Waiting-Line Models

1 Quantitative ModuleLEARNINGOBJECTIVESWhen you complete this module youshould be able toIDENTIFY OR DEFINE:The assumptions of the four basicwaiting- line modelsDESCRIBE OR EXPLAIN:How to apply Waiting-Line modelsHow to conduct an economic analysis of queuesDWaiting- line ModelsModuleOutlineCHARACTERISTICS OF A Waiting-Line SYSTEMA rrival CharacteristicsWaiting- line CharacteristicsService CharacteristicsMeasuring the Queue s PerformanceQUEUING COSTSTHE VARIETY OF QUEUING MODELSM odel A (M/M/1): Single-Channel QueuingModel with Poisson Arrivals andExponential Service-TimesModel B (M/M/S): Multiple-ChannelQueuing ModelModel C (M/D/1): Constant Service-TimeModelModel D: Limited-Population ModelOTHER QUEUING APPROACHESSUMMARYKEYTERMSUSINGSOFTWARE TOSOLVEQUEUINGPROBLEMSSOLVEDPROBLEMSINTE RNET ANDSTUDENTCD-ROM EXERCISESDISCUSSIONQUESTIONSACTIVEMODELE XERCISEPROBLEMSINTERNETHOMEWORKPROBLEMSC ASESTUDIES: NEWENGLANDFOUNDRY; 5/4/05 2:45 PM Page 743744 MODULEDWAITING-LINEMODELSQ ueuing theoryA body of knowledgeabout waiting line (queue)Items or people in a lineawaiting s EuroDisney, Tokyo s Disney Japan, and the s Disney World and Disneyland all have onefeature in common long lines and seemingly endless waits.

2 However, Disney is one of the world sleading companies in the scientific analysis of queuing theory. It analyzes queuing behaviors and canpredict which rides will draw what length crowds. To keep visitors happy, Disney makes lines appear tobe constantly moving forward, entertains people while they wait, and posts signs telling visitors howmany minutes until they reach each Common QueuingSituationsSITUATIONARRIVALS INQUEUESERVICEPROCESSS upermarketGrocery shoppersCheckout clerks at cash registerHighway toll boothAutomobilesCollection of tolls at boothDoctor s officePatientsTreatment by doctors and nursesComputer systemPrograms to be runComputer processes jobsTelephone companyCallersSwitching equipment forwards callsBankCustomersTransactions handled by tellerMachine maintenanceBroken machinesRepair people fix machinesHarborShips and bargesDock workers load and unloadThe body of knowledge about waiting lines, often called queuing theory, is an important part of operations and a valuable tool for the operations manager.

3 waiting linesare a common situation they may, for example, take the form of cars waiting for repair at a Midas Muffler Shop, copying jobs waiting to be completed at a Kinko s print shop, or vacationers waiting to enter Mr. Toad s Wild Ride at Disney. Table lists just a few OM uses of Waiting-Line Models . Waiting-Line Models are useful in both manufacturing and service areas. Analysis of queues interms of Waiting-Line length, average waiting time, and other factors helps us to understand ser-vice systems (such as bank teller stations), maintenance activities (that might repair brokenmachinery), and shop-floor control activities. Indeed, patients waiting in a doctor s office and bro-ken drill presses waiting in a repair facility have a lot in common from an OM perspective. Bothuse human and equipment resources to restore valuable production assets (people and machines)to good 5/4/05 2:45 PM Page 744 CHARACTERISTICS OF AWAITING-LINESYSTEM745 CHARACTERISTICS OF A Waiting-Line SYSTEMIn this section, we take a look at the three parts of a Waiting-Line , or queuing, system (as shown inFigure ) or inputs to the have characteristics such as population size, behav-ior, and a statistical discipline, or the waiting line of the queue include whether itis limited or unlimited in length and the discipline of people or items in service characteristics include its design and the statistical distribution ofservice now examine each of these three CharacteristicsThe input source that generates arrivals or customers for a service system has three the arrival arrivals (statistical distribution).

4 Size of the Arrival (Source) PopulationPopulation sizes are considered either unlimited(essentially infinite) or limited (finite). When the number of customers or arrivals on hand at anygiven moment is just a small portion of all potential arrivals, the arrival population is consideredunlimited, or infinite. Examples of unlimited populations include cars arriving at a big-city car-wash, shoppers arriving at a supermarket, and students arriving to register for classes at a large uni-versity. Most queuing Models assume such an infinite arrival population. An example of a limited,or finite, population is found in a copying shop that has, say, eight copying machines. Each of thecopiers is a potential customer that may break down and require of Arrivals at the SystemCustomers arrive at a service facility either accordingto some known schedule (for example, one patient every 15 minutes or one student every halfhour) or else they arrive randomly.

5 Arrivals are considered random when they are independent ofone another and their occurrence cannot be predicted exactly. Frequently in queuing problems, thenumber of arrivals per unit of time can be estimated by a probability distribution known as theUnlimited,orinfinite,populationA queue in which avirtually unlimited numberof people or items couldrequest the services, or inwhich the number ofcustomers or arrivals onhand at any givenmoment is a very smallportion of ,orfinite,populationA queue in which thereare only a limited numberof potential users of to the systemDave'sCar WashEnterIn the systemExit the ( waiting line )Arrivalsfrom thegeneral population ..Population ofdirty St. SE St. NW St. Exit the systemArrival Characteristics Size of arrival population Behavior of arrivals Statistical distribution of arrivalsWaiting line CharacteristicsService Characteristics Service design Statistical distribution of service Limited vs.

6 Unlimited Queue disciplineFIGURE Three Parts of a waiting line , or Queuing System, at Dave s Car 5/4/05 2:45 PM Page 745746 MODULEDWAITING-LINEMODELS0 Distribution for = 2 PProbability =(x)=e x! xXX0 Distribution for = 4101111 FIGURE Two Examples of thePoisson Distribution forArrival Times The other line alwaysmoves faster. Etorre s Observation If you change lines, theone you just left will startto move faster than theone you are now in. O Brien s Variation1 When the arrival rates follow a Poisson process with mean arrival rate , the time between arrivals follows a negative expo-nential distribution with mean time between arrivals of 1/ . The negative exponential distribution, then, is also representativeof a Poisson process but describes the time between arrivals and specifies that these time intervals are completely any given arrival time (such as 2 customers per hour or 4 trucks perminute), a discrete Poisson distribution can be established by using the formula(D-1)whereP(x) = probability of xarrivalsx= number of arrivals per unit of time = average arrival ratee= (which is the base of the natural logarithms)With the help of the table in Appendix II, which gives the value ofe for use in the Poisson dis-tribution, these values are easy to compute.

7 Figure illustrates the Poisson distribution for =2 and = 4. This means that if the average arrival rate is = 2 customers per hour, the probabil-ity of 0 customers arriving in any random hour is about 13%, probability of 1 customer is about27%, 2 customers about 27%, 3 customers about 18%, 4 customers about 9%, and so on. Thechances that 9 or more will arrive are virtually nil. Arrivals, of course, are not always Poisson dis-tributed (they may follow some other distribution). Patterns, therefore, should be examined tomake certain that they are well approximated by Poisson before that distribution is ()!,, , , ,== for 0 1 2 3 4 KBehavior of ArrivalsMost queuing Models assume that an arriving customer is a patient cus-tomer. Patient customers are people or machines that wait in the queue until they are served and donot switch between lines. Unfortunately, life is complicated by the fact that people have been knownto balk or to renege.

8 Customers who balkrefuse to join the waiting line because it is too long to suittheir needs or interests. Renegingcustomers are those who enter the queue but then become impa-tient and leave without completing their transaction. Actually, both of these situations just serve tohighlight the need for queuing theory and Waiting-Line CharacteristicsThe waiting line itself is the second component of a queuing system. The length of a line can beeither limited or unlimited. A queue is limitedwhen it cannot, either by law or because of physicalrestrictions, increase to an infinite length. A small barbershop, for example, will have only a limitedPoisson distributionA discrete probabilitydistribution that oftendescribes the arrival ratein queuing 5/4/05 2:45 PM Page 746 CHARACTERISTICS OF AWAITING-LINESYSTEM7472 The term FIFS(first-in, first-served) is often used in place of FIFO.

9 Another discipline, LIFS (last-in, first-served) alsocalled last-in, first-out (LIFO), is common when material is stacked or piled so that the items on top are used of waiting chairs. Queuing Models are treated in this module under an assumption ofunlimitedqueue length. A queue is unlimitedwhen its size is unrestricted, as in the case of the tollbooth serving arriving second Waiting-Line characteristic deals with queue discipline. This refers to the rule by which cus-tomers in the line are to receive service. Most systems use a queue discipline known as the first-in, first-out (FIFO) rule. In a hospital emergency room or an express checkout line at a supermarket, however,various assigned priorities may preempt FIFO. Patients who are critically injured will move ahead intreatment priority over patients with broken fingers or noses. Shoppers with fewer than 10 items may beallowed to enter the express checkout queue (but are thentreated as first-come, first-served).

10 Computer-programming runs also operate under priority scheduling. In most large companies, when computer-produced paychecks are due on a specific date, the payroll program gets highest CharacteristicsThe third part of any queuing system are the service characteristics. Two basic properties are impor-tant: (1) design of the service system and (2) the distribution of service Queuing System DesignsService systems are usually classified in terms of theirnumber of channels (for example, number of servers) and number of phases (for example, numberof service stops that must be made). A single-channel queuing system, with one server, is typifiedby the drive-in bank with only one open teller. If, on the other hand, the bank has several tellers onduty, with each customer waiting in one common line for the first available teller, then we wouldhave a multiple-channel queuing system.