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System Approach to a Queue - Queueing Theory

   Posted On :  25.06.2018 02:23 am

A system consists of three components, namely, (i) input, (ii) processes and (iii) output.

System Approach to a Queue

A system consists of three components, namely, (i) input, (ii) processes and (iii) output. As regards a queueing system, the input is constituted by the customers who arrive at a service point in anticipation of a service. The process includes the methods of the service offered by the organization, the behavior of the customers and the discipline of the queue. Hence, the following data are required to understand and analyze a queueing system:
 
i.        The input (the pattern of arrival of the customers to the service point)
 
ii.      The service mechanism (the pattern of service in the organization)
 
iii.    The queue discipline (the principle under which the queue operates) and
 
iv.    The behaviour of the customers

The input (the pattern of arrival)


One has to find out the way in which the customers arrive at a service point and join the queue. Customers normally arrive almost in a random way. It is highly difficult to guess the pattern of arrival of the customers. Therefore we have to associate probabilities with the arrival of the customers and hence the probability distribution for inter-arrival times (the time between two successive arrivals of the customers) has to be found out. We take up a queueing system with the assumption that the customers arrive in a Poisson process. We also assume that the mean arrival rate of the customers is found it to be δ.

The Service Mechanism

 
The term service mechanism refers to the arrangement of service facility to serve the customers. If there is infinite number of servers, then all the customers are served instantaneously as soon as they arrive and a queue will not be found. If the system consists of a finite number of servers, then the customers are served according to a pre-determined rule by making the server service time a constant or a random variable. Distribution of service time follows ‘Exponential distribution’.
 

Queueing Discipline

 
The term queueing discipline refers to a procedure by which the customers are selected from the queue for offering the service. The following disciplines are generally adopted by a queueing system:
 
First Come First Served – (FCFS)
 
First In First Out – (FIFO)
 
Last In First Out – (LIFO)
 
Selection for service In Random Order (SIRO)

The Behaviour of the Customers


In general, one may observe the following modes of behavior of the customers in a queue:
 
Normally, the customers arrive one by one into the system. However, there is a possibility for another phenomenon. The term Bulk arrival refers to the arrival of customers in groups.
 
Consider the case of several service counters in the organization. Then each service counter will have a queue. When there are several queues, the customers from one queue may switch over to another queue if it is of smaller size. Such a behaviour of the customers is referred to as Jockeying.

Sometimes a customer on arrival may not join a queue after observing that the queue length is very large. This behavior of the customers is called Balking of the customers.

In certain cases, a customer already present in a queue may leave the queue thinking that the waiting time may be too much. This behaviour of the customers is called Reneging.

Notations

 
Customers in the system refers to the customers receiving service at the service point and the customers who are waiting to receive the service.

The following notations are used in Queueing theory:
 
n - No of customers in the system
 
C - No of servers in the system
 
P_n(t) – The probability of having n customers in the system at time t
 
P_n  - The steady state probability of having n customers in the system
 
P_0 - The probability of having zero customer in the system L_q - Average number of customers waiting in the queue
 
L_s - Average number of customers waiting in the system (in the queue and in the service points)
 
W_q - Average waiting time of the customers in the queue
 
W_s - Average waiting time of the customers in the system (in the queue and in the service points)
 
δ  - Arrival rate of the customers µ - Service rate of the server
 
ϕ - Utilization factor of the server
 
δ  eff - Effective rate of the arrival of customers M - Poisson distribution
 
N - Maximum number of customers allowed in the system. It also refers to the size of the calling source of the customers
 
GD - General discipline for service in the organization like first in first – served (FIFS), last-in-first served (LIFS), etc.
 

Traffic Intensity (Or Utilization Factor)


The traffic intensity of a queue is denoted by ϕ. It is defined by the rule ϕ = (Mean arrival time)/(Mean service time) = δ/µ (< 1) and the unit of traffic intensity is called Erlang. 
Tags : Operations Management - Game Theory, Goal Programming & Queuing Theory
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