6 - Stochastic Processes

Abstract

Markov processes are powerful analytical tools applicable to the analysis of computer systems. They provide accurate, yet relatively simple means to construct representations of systems and to mathematically analyze a computer system. Markov processes require that we have an understanding of stochastic processes and their analysis. This chapter provides the background necessary to perform the modeling and analysis of such systems. A stochastic process involves the representation of a family of random variables. A random variable is represented as a function on a variable which approximates a number with the result of some experiment. It is shown that stochastic processes have some fundamental properties that allow them to be readily applied to the study of computer systems. An important stochastic process used in computer systems performance evaluation is the Poisson process. A Poisson stochastic process has the property that events are independent, and the interarrival times of events can be described using the exponential distribution. One of these is the concept of the Poisson process and its application to the concept of expected arrival rates or service rates for events within stochastic processes. One special stochastic process is the birth-death process. This process was used to develop the concepts of equilibrium states and balance equations. These were used to determine state probabilities. A further refinement on the birth-death process is the Markov chain. The Markov chain has additional properties that lend it to the application of computer systems modeling.