1. Chapter 6 - Queuing Theory © 2002 South-Western/Thomson Learning™ Slides prepared by Jeff Heyl, Lincoln University

2. © 2002 South-Western/Thomson Slide 6- 2 6.1The Modeling Process for Queuing Studies Step 1: Opportunity/Problem RecognitionStep 1: Opportunity/Problem Recognition Step 2: Model FormulationStep 2: Model Formulation Step 3: Data CollectionStep 3: Data Collection Step 4: Analysis of the ModelStep 4: Analysis of the Model Step 5: ImplementationStep 5: Implementation

3. © 2002 South-Western/Thomson Slide 6- 3 6.2The Queuing Situation Characteristics of Waiting Line SituationsCharacteristics of Waiting Line Situations The Structure of a Queuing SystemThe Structure of a Queuing System The Managerial ProblemThe Managerial Problem The Costs Involved in a Queuing SituationThe Costs Involved in a Queuing Situation

4. © 2002 South-Western/Thomson Slide 6- 4 6.3Modeling Queues Queuing Model NotationQueuing Model Notation Deterministic Queuing SystemsDeterministic Queuing Systems The Arrival ProcessThe Arrival Process The Service ProcessThe Service Process Measures for the ServiceMeasures for the Service The Waiting LineThe Waiting Line

5. © 2002 South-Western/Thomson Slide 6- 5 6.4Analysis of the Basic Queue (M/M/1 FCFS/  /  ) Poisson-Exponential Model CharacteristicsPoisson-Exponential Model Characteristics Measure of Performance (Operating Characteristics)Measure of Performance (Operating Characteristics) Managerial Use of the Measures of PerformanceManagerial Use of the Measures of Performance Using Excel’s Goal Seek FunctionUsing Excel’s Goal Seek Function

6. © 2002 South-Western/Thomson Slide 6- 6 6.5More Complex Queuing Situations Multifacility Queuing Systems (M/M/K FCFS//)Multifacility Queuing Systems (M/M/K FCFS/  /  ) Example: Multichannel QueueExample: Multichannel Queue Example: Multichannel Queue at Macro-MarketExample: Multichannel Queue at Macro-Market Serial (Multiphase) QueuesSerial (Multiphase) Queues Example: Serial Queue—Three- Station ProcessExample: Serial Queue—Three- Station Process

7. © 2002 South-Western/Thomson Slide 6- 7 6.6Detailed Modeling Example Step 1: Opportunity/Problem RecognitionStep 1: Opportunity/Problem Recognition Step 2: Model FormulationStep 2: Model Formulation Step 3: Data CollectionStep 3: Data Collection Step 4: Analysis of the ModelStep 4: Analysis of the Model Step 5: ImplementationStep 5: Implementation

8. © 2002 South-Western/Thomson Slide 6- 8 Questions Experiential Exercises Modeling Exercises Case:City of Help Case:Newtown Maintenance Division

9. © 2002 South-Western/Thomson Slide 6- 9 Printer Wait Problem Printer speed Arrival rate of new rint Waiting time Waiting costs Consultant’s salaries Printer costs Select printer Minimize costs Exhibit 6.1

10. © 2002 South-Western/Thomson Slide 6- 10 Printer Wait Problem Printer speed Arrival rate of new print Waiting time Waiting costs Consultant’s salaries Printer costs Select printer Minimize costs Exhibit 6.1 “Seasonality” in the data Highest average number of jobs submitted in a two-hour block = 162.5 Printer can process 90 jobs per hour

11. © 2002 South-Western/Thomson Slide 6- 11 Printer Wait Problem Printer speed Arrival rate of new print Waiting time Waiting costs Consultant’s salaries Printer costs Select printer Minimize costs Exhibit 6.1 “Seasonality” in the data Highest average number of jobs submitted in a two-hour block = 162.5 Printer can process 90 jobs per hour Average of 8.38 jobs waiting to print Average of 9.29 jobs either waiting or being printed New job will wait 0.10 hours Printer utilization is 90%

12. © 2002 South-Western/Thomson Slide 6- 12 The Modeling Process for Queuing Studies Descriptive toolDescriptive tool Used for predicting operating characteristics or performanceUsed for predicting operating characteristics or performance There must be a waiting line, or queue, which may not always be obviousThere must be a waiting line, or queue, which may not always be obvious

13. © 2002 South-Western/Thomson Slide 6- 13 The Modeling Process for Queuing Studies Models assume a steady state systemModels assume a steady state system The basic type of queuing situation must be describedThe basic type of queuing situation must be described Important data must be collectedImportant data must be collected Qualitative aspects may be difficult to measureQualitative aspects may be difficult to measure

14. © 2002 South-Western/Thomson Slide 6- 14 The Queuing System Cost ($) Optimal levelLevel of service ||||||||||| 012345678910 500 500 – 400 400 – 300 300 – 200 200 – 100 100 – Exhibit 6.4

15. © 2002 South-Western/Thomson Slide 6- 15 The Queuing System Cost ($) Optimal levelLevel of service ||||||||||| 012345678910 500 500 – 400 400 – 300 300 – 200 200 – 100 100 – Exhibit 6.4 Total cost Minimum Cost of providing service Cost of waiting

16. © 2002 South-Western/Thomson Slide 6- 16 The Structure of a Queuing System Exhibit 6.5 Arrival process Population Source xx x x x Waiting areaService facility Service System xxxxxxxxxx Waiting line (queue) x Exit

17. © 2002 South-Western/Thomson Slide 6- 17 The Structure of a Queuing System Exhibit 6.5 Arrival process Population Source xx x x x Waiting areaService facility Service System xxxxxxxxxx Waiting line (queue) x Exit Queues (waiting lines) can be: Single Multiple Priority Queue discipline can be: Random By appointment FCFS Queues can be: Finite Infinite

18. © 2002 South-Western/Thomson Slide 6- 18 The Managerial Problem Waiting for the printer is costing valuable consultant timeWaiting for the printer is costing valuable consultant time Faster service will cost more moneyFaster service will cost more money

19. © 2002 South-Western/Thomson Slide 6- 19 The Managerial Problem Waiting for the printer is costing valuable consultant timeWaiting for the printer is costing valuable consultant time Faster service will cost more moneyFaster service will cost more money What is an ‘appropriate’ level of service?

20. © 2002 South-Western/Thomson Slide 6- 20 The Costs Involved in a Queuing Situation Facility Costs: Cost of constructionCost of construction Cost of operationCost of operation Cost of maintenance and repairCost of maintenance and repair Other costsOther costs

21. © 2002 South-Western/Thomson Slide 6- 21 The Costs Involved in a Queuing Situation Facility Costs: Cost of constructionCost of construction Cost of operationCost of operation Cost of maintenance and repairCost of maintenance and repair Other costsOther costs The Cost of Waiting: Customer ‘ill will’Customer ‘ill will’ Loss of salesLoss of sales Loss of customerLoss of customer

22. © 2002 South-Western/Thomson Slide 6- 22 The Costs Involved in a Queuing Situation Facility Costs: Cost of constructionCost of construction Cost of operationCost of operation Cost of maintenance and repairCost of maintenance and repair Other costsOther costs The Cost of Waiting: Customer ‘ill will’Customer ‘ill will’ Loss of salesLoss of sales Loss of customerLoss of customer Compare alternatives based on total cost TC = C F + C W

23. © 2002 South-Western/Thomson Slide 6- 23 Queuing Model Notation Six necessary items of information: 1.Arrival process: M, Ek, D, N, U, G 2.Service process: M, Ek, D, N, U, G 3.Number of servers: K 4.Queue discipline: FCFS, PRI 5.Maximum size permitted: , n 6.Size of the population: , n

24. © 2002 South-Western/Thomson Slide 6- 24 Notation for Common Queuing Systems Exhibit 6.7 Descriptive LabelComments M/M/IFCFS/  /  Standard single-server model M/M/KFCFS/  /  Standard multiserver model M/Ek/IFCFS/  /  Single Erlang service model M/G/IFCFS/  /  Service time distribution unknown M/M/IPRI/  /  Priority service M/M/KPRI/  /  Multiserver priority service M/M/IFCFS/n/  Finite queue, single server M/M/KFCFS/n/  Finite queue, multiserver M/M/IFCFS/  /nLimited source, single server M/M/KFCFS/  /nLimited source, multiserver

25. © 2002 South-Western/Thomson Slide 6- 25 Deterministic Queuing Systems 1.Arrival Rate Equals Service Rate 100% utilization of server and no waiting lines 2.Arrival Rate Larger than Service Rate Waiting line will continuously build - explosive queue 3.Arrival Rate Smaller than Service Rate Server less than fully utilized and never a queue

26. © 2002 South-Western/Thomson Slide 6- 26 The Arrival Process 1.Finite Verse Infinite Source Infinite (or very large) is typically assumed 2.Batch Verse Individual Arrivals Individual arrivals is typically assumed 3.Scheduled Verses Unscheduled Arrivals Unscheduled arrivals described by average arrival rate or average interarrival time

27. © 2002 South-Western/Thomson Slide 6- 27 Measures for Unscheduled Arrivals: Average Arrival Rate, Average Arrival Rate, Often a Poisson distribution Average Interarrival Time, 1/ Average Interarrival Time, 1/ Often a negative exponential distribution The Arrival Process

28. © 2002 South-Western/Thomson Slide 6- 28 Measures for Unscheduled Arrivals: Average Arrival Rate, Average Arrival Rate, Often a Poisson distribution Average Interarrival Time, 1/ Average Interarrival Time, 1/ Often a negative exponential distribution The Arrival Process ||||| 77:117:337:518 A.M. Time 7:037:137:53 Arrival Arrivals Interarrival time

29. © 2002 South-Western/Thomson Slide 6- 29 The Service Process 1.A single facility 2.Multiple, parallel, identical facilities – a multifacility 3.Multiple, parallel, non-identical facilities 4.Service facilities arranged in a series – a serial arrangement

30. © 2002 South-Western/Thomson Slide 6- 30 The Service Process (a) Single service facility (b) Multiple, parallel identical facilities (c) Multiple facilities, multiple queues (d) Multiple, parallel nonidentical facilities (e) Series of facilities (f) Combination of facilities Waiting line Express line Regular lines Exhibit 6.11 Different Arrangements of Service Facilities

31. © 2002 South-Western/Thomson Slide 6- 31 The Service Process Measures for the Service: Average Length of Service (Service Time), 1/  Most commonly a negative exponential distribution Average Service Rate,  Often a Poisson distribution

32. © 2002 South-Western/Thomson Slide 6- 32 The Waiting Line 1.Queue Discipline Priority SystemPriority System Emergency (Preemptive Priority) SystemEmergency (Preemptive Priority) System Last-Come, First-Served (LCFS)Last-Come, First-Served (LCFS) First-Come, First-Served (FCFS)First-Come, First-Served (FCFS) Queue LengthQueue Length

33. © 2002 South-Western/Thomson Slide 6- 33 The Waiting Line 2.Organization of the Queue 3.Behavior in the Queue BalkingBalking RenegingReneging JockeyingJockeying Combining or DividingCombining or Dividing CyclingCycling

34. © 2002 South-Western/Thomson Slide 6- 34 Analysis of the Basic Queue (M/M/I FCFS/  /  ) 1.Arrival Rate ( ) – random variable, Poisson distribution 2.Service Time (1/  ) – negative exponential distribution 3.Major Assumptions Infinite source populationInfinite source population FCFSFCFS /  < 1 =  /  < 1 =  Steady state systemSteady state system Unlimited queue lengthUnlimited queue length

35. © 2002 South-Western/Thomson Slide 6- 35 Analysis of the Basic Queue (M/M/I FCFS/  /  ) Average Waiting Time, W Average Waiting Time in the Queue, W q Average Number of Customers in the System, L Average Number of Customers in the Queue, L q W = 1  -  - Wq =Wq =Wq =Wq =  (  - ) L =  -  - Lq =Lq =Lq =Lq = 2  (  - ) Measures of Performance:

36. © 2002 South-Western/Thomson Slide 6- 36 Analysis of the Basic Queue (M/M/I FCFS/  /  ) Measures of Performance: Probability of an Empty (Idle) Facility, P(0) Probability of the System Being Busy, P w Probability of Being in the System (Waiting and Being Served) Longer than Time t P(0) = 1 -  P w = 1 - P(0) =  P[T > t ] = e -(  - )t

37. © 2002 South-Western/Thomson Slide 6- 37 Analysis of the Basic Queue (M/M/I FCFS/  /  ) Measures of Performance: Probability of Waiting for Service Longer than Time t q Probability of Finding Exactly N Customers in the System, P(N) Probability that the Number of Customers in the System, N, Exceed a Given Value, n P(N) =  N (1 -  ) P[T q > t q ] =  e -(  - ) tqtqtqtq P[N > n] =  N + 1

38. © 2002 South-Western/Thomson Slide 6- 38 All-American Aviation Co. Toolroom:  = 10 per hour = 12 per hour Toolroom utilization Average waiting time at the toolroom Average waiting time in the line Average number of production employees at the toolroom Average number of production employees in the line  = 10/12 = 0.833 W = 1/(12 - 10) = 0.5 hour W q = 10/(12(12 - 10)) = 0.417 hour L = 0.833/(1 - 0.833) = 5 L q = 0.833 2 /(1 - 0.833) = 4.16

39. © 2002 South-Western/Thomson Slide 6- 39 All-American Aviation Co. Toolroom:  = 10 per hour = 12 per hour Probability the toolroom clerk will be idle Probability the system is busy Probability of waiting longer than 0.5 hour in the system Probability of exactly four production employees in the system Probability of more than three production employees in the system P(0) = 1 - 0.833 = 0.167 P w = 0.833 P[T > 0.5] = e -(12 - 10)0.5 = 0.368 P(4) = 0.833 4 (1 - 0.833) = 0.0804 P[N > 3] = 0.833 4 = 0.481

40. © 2002 South-Western/Thomson Slide 6- 40 Multifacility Queuing Systems (M/M/K FCFS/  /  ) PopulationSource x x x x x Waiting area Single waiting line xxxxxxxx Service facilities S1S1 S2S2 S3S3 S4S4 S5S5 Exit

41. © 2002 South-Western/Thomson Slide 6- 41 Multifacility Queuing Systems (M/M/K FCFS/  /  ) Population Source x x x x x Waiting area Single waiting line xxxxxxxx Service facilities S1S1 S2S2 S3S3 S4S4 S5S5 Exit Assumptions: 1.It is a Poisson-exponential system 2.The service facilities (channels) are identical 3.Only one waiting line exists 4.The arrival rate is smaller than the combined service rate (K  ) of all the service facilities

42. © 2002 South-Western/Thomson Slide 6- 42 Multifacility Queuing Systems (M/M/K FCFS/  /  ) Formulas:  = = K K  Probability of finding no customer in the system: P(0) = 1  K K!(1 -  ) +  IIi!i!IIi!i! K - 1 I = 0 Utilization factor for entire system:

43. © 2002 South-Western/Thomson Slide 6- 43 Multifacility Queuing Systems (M/M/K FCFS/  /  ) Formulas: Probability of finding exactly N customers in the system: P(N) = P(0) when N  K NNN!N!NNN!N! when N > K P(0)  N K K K!

44. © 2002 South-Western/Thomson Slide 6- 44 Multifacility Queuing Systems (M/M/K FCFS/  /  ) Formulas: The average number of customers in the waiting line: Given L q Lq =Lq =Lq =Lq = P(0)  K  K!(1 -  ) 2 P(0) = L q K!(1 -  ) 2  K 

45. © 2002 South-Western/Thomson Slide 6- 45 Multifacility Queuing Systems (M/M/K FCFS/  /  ) Formulas: The average number of customers in the system: The average waiting time in the queue per customer: The average time a customer spends in the system: L = L q +  Wq =Wq =Wq =Wq = L q W = = W q + 1 L

46. © 2002 South-Western/Thomson Slide 6- 46 Multichannel Queue at Macro-Market The Multifacility Solution Process K LqLqLLWqWqWWLqLqLLWqWqWWLW Exhibit 6.16 P(0) Exhibit 6.15

47. © 2002 South-Western/Thomson Slide 6- 47 Multichannel Queue at Macro-Market Service time Rate of arrivals Number of stations to staff Staffing costs Total costs Total revenue Revenue/ customer Customer wait time Customer waiting costs Maximize profit Exhibit 6.17

48. © 2002 South-Western/Thomson Slide 6- 48 Multichannel Queue at Macro-Market Arrival rate: = 16 per hour Service rate:  = 20 per hour For one station: W q = 16/(20(20 - 16)) = 0.2 hour per customer Total profits: Gross income: 16 customers ($15 each)=$240.00 Less operating expense=15.00 Less waiting costs=96.00 Profit=$129.00

49. © 2002 South-Western/Thomson Slide 6- 49 Multichannel Queue at Macro-Market Arrival rate: = 16 per hour Service rate:  = 20 per hour For one station: $129.00 Profit

50. © 2002 South-Western/Thomson Slide 6- 50 Multichannel Queue at Macro-Market Arrival rate: = 16 per hour Service rate:  = 20 per hour For one station: $129.00 Profit For two stations: Total profits: Gross income: 16 customers ($15 each)=$240.00 Less operating expenses: 2(15)=30.00 Less waiting costs : 16(0.0095)(30)=4.60 Profit=$205.40

51. © 2002 South-Western/Thomson Slide 6- 51 Multichannel Queue at Macro-Market Arrival rate: = 16 per hour Service rate:  = 20 per hour For one station: $129.00 Profit For two stations: $205.40 Profit

52. © 2002 South-Western/Thomson Slide 6- 52 Multichannel Queue at Macro-Market Arrival rate: = 16 per hour Service rate:  = 20 per hour For one station: $129.00 Profit For two stations: $205.40 Profit For three stations: Total profits: Gross income: 16 customers ($15 each)=$240.00 Less operating expenses: 3(15)=45.00 Less waiting costs : 16(0.0011)(30)=.53 Profit=$194.47

53. © 2002 South-Western/Thomson Slide 6- 53 Multichannel Queue at Macro-Market Arrival rate: = 16 per hour Service rate:  = 20 per hour For one station: $129.00 Profit For two stations: $205.40 Profit For three stations: $194.47 Profit

54. © 2002 South-Western/Thomson Slide 6- 54 Serial (Multiphase) Queues Arrival rate = 5 per hour K 1 = 1;  1 = 6 (for station 1) K 2 = 3;  2 = 2 (for station 2) K 3 = 2;  3 = 4 (for station 3) Three-Station Process:

55. © 2002 South-Western/Thomson Slide 6- 55 Serial (Multiphase) Queues Arrival rate = 5 per hour K 1 = 1;  1 = 6 (for station 1) K 2 = 3;  2 = 2 (for station 2) K 3 = 2;  3 = 4 (for station 3) x x S3S3S3S3 x x x S1S1S1S1 S2S2S2S2 xx… x Infinite source Station 1 Station 2 Station 3 xxx…xx… Queue 1 Queue 2 Queue 3 Exhibit 6.18 Three-Station Process:

56. © 2002 South-Western/Thomson Slide 6- 56 Serial (Multiphase) Queues Three-Station Process: Station 1: /  = 0.833 L q1 = 4.167 W q1 = 0.833 /  = 0.833 L q1 = 4.167 W q1 = 0.833 Station 2:  2 = 2.5  2 = 0.833 L q2 = 3.333 W q2 = 0.667 Station 3:  3 = 1.25  3 = 0.625 L q3 = 0.815 W q3 = 0.163 Total waiting time: W qSystem = W q1 + W q2 + W q3 = 1.663 hours

57. © 2002 South-Western/Thomson Slide 6- 57 Detailed Modeling Example Truck arrival Service rate Truck waiting time Truck operating costs Truck unloading costs Lease new equipment Equipment costs Minimize costs Exhibit 6.19

58. © 2002 South-Western/Thomson Slide 6- 58 Detailed Modeling Example Truck arrival Service rate Truck waiting time Truck operating costs Truck unloading costs Lease new equipment Equipment costs Minimize costs Exhibit 6.19 Arrival rate, = 2 trucks per hour Service time, 1/  = 20 minutes per truck Truck operating costs = $30 per hour Leasing additional equipment = $200 per day

59. © 2002 South-Western/Thomson Slide 6- 59 Mainland Tech Registration Student arrivals Service rate Number of servers Minimize cost Time in the system

60. © 2002 South-Western/Thomson Slide 6- 60 Mainland Tech Registration Registration is open from 9 A.M. to 6:30 P.M. Monday thought Friday and from 9 A.M. to 12 noon on Saturday for one weekRegistration is open from 9 A.M. to 6:30 P.M. Monday thought Friday and from 9 A.M. to 12 noon on Saturday for one week There are 725 ongoing studentsThere are 725 ongoing students Each registration event takes 15 minutesEach registration event takes 15 minutes Time in the system should be equal to or less than 25 minutesTime in the system should be equal to or less than 25 minutes

61. © 2002 South-Western/Thomson Slide 6- 61 Mainland Tech Registration Registration is open from 9 A.M. to 6:30 P.M. Monday thought Friday and from 9 A.M. to 12 noon on Saturday for one weekRegistration is open from 9 A.M. to 6:30 P.M. Monday thought Friday and from 9 A.M. to 12 noon on Saturday for one week There are 725 ongoing studentsThere are 725 ongoing students Each registration event takes 15 minutesEach registration event takes 15 minutes Time in the system should be equal to or less than 25 minutesTime in the system should be equal to or less than 25 minutes How many staff members does Mainland Tech need at registration to satisfy the student’s criteria at the lowest cost?

62. © 2002 South-Western/Thomson Slide 6- 62 Mainland Tech Registration Using standard M/M/K relationships: Number of students = 725 Available registration hours = 50.5 Service rate per hour (  ) = 4 Arrival rate per hour ( ) = 14.356 Minimum number of servers = 4 K = 4 W = 0.713 hrs or 42.77 minutes K = 5 W = 0.322 hrs or 19.3 minutes