Waiting Line Management Queuing systems: basic framework & key metrics Customer Population Arrival Exit Queue Service 1. Average utilization (% time server busy) 2. Average queuing time 3. Average queue length (# of customers in line) 4. Average system time (queuing + service)

5. Average # of customers in the system (in line + being served) Life without variability Customer Population Arrival Exit Queue Arrival stream: Service time: Average utilization Average queuing time

Average queue length Average system time Average no. in system Service 1 every 5 minutes 1 every 10 minutes Exactly 5 minutes Exactly 5 minutes 100% 0 0 5 1 50%

0 0 5 0.5 Queuing Models Single queue, single server (M/M/1) Arrival rate Service rate Assume

Time between arrivals is Exp()) Time between services is Exp() Queuing Models: Arrivals T = time between arrivals Assume T is an exponential random variable with rate Probability density function for the exponential distribution: Expected time between arrivals: f T (t ) e t , t 0

E[T ] 1 Queuing Models: Arrivals Time between arrivals is Exp() Poisson arrivals at rate If we want to know how many customers arrive in a given time period, we can use the Poisson distribution. N(T) = number of arrivals in T time units n ( T ) e

PN (T ) ( n) n! T , n 0 PN(T)(n) is the probability that the number of arriving customers in any period of length T is exactly n Examples: Customers arrive to a McDonalds according to a Poisson process with rate 2 customers per minute. 1

1 What is the expected time between arrivals? E[T ] min 2 What is the probability that exactly 3 customers arrive in any 5 minute period? ( T ) n e T ( 2 5) 3 e 25 PN ( 5) (3) 0.0076 n! 3!

Queuing Models: Service We assume that service time S is an exponential random variable with rate Example: A bank teller can service customers at a rate of 3 customers per minute. = 3 customers/min What is the expected service time? 1 1 E[ S ] min 3 Utilization for M/M/1

How much of the capacity is being utilized? Arrival rate Utilization = Service rate Number of customers in the system for the M/M/1 queue Ns = number of customers in system PN s (n) (1 ) n , n 0

Average number of customers in the system for the M/M/1 queue On average, how many customers are in the system at any moment in time? Ls = Average # of customers in the system 0 PN s (0) 1PN s (1) 2 PN s (2) Metrics for the M/M/1 queue Utilization

Average # of customers in the system Average # of customers in line Ls 2 Lq

( ) Average time a customer spends in line Wq ( ) 1 Average time a customer spends in the system Ws Example Western National Bank is considering opening a drivethrough window for customer service. Management estimates that customers will arrive at the rate of 15 per hour. The teller who will staff the window can service customers at the rate of one every three minutes. Assuming exponential interarrival and service times,

calculate performance metrics of this queue. Example (continued) Because of limited space availability and a desire to provide an acceptable level of service, the bank manager would like reduce the probability that more than three cars are in the system at any given time. What is the current level of service? What must be service rate at the teller, and what utilization must be achieved, to ensure a 95% service level (probability of having 3 or less cars in the system being 95%). Queuing Models Single queue, multiple servers (M/M/s)

s = # of servers Arrival rate Service rate (for each server) Metrics for the M/M/s queue Utilization s Average # of customers in line

Lq Average # of customers in the system Average time a customer spends in line (see table TN7.11) Ls Lq Wq Average time a customer spends in the system

Lq Lq 1 Ws Example Sharp Discounts Wholesale club has two service desks, one at each entrance of the store. Customers arrive at each service desk at an average of one every six minutes. The service rate at each service desk is four minutes per customer. a. What percentage of time is each service desk idle?

b. What is the probability that both desks are busy? Idle? c. How many customers, on average, are waiting in line? d. How much time does a customer spend at the service desk? (waiting plus service time) e. Should Sharp Discounts consolidate its two service desks? The trade-off in waiting line management Cost of providing faster service vs. cost of waiting Total cost Cost Cost of capacity Cost of waiting

Capacity Example In the service department at Glenn-Mark auto agency, mechanics requiring parts for auto repair or service present their request forms at the parts department counter. The parts clerk fills a request while the mechanic waits. Mechanics arrive according to a Poisson process with rate 40/h, and a clerk can fill requests at the rate of 20/h. The costs for a parts clerk is $6/h, and the cost for a mechanic is $12/h. Assuming there are currently 3 parts clerks, would you add a fourth?