1. 1 Queuing Systems (2)

2. Queueing Models (Henry C. Co)2 Queuing Analysis Cost of service capacity Cost of customers waiting Cost Service capacity Total cost Customer waiting cost Capacity cost =+

3. Queueing Models (Henry C. Co)3 The Basic Model

4. Queueing Models (Henry C. Co)4 A Basic Queue Server Stephen R. Lawrence Leeds School of Business University of Colorado Boulder, CO 80309-0419

5. Queueing Models (Henry C. Co)5 A Basic Queue Customer Arrivals Server

6. Queueing Models (Henry C. Co)6 A Basic Queue Server

7. Queueing Models (Henry C. Co)7 A Basic Queue Customer Departures Server

8. Queueing Models (Henry C. Co)8 A Basic Queue Queue (waiting line) Customer Arrivals Customer Departures Server

9. Queueing Models (Henry C. Co)9 A Basic Queue Queue (waiting line) Customer Arrivals Customer Departures Server Line too long? Customer balks (never enters queue)

10. Queueing Models (Henry C. Co)10 A Basic Queue Queue (waiting line) Customer Arrivals Customer Departures Line too long? Customer reneges (abandons queue) Server Line too long? Customer balks (never enters queue)

11. Queueing Models (Henry C. Co)11 Queuing Analysis Single Channel (or Single Server) Queue

12. Queueing Models (Henry C. Co)12 Queuing Analysis Service Rate ( 

13. Queueing Models (Henry C. Co)13 Queuing Analysis Arrival Rate (  Service Rate ( 

14. Queueing Models (Henry C. Co)14 Queuing Analysis Arrival Rate (  Average Waiting Time in Queue ( W q ) Service Rate ( 

15. Queueing Models (Henry C. Co)15 Queuing Analysis Arrival Rate (  Average Number of People in Queue ( L q ) Average Waiting Time in Queue ( W q ) Service Rate ( 

16. Queueing Models (Henry C. Co)16 Queuing Analysis Arrival Rate (  Average Number of People in Queue ( L q ) Average Time in System ( W ) Average Number in System ( L ) Average Waiting Time in Queue ( W q ) Service Rate ( 

17. Queueing Models (Henry C. Co)17 Characteristics of a Queue

18. Queueing Models (Henry C. Co)18  Source population  Arrival characteristics  Physical features of lines  Selection from the waiting line  Service facility  Exit Elements of Queuing System ArrivalsServiceWaiting line Exit Processing order System

19. Queueing Models (Henry C. Co)19 Source Population  May be finite or infinite.  For practical intent and purposes, when the population is large in comparison to the service system, we assume the source population to be infinite (e.g., in a small barber shop, 200 potential customers per day may be treated as an infinite population).

20. Queueing Models (Henry C. Co)20 Arrival  Pattern of arrivals Controllable arrival pattern  Movie theatres offering Monday specials.  Department stores running sales.  Airlines offering off-season rates.  Overseas telecom rates from 1:00 a.m. To 7:00 a.m. Uncontrollable arrival pattern  Emergency operations.  Fire department.  Size of arrivals: single or batch arrival?  Probability distribution pattern of arrivals. Periodic: constant time-between-arrivals (TBA). Purely random TBA (e.g., exponential distribution).

21. Queueing Models (Henry C. Co)21  Degree of patients A patient arrival is one who waits as long as necessary until the service facility is ready to serve him/her (even if the customer grumble and behave discourteously or Impatiently). Impatient arrivals.  Balking: views the situation (length of queue) and then decides to leave.  Reneging: views the situation, joins the queue, after some time, departs without being served.

22. Queueing Models (Henry C. Co)22 Physical Features of Waiting Line  Length of line: infinite or finite waiting capacity?  Number of lines; configuration of the lines; jockeying.

23. Queueing Models (Henry C. Co)23 Selection from the Waiting Line  Queue discipline: priority rule(s) for determining the order of service to customers in a waiting line FIFO. By reservations/appointment only/first. Emergencies first. Highest profit customer first. Largest orders first. Best customer first. Longest waiting time in line first. Soonest promised date first. Shortest processing time first.  Line structuring: express checkouts (supermarkets); “commercial transactions only” (banks).

24. Queueing Models (Henry C. Co)24 Service Facility  Structure Single-channel single- phase. Single-channel multi- phase. Multi-channel single- phase. Multi-channel multi- phase. Mixed.  Service rate Constant Random (probability distribution). Queuing Systems Multiple channel Multiple phase

25. Queueing Models (Henry C. Co)25 Exit  Return to source population Recurring-common-cold case.  Low probability of re-service Appendectomy-only-once case.

26. Queueing Models (Henry C. Co)26 Steady State

27. Queueing Models (Henry C. Co)27  A stable system: The queue will never increase to infinity. An empty state is reached for sure after some time period.  Condition for Stability: >. This condition MUST be met to make all formulas valid.  The steady state: Probability {n customers in the system} does not depend on the time.

28. Queueing Models (Henry C. Co)28 Waiting Time vs Utilization System Utilization Average number on time waiting in line 0 100%

29. Queueing Models (Henry C. Co)29 M/M/1 Queues 1 st M (for “Markovian) – Arrival Distribution is Exponential 2 nd M – Service Distribution is Exponential 1 – Single Channel

30. Queueing Models (Henry C. Co)30 Population  Time horizon: an infinite horizon.  Source Population: infinite.

31. Queueing Models (Henry C. Co)31 Arrival Process  The inter-arrival time is an exponentially-distributed random variable with average arrival rate =.  If the inter-arrival time is an exponentially-distributed random variable, then the number of arrivals during the fixed period of time is a Poisson distribution.  No balking or reneging

32. Queueing Models (Henry C. Co)32 0 0.05 0.1 0.15 0.2 0.25 0123456789101112 Poisson Distribution

33. Queueing Models (Henry C. Co)33 Service Process  The service time is also assumed to be exponentially distributed with mean service rate .  Only 1 server  First-come-first-served (FCFS) queue priority  Mean length of service = 1/  No limit on the queue size.

34. Queueing Models (Henry C. Co)34 Operating Characteristics Utilization (fraction of time server is busy)

35. Queueing Models (Henry C. Co)35 Operating Characteristics Utilization (fraction of time server is busy) Expected (Average) waiting times

36. Queueing Models (Henry C. Co)36 Operating Characteristics Utilization (fraction of time server is busy) Average waiting times Average numbers

37. Queueing Models (Henry C. Co)37 Fundamental Relationship  Little’s Law: L=W or Lq= Wq

38. Queueing Models (Henry C. Co)38 Example Stephen R. Lawrence Leed School of Business University of Colorado Boulder, CO 80309-0419

39. Queueing Models (Henry C. Co)39 Example Boulder Reservoir has one launching ramp for small boats. On summer weekends, boats arrive for launching at a mean rate of 6 boats per hour. It takes an average of s=6 minutes to launch a boat. Boats are launched FCFS.

40. Queueing Models (Henry C. Co)40 Example Boulder Reservoir has one launching ramp for small boats. On summer weekends, boats arrive for launching at a mean rate of 6 boats per hour. It takes an average of s=6 minutes to launch a boat. Boats are launched FCFS.  = 6/hr  = 1/s =1/6 = 0.167/min or 10/hr

41. Queueing Models (Henry C. Co)41 Example Boulder Reservoir has one launching ramp for small boats. On summer weekends, boats arrive for launching at a mean rate of 6 boats per hour. It takes an average of s=6 minutes to launch a boat. Boats are launched FCFS.  = 6/hr  = 1/s =1/6 = 0.167/min or 10/hr  = 6/10 = 0.6 or 60%

42. Queueing Models (Henry C. Co)42 Example Boulder Reservoir has one launching ramp for small boats. On summer weekends, boats arrive for launching at a mean rate of 6 boats per hour. It takes an average of s=6 minutes to launch a boat. Boats are launched FCFS.  = 6/hr  = 1/s =1/6 = 0.167/min or 10/hr  = 6/10 = 0.6 or 60% L =  = 6/(10-6) = 1.5 boats L q = L  = 1.5(0.6) = 0.9 boats

43. Queueing Models (Henry C. Co)43 Example Boulder Reservoir has one launching ramp for small boats. On summer weekends, boats arrive for launching at a mean rate of 6 boats per hour. It takes an average of s=6 minutes to launch a boat. Boats are launched FCFS.  = 6/hr  = 1/s =1/6 = 0.167/min or 10/hr  = 6/10 = 0.6 or 60% L =  = 6/(10-6) = 1.5 boats L q = L  = 1.5(0.6) = 0.9 boats W = 1/  = 1/(10-6) = 0.25 hrs or 15 mins W q = W  = 0.25(0.6) = 0.15 hrs or 9 mins

44. Queueing Models (Henry C. Co)44 Example (cont.) During the busy Fourth of July weekend, boats are expected to arrive at an average rate of 9 per hour.

45. Queueing Models (Henry C. Co)45 Example (cont.) During the busy Fourth of July weekend, boats are expected to arrive at an average rate of 9 per hour.  = 9/hr  = 1/s =1/6 = 0.167/min or 10/hr

46. Queueing Models (Henry C. Co)46 Example (cont.) During the busy Fourth of July weekend, boats are expected to arrive at an average rate of 9 per hour.  = 9/hr  = 1/s =1/6 = 0.167/min or 10/hr  = 9/10 = 0.9 or 90%

47. Queueing Models (Henry C. Co)47 Example (cont.) During the busy Fourth of July weekend, boats are expected to arrive at an average rate of 9 per hour.  = 9/hr  = 1/s =1/6 = 0.167/min or 10/hr  = 9/10 = 0.9 or 90% L =  = 9/(10-9) = 9.0 boats L q = L  = 9(0.6) = 5.4 boats

48. Queueing Models (Henry C. Co)48 Example (cont.) During the busy Fourth of July weekend, boats are expected to arrive at an average rate of 9 per hour.  = 9/hr  = 1/s =1/6 = 0.167/min or 10/hr  = 9/10 = 0.9 or 90% L =  = 9/(10-9) = 9 boats L q = L  = 9(0.6) = 5.4 boats W = 1/  = 1/(10-9) = 1.0 hrs or 60 mins W q = W  = 1(0.9) = 0.9 hrs or 54 mins

49. Queueing Models (Henry C. Co)49 Resource Utilization service rate  = 

50. Queueing Models (Henry C. Co)50 Resource Utilization service rate  =   =  L q = 

51. Queueing Models (Henry C. Co)51 Resource Utilization Arrival Rate  = 10.0   = 1.0)  = 0.0   = 0.0) service rate  =   =  L q =  LqLq

52. Queueing Models (Henry C. Co)52 Resource Utilization Arrival Rate  = 10.0   = 1.0)  = 0.0   = 0.0) service rate  =   =  L q =  LqLq

53. Queueing Models (Henry C. Co)53 Resource Utilization Arrival Rate  = 10.0   = 1.0)  = 0.0   = 0.0) service rate  =   =  L q =  LqLq

54. Queueing Models (Henry C. Co)54 Flexibility/Utilization Trade-off Utilization   = 1.0  = 0.0

55. Queueing Models (Henry C. Co)55 Flexibility/Utilization Trade-off Utilization   = 1.0  = 0.0 L L q W W q

56. Queueing Models (Henry C. Co)56 Flexibility/Utilization Trade-off Utilization   = 1.0  = 0.0 L L q W W q

57. Queueing Models (Henry C. Co)57 Flexibility/Utilization Trade-off Utilization   = 1.0  = 0.0 L L q W W q High utilization Low flexibility Poor service Low utilization High flexibility Good service

58. Queueing Models (Henry C. Co)58 Queues and Flexibility  Low utilization levels (  < 0.6 ) provide better service levels greater flexibility lower waiting costs (e.g., lost business)  High utilization levels ( > 0.9 ) provide better equipment and employee utilization fewer idle periods lower production/service costs  Must trade off benefits of high utilization levels with benefits of flexibility and service

59. Queueing Models (Henry C. Co)59 Cost Trade-offs Utilization   = 1.0  = 0.0 Cost

60. Queueing Models (Henry C. Co)60 Cost Trade-offs Utilization   = 1.0  = 0.0 Cost Cost of Waiting

61. Queueing Models (Henry C. Co)61 Cost Trade-offs Utilization   = 1.0  = 0.0 Cost Cost of Waiting Cost of Service

62. Queueing Models (Henry C. Co)62 Cost Trade-offs Utilization   = 1.0  = 0.0 Cost CombinedCosts Cost of Waiting Cost of Service

63. Queueing Models (Henry C. Co)63 Queues and Simulation  Only simple queues can be mathematically analyzed  “Real world” queues are often very complex multiple servers, multiple queues balking, reneging, queue jumping machine breakdowns networks of queues,...  Need to analyze, complex or not  Computer simulation !

64. Queueing Models (Henry C. Co)64  Adding an extra server Reduces the expected queue length and waiting time greatly. Reduces the server’s utilization level significantly.  In some cases, a manager wants the expected customer waiting time is below certain critical level. Otherwise, he may lose customers.

65. Questions Queueing Models (Henry C. Co)65