Group members

Hamid Ullah MianMirajuddinSafi Ullah

Queuing Theory

What is Queuing Theory ?

Examples Commercial Queuing Systems Ex. Dentist, bank, ATM, gas stations, plumber and garage.

Transportation service systems Ex. Vehicles waiting at toll stations and traffic lights, trucks or ships waiting to be loaded, taxi cabs, fire engines, elevators and buses.

Business-internal service systems Ex. Inspection stations, conveyor belts, computer support.

Social service systems Ex. Judicial process, the ER at a hospital.

Basic Component of queuing systemINPUT SOURCE OF QUEUE • An input source is characterized by • Size of the calling population

i. According to time ii. According to source iii. According to numbers

• Pattern of arrivals at the system i. static arrival processii. dynamic arrival process

• Behavior of the arrivals i. patientii. impatient.

Service System

Provided by service facility or facilities by aPerson, Machine, Space

Two Aspects of a service system,

• Configuration of the service system• Speed of service system

Configuration of the service system

• Single Server – Single Queue

• Single Server – Several Queues

• Several (Parallel) Servers – Single Queue

• Several Servers – Several Queues

Speed of Service

• Speed can be expressed in two ways,

• Service Rate : The number of customer serviced at particular time.

• Service Time : The service time indicates the amount of time needed to service a customer.

QUEUE CONFIGURATION

• Queuing process refers to the number of Queues and their respective length

• Length (or size) of the queue depends upon the operational situation such as

• physical space, • legal restrictions, and • attitude of the customers.

Queue Discipline

In the queue structure, the important thing to know is the queue discipline. The queue discipline is the order or manner in which customers from the queue are selected for service.

• Static queue disciplines

• Dynamic queue disciplines

Static queue disciplines

• First-come, first-served (FCFS)

• Last-come-first-served (LCFS)

Dynamic queue disciplines

• Service in Random Order (SIRO)

• Priority Service

Queuing Models1. Deterministic queuing model2. Probabilistic queuing model

3. Deterministic queuing model :--

= Mean number of arrivals per time period

µ = Mean number of units served per time period

2.Probabilistic queuing model

Probability that n customers will arrive in the system in time interval T is

!n

etnP

tn

t

Kendall’s notation

• Standard system used to describe queues.• Using 3 factors written in the form A/S/c• A – interarrival time• S – size of job/service distribution time ( G and

M• C – number of servers.

Utilization factor

• Also known as traffic intensity• Mathematicaly it is expressed as the ratio

between arrival rate (λ) and service rate (µ).

– Utilization Ratio = λ /µ

Basic points

• λ /µ > 1 queue is growing w/o end

• λ /µ < 1 queue is deminishing

• λ /µ = 1 queue length remains constant

Single Channel Model

= Mean number of arrivals per time periodµ = Mean number of units served per time period

Ls = Average number of units (customers) in the system (waiting and being served)

=

Ws = Average time a unit spends in the system (waiting time plus service time)

=

µ –

1µ –

Lq = Average number of units waiting in the queue

=

Wq = Average time a unit spends waiting in the queue

=

p = Utilization factor for the system

=

2

µ(µ – )

µ(µ – )

µ

P0 = Probability of 0 units in the system (that is, the service unit is idle)

= 1 –

Pn > k = Probability of more than k units in the system, where n is the number of units in the system

=

µ

µ

k + 1

Question 1.People arrive at a cinema ticket booth in a poisson distributed 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 mean waiting time , the mean number in the system , the mean time in the system and the utilization factor?

Question 2. Assume that at a bank teller window the customer arrives at a average rate of 20 per hour according to poission distribution .Assume also that the bank teller spends an distributed customers who arrive from an infinite population are served on a first come first services basis and there is no limit to possible queue length having service rate of 30 customers.

1.what is the value of utilization factor? 2.What is the expected waiting time in the system per customer? 3.what is the probability of zero customer in the system?

Little’s LawL = λW

Which states that the average number of customers in a queuing system L, is equal to the rate at which customers arrive and enter the system λ, multiplied the average sojour time of a custmer, W,

Example : 300 students getting enroll at ims every year for four year program.

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