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.
TopStochastic 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
| Time | States |
| Discrete | Continuous |
| Discrete | Stochastic chain at discrete time | Stochastic process at discrete time |
| Continuous | Stochastic chain at continuous time | Stochastic process at continuous time |