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1             Technical Note 7                    Waiting Line                    ManagementMcGraw-Hill/Irwin             ...
2          OBJECTIVES        Waiting Line Characteristics        Suggestions for Managing Queues        Examples (Model...
Components of the Queuing                                                     3                 System                    ...
Customer Service Population                                                  4                      Sources               ...
5                    Service Pattern                          Service                          Pattern               Const...
6               The Queuing System                             Length                           Queuing                   ...
7                    Examples of Line                       Structures                           Single                   ...
8                    Degree of Patience             No Way!        No Way!         BALK             RENEGMcGraw-Hill/Irwin...
Suggestions for Managing                                              9                    Queues          1. Determine an...
10      Suggestions for Managing Queues                (Continued)       6. Train your servers to be friendly       7. Enc...
11                    Waiting Line Models                                       Source        Model Layout                ...
12 Notation: Infinite Queuing:λ = Arrival rate              Models 1-3µ = Service rate1  = Average service timeµ1  = Avera...
Infinite Queuing Models 1-3                                                                               13 Ls = Average(...
14                    Example: Model 1Assume a drive-up window at a fast food restaurant.Customers arrive at the rate of 2...
15                    Example: Model 1         A) What is the average utilization of the         employee?          λ = 25...
16                    Example: Model 1B) What is the average number of customers inline?                   λ   2          ...
17                    Example: Model 1   D) What is the average waiting time in line?       Lq  Wq =    = .1667 hrs = 10 m...
18                     Example: Model 1  F) What is the probability that exactly two cars  will be in the system (one bein...
19               Example: Model 2       An automated pizza vending machine       heats and       dispenses a slice of pizz...
20                    Example: Model 2    A) The average number of customers in line.               λ2            (10) 2  ...
21               Example: Model 3     Recall the Model 1 example:     Drive-up window at a fast food restaurant.     Custo...
22                    Example: Model 3               Average number of cars in the system  Lq = 0.176  (Exhibit TN7.11 - -...
23  Notation: Finite Queuing:                 Model 4D = Probability that an arrival must wait in lineF = Efficiency facto...
24     Finite Queuing: Model 4               (Continued)n = Average number of units in queuing system        (including th...
25                    Example: Model 4  The copy center of an electronics firm has four copy  The copy center of an electr...
26                    Example: Model 4N, the number of machines in the population = 4M, the number of repair people = 1T, ...
27                        Queuing                     Approximation           This approximation is quick way to analyze ...
Queue                                                                         28                    Approximation        I...
29                 Approximation                        Example           Consider a manufacturing process (for example m...
30             End of Technical                  Note 7McGraw-Hill/Irwin         ©The McGraw-Hill Companies, Inc., 2006
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  1. 1. 1 Technical Note 7 Waiting Line ManagementMcGraw-Hill/Irwin ©The McGraw-Hill Companies, Inc., 2006
  2. 2. 2 OBJECTIVES  Waiting Line Characteristics  Suggestions for Managing Queues  Examples (Models 1, 2, 3, and 4)McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  3. 3. Components of the Queuing 3 System Servicing System Servers Queue orCustomer Waiting LineArrivals ExitMcGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  4. 4. Customer Service Population 4 Sources Population Source Finite Infinite Example: Number of Example: Number of Example: The Example: The machines needing machines needing number of people number of people repair when a repair when a who could wait in who could wait in company only has company only has a line for a line for three machines. three machines. gasoline. gasoline.McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  5. 5. 5 Service Pattern Service Pattern Constant Variable Example: Items Example: Items Example: People Example: People coming down an coming down an spending time spending time automated automated shopping. shopping. assembly line. assembly line.McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  6. 6. 6 The Queuing System Length Queuing Number of Lines & Queue Discipline System Line Structures Service Time DistributionMcGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  7. 7. 7 Examples of Line Structures Single Multiphase Phase One-person Single Channel Car wash barber shop Bank tellers’ Hospital Multichannel windows admissionsMcGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  8. 8. 8 Degree of Patience No Way! No Way! BALK RENEGMcGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  9. 9. Suggestions for Managing 9 Queues 1. Determine an acceptable waiting time for your customers 2. Try to divert your customer’s attention when waiting 3. Inform your customers of what to expect 4. Keep employees not serving the customers out of sight 5. Segment customersMcGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  10. 10. 10 Suggestions for Managing Queues (Continued) 6. Train your servers to be friendly 7. Encourage customers to come during the slack periods 8. Take a long-term perspective toward getting rid of the queuesMcGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  11. 11. 11 Waiting Line Models Source Model Layout Population Service Pattern 1 Single channel Infinite Exponential 2 Single channel Infinite Constant 3 Multichannel Infinite Exponential 4 Single or Multi Finite Exponential These four models share the following characteristics: • Single phase • Poisson arrival • FCFS • Unlimited queue lengthMcGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  12. 12. 12 Notation: Infinite Queuing:λ = Arrival rate Models 1-3µ = Service rate1 = Average service timeµ1 = Average time between arrivalsλ λρ = = Ratio of total arrival rate to sevice rate µ for a single serverLq = Average number waiting in lineMcGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  13. 13. Infinite Queuing Models 1-3 13 Ls = Average(Continued) number in system (including those being served) Wq = Average time waiting in line Ws = Average total time in system (including time to be served) n = Number of units in the system S = Number of identical service channels Pn = Probability of exactly n units in system Pw = Probability of waiting in lineMcGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  14. 14. 14 Example: Model 1Assume a drive-up window at a fast food restaurant.Customers arrive at the rate of 25 per hour.The employee can serve one customer every twominutes.Assume Poisson arrival and exponential servicerates.Determine:Determine:A) What is the average utilization of the employee?A) What is the average utilization of the employee?B) What is the average number of customers in line?B) What is the average number of customers in line?C) What is the average number of customers in theC) What is the average number of customers in thesystem?system?D) What is the average waiting time in line?D) What is the average waiting time in line?E) What is the average waiting time in the system?E) What is the average waiting time in the system?F) What is the probability that exactly two cars will beF) What is the probability that exactly two cars will be in the system? in the system?McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  15. 15. 15 Example: Model 1 A) What is the average utilization of the employee? λ = 25 cust / hr 1 customer µ = = 30 cust / hr 2 mins (1hr / 60 mins) λ 25 cust / hr ρ = = = .8333 µ 30 cust / hrMcGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  16. 16. 16 Example: Model 1B) What is the average number of customers inline? λ 2 (25) 2 Lq = = = 4.167 µ ( µ - λ ) 30(30 - 25)C) What is the average number of customers in thesystem? λ 25 Ls = = =5 µ - λ (30 - 25)McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  17. 17. 17 Example: Model 1 D) What is the average waiting time in line? Lq Wq = = .1667 hrs = 10 mins λ E) What is the average waiting time in the system? Ls Ws = = .2 hrs = 12 mins λMcGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  18. 18. 18 Example: Model 1 F) What is the probability that exactly two cars will be in the system (one being served and the other waiting in line)? λ λ n pn = (1- )( ) µ µ 25 25 2 p 2 = (1- )( ) = .1157 30 30McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  19. 19. 19 Example: Model 2 An automated pizza vending machine heats and dispenses a slice of pizza in 4 minutes. Customers arrive at a rate of one every 6 minutes with the arrival rate exhibiting a Poisson distribution.Determine:Determine:A)A) The average number of customers in line. The average number of customers in line.B)B) The average total waiting time in the system. The average total waiting time in the system.McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  20. 20. 20 Example: Model 2 A) The average number of customers in line. λ2 (10) 2 Lq = = = .6667 2 µ ( µ - λ ) (2)(15)(15 - 10) B) The average total waiting time in the system. Lq .6667 Wq = = = .06667 hrs = 4 mins λ 10 1 1 Ws = Wq + = .06667 hrs + = .1333 hrs = 8 mins µ 15/hrMcGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  21. 21. 21 Example: Model 3 Recall the Model 1 example: Drive-up window at a fast food restaurant. Customers arrive at the rate of 25 per hour. The employee can serve one customer every two minutes. Assume Poisson arrival and exponential service rates.If an identical window (and an identically trained If an identical window (and an identically trainedserver) were added, what would the effects be on server) were added, what would the effects be onthe average number of cars in the system and the the average number of cars in the system and thetotal time customers wait before being served? total time customers wait before being served?McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  22. 22. 22 Example: Model 3 Average number of cars in the system Lq = 0.176 (Exhibit TN7.11 - -using linear interpolation) λ 25 Ls = Lq + = .176 + = 1.009 µ 30 Total time customers wait before being served Lq .176 customers Wq = = = .007 mins ( No Wait! ) λ 25 customers/minMcGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  23. 23. 23 Notation: Finite Queuing: Model 4D = Probability that an arrival must wait in lineF = Efficiency factor, a measure of the effect of having to wait in lineH = Average number of units being servedJ = Population source less those in queuing system ( N - n)L = Average number of units in lineS = Number of service channelsMcGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  24. 24. 24 Finite Queuing: Model 4 (Continued)n = Average number of units in queuing system (including the one being served)N = Number of units in population sourcePn = Probability of exactly n units in queuing systemT = Average time to perform the serviceU = Average time between customer service requirementsW = Average waiting time in lineX = Service factor, or proportion of service time requiredMcGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  25. 25. 25 Example: Model 4 The copy center of an electronics firm has four copy The copy center of an electronics firm has four copy machines that are all serviced by a single technician. machines that are all serviced by a single technician. Every two hours, on average, the machines require Every two hours, on average, the machines require adjustment. The technician spends an average of 10 adjustment. The technician spends an average of 10 minutes per machine when adjustment is required. minutes per machine when adjustment is required. Assuming Poisson arrivals and exponential service, Assuming Poisson arrivals and exponential service, how many machines are “down” (on average)? how many machines are “down” (on average)?McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  26. 26. 26 Example: Model 4N, the number of machines in the population = 4M, the number of repair people = 1T, the time required to service a machine = 10 minutesU, the average time between service = 2 hours T 10 min X= = = .077 T+ U 10 min + 120 min From Table TN7.11, F = .980 (Interpolation) From Table TN7.11, F = .980 (Interpolation) L, the number of machines waiting to be L, the number of machines waiting to be serviced = N(1-F) = 4(1-.980) = .08 machines serviced = N(1-F) = 4(1-.980) = .08 machines H, the number of machines being H, the number of machines being serviced = FNX = .980(4)(.077) = .302 machines serviced = FNX = .980(4)(.077) = .302 machines Number of machines down = L + H = .382 machines Number of machines down = L + H = .382 machinesMcGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  27. 27. 27 Queuing Approximation  This approximation is quick way to analyze a queuing situation. Now, both interarrival time and service time distributions are allowed to be general.  In general, average performance measures (waiting time in queue, number in queue, etc) can be very well approximated by mean and variance of the distribution (distribution shape not very important).  This is very good news for managers: all you need is mean and standard deviation, to compute average waiting time Define: Standard deviation of X Cx = coefficient of variation for r.v. X = Mean of X Variance Cx = squared coefficient of variation (scv) = ( Cx ) = 2 2 mean2McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  28. 28. Queue 28 Approximation Inputs: S, λ, µ, Ca ,Cs2 2 (Alternatively: S, λ, µ, variances of interarrival and service time distributions) λ Compute ρ = Sµ ρ 2( S +1) Ca + Cs2 2 Lq Lq = ⋅ Ls 1− ρ 2 as before, Wq = , and Ws = λ λ Ls = Lq + S ρMcGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  29. 29. 29 Approximation  Example Consider a manufacturing process (for example making plastic parts) consisting of a single stage with five machines. Processing times have a mean of 5.4 days and standard deviation of 4 days. The firm operates make-to-order. Management has collected date on customer orders, and verified that the time between orders has a mean of 1.2 days and variance of 0.72 days. What is the average time that an order waits before being worked on? Using our “Waiting Line Approximation” spreadsheet we get: Lq = 3.154 Expected number of orders waiting to be completed. Wq = 3.78 Expected number of days order waits. Ρ = 0.9 Expected machine utilization.McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
  30. 30. 30 End of Technical Note 7McGraw-Hill/Irwin ©The McGraw-Hill Companies, Inc., 2006

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