Phases or Proxels: The Decision Factors

Proxel-based simulation is a new and deterministic approach to analysing discrete stochastic models and studying their behaviour. One of its main advantages is the intuitiveness with which it approaches models’ analysis. Even though the proxel-based method is a deterministic approach, unlike the other approaches in that class, it does not set up and solve partial differential equations, and possesses intuitiveness which is comparable to the one that characterises discrete-event simulation. Unfortunately, as it is also the case with other deterministic approaches it suffers from the well-known state-space explosion problem, prohibiting its application on large-scale models. One of the ways to reduce the problem of state-space explosion is the inclusion of discrete phase-type approximations, which is possible because of the fact that the underlying stochastic process of the proxel-based method is a discrete-time Markov chain, in which the probabilities for staying in the same state are zero. The use of phases is not always recommendable, and sometimes their approximation can be very expensive in terms of computation time, so that it more than compensates for the saved proxel simulation time. Therefore, we need a way to decide when it is a good decision to substitute a distribution by a particular phase-type approximation, and when proxels might be the better choice. In this paper we use our recently introduced paradigm of a lifetime of a discrete state as another decision indicator, besides the distributions characteristics, when choosing between phases and proxels. The whole decision making process is supported by a demonstrative example. 1 Goals of the Paper The main goal of this paper is to strengthen the decision of whether to substitute a general distribution by a discrete phase-type one for the proxel-based simulation of a given model. Until now this decision was made based on the support of the probability density function of every state change individually, independent of the characteristics of the competing state changes, and its coefficient of variation, as implemented in the tool described in [IH05b]. The approach, however, can be extended to include other factors as well in order to increase the efficiency of the proxel-based simulation combined with discrete phases. One of the most relevant factors is the lifetime of the corresponding discrete state, which is introduced in [LMH05b] and determines the longest time that the model can reside in a given discrete state, based on the simulation parameters. The meaning of this extension of the decision factors is discussed further in the paper.