More Related Content Similar to BU_FCAI_SCC430_Modeling&Simulation_Ch03.pdf Similar to BU_FCAI_SCC430_Modeling&Simulation_Ch03.pdf (20) 2. • Introduction.
• Description of Queuing System.
• Simulating a Single-Server Queue.
✓ Example.
• Simulating a Queue with Multi-Servers.
✓ Example.
2
©Ahmed Hagag Modeling and Simulation
Chapter 3: Queueing Simulation
3. 3
©Ahmed Hagag
Introduction (1/3)
Modeling and Simulation
Queueing Model:
• Simulation is often used in the analysis of queueing
models. In a simple but typical queueing model,
customers arrive from time to time and join a queue
(waiting line), are eventually served, and finally leave
the system.
4. 4
©Ahmed Hagag
Introduction (2/3)
Modeling and Simulation
• Queueing models, whether solved mathematically or
analyzed through simulation, provide the analyst with a
powerful tool for designing and evaluating the
performance of queueing systems.
• Typical measures of system performance include server
utilization (percentage of time a server is busy), length of
waiting lines, and delays of customers.
5. 5
©Ahmed Hagag
Introduction (3/3)
Modeling and Simulation
• Quite often, when designing to improve a queueing
system, the analyst (or decision maker) is involved in
tradeoffs between server utilization and customer
satisfaction in terms of line lengths and delays.
6. 6
©Ahmed Hagag
Description of Queuing Sys (1/8)
Modeling and Simulation
• The key elements of a queueing system are the
customers and servers.
• The term customer can refer to people, machines, trucks,
mechanics, patients, airplanes, e-mail, cases, or orders—
anything that arrives at a facility and requires service.
• The term server might refer to receptionists, repair
personnel, mechanics, medical personnel, runways at an
airport, automatic packers, order pickers, or CPUs in a
computer—any resource (person, machine, etc.) that
provides the requested service.
7. 7
©Ahmed Hagag
Description of Queuing Sys (2/8)
Modeling and Simulation
Queueing System:
1. The Calling Population.
2. System Capacity.
3. The Arrival Process.
4. Queue Behavior and Queue Discipline.
5. Service Times and the Service Mechanism.
8. 8
©Ahmed Hagag
Description of Queuing Sys (3/8)
Modeling and Simulation
1. The Calling Population (1/3):
• Also called: Input source (source of customers).
• The population of potential customers, referred to as the
calling population, may be assumed to be finite or
infinite.
9. 9
©Ahmed Hagag
Description of Queuing Sys (3/8)
Modeling and Simulation
1. The Calling Population (2/3):
• Infinite size:
➢ Simulation assumptions: If a unit leaves the calling
population and joins the waiting line or enters service,
there is no change in the arrival rate of other units that
may need service.
➢ Most simulated models.
• Finite size:
➢ Less common and complicates the simulation.
10. 10
©Ahmed Hagag
Description of Queuing Sys (3/8)
Modeling and Simulation
1. The Calling Population (3/3):
• The main difference between finite and infinite
population models is how the arrival rate is defined. In
an infinite population model, the arrival rate (i.e., the
average number of arrivals per unit of time) is not
affected by the number of customers who have left the
calling population and joined the queueing system.
11. 11
©Ahmed Hagag
Description of Queuing Sys (4/8)
Modeling and Simulation
2. System Capacity:
• Is the maximum number of units that can be
accommodated in the system.
• In many queueing systems, there is a limit to the
number of customers that may be in the waiting line or
system.
• It’s unlimited capacity if the system can accommodate
any number of units in the waiting line.
12. 12
©Ahmed Hagag
Description of Queuing Sys (5/8)
Modeling and Simulation
3. The Arrival Process:
• The arrival process for infinite-population models is
usually characterized in terms of interarrival times of
successive customers. Arrivals may occur at scheduled
times or at random times. When at random times, the
interarrival times are usually characterized by a
probability distribution.
13. 13
©Ahmed Hagag
Description of Queuing Sys (6/8)
Modeling and Simulation
4. Queue Behavior and Queue Discipline (1/2):
• Queue behavior refers to the actions of customers while
in a queue waiting for service to begin. In some
situations, there is a possibility that incoming customers
will balk (leave when they see that the line is too long),
renege (leave after being in the line when they see that
the line is moving too slowly), or jockey (move from
one line to another if they think they have chosen a slow
line).
14. 14
©Ahmed Hagag
Description of Queuing Sys (6/8)
Modeling and Simulation
4. Queue Behavior and Queue Discipline (2/2):
• Queue discipline refers to the logical ordering of
customers in a queue and determines which customer
will be chosen for service when a server becomes free.
• Common queue disciplines include first-in-first-out
(FIFO), last-in-first-out (LIFO), service in random order
(SIRO), shortest processing time first (SPT), and
service according to priority (PR).
15. 15
©Ahmed Hagag
Description of Queuing Sys (7/8)
Modeling and Simulation
5. Service Times and the Service Mechanism (1/3):
• The service times of successive arrivals are denoted by
𝑆1, 𝑆2, 𝑆3, . . . . They may be constant or of random
duration. In the latter case, {𝑆1, 𝑆2, 𝑆3, . . . } is usually
characterized as a sequence of independent and
identically distributed random variables.
• In addition, in some systems, service times depend upon
the time of day or upon the length of the waiting line.
16. 16
©Ahmed Hagag
Description of Queuing Sys (7/8)
Modeling and Simulation
5. Service Times and the Service Mechanism (2/3):
• A queueing system consists of a
number of service centers and
interconnecting queues. Each
service center consists of some
number of servers, 𝑐, working in
parallel; that is, upon getting to
the head of the line, a customer
takes the first available server.
𝒄 = 𝟑
18. 18
©Ahmed Hagag
Description of Queuing Sys (7/8)
Modeling and Simulation
A notational system for parallel server systems: A/B/c/N/K.
These letters represent the following system characteristics:
19. 19
©Ahmed Hagag
Description of Queuing Sys (8/8)
Modeling and Simulation
Queuing System State (1/3):
• The quantities (variables) that collectively completely
describe the system at any given instant from the
viewpoint of the study objectives:
➢ Number of units in the system: How many units are
currently in the queue or being processed by the server.
➢ Status of server (idle, busy): Is the server(s) busy with
processing some units or idle waiting for a job.
22. 22
©Ahmed Hagag
Sim. a Single-Server Queue (1/14)
Modeling and Simulation
Simulation Table (1/3):
• The following table was designed as a simulation table
specifically for a single-channel queue that serves
customers on a first-in–first-out (FIFO) basis.
25. 25
©Ahmed Hagag
Sim. a Single-Server Queue (2/14)
Modeling and Simulation
Example1
The Grocery Checkout, a Single-Server Queue:
26. 26
©Ahmed Hagag
Sim. a Single-Server Queue (3/14)
Modeling and Simulation
• A small grocery store has one
checkout counter. Customers
arrive at the checkout counter
at random times that range
from 1 to 8 minutes apart. We
assume that interarrival times
are integer-valued with each
of the 8 values having equal
probability; this is a discrete
uniform distribution.
27. 27
©Ahmed Hagag
Sim. a Single-Server Queue (3/14)
Modeling and Simulation
• A small grocery store has one
checkout counter. Customers
arrive at the checkout counter
at random times that range
from 1 to 8 minutes apart. We
assume that interarrival times
are integer-valued with each
of the 8 values having equal
probability; this is a discrete
uniform distribution.
28. 28
©Ahmed Hagag
Sim. a Single-Server Queue (3/14)
Modeling and Simulation
• A small grocery store has one
checkout counter. Customers
arrive at the checkout counter
at random times that range
from 1 to 8 minutes apart. We
assume that interarrival times
are integer-valued with each
of the 8 values having equal
probability; this is a discrete
uniform distribution.
29. 29
©Ahmed Hagag
Sim. a Single-Server Queue (4/14)
Modeling and Simulation
Distribution of Time Between Arrivals
30. 30
©Ahmed Hagag
Sim. a Single-Server Queue (5/14)
Modeling and Simulation
Time-Between-Arrivals Determination
31. 31
©Ahmed Hagag
Sim. a Single-Server Queue (6/14)
Modeling and Simulation
• The service times vary from 1
to 6 minutes (also integer-
valued), with the probabilities
shown in the following table.
34. 34
©Ahmed Hagag
Sim. a Single-Server Queue (9/14)
Modeling and Simulation
Simulation Table for Queueing Problem (20 customers)
35. 35
©Ahmed Hagag
Sim. a Single-Server Queue (9/14)
Modeling and Simulation
Simulation Table for Queueing Problem (20 customers)
36. 36
©Ahmed Hagag
Sim. a Single-Server Queue (9/14)
Modeling and Simulation
Simulation Table for Queueing Problem (20 customers)
37. 37
©Ahmed Hagag
Sim. a Single-Server Queue (9/14)
Modeling and Simulation
Simulation Table for Queueing Problem (20 customers)
38. 38
©Ahmed Hagag
Sim. a Single-Server Queue (9/14)
Modeling and Simulation
Simulation Table for Queueing Problem (20 customers)
39. 39
©Ahmed Hagag
Sim. a Single-Server Queue (9/14)
Modeling and Simulation
Simulation Table for Queueing Problem (20 customers)
40. 40
©Ahmed Hagag
Sim. a Single-Server Queue (9/14)
Modeling and Simulation
Simulation Table for Queueing Problem (20 customers)
41. 41
©Ahmed Hagag
Sim. a Single-Server Queue (9/14)
Modeling and Simulation
Simulation Table for Queueing Problem (20 customers)
42. 42
©Ahmed Hagag
Sim. a Single-Server Queue (9/14)
Modeling and Simulation
Simulation Table for Queueing Problem (20 customers)
Total Run Time of
Simulation
43. 43
©Ahmed Hagag
Sim. a Single-Server Queue (10/14)
Modeling and Simulation
Some of the findings from the simulation are as follows (1/7):
44. 44
©Ahmed Hagag
Sim. a Single-Server Queue (10/14)
Modeling and Simulation
Some of the findings from the simulation are as follows (2/7):
45. 45
©Ahmed Hagag
Sim. a Single-Server Queue (10/14)
Modeling and Simulation
Some of the findings from the simulation are as follows (3/7):
46. 46
©Ahmed Hagag
Sim. a Single-Server Queue (10/14)
Modeling and Simulation
Some of the findings from the simulation are as follows (4/7):
This result can be compared with the expected service time by
finding the mean of the service-time distribution using the
equation:
47. 47
©Ahmed Hagag
Sim. a Single-Server Queue (10/14)
Modeling and Simulation
Some of the findings from the simulation are as follows (5/7):
48. 48
©Ahmed Hagag
Sim. a Single-Server Queue (10/14)
Modeling and Simulation
Some of the findings from the simulation are as follows (6/7):
49. 49
©Ahmed Hagag
Sim. a Single-Server Queue (10/14)
Modeling and Simulation
Some of the findings from the simulation are as follows (7/7):
50. 50
©Ahmed Hagag
Sim. a Single-Server Queue (11/14)
Modeling and Simulation
• Our objective is to analyze the system by simulating the
arrival and service of 100 customers and to compute a
variety of typical measures of performance for queueing
models.
➢ In actuality, 100 customers may be too small a sample size to
draw reliable conclusions. Depending on our objectives, the
accuracy of the results may be enhanced by increasing the
sample size (number of customers), or by running multiple
trials (or replications).
51. 51
©Ahmed Hagag
Sim. a Single-Server Queue (12/14)
Modeling and Simulation
• Our objective is to analyze the system by simulating the
arrival and service of 100 customers and to compute a
variety of typical measures of performance for queueing
models.
➢ A second issue is that of initial conditions. A simulation of a
grocery store that starts with an empty system may or may not
be realistic unless the intention is to model the system from
startup or to model until steady-state operation is reached.
Here, to keep calculations simple, the starting conditions are
an empty grocery, and any concerns are overlooked.
52. 52
©Ahmed Hagag
Sim. a Single-Server Queue (13/14)
Modeling and Simulation
Simulation Table for Queueing Problem (100 customers)
53. 53
©Ahmed Hagag
Sim. a Single-Server Queue (14/14)
Modeling and Simulation
Some of the findings from the simulation are as follows (1/4):
54. 54
©Ahmed Hagag
Sim. a Single-Server Queue (14/14)
Modeling and Simulation
Some of the findings from the simulation are as follows (1/4):
55. 55
©Ahmed Hagag
Sim. a Single-Server Queue (14/14)
Modeling and Simulation
Some of the findings from the simulation are as follows (2/4):
Interval (0, 2]
56. 56
©Ahmed Hagag
Sim. a Single-Server Queue (14/14)
Modeling and Simulation
Some of the findings from the simulation are as follows (2/4):
Interval (2, 4]
57. 57
©Ahmed Hagag
Sim. a Single-Server Queue (14/14)
Modeling and Simulation
Some of the findings from the simulation are as follows (2/4):
58. 58
©Ahmed Hagag
Sim. a Single-Server Queue (14/14)
Modeling and Simulation
• We are now interested in running an experiment to
address the question of how the average time in queue
for the first 100 customers varies from day to day.
• Running once with 100 customers corresponds to one
day. Running it 50 times corresponds to 50 days, where
each trial represents one day.
• The overall average waiting time over 50 trials was 1.32
minutes.
Some of the findings from the simulation are as follows (3/4):
59. 59
©Ahmed Hagag
Sim. a Single-Server Queue (14/14)
Modeling and Simulation
Some of the findings from the simulation are as follows (4/4):
60. 60
©Ahmed Hagag
Sim. a Single-Server Queue (14/14)
Modeling and Simulation
Some of the findings from the simulation are as follows (4/4):
Interval (0, 0.5]
61. 61
©Ahmed Hagag
Sim. a Single-Server Queue (14/14)
Modeling and Simulation
Some of the findings from the simulation are as follows (4/4):
Interval (0.5, 1]
62. 62
©Ahmed Hagag
Sim. a Single-Server Queue (14/14)
Modeling and Simulation
Some of the findings from the simulation are as follows (4/4):
Interval (1, 1.5]
63. 63
©Ahmed Hagag
Sim. a Single-Server Queue (14/14)
Modeling and Simulation
Some of the findings from the simulation are as follows (4/4):
76. 76
©Ahmed Hagag
Review Example (11/11)
Modeling and Simulation
Results (2/2):
And compare with the expected service time
And compare with the expected interarrivals time
77. 77
©Ahmed Hagag
Sim. a Multi-Server Queue (1/18)
Modeling and Simulation
Queueing Model:
• Suppose that there are c channels operating in parallel.
Arrivals will join a single queue and enter the first
available service channel.
79. 79
©Ahmed Hagag
Sim. a Multi-Server Queue (2/18)
Modeling and Simulation
Example: Call Center:
• Consider a Call Center where technical personnel take
calls and provide service.
• Two technical support people (2 server) exists:
➢ Able more experienced, provides service faster,
➢ Baker newbie, provides service slower.
• Rule:
➢ Able gets call if both people are idle.
80. 80
©Ahmed Hagag
Sim. a Multi-Server Queue (3/18)
Modeling and Simulation
Interarrival distribution of calls for technical support
81. 81
©Ahmed Hagag
Sim. a Multi-Server Queue (3/18)
Modeling and Simulation
Interarrival distribution of calls for technical support
82. 82
©Ahmed Hagag
Sim. a Multi-Server Queue (3/18)
Modeling and Simulation
Interarrival distribution of calls for technical support
83. 83
©Ahmed Hagag
Sim. a Multi-Server Queue (4/18)
Modeling and Simulation
Service Distribution of Able
Service Distribution of Baker
84. 84
©Ahmed Hagag
Sim. a Multi-Server Queue (5/18)
Modeling and Simulation
The simulation:
• The problem is to find how well the current arrangement
is working. To estimate the system measures of
performance, a simulation of 1 hour of operation is
made. A longer simulation would yield more reliable
results, but for purposes of illustration a l-hour period
has been selected.
85. 85
©Ahmed Hagag
Sim. a Multi-Server Queue (6/18)
Modeling and Simulation
Simulation Table:
Customer
=
Call
86. 86
©Ahmed Hagag
Sim. a Multi-Server Queue (6/18)
Modeling and Simulation
Simulation Table:
Interarrival
Time
87. 87
©Ahmed Hagag
Sim. a Multi-Server Queue (6/18)
Modeling and Simulation
Simulation Table:
Time of
Arrival
113. 113
©Ahmed Hagag
Sim. a Multi-Server Queue (13/18)
Modeling and Simulation
Simulation Table (62 minutes)
Able
Baker
115. 115
©Ahmed Hagag
Sim. a Multi-Server Queue (14/18)
Modeling and Simulation
The analysis of simulation table results in the following:
1. Over the 62-minute period Able was busy 90% of the
time.
2. Baker was busy only 69% of the time. The seniority
rule keeps Baker less busy (and gives Able more tips).
3. Nine of the 26 arrivals (about 35%) had to wait. The
average waiting time for all customers was only about
0.42 minute (25 seconds), which is very small.
116. 116
©Ahmed Hagag
Sim. a Multi-Server Queue (15/18)
Modeling and Simulation
The analysis of simulation table results in the following:
4. Those nine who did have to wait only waited an
average of 1.22 minutes, which is quite low.
5. In summary, this system seems well balanced. One
server cannot handle all the calls, and three servers
would probably be too many. Adding an additional
server would surely reduce the waiting time to nearly
zero. However, the cost of waiting would have to be
quite high to justify an additional server.
117. 117
©Ahmed Hagag
Sim. a Multi-Server Queue (16/18)
Modeling and Simulation
Simulation run for 100 calls:
• 62% of callers had no delay.
• 12% of callers had a delay
up to 2 minutes.
118. 118
©Ahmed Hagag
Sim. a Multi-Server Queue (16/18)
Modeling and Simulation
Simulation run for 100 calls:
• 62% of callers had no delay.
• 12% of callers had a delay
up to 2 minutes.
For example:
Average waiting time
= 2 minutes
119. 119
©Ahmed Hagag
Sim. a Multi-Server Queue (17/18)
Modeling and Simulation
400 simulation trials of 100 caller:
1. Average waiting time = 2 minutes.
2. Average waiting time = 1.7 minutes.
3. Average waiting time = 1.9 minutes.
4. Average waiting time = 2.5 minutes.
5. Average waiting time = 1.5 minutes.
…
400. Average waiting time = 2.7 minutes.
120. 120
©Ahmed Hagag
Sim. a Multi-Server Queue (18/18)
Modeling and Simulation
400 simulation trials of 100 caller:
• 80.5% of callers had delay up to 1 minute.
• 19.5% of callers had delay more than 1 minute.
121. 121
©Ahmed Hagag
Quiz 1 Question
Modeling and Simulation
Try other rules:
Baker gets call if both are idle
Due date:
