Queuing Theory: Analysis and Simulation of Waiting Lines and Systems, Lecture notes of Mathematical Modeling and Simulation

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Queuing SystemQueuing

System

By

Prof. S. Shakya

y

Simulation of Queuing Systems Introduction 

Waiting line queues are one of the most important areas, wherethe technique of simulation has been extensively employed.



The waiting lines or queues are a common site in real life



The waiting lines or queues are a common site in real life.



People at railway ticket window, vehicles at a petrol pump or at atraffic signal, workers at a tool crib, products at a machiningcenter television sets at a repair shop are a few examples ofcenter, television sets at a repair shop are a few examples ofwaiting lines.

Simulation of Queuing Systems 

The queuing theory its development to anThe

queuing theory its development to an

engineer A.K.Earlang, who in 1920, studiedwaiting line queues of telephone calls inC

h

D

k

Copenhagen, Denmark.



The problem was that during the busy period,t l

h

t

bl

t

h

dl

telephone operators were unable to handlethe calls, there was too much waiting time,which resulted in customer dissatisfactionwhich resulted in customer dissatisfaction.

State Variables

customer

queue

server

State: 

InTheAir

: number of aircraft either landing or

waiting to land



OnTheGround

: number of landed aircraft



RunwayFree

: Boolean, true if runway available

Queuing System 

Elements of Queuing Systems



Elements

of Queuing Systems

Queuing System 

Population of Customers or calling source

can be

considered either limited (closed systems) or unlimitedconsidered either limited (closed systems) or unlimited(open systems).



Unlimited population represents a theoretical model of systems with a large number of possible customers (asystems with a large number of possible customers (abank on a busy street, a motorway petrol station).



Example of a limited population may be a number ofprocesses to be run (served) by a computer or a certainprocesses to be run (served) by a computer or a certainnumber of machines to be repaired by a service man.



It is necessary to take the term "customer" very generally. Customers may be people machines of various nature



Customers

may be people, machines of various nature,

computer processes, telephone calls, etc.

Queuing System 

Queue or waiting line

represents a certain number

of customers waiting for service (of course thequeue may be empty).



Typically the customer being served is considered

yp

y

g

not to be in the queue. Sometimes the customersform a queue literally (people waiting in a line for abank teller).



Sometimes the queue is an abstraction (planeswaiting for a runway to land).



There are two important properties of a queue:



There are two important properties of a queue: Maximum Size

and

Queuing Discipline

Queuing System 

Maximum Queue Size

(also called

System

Maximum

Queue Size

(also

called

System

capacity

) is the maximum number of customers that

may wait in the queue (plus the one(s) being

d)

served).



Queue is always limited, but some theoreticalmodels assume an unlimited queue lengthmodels assume an unlimited queue length.



If the queue length is limited, some customers are forced to renounce without being servedforced to renounce without being served

Example application of queuing theory 

In many retail stores and banks



In

many retail stores and banks



multiple line/multiple checkout system

a

queuing system where customers wait for the nextq

g

y

available cashier



We can prove using queuing theory that :throughput improves increases when queues areused instead of separate lines

Example application of queuing theory

Model Queuing System

Queuing System

Queue

Server

Server System

Queuing System



Use Queuing models to 

Describe the behavior of queuing systems



Evaluate system performance

System Configuration

Servers

CCustomers

Single Queue Configuration

Queuing System 

Queuing Discipline

represents the way the queue is organized

(r les of inserting and remo ing c stomers to/from the q e e)(rules of inserting and removing customers to/from the queue).There are these ways:

1) FIFO (First In First Out) also called FCFS (First Come First

Serve) - orderly queue.

)

y q

2) LIFO (Last In First Out) also called LCFS (Last Come First Serve)

- stack.

3) SIRO (Serve In Random Order).4) Priority Queue, that may be viewed as a number of queues for

various priorities.

5) Many other more complex queuing methods that typically change

the customer’s position in the queue according to the time spentthe customer s position in the queue according to the time spentalready in the queue, expected service duration, and/or priority.These methods are typical for computer multi-access systems

Queuing System 

Most quantitative parameters (like average queue

q

p

g

q

length, average time spent in the system) do notdepend on the queuing discipline.



That’s why most models either do not take the



That s why most models either do not take thequeuing discipline into account at all or assume thenormal FIFO queue.



In fact the only parameter that depends on thequeuing discipline is the variance (or standarddeviation) of the waiting time There is this importantdeviation) of the waiting time. There is this importantrule (that may be used for example to verify resultsof a simulation experiment):