Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Waiting Line ManagementQueuing systems: basic framework & key metricsCustomerPopulationArrivalQueueServiceExit1. Average utilization (% time server busy)2. Average queuing time3. 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 variabilityCustomerPopulationArrivalQueue ServiceExitArrival stream: 1 every 5 minutesService time: Exactly 5 minutesAverage utilizationAverage queuing timeAverage queue lengthAverage system timeAverage no. in system50%0050.5100%00511 every 10 minutesExactly 5 minutesQueuing Models…Arrival rate λService rate μSingle queue, single server (M/M/1)Assume•Time between arrivals is Exp(λ)•Time between services is Exp()Queuing Models: ArrivalsT = time between arrivalsAssume T is an exponential random variable with rate 1][ TEExpected time between arrivals:0,)( tetftTProbability density function for the exponential distribution:Queuing Models: ArrivalsIf we want to know how many customers arrive in a given time period, we can use the Poisson distribution.0,!)()()(nneTnPTnTNPN(T)(n) is the probability that the number of arriving customers in any period of length T is exactly n Time between arrivals is Exp()  Poisson arrivals at rate N(T) = number of arrivals in T time unitsExamples:Customers arrive to a McDonalds according to a Poisson process with rate 2 customers per minute.What is the expected time between arrivals?0076.0!3)52(!)()3(523)5(eneTPTnNmin211][ TEWhat is the probability that exactly 3 customers arrive in any 5 minute period?Queuing Models: ServiceWe 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?min311][ SEUtilization for M/M/1…How much of the capacity is being utilized?Utilization = Arrival rateService rateNumber of customers in the systemfor the M/M/1 queue…Ns = number of customers in system 0,)1()(  nnPnNsAverage number of customers in the systemfor 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  )2(2)1(1)0(0sssNNNPPPMetrics for the M/M/1 queue…)(2qL1sW)(qWUtilizationAverage # of customers in lineAverage # of customers in the systemAverage time a customer spends in the systemAverage time a customer spends in linesLExample …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.Example (continued)Because of limited space availability and a desire to providean acceptable level of service, the bank manager would like reduce 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 (probability of having 3 or less cars in the system being 95%).Queuing Models…Single queue, multiple servers (M/M/s)Arrival rate λService rate μ(for each server)s = # of serversMetrics for the M/M/s queue…UtilizationAverage # of customers in lineAverage # of customers in the systemAverage time a customer spends in the systemAverage time a customer spends in linesqL(see table TN7.11)qsLLqqLW 1qsLWExample …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 rateat 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 waitingCapacityCostTotal costCost of capacityCost of waitingExample …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