Stochastic Event Generation Through Markovian Jumps: Generalized Distribution and Minimal Representations

TL;DR

This paper extends Cox's generalization of Erlang distributions to create a more versatile topology for stochastic event timing, enabling minimal state representations and improved modeling of practical distributions in analytic and simulation contexts.

Contribution

It introduces a more general topology for stochastic event timing based on Markovian jumps, capable of representing any practical distribution and optimizing state complexity.

Findings

Derived minimal topologies for two-moment matching

Compared new topologies with existing literature

Demonstrated application in stochastic discrete-event models

Abstract

Stochastic Event Timing is a fundamental issue in developing both analytic and simulation models for stochastic systems. Generalized Erlang distributions are quite useful for generating those random events in a quite general way by inserting intermediary states with markovian jumps. One very important and celebrated generalization of the Erlang distribution was made by D. R. Cox in the middle 50's. This paper discuss further the Cox generalization and presents an even more general topology, capable of representing any practical distribution. As an application, we revisit the classical problem of the first two moments matching, and derive minimal topologies in terms of number of states, then the results are compared with those found in literature. At the end of the paper, we show how the generalized structure can be use for timing general stochastic discrete-event models for analytic and simulation purposes.