On capturing dependence in point processes: Matching moments and other techniques

Providing probabilistic analysis of queueing models can be difficult when the input distributions are non-Markovian. In response, a plethora of methods have been developed to approximate a general renewal process by a process with the time between renewals being distributed as a phase type random variable, which allows the resulting queueing models to become analytically or numerically tractable. However, from previous studies on the manufacturing sector, and more recently in analysis of telecommunications systems, assumptions of independence do not always hold and efforts have been made to approximate nonrenewal processes with Markovian Arrival Processes. In this paper we survey techniques for deriving the appropriate parameters of a Markovian process to accurately capture relevant characteristics of the original point process.