Queuing TheoryQueuing Theory

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Introduction Introduction The study of waiting lines is called queuing theory.The study of waiting lines is called queuing theory.

In general a queue is formed when either units requiring In general a queue is formed when either units requiring service commonly referred to as customers , wait for the service commonly referred to as customers , wait for the service facility, stand idle & wait for the customer.service facility, stand idle & wait for the customer.

Some customers wait when the total number of customers Some customers wait when the total number of customers requiring service exceeds the number of service facilities.requiring service exceeds the number of service facilities.

Some service facilities stand idle when the total number of Some service facilities stand idle when the total number of service facilities exceeds the number of customers service facilities exceeds the number of customers requiring service.requiring service.

Input source Input source The input source is the source from which customer comes The input source is the source from which customer comes

& joins the queue.& joins the queue.

The input source of the customer to the queuing system The input source of the customer to the queuing system can be finite or infinite.can be finite or infinite.

sourceQueue

Server 1Server 2Server 3

.

.

.Sever m Departure

from the system

Queuing Model

Point of arrival in the system

Arrival characteristics Arrival characteristics Size of the arrival population Size of the arrival population

Pattern of arrival at the queuing systemPattern of arrival at the queuing system

Behavior of arrivals.Behavior of arrivals.

Size of the arrival populationSize of the arrival population Population sizes are considered to be either unlimited i.e Population sizes are considered to be either unlimited i.e

infinite or limited i.e finite.infinite or limited i.e finite.

It is considered to be infinite if the rate at which source It is considered to be infinite if the rate at which source generates the customers is not affected by the number of generates the customers is not affected by the number of customers in the service system.customers in the service system.

Alternatively source is considered finite if the rate is Alternatively source is considered finite if the rate is affected by the number of customers in the system.affected by the number of customers in the system.

Pattern Arrival At the systemPattern Arrival At the system Customers arrive at service facility according to some Customers arrive at service facility according to some

known schedule or else they arrive randomly.known schedule or else they arrive randomly.

In queuing theory the number of arrivals per unit of time In queuing theory the number of arrivals per unit of time can be estimated by probability distribution known as can be estimated by probability distribution known as Poisson distribution.Poisson distribution.

The major feature of this distribution is that number of The major feature of this distribution is that number of people arriving at a given time is not at all influenced by people arriving at a given time is not at all influenced by the number of previous arrivals.the number of previous arrivals.

Arrival rate is given by symbol lamdaArrival rate is given by symbol lamda

Behavior of the arrivalBehavior of the arrival

A customer joining the service , does not get the immediate A customer joining the service , does not get the immediate service , the customer may :service , the customer may :

1.1. stay in the system until servedstay in the system until served

2.2. move from one waiting line to anothermove from one waiting line to another

3.3. estimate the waiting time & decide whenever to leave.estimate the waiting time & decide whenever to leave.

The first type of customer who stay in the system are The first type of customer who stay in the system are called as patient customers.called as patient customers.

The other two are classified as impatient or reneging The other two are classified as impatient or reneging customer.customer.

Queuing LengthQueuing Length The length of a line can be either limited or unlimited.The length of a line can be either limited or unlimited.

In a sales department where the customers orders are In a sales department where the customers orders are received , there is no restriction on the number of orders received , there is no restriction on the number of orders that can come in so that a queue of any size can form. that can come in so that a queue of any size can form. When there is o limit on its size the permissible queue is When there is o limit on its size the permissible queue is said to be infinite.said to be infinite.

When there is a limit on size of queue it is said to be finite.When there is a limit on size of queue it is said to be finite.

Service Mechanism Service Mechanism Once the customer has entered the system for service , the Once the customer has entered the system for service , the

service system has following characteristic:service system has following characteristic:

time taken for servicetime taken for service

service facilityservice facility

service discipline of the systemservice discipline of the system

Service time Service time Service time may be either constant or random.Service time may be either constant or random.

If service time is constant , it takes the same amount of If service time is constant , it takes the same amount of time to service each customer.time to service each customer.

The symbol describing the service rate is µ and the units The symbol describing the service rate is µ and the units are in service per unit time.are in service per unit time.

Service facilitiesService facilities Number of channels : A single channel phase case is where Number of channels : A single channel phase case is where

the arrivals pass through only one station in order to the arrivals pass through only one station in order to complete the required service.complete the required service.

E.g. toll booth , theatre ticket windowE.g. toll booth , theatre ticket window

A multiphase channel involves the situations where the A multiphase channel involves the situations where the units to be served pass through a series of station in a units to be served pass through a series of station in a sequence.sequence.

Queue Service facility

Queue Servicefacility

Servicefacility

Single channel

Single phase

Single channel

Multiple phase

Queue

Service facility

Service facility

Queue

Servicefacility

Service facility

Service facility

Servicefacility

Multiple channel Single phase

Multiple channel multiple phase

Service disciplineService discipline Queue discipline refers to the rule by which customers in Queue discipline refers to the rule by which customers in

the line are to receive service.the line are to receive service.

Most systems use queue discipline known as the first in , Most systems use queue discipline known as the first in , first out rule (FIFO).first out rule (FIFO).

Most queues operate on the principle of first –come – first Most queues operate on the principle of first –come – first served .served .

Other ways of dealing with view can be Other ways of dealing with view can be

last in , first out ( LIFO)last in , first out ( LIFO)

Random serviceRandom service

Alphabetically organized. Alphabetically organized.

Application of Queuing systemApplication of Queuing system Commercial service systemCommercial service system Business industrial systemBusiness industrial system Social service systemSocial service system Transportation service systemTransportation service system

Operating characteristics of Operating characteristics of queuing systemqueuing system

Utilization factor for the systemUtilization factor for the system

Average number of customer in the systemAverage number of customer in the system

Average number of customer in queueAverage number of customer in queue

Average time each customer spends in the queueAverage time each customer spends in the queue

Probability that service will be idle.Probability that service will be idle.

Probability of specific number of customers in the systemProbability of specific number of customers in the system

Notations & TerminologiesNotations & TerminologiesNotation that has been used for summarizing the main characteristics of parallel queue has universal standardized format:

(a/b/c ) : (d/e/f)

A : arrival distribution

B : service time distribution

C = number of parallel servers

D = service discipline ( FCFS , LIFO , SIRO)

E = maximum number of customers allowed in system

F = size of calling source

Basic NotationsBasic Notations Arrival rate(Arrival rate(λλ ) ) : The average rate at which customers arrive & : The average rate at which customers arrive &

join the queue is the arrival rate.join the queue is the arrival rate.

Service time (Service time (μμ ): ): The average time required to service one The average time required to service one customer is service time. The units for the service time should be customer is service time. The units for the service time should be the inverse of the units for arrival rate.the inverse of the units for arrival rate.

Number of customer in the system (LNumber of customer in the system (LSS):): these are the these are the average number of customer in the system both waiting & being average number of customer in the system both waiting & being serviced.serviced.

Average waiting time in the system(WAverage waiting time in the system(WQQ): It is the average ): It is the average time each customer in the system both waiting & being serviced.time each customer in the system both waiting & being serviced.

Average waiting time in the queue (WAverage waiting time in the queue (WSS):): It is the average It is the average time each customer waits in the queue.time each customer waits in the queue.

Model : I (M/M/1): (FIFO/Model : I (M/M/1): (FIFO//∞/∞)/∞/∞)

This is a single channel queuing model.This is a single channel queuing model.

The arrival time has a poison distribution & arrival rate is denoted The arrival time has a poison distribution & arrival rate is denoted byby

Arrivals are from infinite populationArrivals are from infinite population

The queue discipline is FCFS ( i.e FIFO) the customers are served The queue discipline is FCFS ( i.e FIFO) the customers are served on first come first served basis.on first come first served basis.

There is only single service stationThere is only single service station

The mean arrival rate is less than the mean service rateThe mean arrival rate is less than the mean service rate The waiting space available for customers in the queue is infinite.The waiting space available for customers in the queue is infinite.

Model II: (M/M/1): (SIRO/Model II: (M/M/1): (SIRO/∞/∞)∞/∞)

This model is essentially same as model I except that This model is essentially same as model I except that service discipline follows SIRO rule ( service in random service discipline follows SIRO rule ( service in random order)order)

As the probability of n customer in the system does not As the probability of n customer in the system does not depend upon any queue discipline it may be concluded that depend upon any queue discipline it may be concluded that SIRO rule case is same as MODEL ISIRO rule case is same as MODEL I

MODEL III : (M/M/1): (FCFS/N/MODEL III : (M/M/1): (FCFS/N/∞)∞)

This model is different from model I in respect to the This model is different from model I in respect to the capacity of the system.capacity of the system.

Here the capacity of the system is limited to N customers Here the capacity of the system is limited to N customers only.only.

Therefore difference equations derived in model as long as Therefore difference equations derived in model as long as n < Nn < N

Model IV : (M /EModel IV : (M /EKK /1) : /1) :

(FCFS / (FCFS /∞/∞)∞/∞) The queues in which arrivals & or departures may not The queues in which arrivals & or departures may not

follow poison distribution are called as non poison follow poison distribution are called as non poison distribution.distribution.

This model consists of k identical exponential service This model consists of k identical exponential service phases in series each with mean service time of 1/kμ .phases in series each with mean service time of 1/kμ .

A new service does not start until all k phases have been A new service does not start until all k phases have been completed.completed.

A service station in which customer passes through k A service station in which customer passes through k phases shall have the service time following Erlang phases shall have the service time following Erlang distribution.distribution.

Characteristics of Erland Characteristics of Erland DistributionDistribution

The Erlang distribution is useful for defining the service The Erlang distribution is useful for defining the service time of a customer who receives k identical repetitions of a time of a customer who receives k identical repetitions of a particular service.particular service.

The probability function of Erlang distribution is given byThe probability function of Erlang distribution is given by (kμ )(kμ )mm

(k- 1)!(k- 1)!

Where m is the expected number of customers completing Where m is the expected number of customers completing service per unit of timeservice per unit of time

In case of constant distribution of customer interaarival In case of constant distribution of customer interaarival

times k tend to times k tend to ∞∞

Limitation of Single channel Limitation of Single channel Queuing ModelQueuing Model

The waiting space may be limited .The waiting space may be limited .

The arrival rate is dependent . Potential customers are The arrival rate is dependent . Potential customers are discouraged from being entering the queue if they observe discouraged from being entering the queue if they observe along line at the time they arrive.along line at the time they arrive.

Another limitation that arrival process is not stationaryAnother limitation that arrival process is not stationary

The population of customers may be finite .The population of customers may be finite .

Multi Channel Queuing ModelsMulti Channel Queuing Models In a multiple queuing system two or more server are In a multiple queuing system two or more server are

available to handle the customer who arrive for service.available to handle the customer who arrive for service.

By introducing number of servicing units the length of the By introducing number of servicing units the length of the queue & the waiting time are reduced.queue & the waiting time are reduced.

Model v: (M/M/S) : (FCFS /Model v: (M/M/S) : (FCFS /∞/∞)∞/∞)

This a multichannel model in which parallel service stations This a multichannel model in which parallel service stations are serving a single queue.are serving a single queue.

All service stations provide same serviceAll service stations provide same service

Customers are served on first come first basis.Customers are served on first come first basis.

Arrival rate has poison distribution with mean rate as λ Arrival rate has poison distribution with mean rate as λ

Service stations have exponential service times with mean Service stations have exponential service times with mean 1/μ customers entering in queue & served by all service 1/μ customers entering in queue & served by all service stationsstations

Queueing Model in seriesQueueing Model in series In real life situations where the arriving customer passes In real life situations where the arriving customer passes

through successive number of service channels in order to through successive number of service channels in order to complete his service are known as queues in series or complete his service are known as queues in series or Queues in Tandem.Queues in Tandem.

Queues in series with possibility of queuing.Queues in series with possibility of queuing.

Queues in series without possibility of QueueingQueues in series without possibility of Queueing