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Operation Research

Queuing theory Prepared by :

Telecommunication students group # 2

Sana’a University

Faulty of Engineering

Department of Electrical

Engineering

Major of Communication andElectronics

Supervised by :

Dr . Ahmed Al- Arashi

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Queuing theory

Overview .Definition

The Basic Queuing Process.

Application of queuing theory to

telephony.

Queuing Models.Basic points

Example2 4/2/2012



generally considered a branch of operation research

because the results are often used when making business

decisions about the resources needed to provide service.

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British

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Definition

Queuing Theory is a collection of mathematical modelsof various queuing systems. It is used extensively to

analyze production and service processes exhibiting

random variability in market demand (arrival times) and

service times.

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The Basic Queuing Process

Input

Source

(Calling

Population) Queue

Service mechanism

Queueing system

Served Mechanism

Customer

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1. Input Source (Calling Population)An input source is characterized by

• Size of the calling population

• Pattern of arrivals at the system

• Behaviour of the arrivals

2. Queue

The queue is where customers wait before being served.

3. Service Mechanism (Service system)

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continue …

Service Mechanism (Service system) The Service system is provided by a service facility (or

facilities). There are two aspects of a service system:

a) Configuration of the service system

b) Speed of Service

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continue …

Service Mechanism (Service system) a) Configuration of the service system

Some of these Configurations : -

1. Single Server – Single Queue

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Yemen

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continue …


2. Several (Parallel) Servers – Single Queue

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continue …


3 -Several Servers – Several Queues

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Queue for voting

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continue …

Service Mechanism (Service system) b) Speed of Service

can be expressed in either of two ways :-

• The service rate describes the number of customers serviced

during a particular time period.• The service time indicates the amount of time needed to service

a customer.

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Application of queuing theory to telephony

The Public Switched Telephone Networks (PSTNs) aredesigned to accommodate the offered traffic intensity with

only a small loss.

The performance of loss systems is quantified by their

Grade of Service (GoS).The use of queuing in PSTNs allows the systems to queue

their customer's requests until free resources become

available.

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Application of queuing theory

continue….

A queuing discipline determines the manner in which theexchange handles calls from customers. Here are details of

three queuing disciplines :

First in First Out (FIFO) - customers are serviced

according to their order of arrival . Last in First Out (LIFO) - the last customer to arrive on

the queue is the one who is actually serviced first.

Processor Sharing (PS) - customers are serviced equally,

i.e. they experience the same amount of delay.

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Queuing Models

Queuing systems are usually described by three valuesseparated by slashes Arrival distribution / servicedistribution / # of servers where:

• M = Markovian or exponentially distributed

• D = Deterministic or constant.

• G = General or binomial distribution

Common Models

o The simplest queuing model is M/M/1 where both the

arrival time and service time are exponentially distributed.

o The M/D/1 model has exponentially distributed arrivaltimes but fixed service time.

o The M/M/n model has multiple servers .

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Basic points

Customer: (Arrival) The arrival unit that requires someservices to performed.

Queue: The number of Customer waiting to be served.

Arrival Rate (λ): The rate which customer arrive to theservice station.

Service rate (µ) : The rate at which the service unit canprovide services to the customer

If Utilization Ratio Or Traffic intensity:

λ / µ > 1 Queue is growing without end. λ / µ < 1 Length of Queue is go on diminishing.

λ /µ = 1 Queue length remain constant.

λ< µ (system work)

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Formulas

1. Traffic Intensity (P)= λ /µ

2. Probability Of System Is Ideal (P0) =1-P , P0 = 1- (λ /µ)

3. Expected Waiting Time In The System (Ws) = 1/ (µ- λ)

4. Expected Waiting Time In Queue (Wq) = λ / µ(µ- λ)

5. Expected Number Of Customer In The System (Ls)= λ / (µ-λ)Ls=Length Of System

6. Expected Number Of Customers In The Queue (Lq)= λ 2 / µ(µ- λ)

7. Expected Length Of Non-Empty Queue (Lneq)= µ/ (µ- λ)

8. What Is The Probability Track That K Or More Than K CustomersIn The System. P >=K (P Is Greater Than Equal To K) = (λ /µ)K

9. What Is The Probability That More Than K Customers Are In TheSystem

( P>K)= (λ /µ)K+1

10. What Is The Probability That At least One Customer Is Standing InQueue. P=K=(λ /µ)2

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Example

People arrive at a cinema ticket booth in a Poissondistributed arrival rate of 25per hour. Service rate is

exponentially distributed with an average time of 2 per

min.

Calculate the mean number in the waiting line, the meanwaiting time , the mean number in the system , the mean

time in the system and the utilization factor?

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Example continue…………

Solution:

Arrival rate λ=25/hr , Service rate µ = 2/min=30/hr

Length of Queue (Lq) = λ2 / µ(µ- λ ) = 252/(30(30-25))

= 4.17 person

Expected Waiting Time In Queue (Wq) = λ / µ(µ- λ)

=25/(30(30-25)) =1/6 hr= 10 min

Expected Waiting Time In The System (Ws) = 1/ (µ- λ)

=1/(30-25)

=1/5hr= 12 min

Utilization Ratio = λ /µ =25/30 = 0.8334 =

83.34%

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inputs Arrival Rate (λ): 50 per/hr

Service rate (µ) 60 per/hr

outputs

Traffic Intensity (P)= λ /µ 83%

Probability Of System Is Ideal zero customer (P0)=1-P 17%

Expected Waiting Time In The System (Ws) 6 minExpected Waiting Time In Queue Wq (Wq) = λ / µ(µ- λ) 5 min

Expected Number Of Customer In The System (Ls)= λ / (µ-λ) 5 customer

Expected Number Of Customers In The Queue (Lq)= λ 2̂/ µ(µ- λ) 4.166666667 customer

Expected Length Of Non-Empty Queue (Lneq)= µ/ (µ- λ) 6

What Is The Probability That At least One Customer Is Standing In

Queue. P=K=(λ /µ)2

69%



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queuing theory last one

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