Single Stage Systems Markovian Case

Single Stage Systems Markovian Case

Copyright: © 2018 |Pages: 36
DOI: 10.4018/978-1-5225-5264-2.ch003

Abstract

In Chapter 3, the authors show the expressions of queueing theory for Markovian systems with a single stage. The chapter begins with definitions of stochastic processes and Markov chains; then; they present the models for calculating the work in process and the cycle time of systems with a single server, multiple servers, and systems with restriction on queue size. Later, the chapter explores heuristic rules to estimate the capacity of a system. The chapter ends with the monetary analysis of the system and the optimum selection of capacity.
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Stochastic Processes

This is a set X of random variables that appear as a parameter t, whose elements belong to a set T, passes. A stochastic process is defined as X (t). In applications of stochastic processes for modeling real systems it is very common for set T to refer to a time sequence and each element t of the set to represent (discrete or continuous) instants of time within the same sequence; therefore the elements of the time sequence are T= {0, 1, 2,…}. Furthermore set X contains the possible events or results: taking a card from a deck, throwing the dice, the result of an experiment are examples of experimental events. Other situations are, for example, the number of customers that arrive at a box-office or leave a row at any given time or the failure of a piece of equipment in a particular moment.

A stochastic process is defined by:

  • 1.

    The possible events of set X, which are known as a set of possible states or just as states.

  • 2.

    The elements of set T.

  • 3.

    The relationship or connection that exists between every possible state.

The relationship between the variables of a stochastic process is as shown in Table 1.

Table 1.
Classification of stochastic processes accordingly to their variables
TimeStates
DiscreteContinuous
DiscreteStochastic chain at discrete timeStochastic process at discrete time
ContinuousStochastic chain at continuous timeStochastic process at continuous time

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