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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
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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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The Basic Queuing Process
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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The Basic Queuing Process
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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The Basic Queuing Process
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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The Basic Queuing Process
continue …
Service Mechanism (Service system) a) Configuration of the service system
2. Several (Parallel) Servers – Single Queue
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The Basic Queuing Process
continue …
Service Mechanism (Service system) a) Configuration of the service system
3 -Several Servers – Several Queues
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Queue for voting
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The Basic Queuing Process
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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4/2/201222
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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