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BU_FCAI_SCC430_Modeling&Simulation_Ch03.pdf

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MaiGaafar

This document describes a queueing simulation example of a single-server grocery checkout counter. It includes details of the simulation such as the discrete uniform distribution of interarrival times between 1-8 minutes and service times between 1-6 minutes. The document also includes a simulation table tracking the queue over time as 20 customers arrive and are served at the single checkout counter.

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SCC430 Modeling and Simulation Chapter 03 Queueing Simulation Dr. Ahmed Hagag Faculty of Computers and Artificial Intelligence Benha University Fall 2020
‱ 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 ©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 ©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 ©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 ©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 ©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 ©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 ©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 ©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 ©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 ©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 ©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 ©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 ©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 ©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
17 ©Ahmed Hagag Description of Queuing Sys (7/8) Modeling and Simulation 5. Service Times and the Service Mechanism (3/3):
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 ©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.
20 ©Ahmed Hagag Description of Queuing Sys (8/8) Modeling and Simulation Queuing System State (2/3):
21 ©Ahmed Hagag Description of Queuing Sys (8/8) Modeling and Simulation Queuing System State (3/3):
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.
23 ©Ahmed Hagag Sim. a Single-Server Queue (1/14) Modeling and Simulation Simulation Table (2/3):
24 ©Ahmed Hagag Sim. a Single-Server Queue (1/14) Modeling and Simulation Simulation Table (3/3):
25 ©Ahmed Hagag Sim. a Single-Server Queue (2/14) Modeling and Simulation Example1 The Grocery Checkout, a Single-Server Queue:
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 ©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 ©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 ©Ahmed Hagag Sim. a Single-Server Queue (4/14) Modeling and Simulation Distribution of Time Between Arrivals
30 ©Ahmed Hagag Sim. a Single-Server Queue (5/14) Modeling and Simulation Time-Between-Arrivals Determination
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.
32 ©Ahmed Hagag Sim. a Single-Server Queue (7/14) Modeling and Simulation Service Time Distribution
33 ©Ahmed Hagag Sim. a Single-Server Queue (8/14) Modeling and Simulation Service Times Generated
34 ©Ahmed Hagag Sim. a Single-Server Queue (9/14) Modeling and Simulation Simulation Table for Queueing Problem (20 customers)
35 ©Ahmed Hagag Sim. a Single-Server Queue (9/14) Modeling and Simulation Simulation Table for Queueing Problem (20 customers)
36 ©Ahmed Hagag Sim. a Single-Server Queue (9/14) Modeling and Simulation Simulation Table for Queueing Problem (20 customers)
37 ©Ahmed Hagag Sim. a Single-Server Queue (9/14) Modeling and Simulation Simulation Table for Queueing Problem (20 customers)
38 ©Ahmed Hagag Sim. a Single-Server Queue (9/14) Modeling and Simulation Simulation Table for Queueing Problem (20 customers)
39 ©Ahmed Hagag Sim. a Single-Server Queue (9/14) Modeling and Simulation Simulation Table for Queueing Problem (20 customers)
40 ©Ahmed Hagag Sim. a Single-Server Queue (9/14) Modeling and Simulation Simulation Table for Queueing Problem (20 customers)
41 ©Ahmed Hagag Sim. a Single-Server Queue (9/14) Modeling and Simulation Simulation Table for Queueing Problem (20 customers)
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 ©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 ©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 ©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 ©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 ©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 ©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 ©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 ©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 ©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 ©Ahmed Hagag Sim. a Single-Server Queue (13/14) Modeling and Simulation Simulation Table for Queueing Problem (100 customers)
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 ©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 ©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 ©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 ©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 ©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 ©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 ©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 ©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 ©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 ©Ahmed Hagag Sim. a Single-Server Queue (14/14) Modeling and Simulation Some of the findings from the simulation are as follows (4/4):
64 ©Ahmed Hagag Modeling and Simulation LAB What have you learned in the LAB yet?
65 ©Ahmed Hagag Review Example (1/11) Modeling and Simulation Distribution of Time Between Arrivals
66 ©Ahmed Hagag Review Example (2/11) Modeling and Simulation Distribution of Time Between Arrivals (Answer)
67 ©Ahmed Hagag Review Example (3/11) Modeling and Simulation Service-Time Distribution
68 ©Ahmed Hagag Review Example (4/11) Modeling and Simulation Service-Time Distribution (Answer)
69 ©Ahmed Hagag Review Example (5/11) Modeling and Simulation Simulation Table
70 ©Ahmed Hagag Review Example (6/11) Modeling and Simulation Simulation Table (Answer)
71 ©Ahmed Hagag Review Example (7/11) Modeling and Simulation Simulation Table (Answer)
72 ©Ahmed Hagag Review Example (8/11) Modeling and Simulation Simulation Table
73 ©Ahmed Hagag Review Example (9/11) Modeling and Simulation Simulation Table (Answer)
74 ©Ahmed Hagag Review Example (10/11) Modeling and Simulation Simulation Table (Answer)
75 ©Ahmed Hagag Review Example (11/11) Modeling and Simulation Results (1/2):
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 ©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.
78 ©Ahmed Hagag Sim. a Multi-Server Queue (2/18) Modeling and Simulation Example: Call Center:
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 ©Ahmed Hagag Sim. a Multi-Server Queue (3/18) Modeling and Simulation Interarrival distribution of calls for technical support
81 ©Ahmed Hagag Sim. a Multi-Server Queue (3/18) Modeling and Simulation Interarrival distribution of calls for technical support
82 ©Ahmed Hagag Sim. a Multi-Server Queue (3/18) Modeling and Simulation Interarrival distribution of calls for technical support
83 ©Ahmed Hagag Sim. a Multi-Server Queue (4/18) Modeling and Simulation Service Distribution of Able Service Distribution of Baker
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 ©Ahmed Hagag Sim. a Multi-Server Queue (6/18) Modeling and Simulation Simulation Table: Customer = Call
86 ©Ahmed Hagag Sim. a Multi-Server Queue (6/18) Modeling and Simulation Simulation Table: Interarrival Time
87 ©Ahmed Hagag Sim. a Multi-Server Queue (6/18) Modeling and Simulation Simulation Table: Time of Arrival
88 ©Ahmed Hagag Sim. a Multi-Server Queue (6/18) Modeling and Simulation Simulation Table: Service Time
89 ©Ahmed Hagag Sim. a Multi-Server Queue (6/18) Modeling and Simulation Simulation Table: Waiting Time
90 ©Ahmed Hagag Sim. a Multi-Server Queue (7/18) Modeling and Simulation Simulation Table:
91 ©Ahmed Hagag Sim. a Multi-Server Queue (7/18) Modeling and Simulation Simulation Table: Able Baker
92 ©Ahmed Hagag Sim. a Multi-Server Queue (7/18) Modeling and Simulation Simulation Table: Able Baker
93 ©Ahmed Hagag Sim. a Multi-Server Queue (7/18) Modeling and Simulation Simulation Table: Able Baker
94 ©Ahmed Hagag Sim. a Multi-Server Queue (7/18) Modeling and Simulation Simulation Table: Able Baker
95 ©Ahmed Hagag Sim. a Multi-Server Queue (7/18) Modeling and Simulation Simulation Table:
96 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table:
97 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table:
98 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table:
99 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table:
100 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table:
101 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table:
102 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table: Able Baker
103 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table: Able Baker
104 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table: Able Baker
105 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table:
106 ©Ahmed Hagag Sim. a Multi-Server Queue (9/18) Modeling and Simulation Simulation Table:
107 ©Ahmed Hagag Sim. a Multi-Server Queue (10/18) Modeling and Simulation Simulation Table:
108 ©Ahmed Hagag Sim. a Multi-Server Queue (11/18) Modeling and Simulation Simulation Table:
109 ©Ahmed Hagag Sim. a Multi-Server Queue (12/18) Modeling and Simulation Simulation Table:
110 ©Ahmed Hagag Sim. a Multi-Server Queue (13/18) Modeling and Simulation Simulation Table (62 minutes)
111 ©Ahmed Hagag Sim. a Multi-Server Queue (13/18) Modeling and Simulation Simulation Table (62 minutes)
112 ©Ahmed Hagag Sim. a Multi-Server Queue (13/18) Modeling and Simulation Simulation Table (62 minutes)
113 ©Ahmed Hagag Sim. a Multi-Server Queue (13/18) Modeling and Simulation Simulation Table (62 minutes) Able Baker
114 ©Ahmed Hagag Sim. a Multi-Server Queue (13/18) Modeling and Simulation Simulation Table (62 minutes)
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 ©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 ©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 ©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 ©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 ©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 ©Ahmed Hagag Quiz 1 Question Modeling and Simulation Try other rules: Baker gets call if both are idle Due date:
Dr. Ahmed Hagag ahagag@fci.bu.edu.eg

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BU_FCAI_SCC430_Modeling&Simulation_Ch03.pdf

  • 1. SCC430 Modeling and Simulation Chapter 03 Queueing Simulation Dr. Ahmed Hagag Faculty of Computers and Artificial Intelligence Benha University Fall 2020
  • 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
  • 17. 17 ©Ahmed Hagag Description of Queuing Sys (7/8) Modeling and Simulation 5. Service Times and the Service Mechanism (3/3):
  • 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.
  • 20. 20 ©Ahmed Hagag Description of Queuing Sys (8/8) Modeling and Simulation Queuing System State (2/3):
  • 21. 21 ©Ahmed Hagag Description of Queuing Sys (8/8) Modeling and Simulation Queuing System State (3/3):
  • 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.
  • 23. 23 ©Ahmed Hagag Sim. a Single-Server Queue (1/14) Modeling and Simulation Simulation Table (2/3):
  • 24. 24 ©Ahmed Hagag Sim. a Single-Server Queue (1/14) Modeling and Simulation Simulation Table (3/3):
  • 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.
  • 32. 32 ©Ahmed Hagag Sim. a Single-Server Queue (7/14) Modeling and Simulation Service Time Distribution
  • 33. 33 ©Ahmed Hagag Sim. a Single-Server Queue (8/14) Modeling and Simulation Service Times Generated
  • 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):
  • 64. 64 ©Ahmed Hagag Modeling and Simulation LAB What have you learned in the LAB yet?
  • 65. 65 ©Ahmed Hagag Review Example (1/11) Modeling and Simulation Distribution of Time Between Arrivals
  • 66. 66 ©Ahmed Hagag Review Example (2/11) Modeling and Simulation Distribution of Time Between Arrivals (Answer)
  • 67. 67 ©Ahmed Hagag Review Example (3/11) Modeling and Simulation Service-Time Distribution
  • 68. 68 ©Ahmed Hagag Review Example (4/11) Modeling and Simulation Service-Time Distribution (Answer)
  • 69. 69 ©Ahmed Hagag Review Example (5/11) Modeling and Simulation Simulation Table
  • 70. 70 ©Ahmed Hagag Review Example (6/11) Modeling and Simulation Simulation Table (Answer)
  • 71. 71 ©Ahmed Hagag Review Example (7/11) Modeling and Simulation Simulation Table (Answer)
  • 72. 72 ©Ahmed Hagag Review Example (8/11) Modeling and Simulation Simulation Table
  • 73. 73 ©Ahmed Hagag Review Example (9/11) Modeling and Simulation Simulation Table (Answer)
  • 74. 74 ©Ahmed Hagag Review Example (10/11) Modeling and Simulation Simulation Table (Answer)
  • 75. 75 ©Ahmed Hagag Review Example (11/11) Modeling and Simulation Results (1/2):
  • 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.
  • 78. 78 ©Ahmed Hagag Sim. a Multi-Server Queue (2/18) Modeling and Simulation Example: Call Center:
  • 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
  • 88. 88 ©Ahmed Hagag Sim. a Multi-Server Queue (6/18) Modeling and Simulation Simulation Table: Service Time
  • 89. 89 ©Ahmed Hagag Sim. a Multi-Server Queue (6/18) Modeling and Simulation Simulation Table: Waiting Time
  • 90. 90 ©Ahmed Hagag Sim. a Multi-Server Queue (7/18) Modeling and Simulation Simulation Table:
  • 91. 91 ©Ahmed Hagag Sim. a Multi-Server Queue (7/18) Modeling and Simulation Simulation Table: Able Baker
  • 92. 92 ©Ahmed Hagag Sim. a Multi-Server Queue (7/18) Modeling and Simulation Simulation Table: Able Baker
  • 93. 93 ©Ahmed Hagag Sim. a Multi-Server Queue (7/18) Modeling and Simulation Simulation Table: Able Baker
  • 94. 94 ©Ahmed Hagag Sim. a Multi-Server Queue (7/18) Modeling and Simulation Simulation Table: Able Baker
  • 95. 95 ©Ahmed Hagag Sim. a Multi-Server Queue (7/18) Modeling and Simulation Simulation Table:
  • 96. 96 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table:
  • 97. 97 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table:
  • 98. 98 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table:
  • 99. 99 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table:
  • 100. 100 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table:
  • 101. 101 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table:
  • 102. 102 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table: Able Baker
  • 103. 103 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table: Able Baker
  • 104. 104 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table: Able Baker
  • 105. 105 ©Ahmed Hagag Sim. a Multi-Server Queue (8/18) Modeling and Simulation Simulation Table:
  • 106. 106 ©Ahmed Hagag Sim. a Multi-Server Queue (9/18) Modeling and Simulation Simulation Table:
  • 107. 107 ©Ahmed Hagag Sim. a Multi-Server Queue (10/18) Modeling and Simulation Simulation Table:
  • 108. 108 ©Ahmed Hagag Sim. a Multi-Server Queue (11/18) Modeling and Simulation Simulation Table:
  • 109. 109 ©Ahmed Hagag Sim. a Multi-Server Queue (12/18) Modeling and Simulation Simulation Table:
  • 110. 110 ©Ahmed Hagag Sim. a Multi-Server Queue (13/18) Modeling and Simulation Simulation Table (62 minutes)
  • 111. 111 ©Ahmed Hagag Sim. a Multi-Server Queue (13/18) Modeling and Simulation Simulation Table (62 minutes)
  • 112. 112 ©Ahmed Hagag Sim. a Multi-Server Queue (13/18) Modeling and Simulation Simulation Table (62 minutes)
  • 113. 113 ©Ahmed Hagag Sim. a Multi-Server Queue (13/18) Modeling and Simulation Simulation Table (62 minutes) Able Baker
  • 114. 114 ©Ahmed Hagag Sim. a Multi-Server Queue (13/18) Modeling and Simulation Simulation Table (62 minutes)
  • 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: