3Distributions and Random Processes Related to Queueing and Reliability Models
Boyan DIMITROV
Department of Mathematics, Kettering University, Flint, MI, USA
In this chapter, we discuss the similarity between probability distributions and random processes related to queueing and reliability models. Analysis of the properties of these distributions helps us to understand their structure, and extend the area of their applications and modeling power. We see how these properties are used to model various processes in economics, industry, demography and environmental studies, and how their use reflects on insurance, risk and cost analysis.
The chapter is mostly based on results obtained by the author and many collaborators and followers. It is designed to give incentives, analytic skills and inspiration for students and researchers. Some open and challenging new problems for further research directions are marked, as well as various possible areas of practical application of mathematics.
3.1. Some useful notations, relationships and interpretations
Most variables in queueing and reliability are non-negative, since they describe the duration of activities, or counting a random number of events. Therefore, they can be treated as lifetimes.
We use capital letters for notation of random variables and the same subscripts for their characteristics. So, if X is a random variable (r.v.), its cumulative distribution function (cdf) is defined as
Equivalent to the cdf is the knowledge of the ...
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