Continuous-Time, Discrete-Event Simulation from Counting Processes

A continuous-time, discrete-event (CTDE) simulation can be based on any one of a number of stochastic processes, including Generalized Semi-Markov Processes (GSMP) [Glynn 1989], Generalized Stochastic Petri Nets (GSPN) [Haas 2002], or chemical stochastic processes [Gillespie 2007] These processes vary in how they define state and how they specify the distribution of times at which each event will fire, but all of them have the remarkable property that the hazard rate of any event in a realization of the process will match the hazard rate of the distribution of times used to specify that event. In their work on chemical stochastic processes, Anderson and Kurtz [2015] offer a construction of a stochastic process which can serve as an alternative method of CTDE simulation. Because it uses counting processes as the basis for the construction, their work emphasizes the role of hazard rates in defining the process. As a result, certain forms of GSMP and GSPN, those which avoid use of immediate events and simultaneous events, could be considered subclasses for the purposes of computation. The formalism of counting processes also offers the algorithmic advantage of a clear separation between defining a transition kernel and sampling that kernel. Anderson and Kurtz present the more general stochastic process, here called competing clocks, as a step in derivation of stochastic processes for chemical simulation.