1. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-1 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Waiting Lines and Queuing Theory Models

2. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-2 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458Introduction The study of waiting lines, called queuing theory, is one of the oldest and most widely used Operations Research techniques. Waiting lines are an everyday occurrence, affecting people shopping for groceries, buying gasoline, or making a bank deposit. Queues, another term for waiting lines, may also take the form of machines waiting to be repaired, trucks in line to be unloaded, or airplanes lined up on a runway waiting for permission to take off. The three basic components of a queuing process are arrivals, service facilities, and the actual waiting line. In this chapter we discuss how analytical models of waiting lines can help managers evaluate the cost and effectiveness of service systems.

3. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-3 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Waiting Line Costs Most waiting line problems are centered on the question of finding the ideal level of services that a firm should provide. "one of the goals of queuing analysis is finding the best level of service for an organization". In most cases, this level of service is an option over which management has control. An extra teller, for example, can be borrowed from another chore or can be hired and trained quickly if demand warrants it. This may not always be the case, though. A plant may not be able to locate or hire skilled mechanics to repair sophisticated electronic machinery.

4. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-4 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Waiting Line Costs\continued When an organization does have control, its objective is usually to find a happy medium between two extremes: 1. On the one hand, a firm can retain a large staff and provide many service facilities. This may result in excellent customer service, with seldom more than one or two customers in a queue. Customers are kept happy with the quick response and appreciate the convenience. This, however, can become expensive. 2. The other extreme is to have the minimum possible number of checkout lines, gas pumps, or teller windows open. This keeps the service cost down but may result in customer dissatisfaction. "As the average length of the queue increases and poor service results, customers and goodwill may be lost."

5. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-5 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Waiting Line Costs\continued "Managers must deal with the trade-off between the cost of providing good service and the cost of customer waiting time. The latter may be hard to quantify." One means of evaluating a service facility is thus to look at a total expected cost, a concept illustrated in the figure (the next slide). Total expected cost is the sum of expected service costs plus expected waiting costs. Service costs are seen to increase as a firm attempts to raise its level of service. As service improves in speed, however, the cost of time spent waiting in lines decreases. "The goal is to find the service level that minimizes total expected cost."

6. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-6 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Queuing Costs and Service Levels Cost of Operating Service Facility Service Level Total Expected Cost Cost of Providing Service Cost of Waiting Time O ptimal Service Level

7. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-7 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Waiting Line Costs\continued Three Rivers Shipping Company Example Three Rivers run a huge docking facility located on the Ohio River near Pittsburgh. Approximately five ships arrive to unload their cargoes of steel and ore during every 12-hour work shift. Each hour that a ship sits idle in line waiting to be unloaded costs the firm a great deal of money, about $1,000 per hour. From experience, management estimates that if one team of stevedores is on duty to handle the unloading work, each ship will wait an average of 7 hours to be unloaded. If two teams are working, the average waiting time drops to 4 hours; for three teams, it's 3 hours; and four teams of stevedores, only 2 hours. But each additional team of stevedores is also an expensive proposition, due to union contacts. Three rivers' superintendent would like to determine the optimal number of teams of stevedores to have on duty each shift. The objective is to minimize total expected costs. This analysis is summarized in the next table:

8. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-8 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Waiting Line Cost Analysis Three Rivers Shipping Number of Stevedore Teams 1234 5555 7432 35201510 $1,000 35,00029,000$15,000$10,000 $6,000$12,00018,000 $24,000 $41,000$32,000$33,000$34,000 Average waiting time per ship Total ship hours lost Est. cost per hour of idle ship time Value of ships' lost time Stevedore teams salary Total Expected Cost Avg. number of ships arriving per shift

9. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-9 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Characteristics Of A Queuing System We will study three parts of a queuing system: 1. The arrivals or inputs to the system (sometimes referred to as the calling population), 2. The queue or the waiting line itself, and 3. The service facility

10. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-10 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 1. Arrival Characteristics The input source that generates arrivals or customers for the service system has three major characteristics. It is important to consider: the size of the calling population, the pattern of arrivals at the queuing system, and the behavior of the arrivals.

11. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-11 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 1. Arrival Characteristics \continued Size of the Calling Population Population sizes are considered to be either unlimited (essentially infinite) or limited (finite). For practical purposes, examples of unlimited population includes cars arriving at a highway tollbooth, shoppers arriving at a supermarket, or students arriving to register for classes at large university. "Most queuing models assume such an infinite calling population". An example of a finite population is a shop with only eight machines that break down and require service.

12. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-12 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 1. Arrival Characteristics \continued Pattern of Arrivals at the System Customers either arrive at service facility according to some known schedule (for example, one patient every 15 minutes or one student for advising every half hour) or else they arrive randomly. Arrivals are considered random when they are independent of one another and their occurrence cannot be predicted exactly. Frequently in queuing problems, the number of arrivals per unit of time can be estimated by a probability distribution known as the Poisson distribution. "The Poisson probability distribution is used in many queuing models to represent arrival patterns."

13. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-13 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Poisson Distribution for Arrival Times For X=0,1,2,3,4,…….. P(X)= probability of X arrivals X= number of arrivals per unit of time = average arrival rate e= 2.7183

14. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-14 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Poisson Distribution for Arrival Times.00.05.10.15.20.25.30.35 0123456789101 X P(X) P(X), = 2.00.05.10.15.20.25.30 0123456789101 X P(X) P(X), = 4

15. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-15 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 1. Arrival Characteristics \continued Behavior of the Arrival Most queuing models assume that an arriving customer is a patient customer. Patient customers are people or machines that wait in the queue until they are served and do not switch between lines. Unfortunately, life and operations research are complicated by the fact that people have been known to balk or renege. Balking refers to customers who refuse to join the waiting line because it is too long to suit their needs or interests. Reneging customers are those who enter the queue but then become impatient and leave without completing their transaction.

16. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-16 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 2. Waiting Line Characteristics We will study the: Length of the queue Service priority/Queue discipline

17. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-17 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 2. Waiting Line Characteristics \continued Length of the queue The length of a line can be either limited or unlimited. A queue is limited when it cannot increase to an infinite length. This may be the case in a small restaurant that has only 10 tables and can serve no more than 50 diners an evening. Analytic queuing models are treated in this chapter under an assumption of unlimited queue length.

18. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-18 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 2. Waiting Line Characteristics \continued Service priority/Queue discipline Queue discipline refers to the rule by which customers in the line are to receive service. Customers may be served according to one of the next disciplines: 1.first-in, first-out rule (FIFO) 2.priority discipline 3.last-come first-served (LCFS) 4.service in random order (SIRO)

19. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-19 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 3. Service Facility Characteristics It is important to examine two basic properties: 1.the configuration of the service system 2.the pattern of service times

20. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-20 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 3. Service Facility Characteristics \continued Basic Queuing System Configuration Service systems are usually classified in terms of number of channels and number of phases in service. 1. Number of channels A single-channel system, with one server, is typified by the drive-in bank that has only one open teller. A multi-channel system, when the bank has several tellers on duty and each customer waits in one common line for the first available teller.

21. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-21 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 3. Service Facility Characteristics \continued 2. Number of phases in service A single-phase system, in this case the customer receives service from only one station and then exists the system. A multiphase system, for example, if the restaurant require you to place your order at one station, pay at a second, and pick up the food at a third service stop.

22. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-22 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Basic Queuing System Configurations Single Channel, Single Phase Queue Service facility Facility 1 Facility 2 Single Channel, Multi-Phase Queue Service Facility

23. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-23 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Basic Queuing System Configurations Service facility 1 Service facility 2 Service facility 3 Multi-Channel, Single Phase Queue Multi-Channel, Multiphase Phase Queue Type 1 Service Facility Type 2 Service Facility

24. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-24 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 3. Service Facility Characteristics \continued The Pattern of Service Times Service patterns are like arrival patterns in that they can be either constant or random. If service time is constant, it takes the same amount of time to take care of each customer. This is the case in a machine-performed service operation such as an automatic car wash. More often, service times are randomly distributed. In many cases it can be assumed that random service times are described by the negative exponential probability distribution.

25. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-25 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 3. Service Facility Characteristics \continued The next figure illustrates that if service times follow an exponential distribution, the probability of any very long service time is low. Probability (for Intervals of 1 Minute) 30 60 90 120 150 180 Average Service Time of 1 Hour Average Service Time of 20 Minutes X

26. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-26 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Identifying Models Using Kendall Notation D.G. Kendall developed a notation that has been widely accepted for specifying the pattern of arrivals, the service time distribution, and the number of channels in a queuing model. The basic three-symbol Kendall notation is in the form: Arrival distribution/ service time distribution/ number of service channels open Where specific letters are used to represent probability distributions. The following letters are commonly used in Kendall notation: M= Poisson distribution for number of occurrences (or exponential times) D= Constant (deterministic) rate G= General distribution with mean and variance known

27. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-27 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Identifying Models Using Kendall Notation \continued Thus, a single channel model with Poisson arrivals and exponential service times would be represented by M/M/1 When a second channel is added, we would have M/M/2 (An M/M/2 model has Poisson arrivals, exponential service times, and two channels) If there are m distinct service channels in the queuing system with Poisson arrivals and exponential service, the Kendall notation would be M/M/m.

28. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-28 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Identifying Models Using Kendall Notation \continued A three-channel system with Poisson arrivals and constant service time would be identified as M/D/3. A four channel system with Poisson arrivals and service times that are normally distributed would be identified as M/G/4.

29. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-29 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Single-Channel Queuing Model With Poisson Arrivals And Exponential Service Times (M/M/1) Assumptions of the Model The single-channel, single-phase model considered here is one of the most widely used and simplest queuing models. It involves assuming that seven conditions exist: 1.Arrivals are served on a FIFO basis. 2.Every arrival waits to be served regardless of the length of the line; that is, there is, no balking or reneging. 3.Arrivals are independent of preceding arrivals, but the average number of arrivals (the arrival rate) does not change over time. 4.Arrivals are described by a Poisson probability distribution and come from an infinite or very large population.

30. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-30 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Single-Channel Queuing Model With Poisson Arrivals And Exponential Service Times (M/M/1) 5.Service times also vary from one customer to the next and are independent of one another, but their average rate is known. 6.Service times occur according to the negative exponential probability distribution. 7.The average service rate is greater than the average arrival rate. When these seven conditions are met, we can develop a series of equations that define the queue's operating characteristics.

31. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-31 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Single-Channel Queuing Model With Poisson Arrivals And Exponential Service Times (M/M/1) Performance Measures of Queuing Systems: Average time each customer spends in the queue Average length of the queue Average time each customer spends in the system Average number of customers in the system Probability that the service facility will be idle Utilization factor for the system Probability of a specific number of customers in the system

32. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-32 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Single-Channel Queuing Model With Poisson Arrivals And Exponential Service Times (M/M/1) Queuing Equations We let λ = means number of arrivals per time period (for example, per hour) µ = means number of people or items served per time period When determining the arrival rate (λ) and the service rate (µ), the same time period must be used.

33. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-33 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Single-Channel Queuing Model With Poisson Arrivals And Exponential Service Times (M/M/1) Queuing Equations The queuing equations follow:

34. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-34 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Single-Channel Queuing Model With Poisson Arrivals And Exponential Service Times (M/M/1) Queuing Equations The queuing equations follow: (The utilization factor for the system,, that is, the probability that the service facility is being used) (The percent idle time, P0, that is, the probability that no one is in the system) ( The probability that the number of customers in the system is greater than K, Pn>k )

35. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-35 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Example: Arnold's Muffler Shop Case Arnold's mechanic, Reid Blank, is able to install new mufflers at an average rate of 3 per hour, or about 1 every 20 minutes. Customers needing this service arrive at the shop on the average of 2 per hour. Larry Arnold, the shop owner, feels that all seven of the conditions for a single-channel model are met. He proceeds to calculate the performance measures of the systems. Solution: λ = 2 cars arriving per hour µ = 3 cars serviced per hour

36. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-36 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Example: Arnold's Muffler Shop Case/solution cars in the system on the average hour that an average car spends in the system cars waiting on line on the average hour = 40 minutes = average waiting time per car

37. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-37 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Example: Arnold's Muffler Shop Case/solution = probability that there are 0 cars in the system Calculate the probability that more than three cars are in the system: implies that there is a 19.8% chance that more than three cars are in the system.

38. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-38 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Introducing Costs into the Model As stated earlier, the solution to queuing problem may require management to make trade-off between the increased cost of providing better service and the decreased waiting costs derived from providing that service. These two costs are called the waiting cost and the service cost. The total service cost is total service cost = (number of channels) (cost per channel) total service cost = mCs where m = number of channels Cs = service cost (labor cost) of each channel

39. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-39 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Introducing Costs into the Model \ continued The waiting cost when the waiting time cost is based on time in the system is total waiting cost = (total time spent waiting by all arrivals) (cost of waiting) = (number of arrivals) (average wait per arrival) Cw So, total waiting cost = (W)Cw if waiting time cost is based on time in the queue, this becomes total waiting cost = (Wq)Cw

40. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-40 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Introducing Costs into the Model \ continued  The total costs of the queuing system: Total cost = total service cost + total waiting cost Total cost = mCs + (W)Cw (waiting time is based on the time in the system) Total cost = mCs + (Wq)Cw (waiting time is based on the time in the queue)

41. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-41 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Example: Arnold's Muffler Shop Case Arnold estimates that the cost of customer waiting time, in terms of customer dissatisfaction and lost goodwill, is $10 per hour of time spent waiting in line. Because on the average a car has a 2/3 hour wait and there are approximately 16 cars serviced per day (2 per hour times 8 working hours per day), the total number of hours that customers spend waiting for mufflers to be installed each day is 2/3 *16=32/3 hours. Hence, in this case, total daily waiting cost = (8 hours per day) WqCw = (8)(2)(2/3)($10) = $106.67

42. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-42 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Example: Arnold's Muffler Shop Case \ continued the only other cost that Larry Arnold can identify in this queuing situation is the pay rate of Reid Blank, the mechanic Blank is paid $7 per hour total daily service cost = (8 hours per day) mCs = 8(1)($7)=$56  The total daily cost of the system as it is currently configured is the total of the waiting cost and the service cost, which gives us Total daily cost of the queuing system = $106.67 + $56 = $162.67

43. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-43 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Example: Arnold's Muffler Shop Case Now comes a decision. Arnold finds out through the muffler business grapevine that the Rusty Muffler, a cross-town competitor, employs a mechanic named Jimmy Smith who can efficiently install new mufflers at the rate of 4 per hour. Larry Arnold contacts Smith and inquires as to his interest in switching employers. Smith says that he would consider leaving the Rusty Muffler but only if he were paid a $9 per hour salary. Arnold, being a crafty businessman, decides that it may be worthwhile to fire Blank and replace him with the speedier but more expensive Smith. He first recomputes all the operating characteristics using new service rate of 4 mufflers per hour.

44. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-44 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Example: Arnold's Muffler Shop Case/solution cars in the system on the average hour that an average car spends in the system cars waiting on line on the average hour = 15 minutes = average waiting time per car

45. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-45 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Example: Arnold's Muffler Shop Case/solution = probability that there are 0 cars in the system It is quite evident that Smith's speed will result in considerably shorter queues and waiting times. For example, a customer would now spend and average of 1/2 hour in the system and 1/4 hour waiting in the queue, as opposed to 1 hour in the system and 2/3 hour in the queue with Blank as mechanic.

46. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-46 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Example: Arnold's Muffler Shop Case \ solution The total daily waiting time cost with Smith as the mechanic will be total daily waiting cost = (8 hours per day) WqCw = (8)(2)(1/4)($10)=$40 per day service cost of Smith = 8 hours/day * $9/hour = $72 per day total expected cost = waiting cost + service cost = $40 + $72 = $112 per day Because the total daily expected cost with Blank as mechanic was $162, Arnold may very well decide to hire Smith and reduce costs by $162-$112=$50 per day.

47. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-47 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 In this case two or more servers or channels are available to handle arriving customers. An example of such a multichannel, single-phase waiting line is found in many banks today. A common line is formed and the customer at the head of the line proceeds to the first free teller. The multiple-channel system presented here again assumes that arrivals follow a Poisson probability distribution and that service times are distributed exponentially. Service is first come, first served, and all servers are assumed to perform at the same rate. Other assumptions listed earlier for the single-channel model apply as well. Multiple-Channel Queuing Model With Poisson Arrivals And Exponential Service Times (M/M/m)

48. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-48 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Multiple-Channel Queuing Model With Poisson Arrivals And Exponential Service Times (M/M/m) Equations for the Multichannel Queuing Model If we let m = number of channel open λ = average arrival rate µ = average service rate at each channel The following formulas may be used in the waiting line analysis. 1.The probability that there are zero customers or units in the system:

49. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-49 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Multiple-Channel Queuing Model With Poisson Arrivals And Exponential Service Times (M/M/m) 2. The average number of customers or units in the system: 3. The average time a unit spends in the waiting line or being serviced (in the system): 4. The average number of customers or units in line waiting for service

50. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-50 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Multiple-Channel Queuing Model With Poisson Arrivals And Exponential Service Times (M/M/m) 5.The average time a customer or unit spends in the queue waiting for service 5.Utilization factor

51. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-51 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Arnold’s Muffler Shop Example (multichannel) Suppose that Arnold opened a second garage bay in which mufflers can be installed. Instead of firing his first mechanic, Blank, he would hire a second worker. The new mechanic would be expected to install mufflers at the same rate as Blank- about µ = 3 per hour. Customers, who would still arrive at the rate of λ = 2 per hour, would wait in a single line until one of the two mechanics is free. To find out how this option compares with the old single channel waiting line system, Arnold computes several operating characteristics for the m=2 channel system

52. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-52 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Arnold’s Muffler Shop Example (multichannel)/solution

53. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-53 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Arnold’s Muffler Shop Example (multichannel)/solution To complete his economic analysis, Arnold assumes that the second mechanic would be paid the same as the current one, Blank, namely, $7 per hour. The daily waiting cost now will be total daily waiting cost = (8 hours) λWqCw =(8)(2)(0.0415)($10)= $6.64 total daily service cost = (8hours)mCs= (8)(2)($7)=$112 Total daily cost of the queuing system = $6.64 + $112= $118.64

54. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-54 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Effect of service level on Arnold’s Operating Characteristics Operating characteristics Level of Service One mechanic (Blank)µ=3 Two mechanics µ=3 for each One mechanic (Smith)µ=4 P00.330.50 L2 cars0.75 car1 car W60 minutes22.5 min.30 min. Lq1.33 cars0.083 car0.50 Wq40 minutes2.5 min.15 min. Total costs$162$118.64$112

55. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-55 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Constant Service Time Model (M/D/1) Some service systems have constant service times instead of exponentially distributed times. When customers or equipment are processed according to a fixed cycle, as in the case of an automatic car wash, constant service rates are appropriate. Because constant rates are certain, the values for Lq, Wq, L, and W are always less than they would be in the models we have just discussed, which have variable service times.

56. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-56 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Constant Service Time Model (M/D/1) Equations for the Constant Service Time Model

57. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-57 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Constant service model (Grocia-Golding Recycling Example) GGR collects and compacts aluminum cans and glass bottles in New York city. Their truck drivers, who arrive to unload these materials for recycling, currently wait an average of 15 minutes before emptying their loads. The cost while waiting in queue is $ 60 /hr. A new automated compactor can be purchased that will process truck loads at a constant rate of 12 trucks / hr. Trucks arrive according to a Poisson dist. at an average rate of 8 /hr. If the new compactor is put in use, its cost will be amortized at a rate of $3 /truck unloaded. Evaluate the costs versus benefits of the purchase.

58. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-58 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Constant service model (Grocia-Golding Recycling Example)/solution Current waiting cost/trip=(1/4 hour waiting now)($60/hour cost) = $15/trip New system: λ = 8 trucks/hour arriving µ = 12 trucks/hour served Waiting cost/trip with new compactor= (1/12 hr wait) ($ 60/hr cost) = $ 5/trip Savings with new equipment = $ 15 (current) - $ 5 (new system) = $ 10/trip Cost of new equipment amortized =$ 3/trip Net savings = $ 10 - $ 3 =$ 7/trip

59. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-59 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Finite Population Model (M/M/1 with finite source) When there is a limited population of potential customers for a service facility, we need to consider a different queuing model. The limited population model permits any number of servers to be considered. The reason this model differs from the three earlier queuing models is that there is now a dependent relationship between the length of the queue and the arrival rate. To illustrate the extreme situation, if your factory had five machines and all were broken and awaiting repair, the arrival rate would drop to zero. In general, as the waiting line becomes longer in the limited population model, the arrival rate of customers or machines drops lower.

60. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-60 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Finite Population Model (M/M/1 with finite source) In this case, we describe a finite calling population model that has the following assumptions: 1.There is only one server 2.The population of units seeking service is finite 3.Arrivals follow a Poisson distribution, and service times are exponentially distributed. 4.Customers are served on a first-comes, first-served basis.

61. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-61 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Finite Population Model (M/M/1 with finite source) Equations for the Finite Population Model using λ = mean arrival rate, µ = mean service rate, N= size of the population

62. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-62 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Finite Population Model (M/M/1 with finite source)

63. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-63 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Finite Population model (Department of Commerce Example) Past records indicate that each of the five high-speed “page” printers at the U.S. Department of commerce, in Washington, needs repair after about 20 hours of use. Breakdowns have been determined to be Poisson distributed. The one technician on duty can service a printer in average of 2 hours, following an exponential dist. To compute the system’s operation characteristics we first note that the mean arrival rate is λ = 1/20=0.05 printer/hour. The mean service rate is µ = ½=0.50 printer/hour. Then:

64. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-64 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Finite Population model (Department of Commerce Example)

65. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 14-65 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Finite Population model (Department of Commerce Example) If printer downtime costs $ 120/hr and the technician is paid $ 25/hr: Total hurly cost = (average number of printers down) (cost per downtime hour) + cost / technician hr =(0.64)($ 120) + $25= $ 101.8