1. DECISION MODELING WITH MICROSOFT EXCEL Chapter 15 Copyright 2001 Prentice Hall Publishers and Ardith E. Baker Part 1 QUEUING

2. INTRODUCTION ________models are everywhere. For example, airplanes “queue up” in holding patterns, waiting for a runway so they can land. Then, they line up again to take off. People line up for_________, to buy groceries, etc. Jobs line up for machines, orders line up to be filled, and so on. A. K. Erlang (a Danish engineer) is credited with founding queuing _________by studying telephone switchboards in Copenhagen for the Danish Telephone Company. Many of the queuing results used today were developed by____________.

3. A _______________is one in which you have a sequence of times (such as people) _________at a facility for service, as shown below: Consider St. Luke’s Hospital in Philadelphia and the following three queuing models. Model 1: St. Luke’s Hematology Lab St. Luke’s treats a large number of patients on an outpatient basis (i.e., not admitted to the hospital). _____________plus those admitted to the 600-bed hospital produce a large flow of new patients each day. Arrivals00000 Service Facility

4. Most new patients must visit the ___________ laboratory as part of the diagnostic process. Each such patient has to be seen by a_____________. After seeing a doctor, the _________arrives at the laboratory and checks in with a clerk. Patients are assigned on a___________, first-served basis to test rooms as they become available. The technician assigned to that room performs the tests ordered by the_________. When the testing is complete, the patient goes on to the next step in the ________and the technician sees a new patient. We must decide how many _______________to hire.

5. WATS (Wide Area Telephone Service) is an acronym for a special flat-rate, long distance service offered by some phone companies. Model 2: Buying WATS Lines As part of its ____________process, St. Luke’s is designing a new communications system which will include ______ lines. We must decide how many WATS lines the hospital should buy so that a ____________of busy signals will be encountered. When all the phone lines allocated to WATS are in ____, the person dialing out will get a ______signal, indicating that the call can’t be completed.

6. The equipment includes __________devices such as Model 3: Hiring Repairpeople St. Luke’s hires repairpeople to ____________20 individual pieces of electronic equipment. If a piece of equipment _____and all the repairpeople are occupied, it must wait to be repaired. We must decide how many __________to hire and balance their cost against the cost of having ______ equipment. electrocardiogram machines small dedicated computers CAT scanner other equipment

7. All three of these models fit the general description of a queuing model as described below: PROBLEM ARRIVALSSERVICE FACILITY 1 Patients Technicians 2 Telephone Calls Switchboard 3 Broken Equipment Repairpeople These models will be resolved by using a combination of __________and simulation models. To begin, let’s start with the _______queuing model.

8. THE BASIC MODEL Consider the Xerox machine located in the fourth- floor secretarial service suite. Assume that users _______at the machine and form a ______________. Each arrival in turn uses the machine to perform a specific ______which varies from obtaining a copy of a 1-page letter to producing 100 copies of a 25- page report. This system is called a ________________(or single- __________) queue.

9. 2. The number of people in the ________(waiting for service). 3. The ____________in the system (the interval between when an individual enters the system and when he or she leaves the system). 4. The waiting time in the queue (the time between _________the system and the beginning of service). Questions about this or any other queuing system center on four _____________: 1. The number of people in the _________(those being served and waiting in line).

10. ASSUMPTIONS OF THE BASIC MODEL 1. ________Process. Each arrival will be called a “job.” The __________time (the time between arrivals) is not known. Therefore, the ___________probability distribution (or negative exponential distribution) will be used to describe the interarrival times for the basic model. The exponential distribution is completely specified by one______________,, the mean arrival rate (i.e., how many jobs arrive on the average during a specified _______period).

11. ________interarrival time is the average time between two________. Thus, for the exponential distribution Avg. time between jobs = mean interarrival time = 1 Thus, if = 0.05 mean interarrival time = = = 20 1 1 0.05 0.05 2. _________Process. In the basic model, the time that it takes to ____________a job (the service time) is also treated with the _____________distribution. The _________for this exponential distribution is called  (the mean service rate in jobs per minute).

12.  T is the number of ____that would be served (on the________) during a period of T minutes if the machine were ______during that time. The mean or average, _____________(the average time to complete a job) is Avg. service time = 1 Thus, if  = 0.10 mean service time = = = 10 mean service time = = = 101 1 0.10 0.10 3. _______Size. There is no limit on the number of jobs that can ______in the queue (an _________queue length).

13. 4. Queue___________. Jobs are served on a first-come, _________basis (i.e., in the same order as they arrive at the queue). 5. Time_________. The system operates as described _______________over an infinite horizon. 6. Source___________. There is an infinite population available to__________.

14. CHARACTERISTICS OF THE BASIC MODEL The values of the ___________ and  (together with the____________) are all that is needed to calculate several important ________________________of the basic model. NOTE: these formulas hold only if < . CHARACTERISTICSYMBOL FORMULA Utilization -- /  Exp. No. in System L /(  – ) Exp. No. in Queue L q 2 /  (  – ) Exp. Waiting Time W 1 /(  – ) Exp. Time in Queue W q /  (  – ) Prob. System is Empty P 0 1 – ( /  )

15. Consider a specific example where = 0.25 and  = 0.10. Thus, on the average, a job ________every 1/ = 1 / 0.25 = 4 minutes. Similarly, the time it takes to complete a job, on __________, is 1/  = 1 / 0.10 = 10 minutes. In this case, since______, the service operation will get further ________(the queue will grow longer) as time goes by. Now, return to the Xerox model, in which < . Spreadsheets are ideal for crunching the numerical results from such formulas.

16. The following Excel spreadsheet was originally developed by Professor David Ashley. It already has the formulas entered. Here is the introductory page.

17. Plugging the __________values from the Xerox model, = 0.05 and  = 0.10 into the appropriate cells of the _____worksheet yields the following results:

18. Steady-State Result Let’s interpret the results. L is the expected _________of people in the system after the queue has reached steady-state. _______________ means that the probability that you will observe a certain number of people in the system does not depend on the _____at which you count them.

19. Thus, in a steady-state, 1.The system is ________with probability ½ 2.On average, there is _________in the queue 3.On average, an ________must wait 10 min. before starting to use the machine. 4.On average, an arrival will spend 20 minutes in the_________.

20. Using the Results These results hold for the ______ model and the particular values for the ___________ ( = 0.05 and  = 0.10 ). Suppose, for example, _______________makes the following calculations: Since = 0.05, on the average 5 / 100 of a job arrives each minute. During each 8-hour day there are 8 x 60 = 480 __________. Thus, during each day there is on the average a total of (0.05)(480) = 24 arrivals. (24 arrivals/day)(20 minutes/arrival) = 480 minutes or 8 hours We know that on the average each person spends 20 minutes in the _________(W = 20). Thus, the total time spent at the _____________is:

21. If management feels that 8 hours is too ________to spend at the facility, then the following steps might be taken: 1.A new machine might be _____________with a smaller mean service time. 2.Another machine might be purchased and both machines used to satisfy the__________. This would change the system to a _________ queue. 3.Some __________might be sent to a different and less busy copying facility. This would change the __________process. In any case, management must _________the cost of providing service against the cost of waiting.

22. A TAXONOMY OF QUEUING MODELS There are many possible ______________models. Therefore, to facilitate communication among those working on queuing models, D. G. Kendall proposed a _________based on the following notation: A/B/s A = arrival distribution B = service distribution s = number of servers where Different _________are used to designate certain _______________. Placed in the A or the B position, they indicate the arrival or the service distribution, respectively.

23. M = ______________distribution D = deterministic____________ G = any (a general) distribution of _______times GI = any (a general) distribution of ______times The following conventions are in general use: For example, the Xerox model is an M/M/1 model (i.e., a _____________queue with exponential inter- arrival and __________times).

24. LITTLE’S FLOW EQUATION AND RELATED RESULTS It can be proven that in a ________________queuing process L = W In the Xerox model, L = 0.05 x 20 = 1.0 Consider the diagram below: Scene 1: Our Hero Arrives In scene 1 our hero arrives and joins the queue. This equation is often called ________flow equation.

25. Arrived during the time our hero waited and was served Scene 2: Hecompletesservice In scene 2 he has just completed service. Assume the _______is in steady-state. Since in this case the average number of people in the system is _______________of time, let’s measure this quantity when our hero completes being served. At this time, the number of people in the ________is precisely the total number who ________after he did. Therefore, if W is his ____________time and people arrive at a rate of, we would expect L to equal____.

26. Note that Little’s flow equation applies to any ___________queuing process and is thus applicable to a wide variety of models. The proof used to establish Little’s flow equation also shows that L q = W q In the Xerox model, L q = 0.05 x 10 = 0.5 It is essential that represent the _____at which arrivals join the queue. Consider, for example, a _______with an upper limit on the number of items that can wait in the queue (called a______________).

27. In such a system, a person can call, find the system ____(receive a busy signal) and be sent away (_____ up). For example, a modern phone system holds only a certain number of calls (say, 10) in a queue until they are answered by the next ____________service representative. This person does not join the queue. This is called a _________. Similarly, a customer may tire of waiting in line (or being on hold) and ________without being served. Here, the person joined the queue, became impatient and left without completing the transaction. This is called______________.

28. Another important general result depends on the _______________that expected waiting time = expected waiting time in queue + expected service time For the basic model, we found that Expected service time = 1 Putting the general result in __________yields W = W q + 1 For the Xerox model: W = 10 + = 20 1 0.10 0.10 This general result holds for any queuing model in which a ________________occurs.

29. Now we can easily compute the four _________ characteristics L, L q, W, W q : First calculate L:  -  - L =L =L =L = From __________flow equation we know that L = W Thus, after computing L, we can now compute W: W = L/ W = L/ Knowing W, we can now compute W q : W q = W - 1 L q = W q And knowing W q, we can now compute L q :

30. THE M/G/1 QUEUE The ________________distribution may not fit the service process very well. Fortunately, there is a generalization of the ______ model that permits the _____________of the service time to be____________. It is not necessary to know the __________ distribution, only the mean service time, 1/ , and its___________,  2.

31. The operating characteristics for this model are: CHARACTERISTICSYMBOL FORMULA Utilization -- /  Exp. No. in System L L q + /  Exp. No. in Queue L q 2  2 + ( /  ) 2 2(1 – /  ) 2(1 – /  ) Exp. Waiting Time W W q + 1 /  Exp. Time in Queue W q L q / Exp. Time in Queue W q L q / Prob. System is Empty P 0 1 – ( /  ) As  2 increases, L, L q, W, W q all increase. This means that the ____________of a server may be as important as the speed of the server.

32. For example, suppose you must hire a secretary and you have to select one of two candidates. Secretary 1 is very____________, typing any document in exactly 15 minutes. Secretary 2 is somewhat faster, with an average of 14 minutes per___________, but with times varying according to the exponential distribution. With the “MG1” and “MMs” worksheets, we can easily determine which secretary will give shorter average _____________times on documents.

33. Since Secretary 1 types every document in exactly 15 minutes,  2 = 0. In addition, = 3 per hour (or 0.05 per minute) and  = 1 / 15 per minute.

34. For Secretary 2, enter the ______________  = 14, = 0.05, and  = 1 / 14 per minute. Even though Secretary 2 is “________,” her average turnaround times are longer because of the high ____________of her service times.

35. MODEL 1: AN M/M/S QUEUE (HEMATOLOGY LAB) Previously, the stated goal was to attack the three St. Luke’s Hospital models with __________models. Consider the blood-testing model (Model 1): 000 Server 1 Server 2 Server 3 Each patient joins a ________ queue and, on arriving at the head of the line, enters the first ___________examining room.

36. This type of system must not be confused with a system in which a queue forms _________of each server, as in a typical grocery store. Now, assume This implies that a new patient arrives every 5 minutes on average since Mean interarrival = 1/ = 1/0.20 = 5 The __________time is given by an exponential distribution with ____________ = 0.20 per min. Each server is______________

37. This implies that the _______service time is 8 minutes since Mean service time for an individual server = 1/  = 1/0.125 = 8 Each service time is given by an __________ distribution with parameter  = 0.125 per min. Note that if there were only one server, the queue would grow without _________because >  (0.20 > 0.125). However, for a __________queue, a steady-state will exist as long as < s , where s is the number of ___________. For example, if there are 2 servers, a steady-state will be __________because 0.20 < (2*0.125).

38. The Key Equations We want to find the values L, L q,W, and W q. However, since this is a __________ queue, we must use different_________. For this model, the probability that the system is _________is: P 0 = 1 ( /  ) n n! n! ( /  ) s s! s! 1 1 - ( /s  ) +  n=0 s-1 Lq =Lq =Lq =Lq = P0P0P0P0 ( /  ) s+1 (s-1)!(s- /  ) 2 The expected _________of people in the queue is:

39. Assume now that we have decided to hire two technicians. We can input the parameters s = 2, = 0.20, and  = 0.125 into the MMs worksheet to obtain the results.

40. Here are the results when a third technician is added (s = 3). Note that the expected waiting time has been reduced from 22.22 minutes to 9.56 minutes.

41. When a fourth technician is added (s = 4), the expected waiting time has been reduced slightly more.

42. Adding additional servers may _________the waiting time, however, you have to consider not only how much extra you will ____for the additional servers but also if those additional servers will be busy most of the time. The previous results show that as more servers are added, the percentage of ________for the technicians increases, which could lead to ________ and sloppy work.

43. ECONOMIC ANALYSIS OF QUEUING SYSTEMS The cost of hiring ____________technicians is fairly clear. Now, let’s determine the cost of___________. The cost to the patient is __________to the decision, except as it affects the patient’s willingness to use the hospital. Besides the possible effect on demand, the hematology lab could _____the hospital money if it reduced the output of the hospital.

44. Cost Parameters If you are willing and able to _________certain costs, you can build expected cost models of _________systems. Consider, for example, the hematology lab model (in general terms any _________queue with exponential ___________and service times), and suppose the manager is willing to specify two costs: C s = cost per hour of having a server available C W = cost per hour of having a person wait in the system (a very “fuzzy” or qualitative cost) Let’s start by calculating the ___________of hiring 2 servers for an 8-hour day.

45. There are two components: _________cost = (C s )(2)(8) where C s is the cost per hour for one server 2 is the number of servers 8 is the number of hours each server works _________cost = (C W )(L 2 )(8) where L 2 is the number of people in the queue when there are two servers. To calculate the ___________of using 4 servers for a 6-hour day, take (C s )(4)(6) + (C W )(L 4 )(6) or [(C s )(4) + (C W )(L 4 )]6

46. The Total Cost per Hour We now define TC(s) = total cost per hour of using s servers = (C s )(s) + (C W )(L s ) The goal is to find the value of s that will _________ the sum of these two costs. Note that as s, the number of_______, increases, the waiting cost will _________and the server cost will increase. Unfortunately, it is not possible to derive a formula that gives the ___________value of s. However, consider the following worksheet…

47. In this worksheet, specify C s = $50/server/hour and C w = $100/customer/hour. The results show that 3 servers minimizes the Total Cost. an 8- hour shift. Compare the cost over

48. Next, create a data table to determine the sensitivity of this decision to the “fuzzy” cost, C W ($0 to $180). Highlight the block, click on Data – Table. Enter B2 as the Column Input and click OK.

49. Excel automatically fills in the table as shown below.

50. These results can be ______to look for patterns and ________. You can see that 2 servers is __________for C W = 0, while 3 servers is optimal from C W = $20 up to $180. For values of C W > $200, 4 servers is optimal.

51. End of Part 1 Please continue to Part 2