2Queuing Models Queuing is the study of waiting lines, or queues. The objective of queuing analysis is to design systems that enable organizations to perform optimally according to some criterion.Possible CriteriaMaximum Profits.Desired Service Level.
3Elements of the queuing process A queuing system consists of five basic components:Calling populationArrival processQueue configurationQueue disciplineService process
12Possible Service Measurements Average time a customer spends in line.Average length of the waiting line.The probability that an arriving customer must wait for service.
13The Arrival Process The random process is more common in businesses. Under three conditions a Poisson Distribution can describe the random arrival process.
14The three conditions required for the existence of the Poisson arrival process: Orderliness : one customer, at most, will arrive during any time interval.Stationarity : for a given time frame, the probability of arrivals within a certain time interval is the same for all time intervals of equal length.Independence : the arrival of one customer has no influence on the arrival of another.These conditions are unrestrictive and are approximately satisfied in many situations.
15The Poisson Arrival Distribution Xke!=lt)t()-Wherel = mean arrival rate per time unit.t = the length of the interval.e = (the base of the natural logarithm).k! = k (k -1) (k -2) (k -3) … (3) (2) (1).
16HANK’s HARDWARE (An illustration of the Poisson distribution) Customers arrive at Hank’s Hardware according to a Poisson distribution.Between 8:00 a.m. and 9:00 a.m., an average of 6 customers arrive at the store.What is the probability that k = 0, 1, 2, … customers will arrive between 8:00 and 8:30 in the morning.
17SOLUTION 1 2 3 Input to the Poisson distribution SOLUTIONInput to the Poisson distributionl= 6 customers per hour.t = 0.5 hour.l t = (6)(0.5) = 3.123456783211=230!3!2!1!
18Measures of Queuing System Performance P0 = Probability that there are no customers in thesystem.Pn = Probability that there are “n” customers in theL = Average number of customers in the system.Lq = Average number of customers in the queue.W = Average time a customer spends in the system.Wq = Average time a customer spends in the queue.Pw = Probability that an arriving customer must wait for service.r = Utilization rate for each server (the percentage of time that each server is busy).
19Classification of Queues Queuing system can be classified by: Arrival process.Service process.Number of servers.System size (infinite/finite waiting line).Population size.NotationM (Markovian) = Poisson arrivals or exponential service time.D (Deterministic) = Constant arrival rate or service time.G (General) = General probability for arrivals or service time.Example:M / M / 6 / 10 / 20
20M / M / 1 Queuing System Characteristics Poisson arrival process. Exponential service time distribution.A single server.Potentially infinite queue.An infinite population.
21Performance Measures for the M / M /1 Queue P0 = 1- (l / m)Pn = [1 - (l / m)] (l/ m)nL = l / (m - l)Lq = l 2 / [m(m - l)]W = 1 / (m - l)Wq = l / [m(m - l)]Pw = l / mr = l / m
22MARY’s SHOESCustomers arrive at Mary’s Shoes every 12 minutes on the average, according to a Poisson process.Service time is exponentially distributed with an average of 8 minutes per customer.Management is interested in determining the performance measures for this service system.
23SOLUTION P0 = 1- (l / m) = 1 - (5 / 7.5) = 0.3333 Inputl = 1/ 12 customers per minute= 60/ 12 = 5 per hour.m = 1/ 8 customers per minute= 60/ 8 = 7.5 per hour.Performance CalculationsP0 = 1- (l / m) = 1 - (5 / 7.5) =Pn = [1 - (l / m)] (l/ m) = (0.3333)(0.6667)nL = l / (m - l) = 2Lq = l2/ [m(m - l)] =W = 1 / (m - l) = 0.4 hours = 24 minutesWq = l / [m(m - l)] = hours = 16 minutesPw = l / m =r = l / m =
26M / M / k Queuing Systems Characteristics Customers arrive according to a Poisson process at a mean rate l.Service time follow an exponential distribution.There are k servers, each of which works at a rate of m customers.Infinite population, and possibly infinite line.
28The performance measurements L, Lq, Wq,, can be obtained from Little’s formulas.
29Revision Question At a Food Lion Provision Shop, customers spend an average of 25 mins selecting their groceries and checking out by entering a single line queue served by two cashiers. The service times required for the cashiers to check out customers follow an exponential distribution and average four minutes. Customers arrive at the cashier counter according to a Poisson distribution at the average rate of eight customers per hour The table below shows part of the computer calculation:Determine the followings :average time a customer spends in the store (in minutes).average number of customers waiting in line prior to being checked out.proportion of customers who will have to wait in line. What assumption(s) did you make in part (a)? Comment whether the assumption(s) is/are realistic.
30Queuing Process Multiple queue advantages : Service can be differentiatedDivision of labour possibleSelection option for customerDeterred balking
31Types of services clearly stated Queuing ProcessMultiple queue disadvantages :AnxietyLack of fairnessLack of privacyTypes of services clearly stated
32Queuing Process Single queue advantages : First come first serve (fairness)No anxiety to select fastest lineReneging difficultQueue cutting resolvedPrivacy enhancedReducing average waiting time
33Unnecessary held up in waiting line Queuing ProcessSingle queue disadvantages :No specialisationUnnecessary held up in waiting linePossible balking
34Free to wander about and browse items Queuing ProcessTake a number advantages :No need for formal lineFree to wander about and browse itemsMore relaxing
35Require large waiting area Queuing ProcessTake a number disadvantages :RenegingRequire large waiting area
36Queuing Process QUEUE DISCIPLINE Policy of selecting next customer from the queue for serviceFirst come first serve most popular static method (selection depends on position in queue only)
37Queuing ProcessDynamic disciplines selection involve some attribute or status in selectionShortest processing time minimise average time of customers
38Queuing ProcessPreemptive priority interrupt current person service for newly arrived customer with higher priority e.g. fire or ambulance servicesRound robin service give customer partial service and then movers to next waiting customer – alternating between waiting and being served
39Queuing ProcessService person begins to take orders while customers are still waiting in line is a direct approach to avoid reneging
40Queuing Process SERVICE PROCESS Distribution of service times, arrangement of servers, management policies and server behaviour contribute to performance
41WAITING PERCEPTION Unoccupied time feels longer than occupied time Preprocess waits feel longer than in process waitsAnxiety makes waits seem longer
42WAITING PERCEPTION Uncertain waits are longer than known finite waits Unexplained waits are longer than explained waitUnfair waits are longer than equitable waits
43WAITING PERCEPTION People more willing to wait for valuable service Solo waiting feels longer than group waitingCustomer attitudes
44WAITING PERCEPTION Environment Unused facilities and idle staff increase annoyanceUnfamiliar music makes perceived time seem longer than familiar background music
45REVIEW QUESTIONS 1. What does M/D/1 refers to? 2. What are the characteristics ofM / G / 1?3. How can you measure the performance of a queue?
46REVIEW QUESTIONS4. List and discuss the factors affecting queue perception.5. What are the five features of a queue?6. How can you configure queue?