# Queuing Theory

**Topics:**Poisson distribution, Exponential distribution, Probability theory

**Pages:**23 (3138 words)

**Published:**December 1, 2011

INTRODUCTION

Waiting lines are the most frequently encountered problems in everyday life. For example, queue at a cafeteria, library, bank, etc. Common to all of these cases are the arrivals of objects requiring service and the attendant delays when the service mechanism is busy. Waiting lines cannot be eliminated completely, but suitable techniques can be used to reduce the waiting time of an object in the system. A long waiting line may result in loss of customers to an organization. Waiting time can be reduced by providing additional service facilities, but it may result in an increase in the idle time of the service mechanism. Queuing theory is based on mathematical theories and deals with the problems arising due to flow of customers towards the service facility The waiting line models help the management in balancing between the cost associated with waiting and the cost of providing service. Thus, queuing or waiting line models can be applied in such situations where decisions have to be taken to minimize the waiting time with minimum investment cost.

Basic terminology

Queuing Model

It is a suitable model used to represent a service oriented problem, where customers arrive randomly to receive some service, the service time being also a random variable.

Arrival

The statistical pattern of the arrival can be indicated through the probability distribution of the number of the arrivals in an interval.

Service Time

The time taken by a server to complete service is known as service time.

Server

It is a mechanism through which service is offered.

Queue Discipline

It is the order in which the members of the queue are offered service.

Poisson Process

It is a probabilistic phenomenon where the number of arrivals in an interval of length t follows a Poisson distribution with parameter λt, where λ is the rate of arrival.

Queue

A group of items waiting to receive service, including those receiving the service, is known as queue.

Waiting time in queue

Time spent by a customer in the queue before being served.

Waiting time in the system

It is the total time spent by a customer in the system. It can be calculated as follows:

Waiting time in the system = Waiting time in queue + Service time

Queue length

Number of persons in the system at any time.

Average length of line

The number of customers in the queue per unit of time.

Average idle time

The average time for which the system remains idle.

FIFO

It is the first in first out queue discipline.

Bulk Arrivals

If more than one customer enter the system at an arrival event, it is known as bulk arrivals.

Components of Queuing System

1. Input Source: The input source generates customers for the service mechanism. The most important characteristic of the input source is its size. It may be either finite or infinite. Please note that the calculations are far easier for the infinite case, therefore, this assumption is often made even when the actual size is relatively large. If the population size is finite, then the analysis of queuing model becomes more involved. The statistical pattern by which calling units are generated over time must also be specified. It may be Poisson or Exponential probability distribution. Usually the source population is considered as unlimited. 2. Queue: It is characterized by the maximum permissible number of units that it can contain. Queues may be infinite or finite.

3. Service Discipline: It refers to the order in which members of the queue are selected for service. Frequently, the discipline is first come, first served.

Following are some other disciplines:

o LIFO (Last In First Out)

o SIRO (Service In Random Order)

o Priority System

4. Service Mechanism: A specification of the service mechanism includes a description of time to complete a service and the number of customers who are satisfied...

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