Generalized Semi-Markov Process models (GSMP)

Description

Definition

Examples

References

 

 

 

Description:

Here we follow the definition of GSMP model given in [1]. GSMP is a stochastic process in which a transition from one state to another is triggered by one or more events.

                                                                

Note that not all definitions of GSMP available in the literature coincide with this. For example some definitions of GSMP do not allow for multiple events occurring simultaneously.  We treat this as a separate formalism – GSMP without simultaneous events.

 

 

Formal Definition:

A Generalized Semi-Markov Process (GSMP) model has the following components:

 

E – is a finite set of events

X – is a countable  set of states

Γ(x) – is a set of active events defined for all x X, with Γ(x) a subset of E,

p(x1; x, E*) – is a state transition probability, defined for all x, x1 X, E*a subset of E, reflecting probability of going from state x to state x1 through simultaneous occurrence of events  in E*.

p0(x) – is the pmf P[X0=x], xX, of the initial state X0.

G={Gi : iE} – is a stochastic clock structure – a set of distribution functions.

 

Note1: Another feature may be (and usually is) incorporated in this formalism – clock rates that in general depend on states. We are not considering this feature for now.

 

Examples:

 

References:

  1. P. J. Haas, Stochastic Petri Nets, Springer, 2002.
  2. G.S. Shedler, Regeneraive Stochastic Simulation, Academic Press, 1993.