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CS433 Modeling and Simulation Lecture 12 Queueing Theory

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Al-Imam Mohammad Ibn Saud University

CS433Modeling and SimulationLecture 12Queueing Theory

Dr. Anis Koubâa

03 May 2008

- 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

- 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
- …

Queuing System

Queue

Server

Click for Queue Simulator

- 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

- Queue (Buffer): with a finite or infinite size

- (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.”

http://www2.uwindsor.ca/~hlynka/queue.html

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

- 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

http://www.andrews.edu/~calkins/math/webtexts/prod10.htm#PD

http://www.bsbpa.umkc.edu/classes/ashley/Chaptr14/sld006.htm

- 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

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.

Queuing System

Queue

Server

Queuing Time

Service Time

Response Time (or Delay)

Expected number of customers in the system

Expected time in the system

Arrival rate IN the system

λ

Aggregate Arrival rate

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

- 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

- Arrival/Departure
- Customer arrivaland 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

- Customer class differentiation

- 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)

http://en.wikipedia.org/wiki/Kendall's_notation

- 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)

- 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 http://en.wikipedia.org/wiki/Erlang_distribution
- Hk: Hyper-exponential with parameter k
- G: General (anything)

- 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

- 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)

λj-1

λ0

λ1

λj

λ2

λj-2

j-1

j

0

1

2

μj+1

μ3

μj-1

μj

μ1

μ2

Solution exists if

- At steady state, we obtain

- In general

- Making the sum equal to 1