2Queuing systems: basic framework & key metrics CustomerPopulationArrivalQueueServiceExitAverage utilization (% time server busy)Average queuing timeAverage queue length (# of customers in line)Average system time (queuing + service)Average # of customers in the system (in line + being served)
3Life without variability CustomerPopulationExitArrivalQueueServiceArrival stream: 1 every 5 minutesService time: Exactly 5 minutes1 every 10 minutesExactly 5 minutesAverage utilizationAverage queuing timeAverage queue lengthAverage system timeAverage no. in system100%5150%50.5
4Single queue, single server (M/M/1) Queuing Models…Single queue, single server (M/M/1)Arrival rate λService rate μAssumeTime between arrivals is Exp(λ)Time between services is Exp()
5Queuing Models: Arrivals T = time between arrivalsAssume T is an exponential random variable with rate lProbability density functionfor the exponential distribution:Expected time between arrivals:
6Queuing Models: Arrivals Time between arrivals is Exp() Poisson arrivals at rate lIf 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 unitsPN(T)(n) is the probability that the number of arriving customers in any period of length T is exactly n
7Examples:Customers arrive to a McDonalds according to a Poisson process with rate 2 customers per minute.What is the expected time between arrivals?What is the probability that exactly 3 customers arrive in any 5 minute period?
8Queuing 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/minWhat is the expected service time?
9Utilization for M/M/1… How much of the capacity is being utilized? Arrival rateService rate
10Number of customers in the system for the M/M/1 queue… Ns = number of customers in system
11Average 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
12Metrics for the M/M/1 queue… UtilizationAverage # of customers in the systemAverage # of customers in lineAverage time a customer spends in lineAverage time a customer spends in the system
13Example … Western National Bank is considering opening a drive- through window for customer service. Management estimatesthat customers will arrive at the rate of 15 per hour. The tellerwho will staff the window can service customers at the rateof one every three minutes.Assuming exponential interarrival and service times,calculate performance metrics of this queue.
14Example (continued)Because of limited space availability and a desire to providean acceptable level of service, the bank manager would likereduce the probability that more than three cars are in thesystem at any given time.What is the current level of service?What must be service rate at the teller, and what utilizationmust be achieved, to ensure a 95% service level (probabilityof having 3 or less cars in the system being 95%).
16Metrics for the M/M/s queue… UtilizationAverage # of customers in line(see table TN7.11)Average # of customers in the systemAverage time a customer spends in lineAverage time a customer spends in the system
17Example … Sharp Discounts Wholesale club has two service desks, one at each entrance of the store. Customers arrive at each servicedesk at an average of one every six minutes. The service rateat each service desk is four minutes per customer.What percentage of time is each service desk idle?What is the probability that both desks are busy? Idle?How many customers, on average, are waiting in line?How much time does a customer spend at the service desk?(waiting plus service time)e. Should Sharp Discounts consolidate its two service desks?
18Cost of providing faster service vs. “cost” of waiting The trade-off in waiting line management…Cost of providing faster service vs. “cost” of waitingTotal costCostCost of capacityCost of waitingCapacity
19Example …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?