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# Queuing Theory Basics ( PDFDrive ).pdf

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# Queuing Theory Basics ( PDFDrive ).pdf

Waiting Line Models and Queuing Theory. Describes fundamentals of Queuing theory and Models and their application

Waiting Line Models and Queuing Theory. Describes fundamentals of Queuing theory and Models and their application

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### Queuing Theory Basics ( PDFDrive ).pdf

1. 1. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 1 Waiting Line Models: Queuing Theory Basics
2. 2. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 2 When you are in queue?  In the bank, restaurant, supermarket…  In front of restroom during the break of football game How much is your patience?  Waiting costs your patience and your temper and it also costs the business. Time = For the business, they have to find the optimal service level that keeps customers happy and makes them profitable.
3. 3. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 3 INTRODUCTION INTRODUCTION Queuing models are everywhere. For example, Queuing models are everywhere. For example, airplanes airplanes “ “queue up queue up” ” in holding patterns, waiting for in holding patterns, waiting for a runway so they can land. Then, they line up again a runway so they can land. Then, they line up again to take off. to take off. People line up for tickets, to buy groceries, etc. People line up for tickets, to buy groceries, etc. Jobs line up for machines, orders line up to be filled, Jobs line up for machines, orders line up to be filled, and so on. and so on.
4. 4. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 4 INTRODUCTION INTRODUCTION A. K. Erlang (a Danish engineer) is credited with A. K. Erlang (a Danish engineer) is credited with founding queuing theory by studying telephone founding queuing theory by studying telephone switchboards in Copenhagen for the Danish switchboards in Copenhagen for the Danish Telephone Company. Telephone Company. Many of the queuing results used today were Many of the queuing results used today were developed by Erlang. developed by Erlang.
5. 5. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 5 A A queuing model queuing model is one in which you have a is one in which you have a sequence of times (such as people) arriving at a sequence of times (such as people) arriving at a facility for service, as shown below: facility for service, as shown below: Arrivals Arrivals 00000 00000 Service Facility Service Facility
6. 6. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 6 Consider St. Luke Consider St. Luke’ ’s Hospital in Philadelphia and the s Hospital in Philadelphia and the following three queuing models. following three queuing models. Model 1: St. Luke Model 1: St. Luke’ ’s Hematology Lab s Hematology Lab St. Luke St. Luke’ ’s s treats a large number of patients on an outpatient treats a large number of patients on an outpatient basis (i.e., not admitted to the hospital). basis (i.e., not admitted to the hospital). Outpatients plus those admitted to the 600 Outpatients plus those admitted to the 600- -bed bed hospital produce a large flow of new patients each hospital produce a large flow of new patients each day. day.
7. 7. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 7 Most new patients must visit the hematology Most new patients must visit the hematology laboratory as part of the diagnostic process. Each laboratory as part of the diagnostic process. Each such patient has to be seen by a technician. such patient has to be seen by a technician. After seeing a doctor, the patient arrives at the After seeing a doctor, the patient arrives at the laboratory and checks in with a clerk. laboratory and checks in with a clerk. Patients are assigned on a first Patients are assigned on a first- -come, first come, first- -served served basis to test rooms as they become available. basis to test rooms as they become available. The technician assigned to that room performs the The technician assigned to that room performs the tests ordered by the doctor. tests ordered by the doctor. When the testing is complete, the patient goes on to When the testing is complete, the patient goes on to the next step in the process and the technician sees the next step in the process and the technician sees a new patient. a new patient. We must decide how many technicians to hire. We must decide how many technicians to hire.
8. 8. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 8 WATS (Wide Area Telephone Service) is an acronym WATS (Wide Area Telephone Service) is an acronym for a special flat for a special flat- -rate, long distance service offered rate, long distance service offered by some phone companies. by some phone companies. Model 2: Buying WATS Lines Model 2: Buying WATS Lines As part of its As part of its remodeling process, St. Luke remodeling process, St. Luke’ ’s is designing a new s is designing a new communications system which will include WATS communications system which will include WATS lines. lines.
9. 9. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 9 We must decide how many WATS lines the hospital We must decide how many WATS lines the hospital should buy so that a minimum of busy signals will should buy so that a minimum of busy signals will be encountered. be encountered. When all the phone lines allocated to WATS are in When all the phone lines allocated to WATS are in use, the person dialing out will get a busy signal, use, the person dialing out will get a busy signal, indicating that the call can indicating that the call can’ ’t be completed. t be completed.
10. 10. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 10 The equipment includes measuring devices such as The equipment includes measuring devices such as Model 3: Hiring Repairpeople Model 3: Hiring Repairpeople St. Luke St. Luke’ ’s hires s hires repairpeople to maintain repairpeople to maintain 20 20 individual pieces of individual pieces of electronic equipment. electronic equipment. electrocardiogram machines electrocardiogram machines small dedicated computers small dedicated computers CAT scanner CAT scanner other equipment other equipment
11. 11. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 11 If a piece of equipment fails and all the repairpeople If a piece of equipment fails and all the repairpeople are occupied, it must wait to be repaired. are occupied, it must wait to be repaired. We must decide how many repairpeople to hire and We must decide how many repairpeople to hire and balance their cost against the cost of having broken balance their cost against the cost of having broken equipment. equipment.
12. 12. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 12 All three of these models fit the general description All three of these models fit the general description of a queuing model as described below: of a queuing model as described below: PROBLEM PROBLEM ARRIVALS ARRIVALS SERVICE FACILITY SERVICE FACILITY 1 1 Patients Patients Technicians Technicians 2 2 Telephone Calls Telephone Calls Switchboard Switchboard 3 3 Broken Equipment Repairpeople Broken Equipment Repairpeople These models will be resolved by using a These models will be resolved by using a combination of analytic and simulation models. combination of analytic and simulation models. To begin, let To begin, let’ ’s start with the basic queuing model. s start with the basic queuing model.
13. 13. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 13 Queuing Systems ... customers channel (server) system arrival departure waiting line (queue) Single Channel Waiting Line System system arrival departure server 2 server k ... . . . server 1 Multi-Channel Waiting Line System
14. 14. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 14 Bank Customers Teller Deposit etc. Doctor’s Patient Doctor Treatment office Traffic Cars Light Controlled intersection passage Assembly line Parts Workers Assembly Tool crib Workers Clerks Check out/in tools Situation Arrivals Servers Service Process Waiting Line Examples
15. 15. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 15 THE BASIC MODEL THE BASIC MODEL Consider the Xerox machine located in the fourth Consider the Xerox machine located in the fourth- - floor secretarial service suite. Assume that users floor secretarial service suite. Assume that users arrive at the machine and form a single line. arrive at the machine and form a single line. Each arrival in turn uses the machine to perform a Each arrival in turn uses the machine to perform a specific task which varies from obtaining a copy of a specific task which varies from obtaining a copy of a 1 1- -page letter to producing page letter to producing 100 100 copies of a copies of a 25 25- -page page report. report. This system is called a single This system is called a single- -server (or single server (or single- - channel channel) queue. ) queue.
16. 16. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 16 2. 2. The number of people in the queue (waiting The number of people in the queue (waiting for service). for service). Questions about this or any other queuing system Questions about this or any other queuing system center on four quantities: center on four quantities: 1. 1. The number of people in the system (those The number of people in the system (those being served and waiting in line). being served and waiting in line).
17. 17. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 17 3. 3. The waiting time in the system (the interval The waiting time in the system (the interval between when an individual enters the system between when an individual enters the system and when he or she leaves the system). and when he or she leaves the system). 4. 4. The waiting time in the queue (the time The waiting time in the queue (the time between entering the system and the between entering the system and the beginning of service). beginning of service).
18. 18. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 18 ASSUMPTIONS OF THE BASIC MODEL ASSUMPTIONS OF THE BASIC MODEL 1. 1. Arrival Process. Arrival Process. Each arrival will be called a Each arrival will be called a “ “job. job.” ” The The interarrival time interarrival time (the time between (the time between arrivals) is not known. arrivals) is not known.
19. 19. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 19 Therefore, the Therefore, the exponential probability exponential probability distribution distribution (or (or negative exponential negative exponential distribution distribution) will be used to describe the ) will be used to describe the interarrival times for the basic model. interarrival times for the basic model. Probabilidad de que la atención sea completada dentro de “ t “ unidades de tiempo X = t
20. 20. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 20 The exponential distribution is The exponential distribution is completely specified by one completely specified by one parameter, l, the mean arrival rate parameter, l, the mean arrival rate (i.e., how many jobs arrive on the (i.e., how many jobs arrive on the average during a specified time average during a specified time period). period).
21. 21. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 21 Mean interarrival time is the average time Mean interarrival time is the average time between two arrivals. Thus, for the between two arrivals. Thus, for the exponential distribution exponential distribution Avg. time between jobs = mean interarrival time = Avg. time between jobs = mean interarrival time = 1 1 λ λ Thus, if Thus, if λ λ = 0.05 = 0.05 mean interarrival time = = = 20 mean interarrival time = = = 20 1 1 λ λ 1 1 0.05 0.05
22. 22. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 22 2. 2. Service Process. Service Process. In the basic model, the time In the basic model, the time that it takes to complete a job (the that it takes to complete a job (the service service time time) is also treated with the exponential ) is also treated with the exponential distribution. distribution. The parameter for this exponential distribution The parameter for this exponential distribution is called is called μ μ (the (the mean service rate mean service rate in jobs per in jobs per minute). minute).
23. 23. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 23 μ μT T is the number of jobs that would be served is the number of jobs that would be served (on the average) during a period of (on the average) during a period of T T minutes minutes if the machine were busy during that time. if the machine were busy during that time. The The mean mean or or average average, , service time service time (the (the average time to complete a job) is average time to complete a job) is Avg. service time = Avg. service time = 1 1 μ μ Thus, if Thus, if μ μ = 0.10 = 0.10 mean service time = = = 10 mean service time = = = 10 1 1 μ μ 1 1 0.10 0.10
24. 24. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 24 3. 3. Queue Size. Queue Size. There is no limit on the number There is no limit on the number of jobs that can wait in the queue (an infinite of jobs that can wait in the queue (an infinite queue length). queue length).
25. 25. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 25 4. 4. Queue Discipline. Queue Discipline. Jobs are served on a first Jobs are served on a first- - come, first come, first- -serve basis (i.e., in the same order serve basis (i.e., in the same order as they arrive at the queue). as they arrive at the queue).
26. 26. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 26 5. 5. Time Horizon. Time Horizon. The system operates as The system operates as described continuously over an infinite described continuously over an infinite horizon. horizon.
27. 27. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 27 6. 6. Source Population. Source Population. There is an infinite There is an infinite population available to arrive. population available to arrive.
28. 28. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 28 QUEUE DISCIPLINE QUEUE DISCIPLINE In addition to the arrival distribution, service In addition to the arrival distribution, service distribution and number of servers, the queue distribution and number of servers, the queue discipline must also be specified to define a queuing discipline must also be specified to define a queuing system. system. So far, we have always assumed that arrivals were So far, we have always assumed that arrivals were served on a first served on a first- -come, first come, first- -serve basis (often called serve basis (often called FIFO, for FIFO, for “ “first first- -in, first in, first- -out out” ”). ).
29. 29. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 29 QUEUE DISCIPLINE QUEUE DISCIPLINE However, this may not always be the case. For However, this may not always be the case. For example, in an elevator, the last person in is often example, in an elevator, the last person in is often the first out (LIFO). the first out (LIFO). Adding the possibility of selecting a good queue Adding the possibility of selecting a good queue discipline makes the queuing models more discipline makes the queuing models more complicated. complicated. These models are referred to as scheduling models. These models are referred to as scheduling models.
30. 30. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 30 Various Type of Queues • Single Channel/Single Phase • Multi-channel/Single Phase • Single Channel/Multi-phase • Queuing Network
31. 31. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 31 Queuing System Structure Server departure arrival • Arrival characteristic 1. Size of units - Single - Batch 2. Arrival rate - Constant - Probabilistic 3. Degree of patience - Patient - Impatient • Arrival characteristic 1. Size of units - Single - Batch 2. Arrival rate - Constant - Probabilistic 3. Degree of patience - Patient - Impatient • Features of lines 1. Length - Infinite capacity - Limited capacity 2. Number - Single - Multiple 3. Queue discipline - FIFO - Priorities • Features of lines 1. Length - Infinite capacity - Limited capacity 2. Number - Single - Multiple 3. Queue discipline - FIFO - Priorities • Service facility 1. Structure 2. Service rate - Constant - Probabilistic - random services - State-dependent service time • Service facility 1. Structure 2. Service rate - Constant - Probabilistic - random services - State-dependent service time • Population Source - Finite - Infinite • Population Source - Finite - Infinite • Exit 1. Return to service population 2. Do not return to service population • Exit 1. Return to service population 2. Do not return to service population
32. 32. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 32 Deciding on the Optimum Level of Service Total expected cost Negative Cost of waiting time to company Cost Low level of service Optimal service level High level of service Minimum total cost Cost of providing service
33. 33. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 33 Performance Measures P0 = Probability that there are no customers in the system Pn = Probability that there are n customers in the system LS = Average number of customers in the system LQ = Average number of customers in the queue WS = Average time a customer spends in the system WQ = Average time a customer spends in the queue Pw = Probability that an arriving customer must wait for service ρ = Utilization rate of each server (the percentage of time that each server is busy)
34. 34. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 34 X: # of customer arrivals within time interval of length t Pr(X=n) = λ = the mean arrival rate per time unit t = the length of the time interval e = 2.7182818 (the base of the natural logarithm) n! = n(n−1)(n−2) (n−3)… (3)(2)(1) X: # of customer arrivals within time interval of length t Pr(X=n) = λ = the mean arrival rate per time unit t = the length of the time interval e = 2.7182818 (the base of the natural logarithm) n! = n(n−1)(n−2) (n−3)… (3)(2)(1) ( ) ! n e t t n λ λ − Very large population of potential customers • behave independently • in any time instant, at most one arrives • arrive at intervals of average duration 1/λ Arrival Process Very large population of potential customers • behave independently • in any time instant, at most one arrives • arrive at intervals of average duration 1/λ X follows Poisson Distribution(λt) Mean = λt Variance = λt λ = arrival rate = # of arrivals per unit of time t should be expressed in the same time unit as λ X follows Poisson Distribution(λt) Mean = λt Variance = λt λ = arrival rate = # of arrivals per unit of time t should be expressed in the same time unit as λ Important Important
35. 35. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 35 Examples of Poisson Distribution 0.0 0.2 0.4 p(x) Poisson distribution with parameter 1/2 x 0 1 2 3 0.0 0.2 p(x) Poisson distribution with parameter 2 x 0 1 2 3 4 5 Poisson distribution with parameter 1 0.0 0.2 0.4 p(x) 0 1 2 3 4 x
36. 36. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 36 Service Process Assume service time is exponentially distributed(μ) P.D.F. f(t) = μe-μt Pr(Service ≤ t) = 1 − e-μt Mean = 1/μ Variance = 1/μ2 μ= service rate = # of customers served per unit time Assume service time is exponentially distributed(μ) P.D.F. f(t) = μe-μt Pr(Service ≤ t) = 1 − e-μt Mean = 1/μ Variance = 1/μ2 μ= service rate = # of customers served per unit time Properties of exponential distribution 1. Memoryless (The conditional probability is the same as the unconditional probability.) 2. Most customers require short services; few require long service 3. If arrival process follows Possion (λ), then inter-arrival time follows exponential(λ) Properties of exponential distribution 1. Memoryless (The conditional probability is the same as the unconditional probability.) 2. Most customers require short services; few require long service 3. If arrival process follows Possion (λ), then inter-arrival time follows exponential(λ) Important Important
37. 37. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 37 Examples of Exponential Distribution
38. 38. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 38 Queuing Theory Notation  A standard notation is used in queuing theory to denote the type of system we are dealing with.  Typical examples are:  M/M/1 Poisson Input/Poisson Server/1 Server  M/G/1 Poisson Input/General Server/1 Server  D/G/n Deterministic Input/General Server/n Servers  E/G/∞ Erlangian Input/General Server/Inf. Servers  The first letter indicates the input process, the second letter is the server process and the number is the number of servers.  (M = Memoryless = Poisson)
39. 39. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 39 Terminology λ = Arrival rate = 1/ Mean arrival interval μ = Service rate = 1/ Mean service time ρ = λ/ μ k = # of Servers λ = Arrival rate = 1/ Mean arrival interval μ = Service rate = 1/ Mean service time ρ = λ/ μ k = # of Servers P0 = Probability that there are no customers in the system Pn = Probability that there are n customers in the system LS = Average number of customers in the system LQ = Average number of customers in the queue WS = Average time a customer spends in the system WQ = Average time a customer spends in the queue Pw = Probability that an arriving customer must wait for service ρ = Utilization rate of each server (the percentage of time that each server is busy) P0 = Probability that there are no customers in the system Pn = Probability that there are n customers in the system LS = Average number of customers in the system LQ = Average number of customers in the queue WS = Average time a customer spends in the system WQ = Average time a customer spends in the queue Pw = Probability that an arriving customer must wait for service ρ = Utilization rate of each server (the percentage of time that each server is busy) Performance measures
40. 40. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 40 Single Server Case Poisson arrivals, exponential service rate, no priorities, no balking, steady state) STATE ENTRY RATE LEAVING RATE 0 λP0 µP1 1 λP0 + µP2 (λ + µ)P1 2 λP1 + µP3 (λ + µ)P2 3 λP2 + µP4 (λ + µ)P3 : : : : : : Pn = (λ/μ)nP0 = ρnP0 for n = 1,2,3,... 1 = 1 if 1 0 0 0 0 < = = − ∞ = ∞ = ∑ ∑ ρ ρ ρ P n n n n P P 0 1 2 3 4 .... λ μ λ μ λ μ λ μ ONLY IF λ< μ or λ/μ = ρ <1 Steady state exists! ONLY IF λ< μ or λ/μ = ρ <1 Steady state exists!
41. 41. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 41 Single Server Case ⇒ P0 = 1 − ρ ⇒ Pn = ρn (1 − ρ ) LS = E[N] where N = no. of customers in system (denote S) = = ρ /(1 − ρ ) LQ = E[Nq] where Nq = no. of customers in queue (denote Q) = = ρ2/(1− ρ ) WS = LS/ λ WQ = LQ/ λ ∑ ∞ = 0 n n nP ( ) ∑ ∞ = − 0 1 n n P n Little’s Law LS = λ WS LQ= λ WQ Little’s Law LS = λ WS LQ= λ WQ
42. 42. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 42 Single Server Queue Performance (M/M/1) P0 = 1 – λ/μ Pn = (λ/μ)nP0 LQ = LS = LQ + λ/μ = LQ + ρ = ρ/(1− ρ) WQ = LQ / λ WS = WQ + 1/μ Pw = 1 – P0 = ρ P0 = 1 – λ/μ Pn = (λ/μ)nP0 LQ = LS = LQ + λ/μ = LQ + ρ = ρ/(1− ρ) WQ = LQ / λ WS = WQ + 1/μ Pw = 1 – P0 = ρ ( ) ρ ρ λ μ μ λ − = − 1 2 2
43. 43. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 43 The Schips, Inc. Truck Dock Problem Schips, Inc. is a large department store chain that has six branch stores located throughout the city. The company’s Western Hills store, which was built some years ago, has recently been experiencing some problems in its receiving and shipping department because of the substantial growth in the branch’s sales volume. Unfortunately, the store’s truck dock was designed to handle only one truck at a time, and the branch’s increased business volume has led to a bottleneck in the truck dock area. At times, the branch manager has observed as many as five Schips trucks waiting to be loaded or unloaded. As a result, the manager would like to consider various alternatives for improving the operation of the truck dock and reducing the truck waiting times.
44. 44. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 44 The Schips, Inc. Truck Dock Problem One alternative the manager is considering is to speed up the loading/unloading operation by installing a conveyor system at the truck dock. As another alternative, the manager is considering adding a second truck dock so that two trucks could be loaded and/or unloaded simultaneously. What should the manager do to improve the operation of the truck dock? While the alternatives being considered should reduce the truck waiting times, they may also increase the cost of operating the dock. Thus the manager will want to know how each alternative will affect both the waiting times and the cost of operating the dock before making a final decision Truck arrival information: truck arrivals occur at an average rate of three trucks per hour. Service information: the truck dock can service an average of four trucks per hour.
45. 45. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 45 The Schips, Inc. Truck Dock Problem Options: 1. Using conveyor to speed up service rate 2. Add another dock server Assumptions:  The waiting cost is linear  Poisson Arrivals  Exponential service time
46. 46. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 46 Schips, Inc. - Current Situation P0 = Probability that there are no customers in the system Pn = Probability that there are n customers in the system LS = Average number of customers in the system LQ = Average number of customers in the queue WS = Average time a customer spends in the system WQ = Average time a customer spends in the queue Pw = Probability that an arriving customer must wait for service ρ = Utilization rate of each server (the percentage of time that each server is busy) P0 = Probability that there are no customers in the system Pn = Probability that there are n customers in the system LS = Average number of customers in the system LQ = Average number of customers in the queue WS = Average time a customer spends in the system WQ = Average time a customer spends in the queue Pw = Probability that an arriving customer must wait for service ρ = Utilization rate of each server (the percentage of time that each server is busy)
47. 47. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 47 Schips, Inc. - Alternative I  Alternative I: Speed up the loading/unloading operations by installing a conveyor system (costs of different conveyer system are not provided here, but you should consider it when you evaluate the total cost)
48. 48. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 48 M/M/k Queue Server 1 Departure kμ Arrival λ Server 2 Server k Multiple server, single queue (Poisson arrivals, I.I.D. exponential service rate, no priority, no balking, steady state) ONLY IF λ< kμ or λ / kμ = ρ <1 Steady state exists! ONLY IF λ< kμ or λ / kμ = ρ <1 Steady state exists!
49. 49. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 49 M/M/k Queue Performance Measures
50. 50. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 50 The Schips, Inc. Problem (continued) Alternative I Alternative II: k = 2 P0 = 0.4545 LQ = 0.123 LS = 0.873 WQ = 0.041 WS = 0.291 Pw = 0.2045
51. 51. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 51 Economic Analysis of Queuing System Cost of waiting vs. Cost of capacity Cost of waiting vs. Cost of capacity COST CAPACITY CAPACITY WAITING TOTAL
52. 52. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 52 The Schips, Inc. Problem (Cost analysis, FYI) cW = Hourly waiting cost for each customer cServer = Hourly cost for each server LS = Average number of customers in system k = Number of servers cW = Hourly waiting cost for each customer cServer = Hourly cost for each server LS = Average number of customers in system k = Number of servers Total waiting cost/hour = cWL Total server cost/hour = cServerk Total cost per hour = cWL+ cServerk Total waiting cost/hour = cWL Total server cost/hour = cServerk Total cost per hour = cWL+ cServerk Total Hourly Cost Summary for The Schips Truck Dock Problem cW = \$25/hour, cServer = \$30/hour Incremental cost of using conveyor: \$20/hour for every Δμ = 2 Total Hourly Cost Summary for The Schips Truck Dock Problem cW = \$25/hour, cServer = \$30/hour Incremental cost of using conveyor: \$20/hour for every Δμ = 2 System μ Avg. # of Trucks in system (L) Total Cost/Hour cWL + cSk 1-server 4 3 (25)(3)+(30)(1)=\$105 1-server +conveyor 6 1 (25)(1) +(30+20)(1) = \$75 1-server +conveyor 8 0.6 (25)(0.6) + (30+40)(1) = \$85 1-server +conveyor 10 0.43 (25)(0.43) + (30+60)(1) = \$100.71 2-server 4 0.873 (25)(0.873) + (30)(2) = \$81.83
53. 53. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 53 Discrete distribution  Suppose the bank has only one server, the interarrival and service rate are both discrete distribution.  This bank wants to simulate for 150 customers arrival.  This bank wants to know the queuing length and waiting time of their current service. Interarrival distribution Service time distribution Value Prob Value Prob 1 0.05 1 0.15 2 0.15 2 0.15 3 0.35 3 0.25 4 0.35 3 0.20 5 0.10 4 0.10 Cum. Prob. 1 5 0.05 6 0.05 7 0.03 8 0.02 Cum. Prob. 1
54. 54. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 54 Discrete distribution Waiting Times in Queue 0 2 4 6 8 10 12 14 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 Customer Queue Length Versus Time (Shown only at times just after arrivals) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 50 100 150 200 250 300 350 400 450 500 Customer Arrival Times Single server queueing simulation (starting empty and idle) Customer IA_Time Arrival_Time Service_Time Queue_Time Start_Time Depart_Time Before entry After entry Before entry After entry 1 1 1 3 0 1 4 0 3 0 0 2 3 4 3 0 4 7 0 3 0 0 3 3 7 3 0 7 10 0 3 0 0 4 3 10 2 0 10 12 0 2 0 0 5 3 13 2 0 13 15 0 2 0 0 6 4 17 3 0 17 20 0 3 0 0 7 4 21 6 0 21 27 0 6 0 0 8 4 25 3 2 27 30 2 5 0 1 9 1 26 3 4 30 33 4 7 1 2 10 4 30 7 3 33 40 3 10 0 1 11 3 33 3 7 40 43 7 10 0 1 12 2 35 3 8 43 46 8 11 1 2 13 4 39 3 7 46 49 7 10 2 3 14 4 43 3 6 49 52 6 9 1 2 15 3 46 1 6 52 53 6 7 1 2 Number in queue Server work
55. 55. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 55 Discrete distribution  If the arrival rate keeps the same, but the service rate is faster… Service time distribution Value Prob 1 0.25 2 0.25 3 0.20 3 0.20 4 0.10 5 0.00 6 0.00 7 0.00 8 0.00 Cum. Prob. 1 Waiting Times in Queue-Faster Service Rate 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 Customer Queue Length Versus Time- Faster Service (Shown only at times just after arrivals) 0 0.5 1 1.5 2 2.5 0 50 100 150 200 250 300 350 400 450 500 Customer Arrival Times
56. 56. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 56 Discrete distribution : L5-QSim2-3servers Interarrival distribution Service time distribution Value Prob Value Prob 1 0.80 1 0.15 2 0.15 2 0.15 3 0.03 3 0.25 4 0.01 3 0.2 5 0.01 4 0.1 1 5 0.05 6 0.05 7 0.03 8 0.02 1  If the bank has more frequent arrival, they definitely need more servers. Now they have 3 servers.
57. 57. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 57 Discrete distribution : L5-QSim2-3servers (con’t) Waiting Times in Queue-3 servers 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 Customer Queue Length Versus Time- 3 servers (Shown only at times just after arrivals) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 50 100 150 200 250 Customer Arrival Times Queue Length Versus Time- 2 servers (Shown only at times just after arrivals) 0 5 10 15 20 25 30 0 50 100 150 200 250 Customer Arrival Times  If you change to 2 servers, then….  The waiting time and queuing length with 3 servers… Waiting Times in Queue-2 servers 0 5 10 15 20 25 30 35 40 45 50 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 Customer
58. 58. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 58 Waiting Line Models Extensions Notation for Classifying Waiting Line Models M = Designates a Poisson probability distribution for the arrivals or an exponential probability distribution for service time D = Designates that the arrivals or the service time is deterministic or constant G = Designates that the arrivals or the service time has a general probability distribution with a known mean and variance Code indicating arrival distribution Code indicating service time distribution Number of parallel servers others
59. 59. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 59 Waiting Line Models Extensions Notation for Classifying Waiting Line Models M/M/1 M/M/k M/G/1 (M/D/1 is a special case, D for deterministic service time) G/M/1 And more…. G/G/1 G/G/k Code indicating arrival distribution Code indicating service time distribution Number of parallel servers others
60. 60. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 60 M/G/1 Queue Performance Measures M/G/1 System: Steady state results (λ<μ) P0 = 1−ρ (ρ = λ/μ) LQ = LS = LQ + λ/μ = LQ + ρ WQ = LQ / λ WS = WQ + 1/μ Pw = 1 – P0 = ρ ) 1 ( 2 2 2 2 ρ ρ σ λ − + μ = service rate 1/ μ = mean service time σ2 = variance of service time distribution M/D/1 Queue: σ2 = 0 LQ = μ = service rate 1/ μ = mean service time σ2 = variance of service time distribution M/D/1 Queue: σ2 = 0 LQ = ) 1 ( 2 2 ρ ρ −
61. 61. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 61 An Example: Secretary Hiring Suppose you must hire a secretary and you have to select one of two candidates. Secretary 1 is very consistent, typing any document in exactly 15 minutes. Secretary 2 is somewhat faster, with an average of 14 minutes per document, but with times varying according to the exponential distribution. The workload in the office is 3 documents per hour, with interarrival times varying according to the exponential distribution. Which secretary will give you shorter turnaround times on documents?
62. 62. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 62 Secretary Hiring - Queuing Model
63. 63. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 63 M/M/s with Finite Population The number of customers in the system is not permitted to exceed some specified number Example: Machine maintenance problem
64. 64. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 64 M/M/s with Limited Waiting Room Arrivals are turned away when the number waiting in the queue reaches a maximum level Example: Walk-in Dr.s office with limited waiting space
65. 65. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 65  λ = Mean number of arrivals per time period  e.g., 3 units/hour  μ = Mean number of people or items served per time period  e.g., 4 units/hour  1/μ = 15 minutes/unit Remember: λ & μ Are Rates © 1984-1994 T/Maker Co. If average service time is 15 minutes, then μ is 4 customers/hour
66. 66. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 66 Summary  Queuing system design has an important impact on the service provided by an enterprise  Steady state performance measures can provide useful information in assessing service and developing optimal queuing systems  The general procedure of solving a queuing problem:  Many queuing systems do not have closed-form solutions. Simulation is a powerful tool of analyzing those systems. Identify Queue Type Identify Queue Type Estimate Arrival & Service Processes Estimate Arrival & Service Processes Calculate Performance Measures Calculate Performance Measures Conduct Economic Analysis Conduct Economic Analysis
67. 67. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 67 Zapatería Mary’s Los clientes que llegan a la zapatería Mary’s son en promedio 12 por minuto, de acuerdo a la distribución Poisson. El tiempo de atención se distribuye exponencialmente con un promedio de 8 minutos por cliente. La gerencia esta interesada en determinar las medidas de performance para este servicio.
68. 68. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 68 SOLUCION  Datos de entrada λ = 1/ 12 clientes por minuto = 60/ 12 = 5 por hora. μ = 1/ 8 clientes por minuto = 60/ 8 = 7.5 por hora.  Calculo del performance P0 = 1- (λ / μ) = 1 - (5 / 7.5) = 0.3333 Pn = [1 - (λ / μ)] (λ/ μ) = (0.3333)(0.6667)n L = λ / (μ - λ) = 2 Lq = λ2/ [μ(μ - λ)] = 1.3333 W = 1 / (μ - λ) = 0.4 horas = 24 minutos Wq = λ / [μ(μ - λ)] = 0.26667 horas = 16 minutos P0 = 1- (λ / μ) = 1 - (5 / 7.5) = 0.3333 Pn = [1 - (λ / μ)] (λ/ μ) = (0.3333)(0.6667)n L = λ / (μ - λ) = 2 Lq = λ2/ [μ(μ - λ)] = 1.3333 W = 1 / (μ - λ) = 0.4 horas = 24 minutos Wq = λ / [μ(μ - λ)] = 0.26667 horas = 16 minutos Pw = λ / μ = 0.6667 ρ = λ / μ = 0.6667
69. 69. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 69 USANDO WINQSB
70. 70. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 70
71. 71. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 71 Datos de entrada para WINQSB Datos de entrada para WINQSB μ λ
72. 72. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 72 Medidas de desempeño Medidas de desempeño Medidas de desempeño Medidas de desempeño Medidas de desempeño Medidas de desempeño Medidas de desempeño Medidas de desempeño Medidas de desempeño Medidas de desempeño
73. 73. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 73
74. 74. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 74 OFICINA POSTAL TOWN La oficina postal Town atiende público los Sábados entre las 9:00 a.m. y la 1:00 p.m. Datos - En promedio, 100 clientes por hora visitan la oficina postal durante este período. La oficina tiene tres dependientes. - Cada atención dura 1.5 minutos en promedio. - La distribución Poisson y exponencial describen la llegada de los clientes y el proceso de atención de estos respectivamente. La gerencia desea conocer las medidas relevantes al servicio en orden a: – La evaluación del nivel de servicio prestado. – El efecto de reducir el personal en un dependiente. La gerencia desea conocer las medidas relevantes al servicio en orden a: – La evaluación del nivel de servicio prestado. – El efecto de reducir el personal en un dependiente.
75. 75. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 75 SOLUCION Se trata de un sistema de colas M / M / 3 . Datos de entrada λ = 100 clientes por hora. μ = 40 clientes por hora (60 / 1.5). ¿Existe un período estacionario (λ kμ )? λ = 100 kμ = 3(40) = 120.
76. 76. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 76 USANDO EL WINQSB
77. 77. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 77
78. 78. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 78
79. 79. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 79
80. 80. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 80 Sistemas de colas M/G/1 Supuestos - Los clientes llegan de acuerdo a un proceso Poisson con esperanza λ. − El tiempo de atención tiene una distribución general con esperanza μ. − Existe un solo servidor. - Se cuenta con una población infinita y la posibilidad de infinitas filas.
81. 81. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 81 TALLER DE REPARACIONES TED Ted repara televisores y videograbadoras. Datos - El tiempo promedio para reparar uno de estos artefactos es de 2.25 horas. - La desviación estándar del tiempo de reparación es de 45 minutos. - Los clientes llegan a la tienda en promedio cada 2.5 horas, de acuerdo a una distribución Poisson. - Ted trabaja 9 horas diarias y no tiene ayudantes. - El compra todos los repuestos necesarios. + En promedio, el tiempo de reparación esperado debería ser de 2 horas. + La desviación estándar esperada debería ser de 40 minutos.
82. 82. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 82 Ted desea conocer los efectos de usar nuevos equipos para: 1. Mejorar el tiempo promedio de reparación de los artefactos; 2. Mejorar el tiempo promedio que debe esperar un cliente hasta que su artefacto sea reparado. Ted desea conocer los efectos de usar nuevos equipos para: 1. Mejorar el tiempo promedio de reparación de los artefactos; 2. Mejorar el tiempo promedio que debe esperar un cliente hasta que su artefacto sea reparado.
83. 83. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 83 SOLUCION Se trata de un sistema M/G/1 (el tiempo de atención no es exponencial pues σ = 1/μ). Datos  Con el sistema antiguo (sin los nuevos equipos) λ = 1/ 2.5 = 0.4 clientes por hora. μ = 1/ 2.25 = 0.4444 clientes por hora. σ = 45/ 60 = 0.75 horas.  Con el nuevo sistema (con los nuevos equipos) μ = 1/2 = 0.5 clientes por hora. σ = 40/ 60 = 0.6667 horas.
84. 84. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 84 Sistemas de colas M/M/k/F Se deben asignar muchas colas, cada una de un cierto tamaño límite. Cuando una cola es demasiado larga, un modelo de cola infinito entrega un resultado exacto, aunque de todas formas la cola debe ser limitada. Cuando una cola es demasiado pequeña, se debe estimar un límite para la fila en el modelo.
85. 85. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 85 Características del sistema M/M/k/F - La llegada de los clientes obedece a una distribución Poisson con una esperanza λ. - Existen k servidores, para cada uno el tiempo de atención se distribuye exponencialmente, con esperanza μ. − El número máximo de clientes que puede estar presente en el sistema en un tiempo dado es “F”. - Los clientes son rechazados si el sistema se encuentra completo.
86. 86. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 86 Tasa de llegada efectiva. - Un cliente es rechazado si el sistema se encuentra completo. - La probabilidad de que el sistema se complete es PF. - La tasa efectiva de llegada = la tasa de abandono de clientes en el sistema (λe). λe = λ(1 - PF)
87. 87. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 87 COMPAÑÍA DE TECHADOS RYAN Ryan atiende a sus clientes, los cuales llaman y ordenan su servicio. Datos - Una secretaria recibe las llamadas desde 3 líneas telefónicas. - Cada llamada telefónica toma tres minutos en promedio - En promedio, diez clientes llaman a la compañía cada hora.
88. 88. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 88 Cuando una línea telefónica esta disponible, pero la secretaria esta ocupada atendiendo otra llamada, el cliente debe esperar en línea hasta que la secretaria este disponible. Cuando todas las líneas están ocupadas los clientes optan por llamar a la competencia. El proceso de llegada de clientes tiene una distribución Poisson, y el proceso de atención se distribuye exponencialmente.
89. 89. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 89 La gerencia desea diseñar el siguiente sistema con: - La menor cantidad de líneas necesarias. - A lo más el 2% de las llamadas encuentren las líneas ocupadas. La gerencia esta interesada en la siguiente información: El porcentaje de tiempo en que la secretaria esta ocupada. EL número promedio de clientes que están es espera. El tiempo promedio que los clientes permanecen en línea esperando ser atendidos. El porcentaje actual de llamadas que encuentran las líneas ocupadas.
90. 90. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 90 SOLUCION Se trata de un sistema M / M / 1 / 3 Datos de entrada λ = 10 por hora. μ = 20 por hora (1/ 3 por minuto).  WINQSB entrega: P0 = 0.533, P1 = 0.133, P3 = 0.06 6.7% de los clientes encuentran las líneas ocupadas. Esto es alrededor de la meta del 2%. sistema M / M / 1 / 4 P0 = 0.516, P1 = 0.258, P2 = 0.129, P3 = 0.065, P4 = 0.032 3.2% de los clntes. encuentran las líneas ocupadas Aún se puede alcanzar la meta del 2% sistema M / M / 1 / 5 P0 = 0.508, P1 = 0.254, P2 = 0.127, P3 = 0.063, P4 = 0.032 P5 = 0.016 1.6% de los cltes. encuentran las linea ocupadas La meta del 2% puede ser alcanzada.
91. 91. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 91 Datos de entrada para WINQSB Datos de entrada para WINQSB Otros resultados de WINQSB Con 5 líneas telefónicas 4 clientes pueden esperar en línea
92. 92. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 92
93. 93. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 93 Sistemas de colas M/M/1//m En este sistema el número de clientes potenciales es finito y relativamente pequeño. Como resultado, el número de clientes que se encuentran en el sistema corresponde a la tasa de llegada de clientes. Características - Un solo servidor - Tiempo de atención exponencial y proceso de llegada Poisson. - El tamaño de la población es de m clientes (m finito).
94. 94. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 94 CASAS PACESETTER Casas Pacesetter se encuentra desarrollando cuatro proyectos. Datos - Una obstrucción en las obras ocurre en promedio cada 20 días de trabajo en cada sitio. - Esto toma 2 días en promedio para resolver el problema. - Cada problema es resuelto por el V.P. para construcción ¿Cuanto tiempo en promedio un sitio no se encuentra operativo? -Con 2 días para resolver el problema (situación actual) -Con 1.875 días para resolver el problema (situación nueva).
95. 95. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 95 SOLUCION Se trata de un sistema M/M/1//4 Los cuatro sitios son los cuatro clientes El V.P. para construcción puede ser considerado como el servidor. Datos de entrada λ = 0.05 (1/ 20) μ = 0.5 (1/ 2 usando el actual V.P). μ = 0.533 (1/1.875 usando el nuevo V.P).
96. 96. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 96 Medidas del V.P V.P Performance Actual Nuevo Tasa efectiva del factor de utilización del sistema ρ 0,353 0,334 Número promedio de clientes en el sistema L 0,467 0,435 Número promedio de clientes en la cola Lq 0,113 0,100 Número promedio de dias que un cliente esta en el sistema W 2,641 2,437 Número promedio de días que un cliente esta en la cola Wq 0,641 0,562 Probabilidad que todos los servidores se encuentren ociosos Po 0,647 0,666 Probabilidad que un cliente que llega deba esperar en el sist. Pw 0,353 0,334 Medidas del V.P V.P Performance Actual Nuevo Tasa efectiva del factor de utilización del sistema ρ 0,353 0,334 Número promedio de clientes en el sistema L 0,467 0,435 Número promedio de clientes en la cola Lq 0,113 0,100 Número promedio de dias que un cliente esta en el sistema W 2,641 2,437 Número promedio de días que un cliente esta en la cola Wq 0,641 0,562 Probabilidad que todos los servidores se encuentren ociosos Po 0,647 0,666 Probabilidad que un cliente que llega deba esperar en el sist. Pw 0,353 0,334 Resultados obtenidos por WINQSB Resultados obtenidos por WINQSB
97. 97. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 97 Análisis económico de los sistemas de colas Las medidas de desempeño anteriores son usadas para determinar los costos mínimos del sistema de colas. El procedimiento requiere estimar los costos tales como: - Costo de horas de trabajo por servidor - Costo del grado de satisfacción del cliente que espera en la cola. -Costo del grado de satisfacción de un cliente que es atendido.
98. 98. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 98 SERVICIO TELEFONICO DE WILSON FOODS Wilson Foods tiene un línea 800 para responder las consultas de sus clientes Datos - En promedio se reciben 225 llamadas por hora. - Una llamada toma aproximadamente 1.5 minutos. - Un cliente debe esperar en línea a lo más 3 minutos. -A un representante que atiende a un cliente se le paga \$16 por hora. -Wilson paga a la compañía telefónica \$0.18 por minuto cuando el cliente espera en línea o esta siendo atendido. - El costo del grado de satisfacción de un cliente que espera en línea es de \$20 por minuto. -El costo del grado de satisfacción de un cliente que es atendido es de \$0.05. ¿Qué cantidad de representantes para la atención de los clientes deben ser usados para minimizar el costo de las horas de operación? ¿Qué cantidad de representantes para la atención de los clientes deben ser usados para minimizar el costo de las horas de operación?
99. 99. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 99 SOLUCION Costo total del modelo Costo total por horas de trabajo de “k” representantes para la atención de clientes CT(K) = Cwk + CtL + gwLq + gs(L - Lq) Total horas para sueldo Costo total de las llamadas telefónicas Costo total del grado de satisfacción de los clientes que permanecen en línea Costo total del grado de satisfacción de los clientes que son atendidos CT(K) = Cwk + (Ct + gs)L + (gw - gs)Lq
100. 100. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 100 Datos de entrada Cw= \$16 Ct = \$10.80 por hora [0.18(60)] gw= \$12 por hora [0.20(60)] gs = \$0.05 por hora [0.05(60)] Costo total del promedio de horas TC(K) = 16K + (10.8+3)L + (12 - 3)Lq = 16K + 13.8L + 9Lq
101. 101. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 101 Asumiendo una distribución de llegada de los clientes Poisson y una distribución exponencial del tiempo de atención, se tiene un sistema M/M/K λ = 225 llamadas por hora. μ = 40 por hora (60/ 1.5). El valor mínimo posible para k es 6 de forma de asegurar que exista un período estacionario (λKμ). WINQSB puede ser usado para generar los resultados de L, Lq, y Wq.
102. 102. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 102 En resumen los resultados para K= 6,7,8,9,10. K L Lq Wq CT(K) 6 18,1249 12,5 0,05556 458,62 7 7,6437 2,0187 0,00897 235,62 8 6,2777 0,6527 0,0029 220,50 9 5,8661 0,2411 0,00107 227,12 10 5,7166 0,916 0,00041 239,70 K L Lq Wq CT(K) 6 18,1249 12,5 0,05556 458,62 7 7,6437 2,0187 0,00897 235,62 8 6,2777 0,6527 0,0029 220,50 9 5,8661 0,2411 0,00107 227,12 10 5,7166 0,916 0,00041 239,70 Conclusión: se deben emplear 8 representantes para la atención de clientes Conclusión: se deben emplear 8 representantes para la atención de clientes
103. 103. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 103 Sistemas de colas Tándem En un sistema de colas Tándem un cliente debe visitar diversos servidores antes de completar el servicio requerido Se utiliza para casos en los cuales el cliente llega de acuerdo al proceso Poisson y el tiempo de atención se distribuye exponencialmente en cada estación. Tiempo promedio total en el sistema = suma de todos los tiempo promedios en las estaciones individuales Tiempo promedio total en el sistema = suma de todos los tiempo promedios en las estaciones individuales
104. 104. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 104 COMPAÑÍA DE SONIDO BIG BOYS Big Boys vende productos de audio. El proceso de venta es el siguiente: - Un cliente realiza su orden con el vendedor. - El cliente se dirige a la caja para pagar su pedido. - Después de pagar, el cliente debe dirigirse al empaque para obtener su producto.
105. 105. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 105 Datos de la venta de un Sábado normal - Personal + 8 vendedores contando el jefe + 3 cajeras + 2 trabajadores de empaque. - Tiempo promedio de atención + El tiempo promedio que un vendedor esta con un cliente es de 10 minutos. + El tiempo promedio requerido para el proceso de pago es de 3 minutos. + El tiempo promedio en el área de empaque es de 2 minutos. -Distribución + El tiempo de atención en cada estación se distribuye exponencialmente. + La tasa de llegada tiene una distribución Poisson de 40 clientes por hora. Solamente 75% de los clientes que llegan hacen una compra ¿Cuál es la cantidad promedio de tiempo , que un cliente que viene a comprar demora en el local? ¿ ¿Cu Cuá ál es la cantidad promedio de tiempo , l es la cantidad promedio de tiempo , que un cliente que viene a comprar que un cliente que viene a comprar demora en el local? demora en el local?
106. 106. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 106 SOLUCION Estas son las tres estaciones del sistema de colas Tándem M / M / 8 M / M / 3 λ = 4 0 λ = 3 0 λ = 3 0 M / M / 2 W1 = 14 minutos W2 = 3.47 minutos W3=2.67 minutos Total = 20.14 minutos.
107. 107. Metodos Cuantitativos M. En C. Eduardo Bustos Farias 107 Balance de líneas de ensamble Una línea de ensamble puede ser vista como una cola Tándem, porque los productos deben visitar diversas estaciones de trabajo de una secuencia dada. En una línea de ensamble balanceada el tiempo ocupado en cada una de las diferentes estaciones de trabajo es el mismo. El objetivo es maximizar la producción