1. Capacity and Demand
流程的效率：
顧客排隊的時間、
排隊隊伍的人數、
服務人員閒置時間的比率
CHAPTER 8

2. Outline Learning Objectives Characteristics of Queuing Systems (排隊理論)
Kendall Notation
Performance Measures for M/M/1 Systems (效率)
Little’s Flow Equations
Performance Measures for M/M/s Systems
Economic Analysis
Performance Measures for M/G/1 and M/D/1 Systems
Queuing Simulation
Simulating Multistage Systems with ExtendSim
Theory of Constraints (限制理論)
Summary
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8.2

3. Learning Objectives
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8.3

4. After reading this chapter, you will be able to:
Identify queuing characteristics and classify queuing models accordingly
Evaluate the performance of basic queuing systems and compute their costs
Extend the concepts of queuing theory to queuing simulation
Develop and run simulation models
Use simulation to analyze different process configurations and formulate design improvements
Identify bottleneck issues and understand the Theory of Constraints
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8.4

5. Figure 8.1– Mind Map Focus on Delivery
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8.5

6. Characteristics of Queuing Systems
Arrivals
Service
Queues
queue configurations
Queue discipline
population
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8.6

7. Figure 8.2 – General Queuing System
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8.7

8. Arrivals
Flow units’ arrivals in the process or system; demand placed on the system
Arrival rate = number of flow unit arrivals per unit of time (每小時、每日)
Often described by Poisson distribution
Average arrival rate is λ (每小時4人)
Interarrival time = 1/ λ (每15分鐘1人)
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8.8

9. Figure 8.3 – Poisson Distribution for Arrival Rates
過去 1個月 (20天)
每小時 (共160小時)
的來客數
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8.9

10. Figure 8.4 – Negative Exponential Distribution for Interarrival Times
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8.10

11. Service Flow units are “served” or processed; capacity of system
Service rate = number of flow units served per unit of time (每小時、每日)
May be described by Poisson distribution
Average service rate = μ (每小時4人)
Service time = 1/ μ (每15分鐘1人)
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8.11

12. Figure 8.5 – Negative Exponential Distribution for Service Times
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8.12

13. Queues Flow units waiting to be processed are queues
Work-in-process inventory
Passengers waiting in line at airport security check points
Prescriptions waiting to be filled
Patients in waiting room
Also known as “buffers”
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8.13

14. Queue Configurations One queue or multiple queues One queue 公平、神龍見首不見尾
先到、不一定先被服務
等候線感覺比較短
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8.14

15. Queue Configurations (cont.)
One stage or multiple stages
One stage
Multiple stages
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8.15

16. Queue Discipline Order in which flow units are served
First in, first out (FIFO) or first come, first served (FCFS)
Last in, first out (LIFO)
Priority systems
金卡、銀卡
生死交關
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8.16

17. Population of Flow Units
Calling customer population
Infinite: number of potential arrivals is neither restricted nor influenced significantly by the number of flow units in queue
Finite: set number of arrivals in the system
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8.17

18. Kendall Notation Distribution of arrivals Distribution of service
Number of channels
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8.18

19. Kendall Notation Classification scheme (分類法)
The format of the Kendall notation is:
Distribution of Arrivals / Distribution of Service / Number of Channels
To designate the probability distributions of arrivals or service, three letters are used: M, G, and D.
M = Poisson distribution for rates or negative exponential distribution for times
G = General—any—distribution with a known mean and variance
D = Deterministic or constant
Therefore, an M/M/1 would be a queuing model with Poisson arrival rates, negative exponential service times, and 1 channel.
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8.19

20. Performance Measures for M/M/1 Systems
Assumptions
formulas
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8.20

21. Assumptions for M/M/1 Systems
M/M/1 systems rely on the following assumptions:
One channel (either one server or one team of servers)
Poisson arrival rate
Negative exponential service time
FCFS queue discipline
Infinite calling population
λ < s × μ
Since s = 1 in an M/M/1 system, the requirement can be simplified to λ < μ.
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8.21

22. Formulas for M/M/1 Systems
Performance Measure
Formula
System Utilization
Probability that the system is in use
Percentage of time that the server is busy
ρ= λ μ
Probability of no customers in system
Percentage of time that the server is idle
P 0 =1 – λ μ
Average number of customers waiting in line before service
Average number of flow units in the queue
Length of the queue
L q = λ 2 μ (μ − λ)
Average number of customers in the system
Average number of flow units in the system (those waiting for service and those being served)
L = λ μ − λ
= L q + λ μ
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8.22

23. Formulas for M/M/1 System (cont.)
Performance Measure
Formula
Average time spent waiting for service
Average time spent waiting in the queue
W q = λ μ (μ – λ)
Average time spent in the system
Average time spent waiting for service and being served
W= 1 μ − λ
= W q μ
Probability of exactly n customers in the system
P n = P λ μ n
Probability that the number of customers in the system does not exceed k
P n≤k = 1 – λ μ k+1
Probability that an arriving customer would have to wait for service
P w = λ μ
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8.23

24. Little’s Flow Equations
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8.24

25. Little’s Flow Equations
Lq = λWq, [平均排隊人數 = 顧客平均到達人數 * 平均排隊時間] 銀行每小時平均有 20位顧客， 若顧客花 15分鐘 (0.25 小時) 等待時間，然後接受服務 請估計該銀行的平均等待人數？ Lq = λWq = 20 (人 / 小時) * 0.25 小時 = 5 人 The same is true for W (平均系統時間) and L (平均系統人數). L = λW,
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8.25

26. Performance Measures for M/M/s Systems
Formulas
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8.26

27. Formulas for M/M/s Systems
Performance Measure
Formula
System Utilization
Average utilization rate of servers
ρ = λ sμ
Probability of no customers in system
Probability that the system is empty
Percentage of time that servers are idle
P 0 = 1 n=0 s− λ μ n n! λ μ s s! sμ sμ−λ
Average number of customers waiting in line before service
Average number of flow units in the queue
Length of the queue
L q = λ μ s λμ s−1 ! (sμ − λ) 2 P 0
Average number of customers in the system
Average number of flow units in the system (those waiting for service and those being served)
L = L q + λ μ
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8.27

28. Formulas for M/M/s Systems (cont.)
Performance Measure
Formula
Average time spent waiting for service
Average time spent waiting in the queue
W q = L q λ
Average time spent in the system
Average time spent waiting for service and being served
W = W q + 1 μ
Probability of exactly n customers in the system
P n = P λ μ n n! for n ≤ s
P n = P λ μ n s! s n−s for n > s
Probability that the number of customers in the system does not exceed k
P n≤k = n=0 k P n
Probability that an arriving customer would have to wait for service (all servers are busy)
P w = 1 s! λ μ s sμ sμ−λ P 0
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8.28

29. Economic Analysis Goal Procedure
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8.29

30. Goal of Queuing Analysis
Find the optimal amount of capacity to provide an adequate level of service
Total costs = waiting costs + service costs = (Cw × L)
+ (Cs × number of servers)
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8.30

31. Procedure to Find Optimal Capacity
Three steps
Determine the minimum number of servers needed given λ and μ for that system. Compute the total cost.
λ < sμ [每小時來10顧客 、 每小時服務 3顧客]
Increase the number of servers by 1. Compute the total cost.
If the total cost obtained in Step 2 is higher than the cost computed previously, stop.
It means that your total cost is now past the minimum point on the total cost curve.
If the total cost is lower than the cost computed previously, repeat Step 2 until the total cost starts increasing.
The number of servers for which the total cost is lowest is optimal.
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8.31

32. Performance Measures for M/G/1 and M/D/1 Systems
General information
Formulas for m/g/1
Formulas for m/d/1
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8.32

33. Performance Measures for M/G/1 and M/D/1 Systems
M/G/1 is when
Probability of distribution of service times is general with a mean and standard deviation
Number of channels = 1
M/D/1 is when
Service times are constant (σ = 0)
Note: Often applicable when using automated equipment
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8.33

34. Formulas for M/G/1 System
Performance Measure
Formula
System Utilization
Probability that the system is in use
Percentage of time that the server is busy
ρ = λ μ
Probability of no customers in system
Probability that the system is empty
Percentage of time that the server is idle
P 0 = 1 – λ μ
Average number of customers waiting in line before service
Average number of flow units in the queue
Length of the queue
L q = λ 2 σ λ μ − λ μ
Average number of customers in the system
Average number of flow units in the system (those waiting for service and those being served)
L = L q + λ μ
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8.34

35. Formulas for M/G/1 System (cont.)
Performance Measure
Formula
Average time spent waiting for service
Average time spent waiting in the queue
W q = L q λ
Average time spent in the system
Average time spent waiting for service and being served
W = W q + 1 μ
Probability that an arriving customer would have to wait for service
P w = λ μ
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8.35

36. Formulas for M/D/1 System
Same formulas as for M/G/1, except:
Lq = λ 2 2μ (μ − λ)
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8.36

37. Queuing Simulation steps monte carlo simulation example
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8.37

38. Steps Requires the completion of five steps:
Simulation = simplified representation of reality
Risk = uncertainty reduced through knowledge of probability distributions
Requires the completion of five steps:
Define the problem and objectives
Collect data
Validate the model
Generate different model/process configurations
Run the simulations and interpret the performance results
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8.38

39. Monte Carlo Simulation
The game of roulette, popular at the famous casino in Monte Carlo, bears resemblance with many of the random events decision makers face: customer arrivals, service times, consumer demand, costs of supplies, delivery times, and so on.
Monte Carlo simulation is similar to the game of roulette in that it involves drawing a number at random and recording it.
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8.39

40. Interarrival Time (min)
Example
Collecting data and identifying probability distributions
Interarrival times follow an empirical distribution (see below)
Service times are normally distributed with a mean of 7 minutes and a standard deviation of 1 minute
Interarrival Time (min)
Frequency
Probability 3 10
0.05 4 20
0.10 5 60
0.30 6 50
0.25 7 30
0.15 8 9
Total
200
1.00
TABLE 8.4 – Absolute Frequencies and Probability Distribution of Interarrival Times
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8.40

41. Interarrival Time (min) Cumulative Probability Random Number Interval
Example (cont.)
For interarrival times,
Compute the cumulative probabilities and develop random number intervals
Interarrival Time (min)
Probability
Cumulative Probability
Random Number Interval 3
0.05
0.00–< 0.05 4
0.10
0.15
0.05–< 0.15 5
0.30
0.45
0.15–< 0.45 6
0.25
0.70
0.45–< 0.70 7
0.85
0.70–< 0.85 8
0.95
0.85–< 0.95 9
1.00
0.95–< 1.00
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8.41

42. Interarrival Time (min)
Example (cont.)
Select random numbers and match them to the appropriate random number interval and its corresponding interarrival time
Trial
Random Number
Interarrival Time (min) 1
0.3256 5 2
0.2106 3
0.5214 6 4
0.7731 7
0.8102
0.2794
0.5179 8
0.0740 9
0.3920 10
0.0479
Average
5.3
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8.42

43. Example (cont.) For service times, use the Excel function:
=NORM.INV(RAND(),mean,standard deviation)
=NORM.INV(RAND(),7,1)
Trial
Random Number
Service Time (min) 1
0.9846
9.2 2
0.6814
7.5 3
0.1242
5.8 4
0.5581
7.1 5
0.5959
7.2 6
0.7768
7.8 7
0.6770 8
0.8573
8.1 9
0.0769
5.6 10
0.8884
8.2
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8.43

44. Example (cont.) Simulation of 10 prescription orders (one pharmacist)
Cumulative IATs
Max(SEt-1, ATt)
SB+ST
SB-AT
SE-AT
SBt - SEt-1
Order
(t)
Interarrival Time
(IAT)
Arrival Time
(AT)
Service Begins (SB)
Service Time
(ST)
Service Ends
(SE)
Time in Queue
(TQ)
Time in System
(TS)
Server’s Idle Time
(I) 1 5
9.2
14.2 2 10
7.5
21.7
4.2
11.7 3 6 16
5.8
27.5
5.7
11.5 4 7 23
7.1
34.6
4.5
11.6 30
7.2
41.8
4.6
11.8 35
7.8
49.6
6.8
14.6 41
57.1
8.6
16.1 8 45
8.1
65.2
12.1
20.2 9 50
5.6
70.8
15.2
20.8 53
8.2 79
17.8 26
Average
5.3
7.4
8.0
15.4
0.5
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8.44

45. Simulating Multistage Systems with ExtendSim
Input
Setup
output
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8.45

46. Input
Executive block: This block governs the system. It schedules the events and controls the simulation. It is always placed to the left of all other blocks.
Create block: This block generates arrivals according to a distribution specified by the user.
Queue block: This block holds flow units and releases them according to a queue discipline specified by the user.
Activity block: The activity block acts as the server in a queuing system. The distribution of the time to perform the activity, or service time, is specified by the user.
Exit block: This block removes the flow units from the system once their processing is completed. 
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8.46

47. Figure 8.11 – Simulation of Prescription Orders in ExtendSim
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8.47

48. Figure 8.12 – Dialog Box for Create Block
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8.48

49. Figure 8.13 – Dialog Box for Queue Block
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8.49

50. Figure 8.14 – Dialog Box for Activity Block (Pharmacist)
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8.50

51. Figure 8.15 – Dialog Box for Activity Block (Assistant)
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8.51

52. Setup Simulation Setup Duration of the simulation Start time
Time units for the duration of the simulation
Desired random number seed
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8.52

53. Figure 8.16 – Dialog Box for Simulation Setup (Setup Tab)
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8.53

54. Figure 8.17 – Dialog Box for Simulation Setup (Random Number Tab)
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8.54

55. Output Simulation Results Server utilization
Queue statistics (length, wait time, etc.)
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8.55

56. Figure 8.18 – Results (Server Utilization)
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8.56

57. Figure 8.19 – Results (Queue Statistics)
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8.57

58. Figure 8.20 – Simulation of Prescription Orders With Two Pharmacists
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8.58

59. Theory of Constraints Steps Bottleneck Boy scout story
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8.59

60. Theory of Constraints Step 1: Identify the constraint or bottleneck
Step 2: Exploit the constraint
確保 [瓶頸作業] 100% 運作
Step 3: Subordinate everything else to the constraint
其他作業配合 [瓶頸作業]，不要火力全開
Step 4: Elevate the constraint
改善[瓶頸作業]效率，成為非瓶頸
Step 5: If the constraint is broken, do not rest on your laurels. Go back to Step 1
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8.60

61. Figure 8.21 – Bottleneck = Neck of a Bottle
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8.61

62. Figure 8.22(a) – Boy Scouts Walking in File
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8.62

63. Figure 8.22(b) – Gaps Widen; Herbie Is Behind
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8.63

64. Figure 8.22(c) – Entire Troop’s Arrival at Camp
The speed of the process is dictated by the speed of the bottleneck
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8.64

65. Figure 8.23 – Data Entry Process for Audit Claims
Step 3 sets the pace for the entire system
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8.65

66. Summary Connecting concepts
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8.66

67. Figure 8. 24 – Mind Map Showing Links. Between Demand/Capacity and
Figure 8.24 – Mind Map Showing Links Between Demand/Capacity and the Competitive Priorities
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8.67