1. Al-Imam Mohammad Ibn Saud University
CS433 Modeling and Simulation Lecture 12 Queueing Theory
Dr. Anis Koubâa
03 May 2008

2. Goals for Today Understand the Queuing Model and its applications
Understand how to describe a Queue Model
Lean the most important queuing models (Part 02)
Single Queue
Multiple Queues
Multiple Servers

3. Course Outline The Queuing Model and Definitions
Application of Queuing Theory
Little’s Law
Queuing System Notation
Stationary Analysis of Elementary Queueing Systems
M/M/1
M/M/m
M/M/1/K …

4. The Queuing Model Queuing System Queue Server Use Queuing models to
Click for Queue Simulator
Queue
Server
Queuing System
Use Queuing models to
Describe the behavior of queuing systems
Evaluate system performance
A Queue System is characterized by
Queue (Buffer): with a finite or infinite size
The state of the system is described by the Queue Size
Server: with a given processing speed
Events: Arrival (birth) or Departure (death) with given rates

5. Queuing theory definitions
(Bose) “the basic phenomenon of queueing arises whenever a shared facility needs to be accessed for service by a large number of jobs or customers.”
(Wolff) “The primary tool for studying these problems [of congestions] is known as queueing theory.”
(Kleinrock) “We study the phenomena of standing, waiting, and serving, and we call this study Queueing Theory." "Any system in which arrivals place demands upon a finite capacity resource may be termed a queueing system.”
(Mathworld) “The study of the waiting times, lengths, and other properties of queues.”

6. Applications of Queuing Theory

7. Applications of Queuing Theory
Telecommunications
Computer Networks
Predicting computer performance
Health services (eg. control of hospital bed assignments)
Airport traffic, airline ticket sales
Layout of manufacturing systems.

8. Example application of queuing theory
In many stores and banks, we can find:
multiple line/multiple checkout system → a queuing system where customers wait for the next available cashier
We can prove using queuing theory that : throughput improves/increases when queues are used instead of separate lines

9. Example application of queuing theory

10. Queuing theory for studying networks
View network as collections of queues
FIFO data-structures
Queuing theory provides probabilistic analysis of these queues
Examples:
Average length
Average waiting time
Probability queue is at a certain length
Probability a packet will be lost

11. QNAP/Modline Example of a Queue Simulator

12. The Little’s Law
The long-term average number of customers in a stable system N, is equal to the long-term average arrival rate, λ, multiplied by the long- term average time a customer spends in the system, T.

13. The Queuing Times Queue Server Queuing System Queuing Time
Service Time
Response Time (or Delay)

14. Little’s Law Expected number of customers in the system
Expected time in the system
Arrival rate IN the system

15. Generality of Little’s Law
Mean number tasks in system = mean arrival rate x mean response time
Little’s Law is a pretty general result
It does not depend on the arrival process distribution
It does not depend on the service process distribution
It does not depend on the number of servers and buffers in the system.
Applies to any system in equilibrium, as long as nothing in black box is creating or destroying tasks
Queueing
Network λ
Aggregate Arrival rate

16. Specification of Queuing Systems

17. Characteristics of queuing systems
Arrival Process
The distribution that determines how the tasks arrives in the system.
Service Process
The distribution that determines the task processing time
Number of Servers
Total number of servers available to process the tasks

18. Specification of Queueing Systems
Arrival/Departure
Customer arrival and service stochastic models
Structural Parameters
Number of servers: What is the number of servers?
Storage capacity: are buffer finite or infinite?
Operating policies
Customer class differentiation
are all customers treated the same or do some have priority over others?
Scheduling/Queueing policies
which customer is served next
Admission policies
which/when customers are admitted

19. Kendall Notation A/B/m(/K/N/X)
To specify a queue, we use the Kendall Notation.
The First three parameters are typically used, unless specified
A: Arrival Distribution
B: Service Distribution
m: Number of servers
K: Storage Capacity (infinite if not specified)
N: Population Size (infinite)
X: Service Discipline (FCFS/FIFO)

20. Kendall Notation of Queueing System
Arrival Process
M: Markovian
D: Deterministic
Er: Erlang
G: General
Service Process
M: Markovian
D: Deterministic
Er: Erlang
G: General
A/B/m/K/N/X
Number of servers m=1,2,…
Service Discipline
FIFO, LIFO, Round Robin, …
Storage Capacity
K= 1,2,…
(if ∞ then it is omitted)
Number of customers
N= 1,2,…
(for closed networks, otherwise it is omitted)

21. Distributions
M: stands for "Markovian", implying exponential distribution for service times or inter-arrival times.
D: Deterministic (e.g. fixed constant)
Ek: Erlang with parameter k
Hk: Hyper-exponential with parameter k
G: General (anything)
CS352 Fall,2005

22. Kendall Notation Examples
M/M/1 Queue
Poisson arrivals (exponential inter-arrival), and exponential service, 1 server, infinite capacity and population, FCFS (FIFO)
the simplest ‘realistic’ queue
M/M/m Queue
Same, but m servers
M/D/1 Queue
Poisson arrivals and CONSTANT service times, 1 server, infinite capacity and population, FIFO.
G/G/3/20/1500/SPF
General arrival and service distributions, 3 servers, 17 queues (20-3), 1500 total jobs, Shortest Packet First

23. Performance Measures

24. Performance Measures of Interest
We are interested in steady state behavior
Even though it is possible to pursue transient results, it is a significantly more difficult task.
E[S]: average system (response) time (average time spent in the system)
E[W]: average waiting time (average time spent waiting in queue(s))
E[X]: average queue length
E[U]: average utilization (fraction of time that the resources are being used)
E[R]: average throughput (rate that customers leave the system)
E[L]: average customer loss (rate that customers are lost or probability that a customer is lost)

25. Recall the Birth-Death Chain Exampleλ0 1 μ1 λ1 2 μ2
λj-2
j-1
μj-1
λj-1 j μj μ3 λ2 λj
μj+1
At steady state, we obtain
In general
Making the sum equal to 1
Solution exists if

26. End of Part 01