Markov chains theory

This chapter introduces some basic mathematical tools that are widely used in the performance analysis of communication systems. Generally, an accurate performance evaluation of modern communications systems requires sophisticated computer simulations or extensive and expansive testbeds. However, a suitablemathematicalmodel of the systemoftenmakes it possible to investigate in a systematic way the causeeffect relations that govern the system behavior, thus shedding light on the most significant tradeoffs between different performance indices. Furthermore, a mathematical modelmakes possible to investigate the systembehavior in limiting scenarios, such as in the presence of traffic overload or breakdown of parts of the system, that could not be observed in reality nor reproduced in simulations for different reasons (economic, servicecontinuity, complexity, and so on). The comprehension of the fundamental mathematical tools for modeling and analyzing a telecommunication system is, hence, essential for a system designer. To this purpose, this chapter describes the elementary theory of both discretetime and continuoustime Markov chains and of birthdeath processes, which are a special case of Markov chains that play a pivotal role in the queueing theory, presented in Chapter ??. It might be worth remarking that this chapter is not intended to offer a complete and detailed coverage of these subjects, which would require an entire book. Rather, the aim is to provide the principles and the fundamental results that will be used in the analysis developed in later chapters.