Generalized Model Of Birth Death Process In Queueing Theory

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Birth Death Process in Queueing Theory
Ashwani Kumar
Department of Industrial Engineering and Operation research, IIT Bombay
September 20, 2014
1
INTRODUCTION
A flow of customer from infinite/finite population towards the service facil- ity from a queue(waiting line) on account of lack of capability to serve them all at a time. The queues may be of persons waiting at a doctor’s clinic or at railway ticket booking counter, etc. Bt the mean customer we mean the arriving unit that requires some service to be performed. The customer may be persons, machines, vehicles,parts, etc. Queues stands for a number of customers waiting to be served. The queue does not include the customer being serviced.
The mechanism of queueing process is very simple. Customers arrive at a service counter and are attended by one or more servers. As soon as a customer is served, it departs from the system.
The general framework of a queueing system is shown in Fig. 1.1:
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Generalized Model of Birth-Death Process
The theory of Birth-Death process, developed by Feller,comprises part of the subject matter commonly called stochastic processes. We now give the model
1of the Birth-Death process. This model deals with a queueing system having single service channel, Poisson input with no limit on the system capacity.
Arrivals can be considered as births to the system, whereas a departure can be looked upon as a death. Let n = number of customers in the system, λ n = arrival rate of customers given n customers

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