(k), where k is the number of outstanding tokens, i. e. C - k is the number of tokens in queue e(S). o We set g(k) = (k), for k = 1, 2, , C. o The state of this system can be described by the tuple (i, j), where i is the number of customers in queue f(S) and j is the number of tokens in queue e(S). Facultade de Informatica. A Corua. Junio 2005 175

APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Sliding window flow control mechanisms Step 2: Aggregation g(C) g(C) g(C) i,0 1,0 g(C) 0,0 g(C-1) 0,1 Facultade de Informatica. A Corua. Junio 2005 g(C-2) g(C-j) 0,2 g(1)

0,j 0,C 176 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Sliding window flow control mechanisms Step 2: Aggregation o This system is identical to an M/M/1 queue with an arrival rate and a state dependent service rate g(nq) if nq C, and g(C) if nq > C, where nq is the number of customers in this M/M/1 queue. The random variables i and j are related to nq as follows: i = max(0, nq - C) j = max(0, C - nq) o The solution of this system can be obtained by a direct application of classical results. Facultade de Informatica. A Corua. Junio 2005 177 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Sliding window flow control mechanisms Step 2: Aggregation

p i ,0 i p 0 ,0 j p 0 , j j p 0 ,0 where = /g(C) j 1 g C k j k 0 1 Facultade de Informatica. A Corua. Junio 2005 j 0 j 0 178 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Sliding window flow control mechanisms Step 2: Aggregation o The probability p(0, 0) is chosen so that the addition of the state probabilities is equal to 1: 1 p 0,0 1 1 Facultade de Informatica. A Corua. Junio 2005 C j 1

j j 179 APPROXIMATE ANALYTICAL SOLUTIONS Decomposition-aggregation: Sliding window flow control mechanisms Step 2: Aggregation o We obtain the following marginal probabilities for each queue (index 1 is for queue f(S) and 2 for queue e(S)): 1 1 p 0 ,0 p1 0 1 p1 i i p 0 ,0 i 0 p 0 ,0 p2 0 1 j p2 j j p 0 ,0 0 j C Facultade de Informatica. A Corua. Junio 2005 180 APPROXIMATE ANALYTICAL SOLUTIONS Diffusion Method Based on the assumption that very probably the queues are never empty. Under this hypothesis for each queue: o the queue length discrete distribution is studied o this discrete distribution is replaced by a continuous one with the same first two moments (first approximation) o this continuous probability distribution is described by a diffusion equation

o this equation is solved with the appropriate contour conditions for the steady state o the continuous probability distribution of the queue length is discretised by means of some heuristic criteria (second approximation) Facultade de Informatica. A Corua. Junio 2005 181 APPROXIMATE ANALYTICAL SOLUTIONS Diffusion Method To solve a queuing network it is assumed that it will have a product-form. If the system is open it is possible to determine for each node the characteristics of the inter-arrival time distribution and those of the service time are assumed to be known. From the state probabilities, the performance measures can be computed. Facultade de Informatica. A Corua. Junio 2005 182 APPROXIMATE ANALYTICAL SOLUTIONS Diffusion method: Packet communication network 7 D 6 8 A

5 1 C 4 2 B Facultade de Informatica. A Corua. Junio 2005 3 183 APPROXIMATE ANALYTICAL SOLUTIONS Diffusion method: Packet communication network 1 /DECLARE/ QUEUE GEN,LINIA(8,8); 2 REAL TRAB,TRAC,TRAD,TRBA,TRBC,TRBD,TRCA,TRCB,TRCD, ==> TRDA,TRDB,TRDC; 3 INTEGER I; 4 CLASS CLAB,CLAC,CLAD,CLBA,CLBC,CLBD,CLCA,CLCB,CLCD ==> ,CLDA,CLDB,CLDC; 5 Facultade de Informatica. A Corua. Junio 2005 184 APPROXIMATE ANALYTICAL SOLUTIONS Diffusion method: Packet communication network 6 /STATION/ NAME = GEN; 7 TYPE = SOURCE;

8 SERVICE = EXP(60000./930.); 9 TRANSIT = LINIA(1,2),CLAB,60,LINIA(1,2),CLAC,80,L ==> INIA(1,2),CLAD,100, 10 LINIA(3,4),CLBA,75,LINIA(3,4),CLBC,50,L ==> INIA(3,4),CLBD,25, 11 LINIA(5,6),CLCA,80,LINIA(5,6),CLCB,120, ==> LINIA(5,6),CLCD,40, 12 LINIA(7,8),CLDA,100,LINIA(7,8),CLDB,150 ==> ,LINIA(7,8),CLDC,50; 13 Facultade de Informatica. A Corua. Junio 2005 185 APPROXIMATE ANALYTICAL SOLUTIONS Diffusion method: Packet communication network 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 /STATION/ NAME = LINIA;

SERVICE = CST(256.*8./64.); /STATION/ NAME = LINIA(1,2); TRANSIT(CLAB) = LINIA(2,4); TRANSIT(CLAC) = LINIA(2,4); TRANSIT(CLAD) = LINIA(2,8); /STATION/ NAME = LINIA(2,1); TRANSIT = OUT; /STATION/ NAME = LINIA(2,4); TRANSIT(CLAB) = LINIA(4,3); TRANSIT(CLAC) = LINIA(4,6); /STATION/ NAME = LINIA(2,8); TRANSIT(CLAD) = LINIA(8,7); Facultade de Informatica. A Corua. Junio 2005 186 APPROXIMATE ANALYTICAL SOLUTIONS Diffusion method: Packet communication network 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 /STATION/ NAME = LINIA(3,4); TRANSIT(CLBA) = LINIA(4,2); TRANSIT(CLBC) = LINIA(4,6); TRANSIT(CLBD) = LINIA(4,8);

/STATION/ NAME = LINIA(4,2); TRANSIT(CLBA) = LINIA(2,1); /STATION/ NAME = LINIA(4,3); TRANSIT = OUT; /STATION/ NAME = LINIA(4,6); TRANSIT(CLAC) = LINIA(6,5); TRANSIT(CLBC) = LINIA(6,5); /STATION/ NAME = LINIA(4,8); TRANSIT(CLBD) = LINIA(8,7); Facultade de Informatica. A Corua. Junio 2005 187 APPROXIMATE ANALYTICAL SOLUTIONS Diffusion method: Packet communication network 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 /STATION/ NAME = LINIA(5,6); TRANSIT(CLCA) = LINIA(6,8); TRANSIT(CLCB) = LINIA(6,4); TRANSIT(CLCD) = LINIA(6,8); /STATION/ NAME = LINIA(6,4);

TRANSIT(CLCB) = LINIA(4,3); /STATION/ NAME = LINIA(6,5); TRANSIT = OUT; /STATION/ NAME = LINIA(6,8); TRANSIT(CLCA) = LINIA(8,2); TRANSIT(CLCD) = LINIA(8,7); /STATION/ NAME = LINIA(7,8); TRANSIT(CLDA) = LINIA(8,2); TRANSIT(CLDB) = LINIA(8,4); TRANSIT(CLDC) = LINIA(8,6); Facultade de Informatica. A Corua. Junio 2005 188 APPROXIMATE ANALYTICAL SOLUTIONS Diffusion method: Packet communication network 70 71 72 73 74 75 76 77 78 79 80 81 82 /STATION/ NAME = LINIA(8,2); TRANSIT(CLCA) = LINIA(2,1); TRANSIT(CLDA) = LINIA(2,1); /STATION/ NAME = LINIA(8,4); TRANSIT(CLDB) = LINIA(4,3); /STATION/ NAME = LINIA(8,6); TRANSIT(CLDC) = LINIA(6,5); /STATION/ NAME = LINIA(8,7); TRANSIT = OUT; Facultade de Informatica. A Corua. Junio 2005

189 APPROXIMATE ANALYTICAL SOLUTIONS Diffusion method: Packet communication network 83 /EXEC/ 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 BEGIN NETWORK(GEN, LINIA(1,2),LINIA(2,1),LINIA(2,4),LINIA(2,8), LINIA(3,4),LINIA(4,2),LINIA(4,3),LINIA(4,6), LINIA(4,8),LINIA(5,6),LINIA(6,4),LINIA(6,5), LINIA(6,8),LINIA(7,8),LINIA(8,2),LINIA(8,4), LINIA(8,6),LINIA(8,7)); PRINT; SOLVE; TRAB:=MRESPONSE(LINIA(1,2))+ MRESPONSE(LINIA(2,4))+MRESPONSE(LINIA(4,3)); TRAC:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,4))+ MRESPONSE(LINIA(4,6))+MRESPONSE(LINIA(6,5)); TRAD:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,8))+ MRESPONSE(LINIA(8,7)); TRBA:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,2))+ MRESPONSE(LINIA(2,1)); Facultade de Informatica. A Corua. Junio 2005

190 APPROXIMATE ANALYTICAL SOLUTIONS Diffusion method: Packet communication network 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 TRBC:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,6))+ MRESPONSE(LINIA(6,5)); TRBD:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,8))+ MRESPONSE(LINIA(8,7)); TRCA:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+ MRESPONSE(LINIA(8,2))+MRESPONSE(LINIA(2,1)); TRCB:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,4))+ MRESPONSE(LINIA(4,3)); TRCD:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+ MRESPONSE(LINIA(8,7)); TRDA:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,2))+ MRESPONSE(LINIA(2,1)); TRDB:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,4))+ MRESPONSE(LINIA(4,3)); TRDC:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,6))+ MRESPONSE(LINIA(6,5)); Facultade de Informatica. A Corua. Junio 2005

191 APPROXIMATE ANALYTICAL SOLUTIONS Diffusion method: Packet communication network 116 117 118 119 120 PRINT(TRAB,TRAC,TRAD); PRINT(TRBA,TRBC,TRBD); PRINT(TRCA,TRCB,TRCD); PRINT(TRDA,TRDB,TRDC); END; Facultade de Informatica. A Corua. Junio 2005 192 APPROXIMATE ANALYTICAL SOLUTIONS Diffusion method: Packet communication network - APPROXIMATE DIFFUSIONS METHOD ("DIFFU") ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * * * * * * * * GEN * 64.52 * 1.000 * 1.000 * 64.52 *0.1550E-01* * *

* * * * * * LINIA 2 * 32.00 *0.1280 *0.1374 * 34.35 *0.4000E-02* * * * * * * * * LINIA 9 * 32.00 *0.1360 *0.1466 * 34.48 *0.4250E-02* * * * * * * * * LINIA 12 * 32.00 *0.7467E-01*0.7765E-01* 33.28 *0.2333E-02* * * * * * * * * LINIA 16 * 32.00 *0.5333E-01*0.5482E-01* 32.89 *0.1667E-02* *

* * * * * * * LINIA 20 * 32.00 *0.8000E-01*0.8348E-01* 33.39 *0.2500E-02* * * * * * * * * LINIA 26 * 32.00 *0.4000E-01*0.4083E-01* 32.66 *0.1250E-02* * * * * * * * Facultade de Informatica. A Corua. Junio 2005 193 APPROXIMATE ANALYTICAL SOLUTIONS Diffusion method: Packet communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * LINIA 27 * 32.00 *0.1760 *0.1945 * 35.37

*0.5500E-02* * * * * * * * * LINIA 30 * 32.00 *0.6933E-01*0.7190E-01* 33.18 *0.2167E-02* * * * * * * * * LINIA 32 * 32.00 *0.1333E-01*0.1342E-01* 32.22 *0.4167E-03* * * * * * * * * LINIA 38 * 32.00 *0.1280 *0.1374 * 34.35 *0.4000E-02* * * * * * * * * LINIA 44 * 32.00 *0.6400E-01*0.6617E-01* 33.08 *0.2000E-02*

* * * * * * * * LINIA 45 * 32.00 *0.9600E-01*0.1011 * 33.68 *0.3000E-02* * * * * * * * * LINIA 48 * 32.00 *0.6400E-01*0.6617E-01* 33.08 *0.2000E-02* * * * * * * * * LINIA 56 * 32.00 *0.1600 *0.1752 * 35.05 *0.5000E-02* * * * * * * * * LINIA 58 * 32.00 *0.9600E-01*0.1011 * 33.68

*0.3000E-02* * * * * * * * Facultade de Informatica. A Corua. Junio 2005 194 APPROXIMATE ANALYTICAL SOLUTIONS Diffusion method: Packet communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * LINIA 60 * 32.00 *0.8000E-01*0.8343E-01* 33.37 *0.2500E-02* * * * * * * * * LINIA 62 * 32.00 *0.2667E-01*0.2703E-01* 32.44 *0.8333E-03* * * * * * * * * LINIA 63 * 32.00 *0.8800E-01*0.9222E-01* 33.53

*0.2750E-02* ******************************************************************* MEMORY USED: 19812 WORDS OF 4 BYTES ( 2.48 % OF TOTAL MEMORY) 103.0 100.5 135.6 103.2 134.5 100.3 102.8 103.8 100.8 99.14 101.0 101.2 121 122 /END/ Facultade de Informatica. A Corua. Junio 2005 195 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods In this family of methods we establish an iterative computation of the result from a reasonable conjecture. They have neither any theoretical justification of the iteration convergence nor the coincidence of the limit of the iteration with the theoretical result; however the experience shows that there are no counter-examples in the normal domain of use of such methods.

Facultade de Informatica. A Corua. Junio 2005 196 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods They are useful tools to study cases not covered by exact methods, as, for instance: o closed queuing networks with FIFO stations that have with non-exponentially distributed service time o systems with customers seizing simultaneously more than one server (for example, in a disk input-output there are simultaneous seizing of the disk and the path between disks and memory). o systems with customers affected of priorities They are also useful to reduce the computing time in methods derived from the BCMP theorem when they are applied to large dimension systems. Facultade de Informatica. A Corua. Junio 2005 197 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system This case is identical to the previous one with an important difference: o

the disk path to memory is shared by several disks and should be taken into account in the modelling process o the scheduling policy of disks accesses in the control unit is SLTF (Shortest Latency Time First). As in this case the basic assumption of analytical models is not fulfilled, we are obliged to build an iterative model starting with the assumption that there will not be conflicts in the path; from this assumption it is possible to compute the throughput in the disks. With this data we recompute the path conflict time due to the lost rotations. This time is introduced in the disks service time and we restart the iteration. Facultade de Informatica. A Corua. Junio 2005 198 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system 1 /DECLARE/ QUEUE CPU,DISC(4),TERMINAL; 2 3 4 5 6 7 8 9 10 /STATION/ 11 12 13 14 ==> .4; 15 ==> .4; REAL REAL REAL REAL PROB1(4)=(2,1.5,1,0.5);

PROB2(4)=(1.5,2,3,3.5); TR1,TR2; SD(4),VP,VPN,LT,TF,VR=3600,TAU,UC, SK(4)=(13.,12.,11.,10.); REAL EPS; CLASS C1,C2; INTEGER I,N; NAME=CPU; SCHED=PS; SERVICE(C1)=CST(8.52); SERVICE(C2)=CST(12.); TRANSIT(C1)=DISC,PROB1,TERMINAL,C1,0.6,TERMINAL,C2,0 TRANSIT(C2)=DISC,PROB2,TERMINAL,C1,0.6,TERMINAL,C2,0 Facultade de Informatica. A Corua. Junio 2005 199 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 /STATION/ NAME=DISC; TRANSIT=CPU; /STATION/ NAME=DISC(1); SERVICE=EXP(SD(1));

/STATION/ NAME=DISC(2); SERVICE=EXP(SD(2)); /STATION/ NAME=DISC(3); SERVICE=EXP(SD(3)); /STATION/ NAME=DISC(4); SERVICE=EXP(SD(4)); /STATION/ NAME=TERMINAL; TYPE=INFINITE; INIT(C1)=N; SERVICE(C1)=EXP(30000.); SERVICE(C2)=EXP(60000.); TRANSIT=CPU; /CONTROL/ CLASS=ALL QUEUE; Facultade de Informatica. A Corua. Junio 2005 200 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system 33 /EXEC/ 34 35 36 37 38 39 40 41 42 43 44 ==> TF; 45 46 BEGIN TAU:=60.*1000./VR; TF:=TAU/10.; LT:=TAU/2.; FOR N:=150 STEP 150 UNTIL 750 DO BEGIN

PRINT("NOMBRE DUSUARIS =",N); EPS:=1.; VP:=0.; WHILE EPS>=1.E-4 DO BEGIN FOR I:=1 STEP 1 UNTIL 4 DO SD(I):=SK(I)+LT+VP+ PRINT; SOLVE; Facultade de Informatica. A Corua. Junio 2005 201 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system 47 48 ==> 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 UC:=0.; FOR I:=1 STEP 1 UNTIL 4 DO UC := UC + TF * MTH RUPUT(DISC(I)); VPN:=TAU*UC/(1.-UC); EPS:=ABS(VP-VPN)/VPN; VP:=VPN; END;

TR1:=MCUSTNB(CPU,C1); TR2:=MCUSTNB(CPU,C2); FOR I:= 1 STEP 1 UNTIL 4 DO BEGIN TR1:=TR1+MCUSTNB(DISC(I),C1); TR2:=TR2+MCUSTNB(DISC(I),C2); END; TR1:=TR1/MTHRUPUT(TERMINAL,C1); TR2:=TR2/MTHRUPUT(TERMINAL,C2);; PRINT("RESPONSE TIME OF CLASS C1 =",TR1); PRINT("RESPONSE TIME OF CLASS C2 =",TR2); END; END; Facultade de Informatica. A Corua. Junio 2005 202 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system NOMBRE DUSUARIS = 150 - MEAN VALUE ANALYSIS ("MVA") ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* *(C1 )* 8.520 *0.1088 *0.1539 * 12.06 *0.1277E-01* *(C2 )* 12.00 *0.1873 *0.2650 * 16.98 *0.1560E-01* * * *

* * * * * DISC 1 * 23.00 *0.1468 *0.1719 * 26.92 *0.6384E-02* *(C1 )* 23.00 *0.9790E-01*0.1146 * 26.92 *0.4257E-02* *(C2 )* 23.00 *0.4894E-01*0.5729E-01* 26.92 *0.2128E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 149.0 *0.4200E+05*0.3547E-02* *(C1 )*0.3000E+05*0.0000E+00* 63.84 *0.3000E+05*0.2128E-02* *(C2 )*0.6000E+05*0.0000E+00* 85.12 *0.6000E+05*0.1419E-02* * * * * * * * *******************************************************************

Facultade de Informatica. A Corua. Junio 2005 203 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system NOMBRE DUSUARIS = 150 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.2960 *0.4188 * 14.76 *0.2837E-01* *(C1 )* 8.520 *0.1088 *0.1539 * 12.05 *0.1277E-01* *(C2 )* 12.00 *0.1872 *0.2649 * 16.98 *0.1560E-01* * * * * * * * * DISC 1 * 23.72 *0.1514 *0.1782

* 27.91 *0.6383E-02* *(C1 )* 23.72 *0.1009 *0.1188 * 27.91 *0.4256E-02* *(C2 )* 23.72 *0.5046E-01*0.5938E-01* 27.91 *0.2127E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 148.9 *0.4200E+05*0.3546E-02* *(C1 )*0.3000E+05*0.0000E+00* 63.83 *0.3000E+05*0.2128E-02* *(C2 )*0.6000E+05*0.0000E+00* 85.11 *0.6000E+05*0.1418E-02* ******************************************************************* Facultade de Informatica. A Corua. Junio 2005 204 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system NOMBRE DUSUARIS = 150 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * *******************************************************************

* CPU * 10.43 *0.2960 *0.4188 * 14.76 *0.2837E-01* *(C1 )* 8.520 *0.1088 *0.1539 * 12.05 *0.1277E-01* *(C2 )* 12.00 *0.1872 *0.2649 * 16.98 *0.1560E-01* * * * * * * * * DISC 1 * 23.72 *0.1514 *0.1782 * 27.91 *0.6383E-02* *(C1 )* 23.72 *0.1009 *0.1188 * 27.91 *0.4256E-02* *(C2 )* 23.72 *0.5046E-01*0.5938E-01* 27.91 *0.2127E-02* * *

* * * * * * TERMINAL *0.4200E+05*0.0000E+00* 148.9 *0.4200E+05*0.3546E-02* *(C1 )*0.3000E+05*0.0000E+00* 63.83 *0.3000E+05*0.2128E-02* *(C2 )*0.6000E+05*0.0000E+00* 85.11 *0.6000E+05*0.1418E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 204.6 RESPONSE TIME OF CLASS C2 = 439.5 Facultade de Informatica. A Corua. Junio 2005 205 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system NOMBRE DUSUARIS = 300 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.5905 * 1.426 * 25.19 *0.5659E-01* *(C1 )* 8.520 *0.2170 *0.5239

* 20.57 *0.2547E-01* *(C2 )* 12.00 *0.3735 *0.9017 * 28.97 *0.3112E-01* * * * * * * * * DISC 1 * 23.00 *0.2929 *0.4134 * 32.46 *0.1273E-01* *(C1 )* 23.00 *0.1953 *0.2756 * 32.46 *0.8490E-02* *(C2 )* 23.00 *0.9761E-01*0.1378 * 32.46 *0.4244E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 297.1 *0.4200E+05*0.7074E-02* *(C1 )*0.3000E+05*0.0000E+00* 127.3

*0.3000E+05*0.4245E-02* *(C2 )*0.6000E+05*0.0000E+00* 169.8 *0.6000E+05*0.2830E-02* ******************************************************************* Facultade de Informatica. A Corua. Junio 2005 206 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system NOMBRE DUSUARIS = 300 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.5902 * 1.424 * 25.17 *0.5657E-01* *(C1 )* 8.520 *0.2169 *0.5233 * 20.56 *0.2546E-01* *(C2 )* 12.00 *0.3733 *0.9007 * 28.95 *0.3111E-01* * * * * *

* * * DISC 1 * 24.50 *0.3118 *0.4521 * 35.52 *0.1273E-01* *(C1 )* 24.50 *0.2079 *0.3015 * 35.52 *0.8486E-02* *(C2 )* 24.50 *0.1039 *0.1507 * 35.52 *0.4242E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 297.0 *0.4200E+05*0.7071E-02* *(C1 )*0.3000E+05*0.0000E+00* 127.3 *0.3000E+05*0.4243E-02* *(C2 )*0.6000E+05*0.0000E+00* 169.7 *0.6000E+05*0.2828E-02* ******************************************************************* Facultade de Informatica. A Corua. Junio 2005 207 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system

NOMBRE DUSUARIS = 300 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.5902 * 1.424 * 25.17 *0.5657E-01* *(C1 )* 8.520 *0.2169 *0.5233 * 20.56 *0.2546E-01* *(C2 )* 12.00 *0.3733 *0.9007 * 28.95 *0.3111E-01* * * * * * * * * DISC 1 * 24.50 *0.3118 *0.4521 * 35.52 *0.1273E-01* *(C1 )* 24.50 *0.2079 *0.3014 * 35.52

*0.8486E-02* *(C2 )* 24.50 *0.1039 *0.1507 * 35.52 *0.4242E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 297.0 *0.4200E+05*0.7071E-02* *(C1 )*0.3000E+05*0.0000E+00* 127.3 *0.3000E+05*0.4243E-02* *(C2 )*0.6000E+05*0.0000E+00* 169.7 *0.6000E+05*0.2828E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 289.4 RESPONSE TIME OF CLASS C2 = 632.8 Facultade de Informatica. A Corua. Junio 2005 208 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system NOMBRE DUSUARIS = 450 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU

* 10.43 *0.8763 * 6.438 * 76.66 *0.8399E-01* *(C1 )* 8.520 *0.3220 * 2.366 * 62.60 *0.3780E-01* *(C2 )* 12.00 *0.5543 * 4.072 * 88.16 *0.4619E-01* * * * * * * * * DISC 1 * 23.00 *0.4347 *0.7667 * 40.57 *0.1890E-01* *(C1 )* 23.00 *0.2898 *0.5112 * 40.57 *0.1260E-01* *(C2 )* 23.00 *0.1449 *0.2555 * 40.57 *0.6298E-02* *

* * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 440.9 *0.4200E+05*0.1050E-01* *(C1 )*0.3000E+05*0.0000E+00* 189.0 *0.3000E+05*0.6299E-02* *(C2 )*0.6000E+05*0.0000E+00* 252.0 *0.6000E+05*0.4199E-02* ******************************************************************* Facultade de Informatica. A Corua. Junio 2005 209 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system NOMBRE DUSUARIS = 450 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.8754 * 6.393 * 76.20 *0.8390E-01* *(C1 )* 8.520 *0.3217 * 2.349 * 62.22 *0.3776E-01* *(C2

)* 12.00 *0.5537 * 4.044 * 87.64 *0.4614E-01* * DISC 1 * 25.33 *0.4781 *0.9129 * 48.36 *0.1888E-01* *(C1 )* 25.33 *0.3188 *0.6086 * 48.36 *0.1259E-01* *(C2 )* 25.33 *0.1594 *0.3043 * 48.36 *0.6292E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 440.5 *0.4200E+05*0.1049E-01* *(C1 )*0.3000E+05*0.0000E+00* 188.8 *0.3000E+05*0.6293E-02* *(C2 )*0.6000E+05*0.0000E+00* 251.7 *0.6000E+05*0.4195E-02* ******************************************************************* Facultade de Informatica. A Corua. Junio 2005 210

APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system NOMBRE DUSUARIS = 450 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 *0.8754 * 6.393 * 76.20 *0.8390E-01* *(C1 )* 8.520 *0.3217 * 2.349 * 62.22 *0.3776E-01* *(C2 )* 12.00 *0.5537 * 4.044 * 87.64 *0.4614E-01* * * * * * * * * DISC 1 * 25.32 *0.4781 *0.9127 * 48.35 *0.1888E-01* *(C1 )* 25.32

*0.3187 *0.6085 * 48.35 *0.1259E-01* *(C2 )* 25.32 *0.1593 *0.3042 * 48.35 *0.6292E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 440.5 *0.4200E+05*0.1049E-01* *(C1 )*0.3000E+05*0.0000E+00* 188.8 *0.3000E+05*0.6293E-02* *(C2 )*0.6000E+05*0.0000E+00* 251.7 *0.6000E+05*0.4195E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 594.4 RESPONSE TIME OF CLASS C2 = 1376. Facultade de Informatica. A Corua. Junio 2005 211 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system NOMBRE DUSUARIS = 600 ******************************************************************* * NAME

* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 * 1.000 * 93.51 * 975.7 *0.9584E-01* *(C1 )* 8.520 *0.3675 * 34.37 * 796.7 *0.4313E-01* *(C2 )* 12.00 *0.6325 * 59.15 * 1122. *0.5271E-01* * * * * * * * * DISC 1 * 23.00 *0.4960 *0.9841 * 45.63 *0.2157E-01* *(C1 )* 23.00 *0.3307 *0.6561 * 45.63 *0.1438E-01* *(C2 )* 23.00 *0.1653 *0.3280

* 45.63 *0.7187E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 503.2 *0.4200E+05*0.1198E-01* *(C1 )*0.3000E+05*0.0000E+00* 215.7 *0.3000E+05*0.7188E-02* *(C2 )*0.6000E+05*0.0000E+00* 287.5 *0.6000E+05*0.4792E-02* ******************************************************************* Facultade de Informatica. A Corua. Junio 2005 212 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system NOMBRE DUSUARIS = 600 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 * 1.000 * 92.66 * 966.8 *0.9584E-01* *(C1 )* 8.520 *0.3675 * 34.05 * 789.5

*0.4313E-01* *(C2 )* 12.00 *0.6325 * 58.61 * 1112. *0.5271E-01* * * * * * * * * DISC 1 * 25.71 *0.5544 * 1.244 * 57.69 *0.2157E-01* *(C1 )* 25.71 *0.3696 *0.8295 * 57.69 *0.1438E-01* *(C2 )* 25.71 *0.1848 *0.4147 * 57.69 *0.7187E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 503.2 *0.4200E+05*0.1198E-01* *(C1 )*0.3000E+05*0.0000E+00* 215.7

*0.3000E+05*0.7188E-02* *(C2 )*0.6000E+05*0.0000E+00* 287.5 *0.6000E+05*0.4792E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 4997. RESPONSE TIME OF CLASS C2 = 0.1271E+05 Facultade de Informatica. A Corua. Junio 2005 213 APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system NOMBRE DUSUARIS = 750 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 * 1.000 * 243.5 * 2541. *0.9584E-01* *(C1 )* 8.520 *0.3675 * 89.49 * 2075. *0.4313E-01* *(C2 )* 12.00 *0.6325 * 154.0 * 2922. *0.5271E-01* * *

* * * * * * DISC 1 * 23.00 *0.4960 *0.9841 * 45.64 *0.2157E-01* *(C1 )* 23.00 *0.3307 *0.6561 * 45.64 *0.1438E-01* *(C2 )* 23.00 *0.1653 *0.3280 * 45.64 *0.7187E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 503.2 *0.4200E+05*0.1198E-01* *(C1 )*0.3000E+05*0.0000E+00* 215.7 *0.3000E+05*0.7188E-02* *(C2 )*0.6000E+05*0.0000E+00* 287.5 *0.6000E+05*0.4792E-02* ******************************************************************* Facultade de Informatica. A Corua. Junio 2005 214

APPROXIMATE ANALYTICAL SOLUTIONS Iterative methods: Conversational system NOMBRE DUSUARIS = 750 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT * ******************************************************************* * CPU * 10.43 * 1.000 * 242.7 * 2532. *0.9584E-01* *(C1 )* 8.520 *0.3675 * 89.18 * 2067. *0.4313E-01* *(C2 )* 12.00 *0.6325 * 153.5 * 2912. *0.5271E-01* * * * * * * * * DISC 1 * 25.71 *0.5544 * 1.244 * 57.69 *0.2157E-01* *(C1

)* 25.71 *0.3696 *0.8295 * 57.69 *0.1438E-01* *(C2 )* 25.71 *0.1848 *0.4147 * 57.69 *0.7187E-02* * * * * * * * * TERMINAL *0.4200E+05*0.0000E+00* 503.2 *0.4200E+05*0.1198E-01* *(C1 )*0.3000E+05*0.0000E+00* 215.7 *0.3000E+05*0.7188E-02* *(C2 )*0.6000E+05*0.0000E+00* 287.5 *0.6000E+05*0.4792E-02* ******************************************************************* RESPONSE TIME OF CLASS C1 = 0.1266E+05 RESPONSE TIME OF CLASS C2 = 0.3251E+05 Facultade de Informatica. A Corua. Junio 2005 215 OUTLINE INTRODUCTION CONCEPT OF QUEUE CONCEPT OF QUEUEING NETWORK NUMERICAL TECHNIQUES EXACT ANALYTICAL SOLUTIONS APPROXIMATE ANALYTICAL SOLUTIONS

SIMULATION TECHNIQUES Facultade de Informatica. A Corua. Junio 2005 216 SIMULATION TECHNIQUES Computer based simulation implies writing a computer program which depicts the system operations. Running this program permits us to "imitate" the system behaviour in a very short time. Thus, we are able to "observe" the system and collect performance statistics. Facultade de Informatica. A Corua. Junio 2005 217 SIMULATION TECHNIQUES Simulation advantage are: there is no theoretical limitation and it allows us to study systems not able to be studied by means of analytical or numerical techniques. It is easy to learn and to apply. Simulation disadvantages are: the effort (mainly time) to develop and to debug a simulation program. the execution time, that can be quite long to obtain results with enough significance.

Facultade de Informatica. A Corua. Junio 2005 218 SIMULATION TECHNIQUES The complexity of the simulation model depend on the detail of the system behaviour representation. To build up simulation models it is convenient to use: modelling languages as QNAP2, RESQ, PAWS, etc., that make easier to build up queuing network models and that include analytical, numerical and simulation procedures. simulation languages, such as SIMSCRIPT, GPSS, SIMULA, etc., that offer greater simulation capabilities. high level programming language for building up very detailed models. Facultade de Informatica. A Corua. Junio 2005 219 SIMULATION TECHNIQUES The intrinsic problems of any simulation are: random numbers generation simulated time management

extraction of statistical estimations of the simulated behaviour evaluation of the confidence interval of the estimations Facultade de Informatica. A Corua. Junio 2005 220 SIMULATION TECHNIQUES Conversational system Identical to the previously studied by analytical techniques Influence of the memory management by a multiprogramming factor Facultade de Informatica. A Corua. Junio 2005 221 SIMULATION TECHNIQUES Conversational system D is k 1 D is k 2 CPU D is k 3 T e rm in a ls M e m o ry m anagem ent Facultade de Informatica. A Corua. Junio 2005

D is k 4 222 SIMULATION TECHNIQUES Conversational system 1 /DECLARE/ QUEUE CPU,DISC(4),MEM,SMEM,TERMINAL,R(2); 2 REAL PROF1(4)=(2.,3.5,4.5,5.); 3 REAL PROF2(4)=(1.5,3.5,6.5,10.); 4 REAL TR1,TR2; 5 REAL D; 6 CLASS C1,C2; 7 INTEGER I,N,M; 8 REF CUSTOMER C; Facultade de Informatica. A Corua. Junio 2005 223 SIMULATION TECHNIQUES Conversational system 9 /STATION/ NAME=CPU;

10 SCHED=PS; 11 12 13 14 15 16 17 18 19 20 21 22 SERVICE(C1)=BEGIN CST(8.52); D := UNIFORM(0., 6.); FOR I := 1 STEP 1 UNTIL 4 DO IF D <= PROF1(I) THEN TRANSIT(DISC(I)); C:=R(1).FIRST; WHILE C.FATHER <> CUSTOMER DO C:=C.NEXT; TRANSIT(C,OUT); V(SMEM); IF D <= 5.6 THEN TRANSIT(TERMINAL,C1) ELSE TRANSIT(TERMINAL,C2); END; Facultade de Informatica. A Corua. Junio 2005 224 SIMULATION TECHNIQUES Conversational system 23 SERVICE(C2)=BEGIN 24 CST(12.); 25 D := UNIFORM(0., 11.); 26 FOR I := 1 STEP 1 UNTIL 4 DO 27

IF D <= PROF2(I) THEN TRANSIT(DISC(I)); 28 C:=R(2).FIRST; 29 WHILE C.FATHER <> CUSTOMER DO C:=C.NEXT; 30 TRANSIT(C,OUT); 31 V(SMEM); 32 IF D <= 10.6 THEN TRANSIT(TERMINAL,C1) 33 ELSE TRANSIT(TERMINAL,C2); 34 END; 35 /STATION/ NAME=DISC; 36 TRANSIT=CPU; 37 /STATION/ NAME=DISC(1); 38 SERVICE=EXP(23.); 39 /STATION/ NAME=DISC(2); 40 SERVICE=EXP(22.); Facultade de Informatica. A Corua. Junio 2005 225 SIMULATION TECHNIQUES Conversational system 41 /STATION/ NAME=DISC(3); 42 SERVICE=EXP(21.); 43 /STATION/ NAME=DISC(4); 44 SERVICE=EXP(20.); 45 /STATION/ NAME=MEM; 46

47 48 49 SERVICE=BEGIN P(SMEM); TRANSIT(CPU); END; 50 /STATION/ NAME=SMEM; 51 TYPE=SEMAPHORE,MULTIPLE(M); Facultade de Informatica. A Corua. Junio 2005 226 SIMULATION TECHNIQUES Conversational system 52 /STATION/ NAME=TERMINAL; 53 TYPE=INFINITE; 54 INIT(C1)=6*N/10; 55 INIT(C2)=4*N/10; 56 SERVICE(C1)=BEGIN 57 EXP(30000.); 58 TRANSIT(NEW(CUSTOMER),R(1),C1); 59 END; 60 SERVICE(C2)=BEGIN 61 EXP(60000.); 62 TRANSIT(NEW(CUSTOMER),R(2),C2); 63 END;

64 TRANSIT=MEM; 65 /CONTROL/ TMAX=5000000.; 66 CLASS=ALL QUEUE; 67 ACCURACY=ALL QUEUE,ALL CLASS; 68 OPTION=NRESULT; Facultade de Informatica. A Corua. Junio 2005 227 SIMULATION TECHNIQUES Conversational system 69 /EXEC/ BEGIN 70 FOR N:=150 STEP 150 UNTIL 750 DO 71 FOR M:=1 STEP 1 UNTIL 20 DO 72 BEGIN 73 PRINT(" "); 74 PRINT("NUMERO DE USUARIOS =",N); 75 PRINT("FACTOR DE MULTIPROGRAMACION =",M); 76 SIMUL; 77 PRINT("TIEMPO DE RESPUESTA DE LA CLASE C1 =",MRESPONS ==> E(R(1))," +/-"),CRESPONSE(R(1)); 78 PRINT("TIEMPO DE RESPUESTA DE LA CLASE C2 =",MRESPONS ==> E(R(2))," +/-",CRESPONSE(R(2))); 79 PRINT("PRODUCTIVIDAD C1 =",MTHRUPUT(TERMINAL,C1)); 80 PRINT("PRODUCTIVIDAD C2 =",MTHRUPUT(TERMINAL,C2)); 81

PRINT("PRODUCTIVIDAD TOTAL =",MTHRUPUT(TERMINAL)); 82 END; 83 END; Facultade de Informatica. A Corua. Junio 2005 228 SIMULATION TECHNIQUES Conversational system NUMERO DE USUARIOS = 150 FACTOR DE MULTIPROGRAMACION = TIEMPO DE RESPUESTA TIEMPO DE RESPUESTA PRODUCTIVIDAD C1 = PRODUCTIVIDAD C2 = PRODUCTIVIDAD TOTAL NUMERO DE USUARIOS = 150 FACTOR DE MULTIPROGRAMACION = TIEMPO DE RESPUESTA TIEMPO DE RESPUESTA PRODUCTIVIDAD C1 = PRODUCTIVIDAD C2 = PRODUCTIVIDAD TOTAL 1 DE LA CLASE C1 = DE LA CLASE C2 = 0.2059E-02 0.1356E-02 = 0.3415E-02 DE LA CLASE C1 = DE LA CLASE C2 = 0.2131E-02 0.1380E-02 = 0.3511E-02

1203. 1368. +/+/- 132.0 145.4 292.9 499.6 +/+/- 14.44 23.42 2 Facultade de Informatica. A Corua. Junio 2005 229 SIMULATION TECHNIQUES Communication network Identical to the previously studied by analytical techniques Facultade de Informatica. A Corua. Junio 2005 230 SIMULATION TECHNIQUES Communication network 1 /DECLARE/ QUEUE INTEGER J; 2 QUEUE GEN,LINIA(8,8),R(4,4); 3 INTEGER DIREC(8,4)=(0,2,2,2, 4

1,4,4,8, 5 4,0,4,4, 6 2,3,6,8, 7 6,6,0,6, 8 8,4,5,8, 9 8,8,8,0, 10 2,4,6,7); 11 12 REAL TRAB,TRAC,TRAD,TRBA,TRBC,TRBD,TRCA,TRCB,TRCD,TR ==> DA,TRDB,TRDC; 13 INTEGER OD(4,4) = ( 0, 60,140,240, 14 315,315,365,390, 15 470,590,590,630, 16 730,880,930,930); 17 INTEGER INIC(4) = (1, 3, 5, 7); Facultade de Informatica. A Corua. Junio 2005 231 SIMULATION TECHNIQUES Communication network 18 INTEGER I,K; 19 CUSTOMER INTEGER ORIG,DESTI; 20 REF CUSTOMER C; 21 /EXEC/

FOR I := 1 STEP 1 UNTIL 8 DO FOR K := 1 STEP 1 UNTIL ==> 8 DO 22 LINIA(I,K).J := K; 23 /STATION/ NAME = GEN; 24 TYPE = SOURCE; 25 SERVICE = BEGIN 26 EXP(60000./930.); 27 I:= RINT(1,930); 28 ORIG := 1; 29 DESTI := 1; 30 WHILE (I > OD(ORIG,DESTI)) DO 31 IF DESTI < 4 THEN DESTI := DESTI + 1 32 ELSE 33 BEGIN 34 ORIG := ORIG + 1; 35 DESTI := 1; 36 END; 37 C := NEW(CUSTOMER); 38 TRANSIT(C,R(ORIG,DESTI)); 39 TRANSIT(LINIA(INIC(ORIG),DIREC(INIC(ORIG ==> ),DESTI))); 40 END; Facultade de Informatica. A Corua. Junio 2005

232 SIMULATION TECHNIQUES Communication network 42 /STATION/ NAME = LINIA; 43 SERVICE = BEGIN 44 EXP(256.*8./64.); 45 I:=DIREC(J,DESTI); 46 IF I=0 THEN 47 BEGIN 48 C := R(ORIG,DESTI).FIRST; 49 WHILE C <> SON DO C := C.NEXT; 50 TRANSIT(C,OUT); 51 TRANSIT (OUT); 52 END; 53 TRANSIT (LINIA(J,I)); 54 END; 55 56 /CONTROL/ TMAX= 300000; 57 ACCURACY= ALL QUEUE; Facultade de Informatica. A Corua. Junio 2005 233 SIMULATION TECHNIQUES Communication network 59 /EXEC/

60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 BEGIN PRINT; SIMUL; TRAB:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,4))+ MRESPONSE(LINIA(4,3)); TRAC:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,4))+ MRESPONSE(LINIA(4,6))+MRESPONSE(LINIA(6,5)); TRAD:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,8))+ MRESPONSE(LINIA(8,7)); TRBA:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,2))+ MRESPONSE(LINIA(2,1)); TRBC:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,6))+ MRESPONSE(LINIA(6,5)); TRBD:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,8))+ MRESPONSE(LINIA(8,7)); TRCA:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+ MRESPONSE(LINIA(8,2))+MRESPONSE(LINIA(2,1)); Facultade de Informatica. A Corua. Junio 2005 234 SIMULATION TECHNIQUES Communication network 76

77 78 79 80 81 82 83 84 85 86 87 88 89 90 TRCD:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+ MRESPONSE(LINIA(8,7)); TRCB:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,4))+ MRESPONSE(LINIA(4,3)); TRDA:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,2))+ MRESPONSE(LINIA(2,1)); TRDB:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,4))+ MRESPONSE(LINIA(4,3)); TRDC:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,6))+ MRESPONSE(LINIA(6,5)); PRINT(TRAB,TRAC,TRAD); PRINT(TRBA,TRBC,TRBD); PRINT(TRCA,TRCB,TRCD); PRINT(TRDA,TRDB,TRDC); END; Facultade de Informatica. A Corua. Junio 2005 235 SIMULATION TECHNIQUES Communication network ***SIMULATION WITH SPECTRAL METHOD *** .. TIME = 300000.00 , NB SAMPLES = 512 , CONF. LEVEL = 0.95 *******************************************************************

* NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * * * * * * * * GEN * 64.03 * 1.000 * 1.000 * 64.03 * 4684* * +/* 1.829 *0.0000E+00*0.0000E+00* 1.829 * * * * * * * * * * LINIA 2 * 30.88 *0.1246 *0.1477 * 36.61 * 1210* * +/* 2.017 *0.1055E-01*0.1607E-01* 3.188 * * * *

* * * * * * LINIA 9 * 31.34 *0.1347 *0.1531 * 35.63 * 1289* * +/* 1.999 *0.1170E-01*0.1384E-01* 2.597 * * * * * * * * * * LINIA 12 * 31.07 *0.7519E-01*0.8277E-01* 34.20 * 726* * +/* 2.504 *0.7059E-02*0.1367E-01* 3.444 * * * * * * * * * Facultade de Informatica. A Corua. Junio 2005 236

SIMULATION TECHNIQUES Communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * LINIA 16 * 34.91 *0.5632E-01*0.6102E-01* 37.82 * 484* * +/* 4.215 *0.9778E-02*0.1091E-01* 5.125 * * * * * * * * * * LINIA 20 * 32.59 *0.7801E-01*0.8174E-01* 34.15 * 718* * +/* 2.378 *0.7146E-02*0.8014E-02* 2.579 * * * * * * * * * * LINIA 26 * 34.57 *0.4171E-01*0.4353E-01* 36.08

* 362* * +/* 3.707 *0.6404E-02*0.6778E-02* 4.122 * * * * * * * * * * LINIA 27 * 32.92 *0.1857 *0.2277 * 40.37 * 1691* * +/* 1.851 *0.1625E-01*0.2106E-01* 3.039 * * * * * * * * * * LINIA 30 * 35.30 *0.7448E-01*0.8044E-01* 38.12 * 633* * +/* 3.819 *0.9264E-02*0.1115E-01* 4.429 * * Facultade de Informatica. A Corua. Junio 2005

237 SIMULATION TECHNIQUES Communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * LINIA 32 * 28.69 *0.1243E-01*0.1243E-01* 28.69 * 130* * +/* 4.742 *0.2918E-02*0.2918E-02* 4.742 * * * * * * * * * * LINIA 38 * 30.73 *0.1345 *0.1547 * 35.35 * 1313* * +/* 1.734 *0.1065E-01*0.1591E-01* 2.745 * * * * * * * *

* * LINIA 44 * 32.35 *0.6922E-01*0.7420E-01* 34.68 * 642* * +/* 2.659 *0.8011E-02*0.9049E-02* 3.228 * * * * * * * * * * LINIA 45 * 31.92 *0.9321E-01*0.1024 * 35.07 * 876* * +/* 2.950 *0.8071E-02*0.1099E-01* 3.485 * * * * * * * * * * LINIA 48 * 29.55 *0.6609E-01*0.6892E-01* 30.81 * 671* * +/* 2.910 *0.6452E-02*0.6856E-02* 2.968 * *

* * * * * * * Facultade de Informatica. A Corua. Junio 2005 238 SIMULATION TECHNIQUES Communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * LINIA 56 * 32.19 *0.1548 *0.1929 * 40.11 * 1443* * +/* 2.196 *0.1281E-01*0.2053E-01* 3.851 * * * * * * * * * * LINIA 58 * 32.63 *0.1008 *0.1104 * 35.71 * 927*

* +/* 2.415 *0.9855E-02*0.1273E-01* 3.771 * * * * * * * * * * LINIA 60 * 33.53 *0.8171E-01*0.8924E-01* 36.62 * 731* * +/* 2.046 *0.8194E-02*0.9643E-02* 3.040 * * * * * * * * * * LINIA 62 * 35.31 *0.2860E-01*0.2887E-01* 35.64 * 243* * +/* 5.058 *0.5594E-02*0.5755E-02* 5.120 * * * * * * * *

* * LINIA 63 * 34.11 *0.9402E-01*0.1019 * 36.97 * 827* * +/* 2.188 *0.8583E-02*0.9742E-02* 2.344 * * Facultade de Informatica. A Corua. Junio 2005 239 SIMULATION TECHNIQUES Communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * R 2 *0.0000E+00*0.0000E+00*0.1180 * 110.9 * 319* * +/*0.0000E+00*0.0000E+00*0.3193E-01* 9.902 * * * * * * * * * * R 3 *0.0000E+00*0.0000E+00*0.1920 * 141.5 *

407* * +/*0.0000E+00*0.0000E+00*0.2514E-01* 14.12 * * * * * * * * * * R 4 *0.0000E+00*0.0000E+00*0.1836 * 113.8 * 484* * +/*0.0000E+00*0.0000E+00*0.2303E-01* 8.691 * * * * * * * * * * R 5 *0.0000E+00*0.0000E+00*0.1280 * 106.1 * 362* * +/*0.0000E+00*0.0000E+00*0.1609E-01* 7.103 * * * * * * * *

* * R 7 *0.0000E+00*0.0000E+00*0.8297E-01* 110.1 * 226* * +/*0.0000E+00*0.0000E+00*0.1289E-01* 9.790 * * Facultade de Informatica. A Corua. Junio 2005 240 SIMULATION TECHNIQUES Communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * R 8 *0.0000E+00*0.0000E+00*0.4081E-01* 94.18 * 130* * +/*0.0000E+00*0.0000E+00*0.7638E-02* 9.646 * * * * * * * * * * R 9 *0.0000E+00*0.0000E+00*0.2082 * 136.4 * 458* * +/*0.0000E+00*0.0000E+00*0.2374E-01* 6.981

* * * * * * * * * * R 10 *0.0000E+00*0.0000E+00*0.2370 * 110.8 * 642* * +/*0.0000E+00*0.0000E+00*0.2296E-01* 6.225 * * * * * * * * * * R 12 *0.0000E+00*0.0000E+00*0.7255E-01* 102.2 * 213* * +/*0.0000E+00*0.0000E+00*0.7541E-02* 7.903 * * * * * * * * * * R 13 *0.0000E+00*0.0000E+00*0.1787 * 114.3

* 469* * +/*0.0000E+00*0.0000E+00*0.2146E-01* 8.399 * * Facultade de Informatica. A Corua. Junio 2005 241 SIMULATION TECHNIQUES Communication network ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * R 14 *0.0000E+00*0.0000E+00*0.2829 * 116.2 * 730* * +/*0.0000E+00*0.0000E+00*0.3170E-01* 6.388 * * * * * * * * * * R 15 *0.0000E+00*0.0000E+00*0.8914E-01* 110.0 * 243* * +/*0.0000E+00*0.0000E+00*0.1678E-01* 9.066 * * *

* * * * * * ******************************************************************* ... END OF SIMULATION ... 111.2 105.9 137.5 111.4 144.0 107.3 110.4 117.1 111.4 99.80 103.1 110.8 Facultade de Informatica. A Corua. Junio 2005 242 SIMULATION TECHNIQUES Token ring network 8 nodes Uniform traffic Facultade de Informatica. A Corua. Junio 2005 243 SIMULATION TECHNIQUES

Token ring network 1 /DECLARE/ QUEUE INTEGER N, M; 2 QUEUE NUS(8),ESP(8),S,R; 3 INTEGER I; 4 REAL TARR; 5 CUSTOMER REAL TSERV; 6 CUSTOMER INTEGER ORIGEN,DESTI; 7 REF CUSTOMER C,D; 8 FLAG SEM; 9 CLASS TOK,MIS; 10 Facultade de Informatica. A Corua. Junio 2005 244 SIMULATION TECHNIQUES Token ring network 11 /STATION/ NAME = S; 12 TYPE = SOURCE; 13 SERVICE = BEGIN 14 EXP(TARR); 15 ORIGEN := RINT(1,8); 16 DESTI := RINT(1,7); 17 IF DESTI >= ORIGEN THEN DESTI := DESTI + 1; 18 TSERV := EXP(800.); & microsegons

19 C := NEW(CUSTOMER); 20 TRANSIT(C,R,MIS); 21 TRANSIT(ESP(ORIGEN),MIS); 22 END; 23 Facultade de Informatica. A Corua. Junio 2005 245 SIMULATION TECHNIQUES Token ring network 24 /STATION/ NAME = NUS; 25 TYPE = MULTIPLE(2); 26 SERVICE(TOK) = BEGIN 27 WHILE ESP(N).NB > 0 DO 28 BEGIN 29 D := ESP(N).FIRST; 30 CST(D.TSERV); 31 TRANSIT(D,NUS(M)); 32 UNSET(SEM); 33 WAIT(SEM); 34 END; 35 CST(20); & microsegons 36 TRANSIT(NUS(M)); 37

END; Facultade de Informatica. A Corua. Junio 2005 246 SIMULATION TECHNIQUES Token ring network 38 SERVICE(MIS) = BEGIN 39 IF N = ORIGEN THEN 40 BEGIN 41 SET(SEM); 42 TRANSIT(OUT); 43 END; 44 IF N = DESTI THEN 45 BEGIN 46 C := R.FIRST; 47 WHILE SON <> C DO C := C.NEXT; 48 TRANSIT(C,OUT); 49 END; 50 CST(1.6); & microsegons 51 TRANSIT(NUS(M)); 52 END; 53 54 /STATION/ NAME = NUS(1); 55 INIT(TOK) = 1;

Facultade de Informatica. A Corua. Junio 2005 247 SIMULATION TECHNIQUES Token ring network 57 /CONTROL/ TMAX = 10000000; CLASS = ALL QUEUE; 58 ACCURACY = ALL QUEUE, ALL CLASS; 59 60 /EXEC/ BEGIN 61 FOR I := 1 STEP 1 UNTIL 8 DO 62 BEGIN 63 NUS(I).N := I; 64 ESP(I).N := I; 65 NUS(I).M := I + 1; 66 END; 67 NUS(8).M := 1; 68 FOR TARR := 100000, 10000, 1000, 800 DO 69 BEGIN 70 PRINT; 71 PRINT ("TEMPS ENTRE ARRIBADES ",TARR,"MICROSEG"); 72 SIMUL; 73 END; 74 END; Facultade de Informatica. A Corua. Junio 2005

248 SIMULATION TECHNIQUES Token ring network TEMPS ENTRE ARRIBADES 0.1000E+06MICROSEG ***SIMULATION WITH SPECTRAL METHOD *** ... TIME = 10000000.00 , NB SAMPLES = 512 , CONF. LEVEL = 0.95 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * NUS 1 * 20.11 *0.6248E-01*0.1250 * 20.11 * 62141* * +/*0.1347 *0.3563E-03*0.7126E-03*0.1347 * * *(TOK )* 20.14 *0.6247E-01*0.1249 * 20.14 * 62038* * +/*0.1349 *0.3569E-03*0.7137E-03*0.1349 * * *(MIS )* 1.398 *0.7199E-05*0.1440E-04* 1.398 * 103*

* +/*-1.000 *0.1365E-05*0.2730E-05*-1.000 * * Facultade de Informatica. A Corua. Junio 2005 249 SIMULATION TECHNIQUES Token ring network * NUS * +/*(TOK * +/*(MIS * +/* NUS * +/*(TOK * +/*(MIS * +/* NUS * +/*(TOK * +/*(MIS * +/- 2 * 20.07 *0.6237E-01*0.1247 * 20.07 * *0.8790E-01*0.3143E-03*0.6286E-03*0.8790E-01* )* 20.10 *0.6236E-01*0.1247 * 20.10 * *0.8881E-01*0.3151E-03*0.6302E-03*0.8881E-01* )* 1.382

*0.7119E-05*0.1424E-04* 1.382 * *-1.000 *0.1691E-05*0.3381E-05*-1.000 * 3 * 20.09 *0.6242E-01*0.1248 * 20.09 * *0.7185E-01*0.2305E-03*0.4610E-03*0.7185E-01* )* 20.12 *0.6241E-01*0.1248 * 20.12 * *0.7276E-01*0.2315E-03*0.4629E-03*0.7276E-01* )* 1.351 *0.6959E-05*0.1392E-04* 1.351 * *-1.000 *0.1862E-05*0.3724E-05*-1.000 * 4 * 20.08 *0.6239E-01*0.1248 * 20.08 * *0.1148 *0.3504E-03*0.7009E-03*0.1148 * )* 20.11 *0.6238E-01*0.1248 * 20.11 * *0.1140 *0.3514E-03*0.7028E-03*0.1140 * )* 1.522 *0.7839E-05*0.1568E-04* 1.522 * *-1.000 *0.1646E-05*0.3292E-05*-1.000 * Facultade de Informatica. A Corua. Junio 2005

62140* * 62037* * 103* * 62140* * 62037* * 103* * 62140* * 62037* * 103* * 250 SIMULATION TECHNIQUES Token ring network * NUS * +/*(TOK * +/*(MIS * +/* NUS * +/*(TOK * +/*(MIS * +/* NUS * +/*(TOK * +/*(MIS * +/-

5 * 20.19 *0.6274E-01*0.1255 * 20.19 * *0.1325 *0.3989E-03*0.7978E-03*0.1325 * )* 20.22 *0.6273E-01*0.1255 * 20.22 * *0.1470 *0.3994E-03*0.7988E-03*0.1470 * )* 1.382 *0.7119E-05*0.1424E-04* 1.382 * *-1.000 *0.1337E-05*0.2674E-05*-1.000 * 6 * 20.11 *0.6249E-01*0.1250 * 20.11 * *0.1433 *0.4475E-03*0.8951E-03*0.1433 * )* 20.14 *0.6248E-01*0.1250 * 20.14 * *0.1429 *0.4480E-03*0.8960E-03*0.1429 * )* 1.398 *0.7199E-05*0.1440E-04* 1.398 * *-1.000 *0.1526E-05*0.3053E-05*-1.000 * 7 * 20.11 *0.6249E-01*0.1250

* 20.11 * *0.1094 *0.3093E-03*0.6186E-03*0.1094 * )* 20.15 *0.6249E-01*0.1250 * 20.15 * *0.9936E-01*0.3094E-03*0.6187E-03*0.9936E-01* )* 1.398 *0.7199E-05*0.1440E-04* 1.398 * *-1.000 *0.1483E-05*0.2966E-05*-1.000 * Facultade de Informatica. A Corua. Junio 2005 62140* * 62037* * 103* * 62140* * 62037* * 103* * 62140* * 62037* * 103* * 251 SIMULATION TECHNIQUES Token ring network * NUS

* +/*(TOK * +/*(MIS * +/* ESP * +/*(MIS * +/* ESP * +/*(MIS * +/* ESP * +/*(MIS * +/- 8 * 20.17 *0.6268E-01*0.1254 * 20.17 *0.1637 *0.4324E-03*0.8647E-03*0.1637 )* 20.20 *0.6267E-01*0.1253 * 20.20 *0.1642 *0.4327E-03*0.8654E-03*0.1642 )* 1.367 *0.7039E-05*0.1408E-04* 1.367 *-1.000 *0.1334E-05*0.2667E-05*-1.000 1 *0.0000E+00*0.0000E+00*0.9553E-03* 734.8 *-1.000 *0.0000E+00*0.8521E-03*-1.000 )*0.0000E+00*0.0000E+00*0.9553E-03* 734.8 *-1.000 *0.0000E+00*0.8521E-03*-1.000 2 *0.0000E+00*0.0000E+00*0.7404E-03* 528.8 *-1.000 *0.0000E+00*0.6695E-03*-1.000 )*0.0000E+00*0.0000E+00*0.7404E-03* 528.8

*-1.000 *0.0000E+00*0.6695E-03*-1.000 3 *0.0000E+00*0.0000E+00*0.8659E-03* 541.2 *-1.000 *0.0000E+00*0.4874E-03*-1.000 )*0.0000E+00*0.0000E+00*0.8659E-03* 541.2 *-1.000 *0.0000E+00*0.4874E-03*-1.000 Facultade de Informatica. A Corua. Junio 2005 * * * * * * * * * * * * * * * * * * 62140* * 62037* * 103* * 13* * 13* * 14* * 14*

* 16* * 16* * 252 SIMULATION TECHNIQUES Token ring network * ESP * +/*(MIS * +/* ESP * +/*(MIS * +/* ESP * +/*(MIS * +/* ESP * +/*(MIS * +/- 4 *0.0000E+00*0.0000E+00*0.7169E-03* 1434. *-1.000 *0.0000E+00*0.6916E-03*-1.000 )*0.0000E+00*0.0000E+00*0.7169E-03* 1434. *-1.000 *0.0000E+00*0.6916E-03*-1.000 5 *0.0000E+00*0.0000E+00*0.1484E-02* 1060. *-1.000 *0.0000E+00*0.9771E-03*-1.000 )*0.0000E+00*0.0000E+00*0.1484E-02* 1060. *-1.000 *0.0000E+00*0.9771E-03*-1.000 6 *0.0000E+00*0.0000E+00*0.9811E-03* 754.7 *-1.000

*0.0000E+00*0.9433E-03*-1.000 )*0.0000E+00*0.0000E+00*0.9811E-03* 754.7 *-1.000 *0.0000E+00*0.9433E-03*-1.000 7 *0.0000E+00*0.0000E+00*0.9871E-03* 759.3 *-1.000 *0.0000E+00*0.6366E-03*-1.000 )*0.0000E+00*0.0000E+00*0.9871E-03* 759.3 *-1.000 *0.0000E+00*0.6366E-03*-1.000 Facultade de Informatica. A Corua. Junio 2005 * * * * * * * * * * * * * * * * 5* * 5* * 14* * 14* * 13* * 13* * 13*

* 13* * 253 SIMULATION TECHNIQUES Token ring network * ESP 8 *0.0000E+00*0.0000E+00*0.1368E-02* 912.0 * 15* * +/*-1.000 *0.0000E+00*0.1041E-02*-1.000 * * *(MIS )*0.0000E+00*0.0000E+00*0.1368E-02* 912.0 * 15* * +/*-1.000 *0.0000E+00*0.1041E-02*-1.000 * * * S *0.9609E+05* 1.000 * 1.000 *0.9609E+05* 103* * +/*-1.000 *0.0000E+00*0.0000E+00*-1.000 * * * * * * * * * * R

*0.0000E+00*0.0000E+00*0.8148E-02* 791.0 * 103* * +/*-1.000 *0.0000E+00*0.2265E-02*-1.000 * * *(MIS )*0.0000E+00*0.0000E+00*0.8148E-02* 791.0 * 103* * +/*-1.000 *0.0000E+00*0.2265E-02*-1.000 * * * * * * * * * ******************************************************************* ... END OF SIMULATION ... Facultade de Informatica. A Corua. Junio 2005 254 SIMULATION TECHNIQUES Token ring network TEMPS ENTRE ARRIBADES 0.1000E+05MICROSEG ***SIMULATION WITH SPECTRAL METHOD *** ... TIME = 10000000.00 , NB SAMPLES = 512 , CONF. LEVEL = 0.95 ******************************************************************* * NAME

* SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * NUS 1 * 21.31 *0.6245E-01*0.1249 * 21.31 * 58612* * +/*0.4077 *0.1228E-02*0.2456E-02*0.4077 * * *(TOK )* 21.66 *0.6238E-01*0.1248 * 21.66 * 57608* * +/*0.4775 *0.1230E-02*0.2460E-02*0.4775 * * *(MIS )* 1.402 *0.7039E-04*0.1408E-03* 1.402 * 1004* * +/*0.3061E-01*0.5425E-05*0.1085E-04*0.3061E-01* * Facultade de Informatica. A Corua. Junio 2005 255 SIMULATION TECHNIQUES Token ring network * NUS * +/*(TOK *

+/*(MIS * +/* NUS * +/*(TOK * +/*(MIS * +/* NUS * +/*(TOK * +/*(MIS * +/- 2 * 21.70 *0.6360E-01*0.1272 * 21.70 * *0.5427 *0.1433E-02*0.2866E-02*0.5427 * )* 22.06 *0.6353E-01*0.1271 * 22.06 * *0.5847 *0.1434E-02*0.2868E-02*0.5847 * )* 1.381 *0.6935E-04*0.1387E-03* 1.381 * *0.4022E-01*0.4872E-05*0.9745E-05*0.4022E-01* 3 * 21.54 *0.6314E-01*0.1263 * 21.54 * *0.5847 *0.1705E-02*0.3410E-02*0.5847 * )* 21.90 *0.6307E-01*0.1261

* 21.90 * *0.5530 *0.1706E-02*0.3412E-02*0.5530 * )* 1.393 *0.6991E-04*0.1398E-03* 1.393 * *0.3717E-01*0.4980E-05*0.9959E-05*0.3717E-01* 4 * 21.30 *0.6241E-01*0.1248 * 21.30 * *0.4710 *0.1174E-02*0.2348E-02*0.4710 * )* 21.64 *0.6234E-01*0.1247 * 21.64 * *0.4629 *0.1175E-02*0.2349E-02*0.4629 * )* 1.399 *0.7023E-04*0.1405E-03* 1.399 * *0.3573E-01*0.4258E-05*0.8516E-05*0.3573E-01* Facultade de Informatica. A Corua. Junio 2005 58612* * 57608* * 1004* * 58612* * 57608* * 1004* * 58611*

* 57607* * 1004* * 256 SIMULATION TECHNIQUES Token ring network * NUS * +/*(TOK * +/*(MIS * +/* NUS * +/*(TOK * +/*(MIS * +/* NUS * +/*(TOK * +/*(MIS * +/- 5 * 21.02 *0.6161E-01*0.1232 * 21.02 * *0.4159 *0.9359E-03*0.1872E-02*0.4159 * )* 21.37 *0.6154E-01*0.1231 * 21.37 * *0.4241 *0.9370E-03*0.1874E-02*0.4241

* )* 1.415 *0.7103E-04*0.1421E-03* 1.415 * *0.3856E-01*0.7813E-05*0.1563E-04*0.3856E-01* 6 * 21.06 *0.6171E-01*0.1234 * 21.06 * *0.4502 *0.1248E-02*0.2497E-02*0.4502 * )* 21.40 *0.6164E-01*0.1233 * 21.40 * *0.4786 *0.1249E-02*0.2499E-02*0.4786 * )* 1.413 *0.7095E-04*0.1419E-03* 1.413 * *0.3104E-01*0.4489E-05*0.8978E-05*0.3104E-01* 7 * 21.39 *0.6269E-01*0.1254 * 21.39 * *0.4933 *0.1266E-02*0.2532E-02*0.4933 * )* 21.74 *0.6262E-01*0.1252 * 21.74 * *0.3923 *0.1268E-02*0.2537E-02*0.3923 * )* 1.410 *0.7079E-04*0.1416E-03* 1.410 * *0.3545E-01*0.7663E-05*0.1533E-04*0.3545E-01* Facultade de Informatica. A Corua. Junio 2005

58611* * 57607* * 1004* * 58611* * 57607* * 1004* * 58611* * 57607* * 1004* * 257 SIMULATION TECHNIQUES Token ring network * NUS * +/*(TOK * +/*(MIS * +/* ESP * +/*(MIS * +/* ESP * +/*(MIS * +/* ESP * +/*(MIS * +/-

8 * 21.48 *0.6294E-01*0.1259 * 21.48 * *0.6782 *0.1235E-02*0.2471E-02*0.6782 * )* 21.83 *0.6287E-01*0.1257 * 21.83 * *0.7549 *0.1235E-02*0.2470E-02*0.7549 * )* 1.385 *0.6951E-04*0.1390E-03* 1.385 * *0.3685E-01*0.4594E-05*0.9188E-05*0.3685E-01* 1 *0.0000E+00*0.0000E+00*0.1149E-01* 926.7 * *-1.000 *0.0000E+00*0.3546E-02*-1.000 * )*0.0000E+00*0.0000E+00*0.1149E-01* 926.7 * *-1.000 *0.0000E+00*0.3546E-02*-1.000 * 2 *0.0000E+00*0.0000E+00*0.1344E-01* 980.8 * *0.0000E+00*0.0000E+00*0.3167E-02* 125.4 * )*0.0000E+00*0.0000E+00*0.1344E-01* 980.8 * *0.0000E+00*0.0000E+00*0.3167E-02* 125.4 * 3 *0.0000E+00*0.0000E+00*0.1317E-01* 1013. * *0.0000E+00*0.0000E+00*0.3675E-02* 180.5 * )*0.0000E+00*0.0000E+00*0.1317E-01* 1013. *

*0.0000E+00*0.0000E+00*0.3675E-02* 180.5 * Facultade de Informatica. A Corua. Junio 2005 58611* * 57607* * 1004* * 124* * 124* * 137* * 137* * 130* * 130* * 258 SIMULATION TECHNIQUES Token ring network * ESP * +/*(MIS * +/* ESP * +/*(MIS * +/* ESP * +/*(MIS * +/* ESP *

+/*(MIS * +/- 4 *0.0000E+00*0.0000E+00*0.1155E-01* 916.4 *-1.000 *0.0000E+00*0.3341E-02*-1.000 )*0.0000E+00*0.0000E+00*0.1155E-01* 916.4 *-1.000 *0.0000E+00*0.3341E-02*-1.000 5 *0.0000E+00*0.0000E+00*0.1060E-01* 914.1 *-1.000 *0.0000E+00*0.3776E-02*-1.000 )*0.0000E+00*0.0000E+00*0.1060E-01* 914.1 *-1.000 *0.0000E+00*0.3776E-02*-1.000 6 *0.0000E+00*0.0000E+00*0.9766E-02* 834.7 *-1.000 *0.0000E+00*0.2652E-02*-1.000 )*0.0000E+00*0.0000E+00*0.9766E-02* 834.7 *-1.000 *0.0000E+00*0.2652E-02*-1.000 7 *0.0000E+00*0.0000E+00*0.1208E-01* 1015. *-1.000 *0.0000E+00*0.3174E-02*-1.000 )*0.0000E+00*0.0000E+00*0.1208E-01* 1015. *-1.000 *0.0000E+00*0.3174E-02*-1.000 Facultade de Informatica. A Corua. Junio 2005 * * * * * * * * * * * *

* * * * 126* * 126* * 116* * 116* * 117* * 117* * 119* * 119* * 259 SIMULATION TECHNIQUES Token ring network * ESP 8 *0.0000E+00*0.0000E+00*0.1295E-01* 959.5 * 135* * +/*0.0000E+00*0.0000E+00*0.3163E-02* 200.6 * * *(MIS )*0.0000E+00*0.0000E+00*0.1295E-01* 959.5 * 135* * +/*0.0000E+00*0.0000E+00*0.3163E-02* 200.6 * * * S * 9958.

* 1.000 * 1.000 * 9958. * 1004* * +/* 670.0 *0.0000E+00*0.0000E+00* 670.0 * * * * * * * * * * R *0.0000E+00*0.0000E+00*0.9552E-01* 951.4 * 1004* * +/*0.0000E+00*0.0000E+00*0.9458E-02* 75.77 * * *(MIS )*0.0000E+00*0.0000E+00*0.9552E-01* 951.4 * 1004* * +/*0.0000E+00*0.0000E+00*0.9458E-02* 75.77 * * * * * * * * * ******************************************************************* ... END OF SIMULATION ...

Facultade de Informatica. A Corua. Junio 2005 260 SIMULATION TECHNIQUES Token ring network TEMPS ENTRE ARRIBADES 1000. MICROSEG ***SIMULATION WITH SPECTRAL METHOD *** ... TIME = 10000000.00 , NB SAMPLES = 512 , CONF. LEVEL = 0.95 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * * * * * * * * NUS 1 * 61.63 *0.6358E-01*0.1272 * 61.63 * 20631* * +/* 6.193 *0.3718E-02*0.7435E-02* 6.193 * * *(TOK )* 119.1 *0.6288E-01*0.1258 * 119.1 *

10562* * +/* 20.65 *0.3716E-02*0.7433E-02* 20.65 * * *(MIS )* 1.390 *0.6999E-03*0.1400E-02* 1.390 * 10069* * +/*0.9350E-02*0.1776E-04*0.3552E-04*0.9350E-02* * Facultade de Informatica. A Corua. Junio 2005 261 SIMULATION TECHNIQUES Token ring network * NUS * +/*(TOK * +/*(MIS * +/* NUS * +/*(TOK * +/*(MIS * +/* NUS * +/*(TOK * +/*(MIS * +/- 2 * 63.39 *0.6539E-01*0.1308

* 63.39 * * 4.534 *0.3894E-02*0.7787E-02* 4.534 * )* 122.5 *0.6468E-01*0.1294 * 122.5 * * 16.47 *0.3904E-02*0.7809E-02* 16.47 * )* 1.396 *0.7026E-03*0.1405E-02* 1.396 * *0.1102E-01*0.2228E-04*0.4456E-04*0.1102E-01* 3 * 62.98 *0.6497E-01*0.1299 * 62.98 * * 5.727 *0.3575E-02*0.7150E-02* 5.727 * )* 121.7 *0.6427E-01*0.1285 * 121.7 * * 29.96 *0.3579E-02*0.7158E-02* 29.96 * )* 1.389 *0.6994E-03*0.1399E-02* 1.389 * *0.9545E-02*0.1971E-04*0.3942E-04*0.9545E-02* 4 * 60.30 *0.6220E-01*0.1244 * 60.30 * * 5.706 *0.4437E-02*0.8874E-02* 5.706 * )* 116.4 *0.6149E-01*0.1230

* 116.4 * * 19.09 *0.4432E-02*0.8865E-02* 19.09 * )* 1.409 *0.7095E-03*0.1419E-02* 1.409 * *0.1115E-01*0.2008E-04*0.4017E-04*0.1115E-01* Facultade de Informatica. A Corua. Junio 2005 20631* * 10562* * 10069* * 20631* * 10562* * 10069* * 20631* * 10562* * 10069* * 262 SIMULATION TECHNIQUES Token ring network * NUS * +/*(TOK * +/*(MIS * +/* NUS *

+/*(TOK * +/*(MIS * +/* NUS * +/*(TOK * +/*(MIS * +/- 5 * 60.82 *0.6274E-01*0.1255 * 60.82 * * 7.048 *0.4138E-02*0.8275E-02* 7.048 * )* 117.5 *0.6203E-01*0.1241 * 117.5 * * 19.66 *0.4143E-02*0.8285E-02* 19.66 * )* 1.404 *0.7069E-03*0.1414E-02* 1.404 * *0.1153E-01*0.1863E-04*0.3726E-04*0.1153E-01* 6 * 58.71 *0.6056E-01*0.1211 * 58.71 * * 6.512 *0.4101E-02*0.8203E-02* 6.512 * )* 113.3 *0.5985E-01*0.1197 * 113.3 * * 30.61 *0.4105E-02*0.8210E-02* 30.61

* )* 1.411 *0.7102E-03*0.1420E-02* 1.411 * *0.1168E-01*0.2026E-04*0.4051E-04*0.1168E-01* 7 * 57.14 *0.5913E-01*0.1183 * 57.14 * * 6.698 *0.3162E-02*0.6325E-02* 6.698 * )* 110.3 *0.5842E-01*0.1168 * 110.3 * * 18.34 *0.3164E-02*0.6328E-02* 18.34 * )* 1.409 *0.7096E-03*0.1419E-02* 1.409 * *0.9951E-02*0.2002E-04*0.4005E-04*0.9951E-02* Facultade de Informatica. A Corua. Junio 2005 20631* * 10562* * 10069* * 20631* * 10562* * 10069* * 20630* * 10561* * 10069*

* 263 SIMULATION TECHNIQUES Token ring network * NUS * +/*(TOK * +/*(MIS * +/* ESP * +/*(MIS * +/* ESP * +/*(MIS * +/* ESP * +/*(MIS * +/- 8 * 65.03 *0.6708E-01*0.1342 * 65.03 * * 6.675 *0.3661E-02*0.7321E-02* 6.675 * )* 125.7 *0.6638E-01*0.1328 * 125.7 * * 32.18 *0.3660E-02*0.7320E-02* 32.18 * )* 1.390 *0.6998E-03*0.1400E-02* 1.390 *

*0.1438E-01*0.1897E-04*0.3795E-04*0.1438E-01* 1 *0.0000E+00*0.0000E+00*0.7598 * 5756. * *0.0000E+00*0.0000E+00*0.1753 * 1254. * )*0.0000E+00*0.0000E+00*0.7598 * 5756. * *0.0000E+00*0.0000E+00*0.1753 * 1254. * 2 *0.0000E+00*0.0000E+00*0.7163 * 5564. * *0.0000E+00*0.0000E+00*0.1420 * 1172. * )*0.0000E+00*0.0000E+00*0.7163 * 5564. * *0.0000E+00*0.0000E+00*0.1420 * 1172. * 3 *0.0000E+00*0.0000E+00*0.7218 * 5436. * *0.0000E+00*0.0000E+00*0.1340 * 1028. * )*0.0000E+00*0.0000E+00*0.7218 * 5436. * *0.0000E+00*0.0000E+00*0.1340 * 1028. * Facultade de Informatica. A Corua. Junio 2005 20630* * 10561*

* 10069* * 1319* * 1319* * 1285* * 1285* * 1325* * 1325* * 264 SIMULATION TECHNIQUES Token ring network * ESP * +/*(MIS * +/* ESP * +/*(MIS * +/* ESP * +/*(MIS * +/* ESP * +/*(MIS * +/- 4 *0.0000E+00*0.0000E+00*0.6485 *0.0000E+00*0.0000E+00*0.1442 )*0.0000E+00*0.0000E+00*0.6485 *0.0000E+00*0.0000E+00*0.1442

5 *0.0000E+00*0.0000E+00*0.7021 *0.0000E+00*0.0000E+00*0.1777 )*0.0000E+00*0.0000E+00*0.7021 *0.0000E+00*0.0000E+00*0.1777 6 *0.0000E+00*0.0000E+00*0.6706 *0.0000E+00*0.0000E+00*0.1593 )*0.0000E+00*0.0000E+00*0.6706 *0.0000E+00*0.0000E+00*0.1593 7 *0.0000E+00*0.0000E+00*0.6491 *0.0000E+00*0.0000E+00*0.1391 )*0.0000E+00*0.0000E+00*0.6491 *0.0000E+00*0.0000E+00*0.1391 Facultade de Informatica. A Corua. Junio 2005 * * * * * * * * * * * * * * * * 5402. 1177. 5402. 1177. 5699. 1054. 5699. 1054. 5635. 1203. 5635.

1203. 5413. 1111. 5413. 1111. * * * * * * * * * * * * * * * * 1199* * 1199* * 1232* * 1232* * 1190* * 1190* * 1198* * 1198* * 265 SIMULATION TECHNIQUES Token ring network

* ESP 8 *0.0000E+00*0.0000E+00*0.6883 * 5200. * 1321* * +/*0.0000E+00*0.0000E+00*0.1291 * 907.1 * * *(MIS )*0.0000E+00*0.0000E+00*0.6883 * 5200. * 1321* * +/*0.0000E+00*0.0000E+00*0.1291 * 907.1 * * * S * 992.0 * 1.000 * 1.000 * 992.0 * 10079* * +/* 23.59 *0.0000E+00*0.0000E+00* 23.59 * * * * * * * * * * R *0.0000E+00*0.0000E+00* 5.561 * 5517. *

10069* * +/*0.0000E+00*0.0000E+00* 1.161 * 1063. * * *(MIS )*0.0000E+00*0.0000E+00* 5.561 * 5517. * 10069* * +/*0.0000E+00*0.0000E+00* 1.161 * 1063. * * * * * * * * * ******************************************************************* ... END OF SIMULATION ... Facultade de Informatica. A Corua. Junio 2005 266 SIMULATION TECHNIQUES Token ring network TEMPS ENTRE ARRIBADES 800.0 MICROSEG ***SIMULATION WITH SPECTRAL METHOD *** ... TIME = 10000000.00 , NB SAMPLES = 512 , CONF. LEVEL = 0.95 ******************************************************************* *

NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * NUS 1 * 109.5 *0.6738E-01*0.1348 * 109.5 * 12306* * +/* 6.905 *0.3586E-02*0.7173E-02* 6.905 * * *(TOK )*0.2511E+05*0.6653E-01*0.1331 *0.2511E+05* 53* * +/*-1.000 *0.3590E-02*0.7179E-02*-1.000 * * *(MIS )* 1.388 *0.8504E-03*0.1701E-02* 1.388 * 12253* * +/*0.6208E-02*0.9743E-05*0.1949E-04*0.6208E-02* * Facultade de Informatica. A Corua. Junio 2005 267 SIMULATION TECHNIQUES Token ring network * NUS * +/*(TOK * +/*(MIS

* +/* NUS * +/*(TOK * +/*(MIS * +/* NUS * +/*(TOK * +/*(MIS * +/- 2 * 101.4 *0.6239E-01*0.1248 * 101.4 * * 5.621 *0.2796E-02*0.5592E-02* 5.621 * )*0.2322E+05*0.6154E-01*0.1231 *0.2322E+05* *-1.000 *0.2801E-02*0.5602E-02*-1.000 * )* 1.401 *0.8583E-03*0.1717E-02* 1.401 * *0.6746E-02*0.1243E-04*0.2486E-04*0.6746E-02* 3 * 96.70 *0.5950E-01*0.1190 * 96.70 * * 5.425 *0.3074E-02*0.6148E-02* 5.425 * )*0.2213E+05*0.5864E-01*0.1173 *0.2213E+05* *-1.000 *0.3074E-02*0.6148E-02*-1.000 *

)* 1.409 *0.8635E-03*0.1727E-02* 1.409 * *0.6752E-02*0.1281E-04*0.2563E-04*0.6752E-02* 4 * 102.6 *0.6311E-01*0.1262 * 102.6 * * 5.116 *0.2576E-02*0.5152E-02* 5.116 * )*0.2349E+05*0.6225E-01*0.1245 *0.2349E+05* *-1.000 *0.2576E-02*0.5152E-02*-1.000 * )* 1.407 *0.8622E-03*0.1724E-02* 1.407 * *0.6784E-02*0.1131E-04*0.2261E-04*0.6784E-02* Facultade de Informatica. A Corua. Junio 2005 12306* * 53* * 12253* * 12306* * 53* * 12253* * 12306* * 53* * 12253* * 268

SIMULATION TECHNIQUES Token ring network * NUS * +/*(TOK * +/*(MIS * +/* NUS * +/*(TOK * +/*(MIS * +/* NUS * +/*(TOK * +/*(MIS * +/- 5 * 107.5 *0.6615E-01*0.1323 * 107.5 * * 3.395 *0.1742E-02*0.3485E-02* 3.395 * )*0.2464E+05*0.6529E-01*0.1306 *0.2464E+05* *-1.000 *0.1747E-02*0.3494E-02*-1.000 * )* 1.399 *0.8573E-03*0.1715E-02* 1.399 * *0.5035E-02*0.1366E-04*0.2733E-04*0.5035E-02* 6 * 97.64 *0.6008E-01*0.1202 * 97.64 *

* 3.719 *0.3193E-02*0.6385E-02* 3.719 * )*0.2235E+05*0.5922E-01*0.1184 *0.2235E+05* *-1.000 *0.3198E-02*0.6396E-02*-1.000 * )* 1.402 *0.8588E-03*0.1718E-02* 1.402 * *0.6413E-02*0.1194E-04*0.2389E-04*0.6413E-02* 7 * 104.8 *0.6473E-01*0.1295 * 104.8 * * 4.550 *0.3379E-02*0.6758E-02* 4.550 * )*0.2446E+05*0.6387E-01*0.1277 *0.2446E+05* *-1.000 *0.3384E-02*0.6769E-02*-1.000 * )* 1.396 *0.8553E-03*0.1711E-02* 1.396 * *0.4878E-02*0.1412E-04*0.2824E-04*0.4878E-02* Facultade de Informatica. A Corua. Junio 2005 12306* * 53* * 12253* * 12306* * 53* * 12253* *

12305* * 52* * 12253* * 269 SIMULATION TECHNIQUES Token ring network * NUS * +/*(TOK * +/*(MIS * +/* ESP * +/*(MIS * +/* ESP * +/*(MIS * +/* ESP * +/*(MIS * +/- 8 * 103.2 *0.6352E-01*0.1270 * 103.2 * * 4.519 *0.2117E-02*0.4234E-02* 4.519 * )*0.2410E+05*0.6266E-01*0.1253 *0.2410E+05* *-1.000 *0.2115E-02*0.4230E-02*-1.000 *

)* 1.396 *0.8551E-03*0.1710E-02* 1.396 * *0.5278E-02*0.1169E-04*0.2338E-04*0.5278E-02* 1 *0.0000E+00*0.0000E+00* 20.89 *0.1285E+06* *0.0000E+00*0.0000E+00* 2.330 *0.1118E+05* )*0.0000E+00*0.0000E+00* 20.89 *0.1285E+06* *0.0000E+00*0.0000E+00* 2.330 *0.1118E+05* 2 *0.0000E+00*0.0000E+00* 19.79 *0.1297E+06* *0.0000E+00*0.0000E+00* 2.213 *0.1044E+05* )*0.0000E+00*0.0000E+00* 19.79 *0.1297E+06* *0.0000E+00*0.0000E+00* 2.213 *0.1044E+05* 3 *0.0000E+00*0.0000E+00* 18.60 *0.1274E+06* *0.0000E+00*0.0000E+00* 1.609 *0.1332E+05* )*0.0000E+00*0.0000E+00* 18.60 *0.1274E+06* *0.0000E+00*0.0000E+00* 1.609 *0.1332E+05* Facultade de Informatica. A Corua. Junio 2005 12305* * 52* * 12253* * 1622* * 1622* * 1523* *

1523* * 1458* * 1458* * 270 SIMULATION TECHNIQUES Token ring network * ESP * +/*(MIS * +/* ESP * +/*(MIS * +/* ESP * +/*(MIS * +/* ESP * +/*(MIS * +/- 4 *0.0000E+00*0.0000E+00* *0.0000E+00*0.0000E+00* )*0.0000E+00*0.0000E+00* *0.0000E+00*0.0000E+00* 5 *0.0000E+00*0.0000E+00* *0.0000E+00*0.0000E+00* )*0.0000E+00*0.0000E+00* *0.0000E+00*0.0000E+00* 6 *0.0000E+00*0.0000E+00* *0.0000E+00*0.0000E+00* )*0.0000E+00*0.0000E+00* *0.0000E+00*0.0000E+00* 7 *0.0000E+00*0.0000E+00*

*0.0000E+00*0.0000E+00* )*0.0000E+00*0.0000E+00* *0.0000E+00*0.0000E+00* 18.17 1.572 18.17 1.572 19.04 2.000 19.04 2.000 19.32 2.245 19.32 2.245 20.31 1.996 20.31 1.996 Facultade de Informatica. A Corua. Junio 2005 *0.1232E+06* *0.1280E+05* *0.1232E+06* *0.1280E+05* *0.1240E+06* *0.1414E+05* *0.1240E+06* *0.1414E+05* *0.1273E+06* *0.1378E+05* *0.1273E+06* *0.1378E+05* *0.1297E+06* *0.1299E+05* *0.1297E+06* *0.1299E+05* 1474* * 1474*

* 1535* * 1535* * 1517* * 1517* * 1561* * 1561* * 271 SIMULATION TECHNIQUES Token ring network * ESP 8 *0.0000E+00*0.0000E+00* 19.63 *0.1249E+06* 1563* * +/*0.0000E+00*0.0000E+00* 1.951 *0.1021E+05* * *(MIS )*0.0000E+00*0.0000E+00* 19.63 *0.1249E+06* 1563* * +/*0.0000E+00*0.0000E+00* 1.951 *0.1021E+05* * * S * 811.1 * 1.000 * 1.000 * 811.1 * 12328* * +/* 14.66 *0.0000E+00*0.0000E+00* 14.66

* * * * * * * * * * R *0.0000E+00*0.0000E+00* 155.7 *0.1269E+06* 12253* * +/*0.0000E+00*0.0000E+00* 24.78 *0.2250E+05* * *(MIS )*0.0000E+00*0.0000E+00* 155.7 *0.1269E+06* 12253* * +/*0.0000E+00*0.0000E+00* 24.78 *0.2250E+05* * * * * * * * * ******************************************************************* ... END OF SIMULATION ... Facultade de Informatica. A Corua. Junio 2005 272 SIMULATION TECHNIQUES Ethernet network

8 nodes Uniform traffic Facultade de Informatica. A Corua. Junio 2005 273 SIMULATION TECHNIQUES Ethernet network 1 /DECLARE/ QUEUE S, EST(8), BUS, R, SF; 2 CUSTOMER INTEGER I; 3 CUSTOMER REAL TSERV, TESP; 4 REAL TARR, SERV, ESP, TEMPS, TPROP, TEMPSC; 5 REF CUSTOMER C,D; 6 LABEL L; 7 FLAG FL, BL; 8 INTEGER CONF = 0; 9 Facultade de Informatica. A Corua. Junio 2005 274 SIMULATION TECHNIQUES Ethernet network 1 /DECLARE/ QUEUE S, EST(8), BUS, R, SF; 2 CUSTOMER INTEGER I; 3 CUSTOMER REAL TSERV, TESP; 4 REAL TARR, SERV, ESP, TEMPS, TPROP, TEMPSC; 5

REF CUSTOMER C,D; 6 LABEL L; 7 FLAG FL, BL; 8 INTEGER CONF = 0; 9 Facultade de Informatica. A Corua. Junio 2005 275 SIMULATION TECHNIQUES Ethernet network 10 /STATION/ NAME = S; 11 TYPE = SOURCE; 12 SERVICE = BEGIN 13 EXP(TARR); 14 I := RINT(1,8); 15 TSERV := EXP(SERV); 16 IF TSERV < 3.*TPROP THEN TSERV := 3.*TPROP; 17 C := NEW(CUSTOMER); 18 TRANSIT(C,R); 19 TRANSIT(EST(I)); 20 END; 21 Facultade de Informatica. A Corua. Junio 2005 276 SIMULATION TECHNIQUES

Ethernet network 22 /STATION/ NAME = EST; 23 SERVICE = BEGIN 24 L: IF BUS.NB = 0 THEN 25 BEGIN 26 C := NEW(CUSTOMER); 27 C.TSERV := TSERV; 28 UNSET(FL); 29 UNSET(BL); 30 TEMPS := TIME + TPROP; 31 TRANSIT(C, BUS); 32 WAIT(FL); Facultade de Informatica. A Corua. Junio 2005 277 SIMULATION TECHNIQUES Ethernet network 33 34 35 36 37 38 39 40 41 42 43 44 IF CONF = 0 THEN

BEGIN SET(BL); TRANSIT(OUT); END; SET(BL); CONF := 0; TESP := EXP(ESP); IF TESP<2.*TPROP THEN TESP := 2. * TPROP; CST(TESP); GOTO L; END; Facultade de Informatica. A Corua. Junio 2005 278 SIMULATION TECHNIQUES Ethernet network 45 46 47 48 49 50 51 52 53 54 55 56 57 IF (TIME < TEMPS) AND (CONF = 0) THEN BEGIN CONF := 1; D := BUS.FIRST; TEMPSC := TEMPS - TIME; CST(TEMPSC); TRANSIT(D, OUT); SET(FL); TESP := EXP(ESP); IF TESP<2.*TPROP THEN TESP := 2. * TPROP; CST(TESP);

GOTO L; END; Facultade de Informatica. A Corua. Junio 2005 279 SIMULATION TECHNIQUES Ethernet network 58 59 60 61 62 63 64 65 66 67 68 IF (TIME < TEMPS) AND (CONF = 1) THEN BEGIN TESP := EXP(ESP); IF TESP<2.*TPROP THEN TESP := 2. * TPROP; CST(TESP); GOTO L; END; WAIT(BL); GOTO L; END; Facultade de Informatica. A Corua. Junio 2005 280 SIMULATION TECHNIQUES Ethernet network 69 /STATION/ NAME = BUS; 70 SERVICE = BEGIN 71 CST(TSERV + 2.*TPROP);

72 C := R.FIRST; 73 WHILE C.FATHER <> FATHER DO C := C.NEXT; 74 TRANSIT(C, OUT); 75 SET(FL); 76 TRANSIT(OUT); 77 END; 78 Facultade de Informatica. A Corua. Junio 2005 281 SIMULATION TECHNIQUES Ethernet network 79 /STATION/ NAME = SF; 80 INIT = 1; 81 SERVICE = BEGIN 82 SET(FL); 83 SET(BL); 84 TRANSIT(OUT); 85 END; 86 87 /CONTROL/ TMAX = 100000.; ACCURACY = ALL QUEUE; 88 89 /EXEC/ BEGIN 90 TPROP := 0.01; 91 SERV := 0.8; 92 ESP := 0.1;

93 TEMPS := -TPROP; 94 FOR TARR := 5, 2.5, 1.25 DO SIMUL; 95 END; Facultade de Informatica. A Corua. Junio 2005 282 SIMULATION TECHNIQUES Ethernet network ***SIMULATION WITH SPECTRAL METHOD *** ... TIME = 100000.00 , NB SAMPLES = 512 , CONF. LEVEL = 0.95 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * S * * 5.073 +/- * EST *

+/- * EST * +/- * EST * +/- * EST * +/- * EST * +/- * 19713* *0.7027E-01*0.0000E+00*0.0000E+00*0.7027E-01* * 1 *0.9595 * 1.000 * 1.000 * 5.073 * 2491* *0.4293E-01*0.1102E-02*0.1180E-02*0.5266E-01*

* 2 *0.9714 *0.2390E-01*0.2459E-01*0.9873 * 2433* *0.3115E-01*0.1343E-02*0.1491E-02*0.3891E-01* * 3 *0.9993 *0.2364E-01*0.2425E-01*0.9966 * 2536* *0.4504E-01*0.1101E-02*0.1159E-02*0.4904E-01* * 4 *0.9660 *0.2534E-01*0.2609E-01* 1.029 * 2433* *0.4483E-01*0.1456E-02*0.1518E-02*0.7335E-01* * 5 *0.9709 *0.2350E-01*0.2407E-01*0.9892 *0.2321E-01*0.2356E-01*0.9856

* 2390* *0.4695E-01*0.1439E-02*0.1497E-02*0.4730E-01* * Facultade de Informatica. A Corua. Junio 2005 283 SIMULATION TECHNIQUES Ethernet network * EST 6 *0.9677 *0.2312E-01*0.2366E-01*0.9905 * 2389* * +/*0.3450E-01*0.9612E-03*0.1328E-02*0.4168E-01* * * EST 7 *0.9455 *0.2308E-01*0.2367E-01*0.9696 * 2441* * +/*0.3795E-01*0.1200E-02*0.1322E-02*0.3643E-01* * * EST 8 *0.9635 *0.2505E-01*0.2576E-01*0.9906 * 2600* * +/*0.3988E-01*0.1589E-02*0.1727E-02*0.4196E-01*

* * BUS *0.7994 *0.1624 *0.1624 *0.7994 * 20314* * +/*0.9631E-02*0.2830E-02*0.2830E-02*0.9631E-02* * * R *0.0000E+00*0.0000E+00*0.1956 *0.9924 * 19713* * +/*0.0000E+00*0.0000E+00*0.5210E-02*0.2011E-01* * ******************************************************************* ... END OF SIMULATION ... Facultade de Informatica. A Corua. Junio 2005 284 SIMULATION TECHNIQUES Ethernet network ***SIMULATION WITH SPECTRAL METHOD *** ... TIME = 100000.00 , NB SAMPLES = 512 , CONF. LEVEL = 0.95 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * S * 2.523 * 1.000 * 1.000 * 2.523 *

39639* * +/*0.3238E-01*0.0000E+00*0.0000E+00*0.3238E-01* * * EST 1 * 1.190 *0.6058E-01*0.6443E-01* 1.266 * 5091* * +/*0.3643E-01*0.1890E-02*0.2066E-02*0.4254E-01* * * EST 2 * 1.189 *0.5951E-01*0.6372E-01* 1.274 * 5003* * +/*0.3580E-01*0.4234E-02*0.3192E-02*0.4344E-01* * * EST 3 * 1.174 *0.5754E-01*0.6086E-01* 1.242 * 4901* * +/*0.4252E-01*0.2450E-02*0.2627E-02*0.4427E-01* * * EST 4 * 1.191 *0.5883E-01*0.6312E-01* 1.278 * 4937* * +/*0.3555E-01*0.2305E-02*0.2675E-02*0.4592E-01* * * EST 5 * 1.198 *0.5927E-01*0.6323E-01* 1.278 * 4949* * +/*0.4266E-01*0.2633E-02*0.2816E-02*0.4674E-01*

* Facultade de Informatica. A Corua. Junio 2005 285 SIMULATION TECHNIQUES Ethernet network * EST 6 * 1.185 *0.5835E-01*0.6266E-01* 1.272 * 4926* * +/*0.3425E-01*0.2123E-02*0.2498E-02*0.4209E-01* * * EST 7 * 1.181 *0.5895E-01*0.6315E-01* 1.265 * 4993* * +/*0.3740E-01*0.2661E-02*0.3207E-02*0.5317E-01* * * EST 8 * 1.182 *0.5717E-01*0.6104E-01* 1.262 * 4837* * +/*0.3262E-01*0.2484E-02*0.3251E-02*0.4343E-01* * * BUS *0.7520 *0.3316 *0.3316 *0.7520 * 44091* * +/*0.1085E-01*0.4436E-02*0.4436E-02*0.1085E-01* * * R

*0.0000E+00*0.0000E+00*0.5022 * 1.267 * 39637* * +/*0.0000E+00*0.0000E+00*0.1750E-01*0.2458E-01* * ******************************************************************* ... END OF SIMULATION ... Facultade de Informatica. A Corua. Junio 2005 286 SIMULATION TECHNIQUES Ethernet network ***SIMULATION WITH SPECTRAL METHOD *** ... TIME = 100000.00 , NB SAMPLES = 512 , CONF. LEVEL = 0.95 ******************************************************************* * NAME * SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB * ******************************************************************* * S * 1.257 * 1.000 * 1.000 * 1.257 * 79568* * +/*0.9795E-02*0.0000E+00*0.0000E+00*0.9795E-02* * * EST 1 * 2.045 *0.2052 *0.2709 * 2.699 * 10034* *

+/*0.5773E-01*0.8730E-02*0.1612E-01*0.1282 * * * EST 2 * 2.020 *0.2039 *0.2667 * 2.642 * 10096* * +/*0.7101E-01*0.8998E-02*0.1768E-01*0.1390 * * * EST 3 * 2.011 *0.1991 *0.2592 * 2.617 * 9904* * +/*0.6745E-01*0.8805E-02*0.1469E-01*0.1115 * * * EST 4 * 2.028 *0.2000 *0.2602 * 2.637 * 9864* * +/*0.5786E-01*0.6761E-02*0.1179E-01*0.1073 * * * EST 5 * 2.001 *0.2017 *0.2622 * 2.601 * 10079*

* +/*0.6415E-01*0.6065E-02*0.1510E-01*0.1624 * * Facultade de Informatica. A Corua. Junio 2005 287 SIMULATION TECHNIQUES Ethernet network * EST 6 * 2.035 *0.1993 *0.2610 * 2.664 * 9797* * +/*0.7039E-01*0.8648E-02*0.1688E-01*0.1333 * * * EST 7 * 2.021 *0.2007 *0.2594 * 2.611 * 9934* * +/*0.6259E-01*0.7496E-02*0.1742E-01*0.2369 * * * EST 8 * 2.014 *0.1985 *0.2571 * 2.608 * 9858* * +/*0.7488E-01*0.7814E-02*0.1403E-01*0.1413 *

* * BUS *0.5523 *0.6622 *0.6622 *0.5523 * 119899* * +/*0.5897E-02*0.7369E-02*0.7369E-02*0.5897E-02* * * R *0.0000E+00*0.0000E+00* 2.096 * 2.635 * 79566* * +/*0.0000E+00*0.0000E+00*0.8279E-01*0.1023 * * ******************************************************************* ... END OF SIMULATION ... Facultade de Informatica. A Corua. Junio 2005 288

Recently Viewed Presentations

  • Fast edge-directed single-image super-resolution Mushfiqur Rouf1 Dikpal Reddy2

    Fast edge-directed single-image super-resolution Mushfiqur Rouf1 Dikpal Reddy2

    Inverse problem formulation. Latent image. Anti-aliasing filter. Observed image. Downsampling. Convolution. Priors. Sparse gradient. Smooth contour. Only sharpens ...
  • SCHIZOPHRENIA AND RELATED DISORDERS CHAPTER 9  2013 by

    SCHIZOPHRENIA AND RELATED DISORDERS CHAPTER 9 2013 by

    For treatment-resistant clients, clozapine is the only approach that has empirical support. * Token economy: A form of contingency management in which a client who performs desired activities earns chips or tokens that can later be exchanged for tangible benefits....
  • RETORIKA DAN PROTOKOLER (Bidang Keorganisaasian)

    RETORIKA DAN PROTOKOLER (Bidang Keorganisaasian)

    Pembawa acara : adalah orang yang menyajikan atau menyampaikan satu per satu acara, memandu acara dengan menggunakan suara, mimik serta juga memperhatikan tata rias, busana, bahasa dan etika. Pembawa acara sesuai dengan tugas dan fungsinya lebih cocok dilekatkan pada acara-acara...
  • Where did you go to College? - garlandisdschools.net

    Where did you go to College? - garlandisdschools.net

    Where did you go to College? ... I chose TWU because it has a respected MLS program and the majority of the courses are offered online. ... I enjoyed receiving course work assistance from my professors when I needed it.
  • The Food Pyramid

    The Food Pyramid

    THE FOOD PYRAMID By Leah Starck TEKS 115.5. Health Education, Grade 3. A) Knowledge and Skills 1. Health Behaviors. The Student explains ways to enhance and maintain health throughout their lifespan.
  • Knight Sweeper 4200 Group 9 - UCF Department of EECS

    Knight Sweeper 4200 Group 9 - UCF Department of EECS

    Knight Sweeper. Knight Sweeper will be able to operate in autonomous and manual mode. During autonomous mode Knight Sweeper will be able to navigate itself from a start to end point . During the autonomous operation Knight Sweeper will be...
  • Classical Conditioning - Methacton School District

    Classical Conditioning - Methacton School District

    In A Clockwork Orange, a brutal sociopath, a mass murderer, is strapped to a chair and forced to watch violent movies while he is injected with a drug that nauseates him.So he sits and gags and retches as he watches...
  • Welcome to 890 Wolf Grove Road Mississippi Mills

    Welcome to 890 Wolf Grove Road Mississippi Mills

    A maple staircase leads you to the upper loft at the top of the stairs is open to the main floor living area. This space is perfect for enjoying a book or quiet relaxation. Vaulted ceilings, maple hardwood floors, pot...