Stochastic Timed Automata
In practice, systems always operate in environments which are constantly plagued by uncertainty. This is especially true in dealing with DES, which, by their nature, often involve unpredictable human actions and machine failures. The process of resource sharing (which provides an important motivation for studying DES) is inherently characterized by such unpredictability: changing user demand, computer breakdowns, inconsistencies in human decision making, etc. While the untimed (or logical) models considered in Chaps. 2–4 do account for “all possible behaviors” of the system, their use is limited to logical (or qualitative) performance objectives. Deterministic timed models of the type considered in Chap. 5 certainly contribute to our basic understanding of some quantitative properties of the dynamic behavior of a system (for instance, the periodic behavior of systems that can be modeled as marked graphs). But their use is limited since models with deterministic clock structures only capture a single timed string of events (or states), or, in other words, a single sample path of the system. If we are to develop either descriptive or prescriptive techniques for evaluating performance and for “optimally” controlling timed DES with respect to quantitative performance measures and in the presence of uncertainty, more refined models that incorporate stochastic elements are required.
In this chapter we introduce some basic stochastic models which will allow us to extend the framework developed thus far, and to derive quantitative means of analysis. In particular, we concentrate on timed automata models. We saw that the input for these models is a clock structure, which determines the actual event sequence driving the underlying automaton. Our main effort will be to develop stochastic models for the clock structure, and to study the properties of the resulting DES.
KeywordsPoisson Process Queue Length Sample Path Interarrival Time Counting Process
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∎ Review of Probability and Stochastic Processes
- — Gallager, R.G., Discrete Stochastic Processes, Kluwer Academic Publishers, Boston, 1996.Google Scholar
∎ Stochastic Timed Automata
∎ Stochastic Hybrid Automata
- — Cassandras, C.G., and Lygeros, J., Stochastic Hybrid Systems, Taylor and Francis, Boca Raton, 2006.Google Scholar