1. 1 1 Slide © 2005 Thomson/South-Western Chapter 12, Part A Waiting Line Models n Structure of a Waiting Line System n Queuing Systems n Queuing System Input Characteristics n Queuing System Operating Characteristics n Analytical Formulas n Single-Channel Waiting Line Model with Poisson Arrivals and Exponential Service Times n Multiple-Channel Waiting Line Model with Poisson Arrivals and Exponential Service Times n Economic Analysis of Waiting Lines

2. 2 2 Slide © 2005 Thomson/South-Western n Queuing theory is the study of waiting lines. n Four characteristics of a queuing system are: the manner in which customers arrive the manner in which customers arrive the time required for service the time required for service the priority determining the order of service the priority determining the order of service the number and configuration of servers in the system. the number and configuration of servers in the system. Structure of a Waiting Line System

3. 3 3 Slide © 2005 Thomson/South-Western Structure of a Waiting Line System n Distribution of Arrivals Generally, the arrival of customers into the system is a random event. Generally, the arrival of customers into the system is a random event. Frequently the arrival pattern is modeled as a Poisson process. Frequently the arrival pattern is modeled as a Poisson process. n Distribution of Service Times Service time is also usually a random variable. Service time is also usually a random variable. A distribution commonly used to describe service time is the exponential distribution. A distribution commonly used to describe service time is the exponential distribution.

4. 4 4 Slide © 2005 Thomson/South-Western Structure of a Waiting Line System n Queue Discipline Most common queue discipline is first come, first served (FCFS). Most common queue discipline is first come, first served (FCFS). An elevator is an example of last come, first served (LCFS) queue discipline. An elevator is an example of last come, first served (LCFS) queue discipline. Other disciplines assign priorities to the waiting units and then serve the unit with the highest priority first. Other disciplines assign priorities to the waiting units and then serve the unit with the highest priority first.

5. 5 5 Slide © 2005 Thomson/South-Western Structure of a Waiting Line System n Single Service Channel n Multiple Service Channels S1S1S1S1 S1S1S1S1 S1S1S1S1 S1S1S1S1 S2S2S2S2 S2S2S2S2 S3S3S3S3 S3S3S3S3 Customerleaves Customerleaves Customerarrives Customerarrives Waiting line System System

6. 6 6 Slide © 2005 Thomson/South-Western Queuing Systems n A three part code of the form A / B / k is used to describe various queuing systems. n A identifies the arrival distribution, B the service (departure) distribution and k the number of channels for the system.

7. 7 7 Slide © 2005 Thomson/South-Western Queuing Systems n Symbols used for the arrival and service processes are: M - Markov distributions (Poisson/exponential), D - Deterministic (constant) and G - General distribution (with a known mean and variance). n For example, M / M / k refers to a system in which arrivals occur according to a Poisson distribution, service times follow an exponential distribution and there are k servers working at identical service rates.

8. 8 8 Slide © 2005 Thomson/South-Western Queuing System Input Characteristics  = the average arrival rate 1/ = the average time between arrivals 1/ = the average time between arrivals µ = the average service rate for each server µ = the average service rate for each server 1/ µ = the average service time 1/ µ = the average service time  = the standard deviation of the service time  = the standard deviation of the service time

9. 9 9 Slide © 2005 Thomson/South-Western Queuing System Operating Characteristics P 0 = probability the service facility is idle P 0 = probability the service facility is idle P n = probability of n units in the system P n = probability of n units in the system P w = probability an arriving unit must wait for service P w = probability an arriving unit must wait for service L q = average number of units in the queue awaiting service L q = average number of units in the queue awaiting service L = average number of units in the system L = average number of units in the system W q = average time a unit spends in the queue awaiting service W q = average time a unit spends in the queue awaiting service W = average time a unit spends in the system W = average time a unit spends in the system

10. 10 Slide © 2005 Thomson/South-Western Analytical Formulas n For nearly all queuing systems, there is a relationship between the average time a unit spends in the system or queue and the average number of units in the system or queue. n These relationships, known as Little's flow equations are: L = W and L q = W q L = W and L q = W q

11. 11 Slide © 2005 Thomson/South-Western Analytical Formulas n When the queue discipline is FCFS, analytical formulas have been derived for several different queuing models including the following: M / M /1 M / M /1 M / M / k M / M / k M / G /1 M / G /1 M / G / k with blocked customers cleared M / G / k with blocked customers cleared M / M /1 with a finite calling population M / M /1 with a finite calling population n Analytical formulas are not available for all possible queuing systems. In this event, insights may be gained through a simulation of the system.

12. 12 Slide © 2005 Thomson/South-Western M/M/1 Queuing System n Single channel n Poisson arrival-rate distribution n Exponential service-time distribution n Unlimited maximum queue length n Infinite calling population n Examples: Single-window theatre ticket sales booth Single-window theatre ticket sales booth Single-scanner airport security station Single-scanner airport security station

13. 13 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (A) n M / M /1 Queuing System Joe Ferris is a stock trader on the floor of the New York Stock Exchange for the firm of Smith, Jones, Johnson, and Thomas, Inc. Stock transactions arrive at a mean rate of 20 per hour. Each order received by Joe requires an average of two minutes to process.

14. 14 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (A) n M / M /1 Queuing System Orders arrive at a mean rate of 20 per hour or one order every 3 minutes. Therefore, in a 15 minute interval the average number of orders arriving will be = 15/3 = 5. = 15/3 = 5.

15. 15 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (A) n Arrival Rate Distribution Question What is the probability that no orders are received within a 15-minute period? Answer P ( x = 0) = (5 0 e -5 )/0! = e -5 =.0067 P ( x = 0) = (5 0 e -5 )/0! = e -5 =.0067

16. 16 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (A) n Arrival Rate Distribution Question What is the probability that exactly 3 orders are received within a 15-minute period? Answer P ( x = 3) = (5 3 e -5 )/3! = 125(.0067)/6 =.1396

17. 17 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (A) n Arrival Rate Distribution Question What is the probability that more than 6 orders arrive within a 15-minute period? Answer P ( x > 6) = 1 - P ( x = 0) - P ( x = 1) - P ( x = 2) P ( x > 6) = 1 - P ( x = 0) - P ( x = 1) - P ( x = 2) - P ( x = 3) - P ( x = 4) - P ( x = 5) - P ( x = 3) - P ( x = 4) - P ( x = 5) - P ( x = 6) - P ( x = 6) = 1 -.762 =.238 = 1 -.762 =.238

18. 18 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (A) n Service Rate Distribution Question What is the mean service rate per hour? Answer Since Joe Ferris can process an order in an average time of 2 minutes (= 2/60 hr.), then the mean service rate, µ, is µ = 1/(mean service time), or 60/2.  = 30/hr.  = 30/hr.

19. 19 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (A) n Service Time Distribution Question What percentage of the orders will take less than one minute to process? Answer Since the units are expressed in hours, P ( T < 1 minute) = P ( T < 1/60 hour). P ( T < 1 minute) = P ( T < 1/60 hour). Using the exponential distribution, P ( T < t ) = 1 - e -µt. Hence, P ( T < 1/60) = 1 - e -30(1/60) = 1 -.6065 =.3935 = 39.35% = 1 -.6065 =.3935 = 39.35%

20. 20 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (A) n Service Time Distribution Question What percentage of the orders will be processed in exactly 3 minutes? Answer Since the exponential distribution is a continuous distribution, the probability a service time exactly equals any specific value is 0.

21. 21 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (A) n Service Time Distribution Question What percentage of the orders will require more than 3 minutes to process? Answer The percentage of orders requiring more than 3 minutes to process is: P ( T > 3/60) = e -30(3/60) = e -1.5 =.2231 = 22.31% P ( T > 3/60) = e -30(3/60) = e -1.5 =.2231 = 22.31%

22. 22 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (A) n Average Time in the System Question What is the average time an order must wait from the time Joe receives the order until it is finished being processed (i.e. its turnaround time)? Answer This is an M / M /1 queue with = 20 per hour and  = 30 per hour. The average time an order waits in the system is: W = 1/(µ - ) = 1/(30 - 20) = 1/(30 - 20) = 1/10 hour or 6 minutes = 1/10 hour or 6 minutes

23. 23 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (A) n Average Length of Queue Question What is the average number of orders Joe has waiting to be processed? Answer Average number of orders waiting in the queue is: L q = 2 /[µ(µ - )] L q = 2 /[µ(µ - )] = (20) 2 /[(30)(30-20)] = (20) 2 /[(30)(30-20)] = 400/300 = 400/300 = 4/3 = 4/3

24. 24 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (A) n Utilization Factor Question What percentage of the time is Joe processing orders? Answer The percentage of time Joe is processing orders is equivalent to the utilization factor, / . Thus, the percentage of time he is processing orders is: /  = 20/30 /  = 20/30 = 2/3 or 66.67% = 2/3 or 66.67%

25. 25 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (A) n Formula Spreadsheet

26. 26 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (A) n Spreadsheet Solution

27. 27 Slide © 2005 Thomson/South-Western M / M / k Queuing System n Multiple channels (with one central waiting line) n Poisson arrival-rate distribution n Exponential service-time distribution n Unlimited maximum queue length n Infinite calling population n Examples: Four-teller transaction counter in bank Four-teller transaction counter in bank Two-clerk returns counter in retail store Two-clerk returns counter in retail store

28. 28 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (B) n M / M /2 Queuing System Smith, Jones, Johnson, and Thomas, Inc. has begun a major advertising campaign which it believes will increase its business 50%. To handle the increased volume, the company has hired an additional floor trader, Fred Hanson, who works at the same speed as Joe Ferris. Note that the new arrival rate of orders,, is 50% higher than that of problem (A). Thus, = 1.5(20) = 30 per hour.

29. 29 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (B) n Sufficient Service Rate Question Why will Joe Ferris alone not be able to handle the increase in orders? Answer Since Joe Ferris processes orders at a mean rate of µ = 30 per hour, then = µ = 30 and the utilization factor is 1. This implies the queue of orders will grow infinitely large. Hence, Joe alone cannot handle this increase in demand. This implies the queue of orders will grow infinitely large. Hence, Joe alone cannot handle this increase in demand.

30. 30 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (B) n Probability of n Units in System Question What is the probability that neither Joe nor Fred will be working on an order at any point in time?

31. 31 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (B) n Probability of n Units in System (continued) Answer Given that = 30, µ = 30, k = 2 and ( /µ) = 1, the probability that neither Joe nor Fred will be working is: = 1/[(1 + (1/1!)(30/30)1] + [(1/2!)(1)2][2(30)/(2(30)-30)] = 1/[(1 + (1/1!)(30/30)1] + [(1/2!)(1)2][2(30)/(2(30)-30)] = 1/(1 + 1 + 1) = 1/3 =.333 = 1/(1 + 1 + 1) = 1/3 =.333

32. 32 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (B) n Average Time in System Question What is the average turnaround time for an order with both Joe and Fred working?

33. 33 Slide © 2005 Thomson/South-Western n Average Time in System (continued) Answer The average turnaround time is the average waiting time in the system, W. L = L q + ( / µ ) = 1/3 + (30/30) = 4/3 L = L q + ( / µ ) = 1/3 + (30/30) = 4/3 W = L /  (4/3)/30 = 4/90 hr. = 2.67 min. W = L /  (4/3)/30 = 4/90 hr. = 2.67 min. Example: SJJT, Inc. (B)

34. 34 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (B) n Average Length of Queue Question What is the average number of orders waiting to be filled with both Joe and Fred working? Answer The average number of orders waiting to be filled is L q. This was calculated earlier as 1/3.

35. 35 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (B) n Formula Spreadsheet

36. 36 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (B) n Spreadsheet Solution

37. 37 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (B) n Creating Special Excel Function to Compute P 0 Select the Tools pull-down menu Select the Macro option Choose the Visual Basic Editor When the Visual Basic Editor appears Select the Insert pull-down menu Select the Insert pull-down menu Choose the Module option Choose the Module option When the Module sheet appears Enter Function Po (k,lamda,mu) Enter Function Po (k,lamda,mu) Enter Visual Basic program (on next slide) Enter Visual Basic program (on next slide) Select the File pull-down menu Choose the Close and Return to MS Excel option

38. 38 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (B) n Visual Basic Module for P 0 Function Function Po(k, lamda, mu) Sum = 0 For n = 0 to k - 1 Sum = Sum + (lamda/mu) ^ n / Application.Fact(n) Sum = Sum + (lamda/mu) ^ n / Application.Fact(n)Next Po = 1/(Sum+(lamda/mu)^k/Application.Fact(k))* (k*mu/(k*mu-lamda))) (k*mu/(k*mu-lamda))) End Function

39. 39 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (C) n Economic Analysis of Queuing Systems The advertising campaign of Smith, Jones, Johnson and Thomas, Inc. (see problems (A) and (B)) was so successful that business actually doubled. The mean rate of stock orders arriving at the exchange is now 40 per hour and the company must decide how many floor traders to employ. Each floor trader hired can process an order in an average time of 2 minutes.

40. 40 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (C) n Economic Analysis of Queuing Systems Based on a number of factors the brokerage firm has determined the average waiting cost per minute for an order to be $.50. Floor traders hired will earn $20 per hour in wages and benefits. Using this information compare the total hourly cost of hiring 2 traders with that of hiring 3 traders.

41. 41 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (C) n Economic Analysis of Waiting Lines Total Hourly Cost = (Total salary cost per hour) = (Total salary cost per hour) + (Total hourly cost for orders in the system) + (Total hourly cost for orders in the system) = ($20 per trader per hour) x (Number of traders) = ($20 per trader per hour) x (Number of traders) + ($30 waiting cost per hour) x (Average number of orders in the system) + ($30 waiting cost per hour) x (Average number of orders in the system) = 20 k + 30 L. = 20 k + 30 L. Thus, L must be determined for k = 2 traders and for k = 3 traders with = 40/hr. and  = 30/hr. (since the average service time is 2 minutes (1/30 hr.). Thus, L must be determined for k = 2 traders and for k = 3 traders with = 40/hr. and  = 30/hr. (since the average service time is 2 minutes (1/30 hr.).

42. 42 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (C) n Cost of Two Servers P 0 = 1 / [1+(1/1!)(40/30)]+[(1/2!)(40/30)2(60/(60-40))] P 0 = 1 / [1+(1/1!)(40/30)]+[(1/2!)(40/30)2(60/(60-40))] = 1 / [1 + (4/3) + (8/3)] = 1 / [1 + (4/3) + (8/3)] = 1/5 = 1/5

43. 43 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (C) n Cost of Two Servers (continued) Thus, L = L q + ( / µ ) = 16/15 + 4/3 = 2.40 L = L q + ( / µ ) = 16/15 + 4/3 = 2.40 Total Cost = (20)(2) + 30(2.40) = $112.00 per hour Total Cost = (20)(2) + 30(2.40) = $112.00 per hour

44. 44 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (C) n Cost of Three Servers P 0 = 1/[[1+(1/1!)(40/30)+(1/2!)(40/30)2]+ [(1/3!)(40/30)3(90/(90-40))] ] [(1/3!)(40/30)3(90/(90-40))] ] = 1 / [1 + 4/3 + 8/9 + 32/45] = 1 / [1 + 4/3 + 8/9 + 32/45] = 15/59 = 15/59

45. 45 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (C) n Cost of Three Servers (continued) Thus, L =.1446 + 40/30 = 1.4780 Thus, L =.1446 + 40/30 = 1.4780 Total Cost = (20)(3) + 30(1.4780) = $104.35 per hour Total Cost = (20)(3) + 30(1.4780) = $104.35 per hour

46. 46 Slide © 2005 Thomson/South-Western Example: SJJT, Inc. (C) n System Cost Comparison WageWaiting Total WageWaiting Total Cost/HrCost/HrCost/Hr 2 Traders $40.00 $82.00 $112.00 3 Traders 60.00 44.35 104.35 Thus, the cost of having 3 traders is less than that of 2 traders. Thus, the cost of having 3 traders is less than that of 2 traders.

47. 47 Slide © 2005 Thomson/South-Western M / D /1 Queuing System n Single channel n Poisson arrival-rate distribution n Constant service time n Unlimited maximum queue length n Infinite calling population n Examples: Single-booth automatic car wash Single-booth automatic car wash Coffee vending machine Coffee vending machine

48. 48 Slide © 2005 Thomson/South-Western Example: Ride ‘Em Cowboy! n M / D /1 Queuing System The mechanical pony ride machine at the entrance to a very popular J-Mart store provides 2 minutes of riding for $.50. Children wanting to ride the pony arrive (accompanied of course) according to a Poisson distribution with a mean rate of 15 per hour.

49. 49 Slide © 2005 Thomson/South-Western Example: Ride ‘Em Cowboy! n What fraction of the time is the pony idle? = 15 per hour = 15 per hour  = 60/2 = 30 per hour Utilization = /  = 15/30 =.5 Idle fraction = 1 – Utilization = 1 -.5 =.5

50. 50 Slide © 2005 Thomson/South-Western n What is the average number of children waiting to ride the pony? n What is the average time a child waits for a ride? Example: Ride ‘Em Cowboy! (or 1 minute)

51. 51 Slide © 2005 Thomson/South-Western M / G / k Queuing System n Multiple channels n Poisson arrival-rate distribution n Arbitrary service times n No waiting line n Infinite calling population n Example: Telephone system with k lines. (When all k lines are being used, additional callers get a busy signal.) Telephone system with k lines. (When all k lines are being used, additional callers get a busy signal.)

52. 52 Slide © 2005 Thomson/South-Western Example: Allen-Booth n M / G / k Queuing System Allen-Booth (A-B) is an OTC market maker. A broker wishing to trade a particular stock for a client will call on a firm like Allen-Booth to execute the order. If the market maker's phone line is busy, a broker will immediately try calling another market maker to transact the order.

53. 53 Slide © 2005 Thomson/South-Western Example: Allen-Booth n M / G / k Queuing System A-B estimates that on the average, a broker will try to call to execute a stock transaction every two minutes. The time required to complete the transaction averages 75 seconds. A-B has four traders staffing its phones. Assume calls arrive according to a Poisson distribution.

54. 54 Slide © 2005 Thomson/South-Western Example: Allen-Booth This problem can be modeled as an M / G / k system with block customers cleared with: 1/ = 2 minutes = 2/60 hour 1/ = 2 minutes = 2/60 hour = 60/2 = 30 per hour = 60/2 = 30 per hour 1/ µ = 75 sec. = 75/60 min. = 75/3600 hr. 1/ µ = 75 sec. = 75/60 min. = 75/3600 hr. µ = 3600/75 = 48 per hour µ = 3600/75 = 48 per hour

55. 55 Slide © 2005 Thomson/South-Western Example: Allen-Booth n % of A-B’s Customers Lost Due to Busy Line First, we must solve for P 0 where k = 4 where k = 4 1 1 P 0 = P 0 = 1 + (30/48) + (30/48)2/2! + (30/48)3/3! + (30/48)4/4! 1 + (30/48) + (30/48)2/2! + (30/48)3/3! + (30/48)4/4! P 0 =.536 P 0 =.536continued

56. 56 Slide © 2005 Thomson/South-Western n % of A-B’s Customers Lost Due to Busy Line Now, Thus, with four traders 0.3% of the potential customers are lost. Example: Allen-Booth

57. 57 Slide © 2005 Thomson/South-Western End of Chapter 12, Part A