1. Queueing Analysis of
Production Systems
(Factory Physics)

2. Reading Material Chapter 8 from textbook
Handout: Single Server Queueing Model by Wallace Hopp (available for download from class website)

3. Queueing analysis is a tool for evaluating operational performance
Utilization
Time-in-system (flow time, leadtime)
Throughput rate (production rate, output rate)
Waiting time (queueing time)
Work-in-process (number of parts or batches in the systems)

4. A Single Stage System
Finished parts
Raw
material
Processing unit

5. The Queueing Perspective
Departure (completion) of jobs
Arrival (release) of jobs
Queue (logical or physical) of jobs
Server (production facility)

6. System Parameters
E[A]: average inter-arrival time between consecutive jobs
l: arrival rate (average number of jobs that arrive per unit
time), l = 1/E[A]
E[S]: average processing time
m: processing rate (maximum average number of jobs that can be processed per unit time), m = 1/E[S]
r: average utilization, r = E[S]/E[A] = l/m

7. Performance Measures E[W]: average time a job spends in the system
E[Wq]: average time a job spends in the queue
E[N]: average number of job in the system (average WIP in the system)
E[Nq]: average number of jobs in the queue (average WIP in the queue)
TH: throughput rate (average number of jobs produced per unit time)

8. Performance Measures (Continued…)
E[W] = E[Wq] + E[S]
E[N] = E[Nq] + r

9. Little’s Law

10. Little’s Law
E[N] = lE[W]
E[Nq] = lE[Wq]
r = lE[S]

11. Example 1 Jobs arrive at regular & constant intervals
Processing times are constant
Arrival rate < processing rate (l < m)

12. Example 1 Jobs arrive at regular & constant intervals
Processing times are constant
Arrival rate < processing rate (l < m)
E[Wq] = 0
E[W] = E[Wq] + E[S] = E[S]
r = l/m
E[N] = lE[W] = lE[S] = r
E[Nq] = lE[Wq] = 0
TH = l

13. Case 2 Jobs arrive at regular & constant intervals
Processing times are constant
Arrival rate > processing rate (l > m)

14. Case 2 Jobs arrive at regular & constant intervals
Processing times are constant
Arrival rate > processing rate (l > m)
E[Wq] = 
E[W] = E[Wq] + E[S] = 
r = 1
E[N] = lE[W] = 
E[Na] = lE[Wa] = 
TH = m

15. Case 3 Job arrivals are subject to variability
Processing times are subject to variability
Arrival rate < processing rate (l < m)
Example:
Average processing time = 6 min
Inter-arrival time = 8 min

16. Case 3 (Continued…) r = 6/8 = 0.75 TH = 1/8 job/min = 7.5 job/hour
E[Wq] > 0
E[W] > E[S]
E[Nq] > 0
E[N] > r

17. In the presence of variability, jobs may wait for processing and a queue in front of the processing unit may build up.
Jobs should not be released to the system at a faster rate than the system processing rate.

18. Sources of Variability

19. Sources of Variability
Sources of variability include:
Demand variability
Processing time variability
Batching
Setup times
Failures and breakdowns
Material shortages
Rework

20. Measuring Variability

21. Variability Classes CV Low variability (LV) Moderate variability (MV)
High variability
(HV) CV
0.75
1.33

22. Illustrating Processing Time Variability

23. Illustrating Arrival Variability
Low variability arrivals t
High variability arrivals t

24. The G/G/1 Queue
If (1) l < m, (2) the distributions of job processing and inter-arrival times are independent and identically distributed (iid), and (3) jobs are processed on a first come, first served (FCFS) basis, then average waiting time in the queue can be approximated by the “VUT” formula:

25. Example CA = CS = 1 E[S] = 1 Case 1: r = 0.50  E[W] = 2, E[N] = 1

26. Example CA = 1 E[S] = 1 r = 0.8 Case 1: CS = 0  E[W] = 3, E[N] = 2.4

27. Facilities should not be operated near full capacity.
To reduce time in system and WIP, we should allow for excess capacity or reduce variability (or both).

28. A Single Stage System with
Parallel facilities
Departure (completion) of jobs
Arrival (release) of jobs
Queue (logical or physical) of jobs
Servers (production facilities)

29. The G/G/m Queue
If (1) l < mm, (2) the distributions of job processing and inter-arrival times are independent and identically distributed (iid), and (3) jobs are processed on a first come, first served (FCFS) basis, then average waiting time in the queue can be approximated by the “VUT” formula:

30. Increasing Capacity
Capacity can be increased by either increasing the production rate (decreasing processing times) or increasing the number of production facilities

31. Increasing Capacity
Capacity can be increased by either increasing the production rate (decreasing processing times) or increasing the number of production facilities
In a system with multiple parallel production facilities, maximum throughput equals the sum of the production rates

32. Dedicated versus Pooled Capacity
Dedicated system: m production facilities, each with a single processor with production rate m and arrival rate l
Pooled system: A single production facility with m parallel processors, with production rate m per processor, and arrival rate ml

33. Dedicated versus Pooled Capacity
Dedicated system:
Pooled system:

34. Pooling reduces expected waiting time by more than a factor of m
Pooling makes better use of existing capacity by continuously balancing the load among different processors

35. The M/M/1 Queue

36. A Common Notation GX/GY/k/N N: maximum number of customers allowed
G: distribution of inter-arrival times
G: distribution of service times
k: number of servers
X: distribution of arrival batch (group) size
Y: distribution of service batch size

37. Common examples
M/M/1
M/G/1
M/M/k
M/M/1/N
MX/M/1
GI/M/1
M/M/k/k

38. Notation in the Book versus Notation in the Lecture Notes
CT (cycle time): E(W)
CTq (cycle time in the queue): E(Wq)
WIP: E(N)
WIPq (WIP in the queue): E(Nq)
u: U (r =l/m)
ra: l
te: E(S); ts: E(X);
ca: cA
ce: cS

39. Assumptions A single server queue
The distribution of inter-arrival times is exponential (Markovian arrivals)
The distribution of processing times is exponential (Markovian processing times)

40. Distribution of Inter-arrival Times

41. The Memoryless Property

42. The Taylor Series Expansion

43. Exponential Inter-arrival Times and the Poisson Process
Poisson distribution

44. Distribution of Processing Times

45. Similarly, when h is small,

46. The Distribution of the Number of Jobs in the System

47. The Distribution of the Number of Jobs in the System

50. The Birth-Death Model l l l 1 2 3 m m m

54. Applying Little’s law

55. The M/G/1 Queue A single server queue
The distribution of inter-arrival times is exponential (Markovian arrivals)
The distribution of processing times is general

56. The M/G/1 Queue (Continued…)

57. The G/G/1 Queue Revisited

58. The M/M/m Queue A queue with m servers
The distribution of inter-arrival times is exponential (Markovian arrivals)
The distribution of processing times is exponential (Markovian arrivals)

59. The Balance Equations
Using analysis similar to the one for the M/M/1 queue:

60. The Birth-Death Model l0 l1 l2 1 2 3 m1 m2 m3

64. The G/G/m Queue
For a queue with a general distribution for arrivals and processing times, average time in the queue can be approximated as

65. Notation in the Book versus Notation in the Lecture Notes
CT (cycle time): E(W)
CTq (cycle time in the queue): E(Wq)
WIP: E(N)
WIPq (WIP in the queue): E(Nq)
u: U
ra: l
te: E(S); ts: E(X);
ca: cA
ce: cS

66. Propagation of Variability
CS(i)
Single server queue:
Multi-server queue:
CA(i)
CD(i) = CA(i+1) i
i+1

67. Propagation of Variability
High Utilization Station
High Process Var
Low Flow Var
High Flow Var
Low Utilization Station
High Process Var
Low Flow Var
Low Flow Var

68. Propagation of Variability (Continued…)
High Utilization Station
Low Process Var
High Flow Var
Low Flow Var
Low Utilization Station
Low Process Var
High Flow Var
High Flow Var

69. Variability Relationships
E(S), CS2
l , CA2
l , CD2

70. If utilization is low, reduce arrival arrival variability; if utilization is high, reduce process variability.
Operations with the highest variability should be done as late as possible in the production process.

71. A Production Line N: number of stages in the production line
Si: processing time in stage i (a random variable), i=1,…, N
U(i): utilization at stage i
CA(i): coefficient of variation in inter-arrival times to stage i
CS(i): coefficient of variation in processing time at stage i

72. Time in System for a Production Line
E[Wq(i)]= V(i) U(i) E[S(i)]

73. Time in System for a Production Line

74. Reducing Time in System (Cycle Time)
Reduce Variability
failures
setup times
uneven arrivals
process control
worker training
Reduce Utilization
arrival rate (yield, rework, etc.)
processing time (processing
speed, availability)
capacity (number of machines)

75. Expected WIP in System for a Production Line