1. ADVANCED MANAGEMENT SCIENCE (MSC551D) INSTRUCTOR: Dr. Jose Ramon Albert Statistical Research & Training Center www.srtc.gov.ph Email: srtcres@srtc.gov.ph

2. QUEUING MODELS

3. 3 Advanced Management Science INTRODUCTION The study of queues (or waiting lines) is one of the oldest and widely used quantitative methods. The study of queues (or waiting lines) is one of the oldest and widely used quantitative methods. Queues are a common everyday experience Queues are a common everyday experience banks/supermarkets - waiting for servicebanks/supermarkets - waiting for service computers - waiting for a responsecomputers - waiting for a response failure situations - waiting for a failure to occur e.g. in a piece of machineryfailure situations - waiting for a failure to occur e.g. in a piece of machinery business/industrial service systemsbusiness/industrial service systems social service systems, e.g. judicial systemsocial service systems, e.g. judicial system public transport - waiting for a train, LRT/MRT, jeepney or a buspublic transport - waiting for a train, LRT/MRT, jeepney or a bus

4. 4 Advanced Management Science INTRODUCTION Queues form because resources are limited. Queues form because resources are limited. It makes economic sense to have queues. How many buses or trains would be needed if queues were to be avoided/eliminated? How many employees would a bank have to hire if the bank needs to avoid queuing? It makes economic sense to have queues. How many buses or trains would be needed if queues were to be avoided/eliminated? How many employees would a bank have to hire if the bank needs to avoid queuing? In designing queuing systems we need to aim for a balance between service to customers (short queues implying many servers) and economic considerations (not too many servers). In designing queuing systems we need to aim for a balance between service to customers (short queues implying many servers) and economic considerations (not too many servers).

5. 5 Advanced Management Science INTRODUCTION Decisions regarding the amount of capacity to provide are needed. Decisions regarding the amount of capacity to provide are needed. It is often impossible to predict accurately when units will arrive to seek service and/or how much time will be required to provide that service. This makes decision making difficult. It is often impossible to predict accurately when units will arrive to seek service and/or how much time will be required to provide that service. This makes decision making difficult. Queuing theory contributes vital information required for such decisions by predicting various operating characteristics of the waiting line such as the average waiting time. Queuing theory contributes vital information required for such decisions by predicting various operating characteristics of the waiting line such as the average waiting time.

6. 6 Advanced Management Science INTRODUCTION In essence all queuing systems can be broken down into individual sub- systems consisting of entities queuing for some activity. Typically we can talk of this individual sub-system as dealing with customers queuing for service.

7. 7 Advanced Management Science INTRODUCTION To analyze a sub-system we need information on: Arrival process Arrival process Service mechanism (including Calling Popn) Service mechanism (including Calling Popn) Queuing characteristics Queuing characteristics QueueServer Customer Arrivals Calling Population (Input Source) Served customers Queuing system

8. 8 Advanced Management Science INTRODUCTION Queue Server Arrivals SINGLE CHANNEL SINGLE PHASE SYSTEM

9. 9 Advanced Management Science INTRODUCTION QueueType 1 Server Arrivals SINGLE CHANNEL MULTIPHASE SYSTEM Type 2 Server

10. 10 Advanced Management Science INTRODUCTION Queue Server 1 Arrivals MULTICHANNEL SINGLE PHASE SYSTEM Server 3 Server 2

11. 11 Advanced Management Science INTRODUCTION Queue Type 1 Server 1 Arrivals MULTICHANNEL MULTI PHASE SYSTEM Type 1 Server 2 Type 2 Server 2 Type 2 Server 1

12. 12 Advanced Management Science Arrival Process how big is the size of the calling population (the total number of customers that arrive in the system) how big is the size of the calling population (the total number of customers that arrive in the system) how customers arrive e.g. singly or in groups (batch or bulk arrivals) how customers arrive e.g. singly or in groups (batch or bulk arrivals) how the arrivals are distributed in time, e.g. what is the probability distribution of time between successive arrivals (the interarrival time distribution). how the arrivals are distributed in time, e.g. what is the probability distribution of time between successive arrivals (the interarrival time distribution). Typical assumption : exponential distributionTypical assumption : exponential distribution

13. 13 Advanced Management Science The Exponential Distn (w/ parameter  Parrival time < t()  1 - e e = the mathematical constant 2.71828 - t  = the population mean of arrivals t = any value of the continuous random variable e.g. Drivers Arriving at a Toll Bridge Customers Arriving at an ATM Machine

14. 14 Advanced Management Science Describes time or distance between events Describes time or distance between events Used for queuesUsed for queues Density function Density function Parameters Parameters f(t) = e - t  f(t) t = 2.0 = 0.5 The Exponential Distn (w/ parameter 

15. 15 Advanced Management Science The Exponential Distn (w/ parameter  f(t) = exp{- t}where t>0 The mean and variance of the exp distn with parameter are 1/; 1/ 2 The mean and variance of the exp distn with parameter are 1/; 1/ 2 The distn is the only memoryless distn, i.e. The distn is the only memoryless distn, i.e. P{ T > s+t | T > t} = P{T > s} Related to the Poisson distn. Related to the Poisson distn. If T ~ exp distn with parameter then X(t) has a Poisson distn with mean t

16. 16 Advanced Management Science The Poisson Distn Poisson Process: Discrete Events in an ‘Interval’ Discrete Events in an ‘Interval’ The Probability of One Success in Interval is Stable The Probability of One Success in Interval is Stable The Probability of More than One Success in this Interval is 0 The Probability of More than One Success in this Interval is 0 Probability of Success is Independent from Interval to Probability of Success is Independent from Interval toInterval # Customers Arriving in 15 min. # Defects Per Case of Light Bulbs. PXx x x (| !  e -

17. 17 Advanced Management Science P(x ) = probability of x successes given =expected (mean) number of ‘successes’ e=2.71828 (base of natural logs) x=number of ‘successes’ per unit Px x x () !   e e.g. Find the probability of 4 customers arriving in 3 minutes when the mean is 3.6. P(X) = e -3.6 3.6 4! 4 =.1912 The Poisson Distn

18. 18 Advanced Management Science  = 0.5  = 6 Mean Standard Deviation   i i N i EX xPx      () () 1 0.2.4.6 012345 X P(X) 0.2.4.6 0246810 X P(X) The Poisson Distn

19. 19 Advanced Management Science Arrival Process whether there is a finite population of customers or (effectively) an infinite number that could arrive in a certain time interval. whether there is a finite population of customers or (effectively) an infinite number that could arrive in a certain time interval. The simplest arrival process is one where we have completely regular arrivals (i.e. the same constant time interval between successive arrivals).The simplest arrival process is one where we have completely regular arrivals (i.e. the same constant time interval between successive arrivals).

20. 20 Advanced Management Science Arrival Process A Poisson stream of arrivals corresponds to arrivals at random. Here, inter-arrival time of customers are independent, exponentially distributed. The Poisson stream is described by a single parameter - the average arrival rate.A Poisson stream of arrivals corresponds to arrivals at random. Here, inter-arrival time of customers are independent, exponentially distributed. The Poisson stream is described by a single parameter - the average arrival rate. Other important arrival processes are scheduled arrivals; batch arrivals; and time dependent arrival rates (i.e. the arrival rate varies according to the time of day).Other important arrival processes are scheduled arrivals; batch arrivals; and time dependent arrival rates (i.e. the arrival rate varies according to the time of day).

21. 21 Advanced Management Science Service Mechanism a description of the resources needed for service to begin a description of the resources needed for service to begin how long the service will take (the service time distribution) how long the service will take (the service time distribution) the number of servers available the number of servers available whether the servers are in series (each server has a separate queue) or in parallel (one queue for all servers) whether the servers are in series (each server has a separate queue) or in parallel (one queue for all servers) whether preemption is allowed (a server can stop processing a customer to deal with another "emergency" customer) whether preemption is allowed (a server can stop processing a customer to deal with another "emergency" customer)

22. 22 Advanced Management Science Service Mechanism Assuming that the service times for customers are independent and do not depend upon the arrival process is common. Another common assumption about service times is that they are exponentially distributed. Assuming that the service times for customers are independent and do not depend upon the arrival process is common. Another common assumption about service times is that they are exponentially distributed.

23. 23 Advanced Management Science Queue Characteristics Is the queue infinite or finite? Is the queue infinite or finite? The standard model assumes an infinite queue even for situations where there is a finite upper bound on the no. of permissible customers (as this bound may be rather large)The standard model assumes an infinite queue even for situations where there is a finite upper bound on the no. of permissible customers (as this bound may be rather large) The queue discipline: How, from the set of customers waiting for service, do we choose the one to be served next The queue discipline: How, from the set of customers waiting for service, do we choose the one to be served next FIFO (first-in first-out)FIFO (first-in first-out) LIFO (last-in first-out)LIFO (last-in first-out) randomlyrandomly

24. 24 Advanced Management Science Queue Characteristics Do we have: Do we have: balking (customers deciding not to join the queue if it is too long)balking (customers deciding not to join the queue if it is too long) reneging (customers leave the queue if they have waited too long for service)reneging (customers leave the queue if they have waited too long for service) jockeying (customers switch between queues if they think they will get served faster by so doing)jockeying (customers switch between queues if they think they will get served faster by so doing)

25. 25 Advanced Management Science Queue Characteristics Changing the queue discipline (the rule by which we select the next customer to be served) can often reduce congestion. Changing the queue discipline (the rule by which we select the next customer to be served) can often reduce congestion. Often the queue discipline "choose the customer with the lowest service time" results in the smallest value for the time (on average) a customer spends queuing. Often the queue discipline "choose the customer with the lowest service time" results in the smallest value for the time (on average) a customer spends queuing.

26. 26 Advanced Management Science Typical Questions of Interest How long does a customer expect to wait in the queue before they are served, and how long will they have to wait before the service is complete? How long does a customer expect to wait in the queue before they are served, and how long will they have to wait before the service is complete? What is the probability of a customer having to wait longer than a given time interval before they are served? What is the probability of a customer having to wait longer than a given time interval before they are served? What is the average length of the queue? What is the average length of the queue?

27. 27 Advanced Management Science Typical Questions of Interest What is the probability that the queue will exceed a certain length? What is the probability that the queue will exceed a certain length? What is the expected utilization of the server and the expected time period during which he will be fully occupied (remember servers cost us money so we need to keep them busy). In fact if we can assign costs to factors such as customer waiting time and server idle time then we can investigate how to design a system at minimum total cost. What is the expected utilization of the server and the expected time period during which he will be fully occupied (remember servers cost us money so we need to keep them busy). In fact if we can assign costs to factors such as customer waiting time and server idle time then we can investigate how to design a system at minimum total cost.

28. 28 Advanced Management Science Typical Problems Is it worthwhile to invest effort in reducing the service time? Is it worthwhile to invest effort in reducing the service time? How many servers should be employed? How many servers should be employed? Should priorities for certain types of customers be introduced? Should priorities for certain types of customers be introduced? Is the waiting area for customers adequate? Is the waiting area for customers adequate?

29. 29 Advanced Management Science Basic Approaches to Solving Problems/Questions analytic methods or queuing theory (formula based) analytic methods or queuing theory (formula based) this is only available for relatively simple queuing systems simulation (computer based) simulation (computer based) Complex queuing systems are almost always analyzed using discrete event simulation.

30. 30 Advanced Management Science Queuing Notations It is common to use to use the symbols: lambda to be the mean (or average) number of arrivals per time period, i.e. the mean arrival rate lambda to be the mean (or average) number of arrivals per time period, i.e. the mean arrival rate mu µ to be the mean (or average) number of customers served per time period, i.e. the mean service rate mu µ to be the mean (or average) number of customers served per time period, i.e. the mean service rate

31. 31 Advanced Management Science Queuing Notations Standard notation system A/B/C/D/E to classify queuing systems where: A represents the prob’ty distribution for the arrival process A represents the prob’ty distribution for the arrival process B represents the probability distribution for the service process B represents the probability distribution for the service process C represents the number of channels (servers) C represents the number of channels (servers) D represents the maximum number of customers allowed in the queuing system (either being served or waiting for service) D represents the maximum number of customers allowed in the queuing system (either being served or waiting for service) E represents the maximum number of customers in total E represents the maximum number of customers in total

32. 32 Advanced Management Science Queuing Notations Common options for A and B are: M for an exponential interarrival (Poisson arrival distribution) or an exponential service time distribution M for an exponential interarrival (Poisson arrival distribution) or an exponential service time distribution D for a deterministic or constant value (called also a degenerate distribution) D for a deterministic or constant value (called also a degenerate distribution) E k for an Erlang distribution E k for an Erlang distribution G for a general distribution (but with a known mean and variance) G for a general distribution (but with a known mean and variance) If D and E are not specified then it is assumed that they are infinite.

33. 33 Advanced Management Science Single Server Queue The M/M/1 queuing system, the simplest queuing system the simplest queuing system has a Poisson arrival distribution, an exponential service time distribution and a single channel (one server). has a Poisson arrival distribution, an exponential service time distribution and a single channel (one server). Since D and E are not specified, then it is assumed that they are infinite. Since D and E are not specified, then it is assumed that they are infinite.

34. 34 Advanced Management Science Single Server Queue Example Suppose we have a single server in a shop and customers arrive in the shop with a Poisson arrival distribution at a mean rate of lambda=24 customers per hour, i.e. on average one customer appears every 1/lambda = 1/24 hours=2.5 minutes. Suppose we have a single server in a shop and customers arrive in the shop with a Poisson arrival distribution at a mean rate of lambda=24 customers per hour, i.e. on average one customer appears every 1/lambda = 1/24 hours=2.5 minutes. Interarrival times have an exponential distribution with an average interarrival time of 2.5 minutes. Interarrival times have an exponential distribution with an average interarrival time of 2.5 minutes.

35. 35 Advanced Management Science Single Server Queue Example The server has an exponential service time distribution with a mean service rate of 30 customers per hour, i.e. the service rate µ=30 customers per hour. The server has an exponential service time distribution with a mean service rate of 30 customers per hour, i.e. the service rate µ=30 customers per hour. As we have a Poisson arrival rate, an exponential service time, and a single server we have a M/M/1 queue in terms of the standard notation. As we have a Poisson arrival rate, an exponential service time, and a single server we have a M/M/1 queue in terms of the standard notation.

36. 36 Advanced Management Science Formulas for Single-Server Model L =   -   - Average number of customers in the system Probability that no customers are in the system (either in the queue or being served) P 0 = 1 -  Probability of exactly n customers in the system P n = P 0 n = 1 -  n  Average number of customers in the waiting line L q =   (  - )

37. 37 Advanced Management Science Formulas for Single-Server Model  =  Probability that the server is busy and the customer has to wait Average time a customer spends in the queuing system W = = 1  -  L Probability that the server is idle and a customer can be served = 1 -  = 1 -  = 1 - = P 0 Average time a customer spends waiting in line to be served W q =  (  - )

38. 38 Advanced Management Science A Single-Server Model Given = 24 per hour,  = 30 customers per hour Probability of no customers in the system P 0 = 1 - = 1 - = 0.20  2430 L = = = 4 Average number of customers in the system  -  24 30 - 24 Average number of customers waiting in line L q = = = 3.2 (24) 2 30(30 - 24) 2  (  - )

39. 39 Advanced Management Science A Single-Server Model Given = 24 per hour,  = 30 customers per hour Average time in the system per customer W = = = 0.167 hour 1  -  1 30 - 24 Average time waiting in line per customer W q = = = 0.133  (  -  ) 24 30(30 - 24) Probability that the server will be busy and the customer must wait  = = = 0.80  2430 Probability the server will be idle 1 -  = 1 - 0.80 = 0.20

40. 40 Advanced Management Science Single Server Queue Example

41. 41 Advanced Management Science Single Server Queue Example

42. 42 Advanced Management Science Single Server Queue Example Note the first line of the output says that the results are from a formula. For this very simple queuing system, there are exact formulae that give these statistics in the output under the assumption that the system has reached a steady state - that is that the system has been running long enough so as to settle down into some kind of equilibrium position.

43. 43 Advanced Management Science Single Server Queue Example If customers are served faster than they arrive, i.e.,  1. System utilization or traffic intensity U= (arrival rate)/(departure rate) = Traffic intensity is a measure of the congestion of the system. If it is near to zero there is very little queuing and in general as the traffic intensity increases (to near 1) the amount of queuing increases. For the system we have considered above the arrival rate is 24 and the departure rate is 30 so the traffic intensity is 24/30 = 0.80

44. 44 Advanced Management Science Single Server Queue Example 2. Average number of customers in the queuing system (customers being serviced and in the waiting line) is L=  Here, 24/(30-24) = 4.0 3. Average number of customers in the waiting line is L q = (U)  Here, (0.8) 24/(30-24) = 3.2

45. 45 Advanced Management Science Single Server Queue Example 4. Average time a customer spends in the queuing system is W= L/ Here, 4.0/24=0.1667 5. Average time a customer spends in the waiting line is W q = L q / W q = L q / Here, (3.2)/24=0.1333 Note: The relationship L= W is called Little’s formula. (John D. C. Little, 1961) Note: The relationship L= W is called Little’s formula. (John D. C. Little, 1961)

46. 46 Advanced Management Science Single Server Queue Example 6. Probability server is idle (i.e. an arriving customer can be served) is I= (1- / Here, 1-24/30=0.20 Note this is the same as the probty that no customers are in the queuing system. 7. Probability an arriving customer has to wait 1-I Here, 1-0.2 =0.8

47. 47 Advanced Management Science Single Server Queue Example

48. 48 Advanced Management Science With QM for Windows

49. 49 Advanced Management Science Soln with QM for Windows

50. 50 Advanced Management Science Soln with QM for Windows

51. 51 Advanced Management Science Soln with QM for Windows

52. 52 Advanced Management Science Faster Server or More Servers? Consider the M/M/1situation, which would we prefer: one server working twice as fast; or one server working twice as fast; or two servers each working at the original rate? two servers each working at the original rate? The simple answer is that we can analyze this problem using the WINQSB. For the first situation one server working twice as fast corresponds to a service rate µ=60 customers per minute yielding the following performance.

53. 53 Advanced Management Science Faster Server

54. 54 Advanced Management Science Two Servers (M/M/2)

55. 55 Advanced Management Science Two Servers (M/M/2)

56. 56 Advanced Management Science Faster Server or More Servers? It can be seen that with one server working twice as fast, customers spend less time in the system on average, but have to wait longer for service and also have a higher probability of having to wait for service.

57. 57 Advanced Management Science Faster Server or More Servers? Of the figures in the outputs for the fast M/M/1 and the M/M/2, some are identical. Extracting key figures which are different we have: Fast M/M/1M/M/2 Ave.time in system.0278.0397 Ave time in queue.0111.0063 Probty of having to 40% 22.8571% wait for service wait for service

58. 58 Advanced Management Science Extending the example: M/M/2 with costs Consider the M/M/2 system we had but now the service rate is at 40 customers served per hour. We have also entered a queue capacity (waiting space) of 3 - i.e. if all servers are occupied and 3 customers are waiting when a new customer appears then they go away - this is known as balking. Consider the M/M/2 system we had but now the service rate is at 40 customers served per hour. We have also entered a queue capacity (waiting space) of 3 - i.e. if all servers are occupied and 3 customers are waiting when a new customer appears then they go away - this is known as balking.

59. 59 Advanced Management Science Extending the example: M/M/2 with costs We have also added cost information relating to the server and customers: We have also added cost information relating to the server and customers: each hour a server is idle costs us 10 dollars each hour a server is idle costs us 10 dollars each hour a customer waits for a server costs us 20 dollars each hour a customer waits for a server costs us 20 dollars each customer who is balked (goes away without being served) costs us 100 dollars each customer who is balked (goes away without being served) costs us 100 dollars

60. 60 Advanced Management Science M/M/2/5 with Costs

61. 61 Advanced Management Science M/M/2/5 with Costs

62. 62 Advanced Management Science Simulation on M/M/2/5

63. 63 Advanced Management Science Simulation Performance on M/M/2/5

64. 64 Advanced Management Science Constant Service Times Constant service times occur with machinery and automated equipment Constant service times occur with machinery and automated equipment Constant service times are a special case of the single-server model with general or undefined service times Constant service times are a special case of the single-server model with general or undefined service times

65. 65 Advanced Management Science Operating Characteristics for Constant Service Times  P 0 = 1 - Probability that no customers are in system Average number of customers in system L = L q +  Average number of customers in queue Lq =Lq =Lq =Lq = 2 2  (  - )

66. 66 Advanced Management Science Operating Characteristics for Constant Service Times   = Probability that the server is busy Average time customer spends in the system W = W q + 1 Average time customer spends in queue Wq =Wq =Wq =Wq =  L q

67. 67 Advanced Management Science L q = = = 1.14 cars waiting (10) 2 2(13.3)(13.3 - 10) Constant Service Times Automated car wash with service time = 4.5 min Cars arrive at rate = 10/hour (Poisson)  = 60/4.5 = 13.3/hour W q = = 1.14/10 =.114 hour or 6.84 minutes L q 2 2  (  - )

68. 68 Advanced Management Science Finite Queue Length A physical limit exists on length of queue A physical limit exists on length of queue M = maximum number in queue M = maximum number in queue Service rate does not have to exceed arrival rate (  ) to obtain steady-state conditions Service rate does not have to exceed arrival rate (  ) to obtain steady-state conditions P 0 = Probability that no customers are in system 1 - /  1 - ( /  ) M + 1 Probability of exactly n customers in system  P n = (P 0 ) for n ≤ M n L = - Average number of customers in system /  /  1 - /  (M + 1)( /  ) M + 1 1 - ( /  ) M + 1

69. 69 Advanced Management Science Finite Queue Length Let P M = probability a customer will not join system Average time customer spends in system W =W =W =W = L  (1 - P M ) L q = L -  (1- P M )  Average number of customers in queue Average time customer spends in queue W q = W - 1

70. 70 Advanced Management Science Finite Queue Quick Lube has waiting space for only 3 cars. = 20,  = 30, M = 4 cars (1 in service + 3 waiting) = 20,  = 30, M = 4 cars (1 in service + 3 waiting) Probability that no cars are in the system P 0 = = = 0.38 1 - 20/30 1 - (20/30) 5 1 - /  1 - ( /  ) M + 1 P n = (P 0 ) = (0.38) = 0.076 Probability of exactly 4 cars in the system 2030 4  n=Mn=Mn=Mn=M L = - = 1.24 Average number of cars in the system /  /  1 - /  (M + 1)( /  ) M + 1 1 - ( /  ) M + 1

71. 71 Advanced Management Science Finite Queue Average time a car spends in the system W = = 0.067 hr L  (1 - P M ) L q = L - = 0.62  (1- P M )  Average number of cars in the queue Quick Lube has waiting space for only 3 cars. = 20,  = 30, M = 4 cars (1 in service + 3 waiting) = 20,  = 30, M = 4 cars (1 in service + 3 waiting) Average time a car spends in the queue W q = W - = 0.033 hr 1

72. 72 Advanced Management Science Finite Calling Population Arrivals originate from a finite (countable) population Arrivals originate from a finite (countable) population N = population size N = population size Probability of exactly n customers in system P n = P 0 where n = 1, 2,..., N n  N! (N - n)! Average number of customers in queue L q = N - (1- P 0 )  +  Probability that no customers are in system P 0 = n = 0 n = 0 N! (N - n)!  N n 1

73. 73 Advanced Management Science Finite Calling Population Arrivals originate from a finite (countable) population Arrivals originate from a finite (countable) population N = population size N = population size Wq =Wq =Wq =Wq = L q L q (N - L) (N - L) Average time customer spends in queue L = L q + (1 - P 0 ) Average number of customers in system W = W q + Average time customer spends in system 1

74. 74 Advanced Management Science 20 machines which operate an average of 200 hrs before breaking down ( = 1/200 hr = 0.005/hr) Mean repair time = 3.6 hrs (  = 1/3.6 hr = 0.2778/hr) Probability that no machines are in the system P 0 = 0.652 Average number of machines in the queue L q = 0.169 Average number of machines in system L = 0.169 + (1 - 0.652) =.520 Finite Calling Population

75. 75 Advanced Management Science 20 machines which operate an average of 200 hrs before breaking down ( = 1/200 hr = 0.005/hr) Mean repair time = 3.6 hrs (  = 1/3.6 hr = 0.2778/hr) Finite Calling Population Average time machine spends in queue W q = 1.74 hrs Average time machine spends in system W = 5.33 hrs

76. 76 Advanced Management Science Capacity Performance WinQSB can automatically perform an analysis for us of how total cost varies with the number of servers

77. 77 Advanced Management Science Capacity Performance Approx.

78. 78 Advanced Management Science General Queuing The screen on the right shows the possible input parameters to the package in the case of a general queueing model (i.e. not a M/M/r system). The screen on the right shows the possible input parameters to the package in the case of a general queueing model (i.e. not a M/M/r system).

79. 79 Advanced Management Science General Queuing Here we have a number of possible choices for the service time distribution and the inter-arrival time distribution. In fact the package recognizes some 15 different distributions! Other items mentioned above are: Here we have a number of possible choices for the service time distribution and the inter-arrival time distribution. In fact the package recognizes some 15 different distributions! Other items mentioned above are: service pressure coefficient - indicates how servers speed up service when the system is busy, i.e. when all servers are busy the service rate is increased. If this coefficient is s and we have r servers each with service rate µ then the service rate changes from µ to (n/r) s µ when there are n customers in the system and n>=r. service pressure coefficient - indicates how servers speed up service when the system is busy, i.e. when all servers are busy the service rate is increased. If this coefficient is s and we have r servers each with service rate µ then the service rate changes from µ to (n/r) s µ when there are n customers in the system and n>=r.

80. 80 Advanced Management Science General Queuing arrival discourage coefficient - indicates how customer arrivals are discouraged when the system is busy, i.e. when all servers are busy the arrival rate is decreased. If this coefficient is s and we have r servers with the arrival rate being lambda then the arrival rate changes from lambda to (r/(n+1)) s lambda when there are n customers in the system and n>=r. arrival discourage coefficient - indicates how customer arrivals are discouraged when the system is busy, i.e. when all servers are busy the arrival rate is decreased. If this coefficient is s and we have r servers with the arrival rate being lambda then the arrival rate changes from lambda to (r/(n+1)) s lambda when there are n customers in the system and n>=r. batch (bulk) size distribution - customers can arrive together (in batches, also known as in bulk) and this indicates the distribution of size of such batches. batch (bulk) size distribution - customers can arrive together (in batches, also known as in bulk) and this indicates the distribution of size of such batches.

81. 81 Advanced Management Science General Queuing Example

82. 82 Advanced Management Science General Queuing Example

83. 83 Advanced Management Science General Queuing Example

84. 84 Advanced Management Science General Queuing Clearly the longer we simulate, the more confidence we may have in the statistics/probabilities obtained. Clearly the longer we simulate, the more confidence we may have in the statistics/probabilities obtained. As before we can investigate how the system might behave with more servers. Simulating for 1000 hours (to reduce the overall elapsed time required) and looking at just the total system cost per hour (item 22 in the above outputs) we get the following results: As before we can investigate how the system might behave with more servers. Simulating for 1000 hours (to reduce the overall elapsed time required) and looking at just the total system cost per hour (item 22 in the above outputs) we get the following results:

85. 85 Advanced Management Science General Queuing No. of Servers (Total system cost) No. of Servers (Total system cost) 1 (4452)2 (3314) 3 (2221) 4 (1614)5 (1257) 6 ( 992) 7 (832) 8 (754)9 (718) 10 (772) 11(833) 12(902) Hence here the number of servers associated with the minimum total system cost is 9