In the stochastic processes literature, the resulting process of several concurrent temporal processes is called a generalized process. The main example of such a process is the frame- work of Generalized Semi-Markov Processes introduced by [Glynn, 1989]. These processes were briefly introduced in chapter 2 and we provide more details here.
A GSMP represents the concurrent execution of several semi-Markov processes (SMPs). All these processes have stochastic transition destinations and stochastic sojourn times. Moreover, there is a strong coupling between the processes because they all affect the same random variables. Consequently, the overall process is a discrete event process resulting from the successive triggering of transitions in the different individual SMPs.
Formally, a GSMP is described by a set S of states and a set E of events. Each event can be described as an independent semi-Markov process over the random variables of the state space. At any time, the process is in a state s and there exists a subset Es of events that
11.2. Generalized Semi-Markov Processes are called active or enabled. These events represent the different concurrent processes that compete for the next transition. To each active event e, we associate a clock ce representing
the duration before this event triggers a transition as presented on figure 11.1. This duration would be the sojourn time in state s if event e was the only active event and thus corresponds to the associated SMP’s sojourn time in state s. The event e∗ with the smallest clock ce∗
(the first to trigger) is the one that takes the process to a new state. The transition is then described by the transition model of the triggering event: the next state s0 is picked
according to the probability distribution Pe∗(s0|s). In the new state s0, events that are not
in Es0 are disabled (which actually implies setting their clocks to +∞). For the events of
Es0, clocks are updated the following way:
• If e ∈ Es\ {e∗}, then ce← ce− ce∗
• If e 6∈ Es or if e = e∗, pick ce according to Fe(τ|s0)
The first active event to trigger then takes the process to a new state where the above oper- ations are repeated.
Definition (Generalized Semi-Markov Process, [Glynn, 1989]). A GSMP is given by the 4-tuplehS, E, F, P i, where:
• S is the set of possible values for the process’ state.
• E is the set of events describing the process. This set can be reduced to a subset Es of active events in each state s.
• F is the cumulative distribution function giving the duration before an event triggers. The duration τ before e triggers is drawn according to F (τ|s, e) = Fe(τ|s).
• P is the transition function of the process. When event e triggers, the new state s0 of
the process is drawn according toP (s0|s, e) = Pe(s0|s).
The framework of GSMPs could be compared with the (deterministic) framework of Timed Automata introduced in [Alur and Dill, 1994] which uses a similar description of the temporal behaviour of a system involving concurrency.
s1 Es1 : e2 e4 e5 e7 s2 Pe4(s′|s1) Es2 : e2 e3 e7 Pe7(s′|s2) Figure 11.1: Illustration of a GSMP
A GSMP is an event-driven stochastic process, summarizing the concurrent effects of several semi-Markov processes on a common state space.
Chapter 11. Concurrency: an origin for complexity
One can notice that — as in the SMP case — one can let the transition model depend on the clock ce∗, thus yielding a Pe∗(s0|s, ce∗) transition function.
Since GMSPs represent the overall process by factoring it through its separate concur- rent events, it provides a much simpler description than a monolithic model of the global process. In fact, each of the individual SMPs constituting the GMSP might have rather simple transition and duration probability functions and thus can be easy to model. Writing the corresponding GSMP avoids the heavy task of explicitly integrating all these concurrent processes into one large, explicit stochastic process. The drawback of this situation is that we do not have an explicit formalization of the overall process anymore but rather a compact description of its dynamics.
Consequently — as exposed by Glynn in his introductory paper in 1989 — GMSPs pro- vide both a “precise language for describing discrete event systems, and a mathematical setting within which to analyze discrete event processes”; the core idea being to capture the essential dynamical structure of a (stochastic) discrete event system. The analysis of GMSPs clarifies the connections between continuous variable dynamic systems and discrete event dynamic systems by considering GSMPs as event-driven stochastic processes.
The specialization of GSMPs to time-homogeneous sojourn times yields the time-homogeneous GSMP setting which can be reduced and analyzed as a continuous time Markov chain and thus as a standard Markov process through the operation of uniformization. This raises a similar question for the general case of GSMPs: does the stochastic process corresponding to the evolution of the SMPs’ common state space random variables s still retain Markov’s property?
As for SMPs, the answer is no. It is rather straightforward to provide a physical ex- planation to this: when considering the above process, defined on the common state space random variables s — which we will call the natural process — from an external point of view, an observer does not have enough information to predict which event will trigger next, and hence, which is the probability distribution on the next state of the process. This also implies the GSMP does not even retain the semi-Markov behaviour of the underlying SMPs. In his 1998 paper, Nilsen presents an implementation of a GSMP modeling and simu- lation tool (GMSim). In order to build the simulator’s underlying process, Nilsen uses the supplementary variable technique(presented, for instance, in [Cox and Miller, 1965]) in order to insure the semi-Markov behaviour of the global process, namely, to be able to predict the future state by only looking at the current state. However, as expected from a collection of SMPs, the sojourn times remain time inhomogeneous. The supplementary variable tech- nique is used to construct an augmented state containing both the state of the natural process (natural state) and the active events’ clocks. With this information, it is possible to write the probability distribution on the next augmented state of the process without information about its past history. We leave the notation details to [Nielsen, 1998] and simply conclude that:
The stochastic process described by the natural state variables of a GSMP does not retain the semi-Markov behaviour of the individual underlying semi-Markov processes. By including the events’ clocks in an augmented state, we are able to build a process over the random variables (s, c) which regains this semi-Markov behaviour.
11.3. DEVS modeling