Elements of Probability and Statistics by Francesca Biagini & Massimo Campanino

Author:Francesca Biagini & Massimo Campanino
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

If we indicate with the probability that Poisson process at time t is in the state s, then we have

where is the initial distribution. It follows that for every initial distribution the functions satisfy the same system of differential equations.

The functions can be considered as particular cases in which and for .

7.4 Queueing Processes

We now consider some examples of continuous time Markov chains that serve as models of queueing processes. As we have said in Sect. 7.1, in queueing theory there is a symbolic notation to indicate the type of a queueing system. In the examples we consider the flow of incoming clients follows a Poisson process with parameter . Clients who find a free server start a service time and after service leave the system. When an arriving client finds all servers engaged, he is put in a queue. When a server becomes free, if there are clients waiting in queue, one of them starts its service time.

For what we are interested in, the order in which clients access the service does not matter; we can assume, for example, that the order is randomly chosen, but other possible choices would not change the results. We assume that service times are stochastically independent, identically distributed and stochastically independent from the Poisson process ruling the flow of arrivals. We also assume that service times are exponentially distributed with some parameter .

A process of this type will be indicated with the symbol M / M / n. The first M means that the flow of arrivals is Poisson, the second M means that service times are exponentially distributed, while n denotes the number of servers and can vary from 1 to ( is an admissible value).

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.