Reconciling real and stochastic time: the need for probabilistic refinement

E P De Vink,P R D’Argenio,J C M Baeten,J Markovski

doi:10.1007/s00165-012-0230-y

E P De Vink, P R D’Argenio + Show 2 more

Open Access

https://doi.org/10.1007/s00165-012-0230-y

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Abstract

Abstract We conservatively extend an ACP-style discrete-time process theory with discrete stochastic delays. The semantics of the timed delays relies on time additivity and time determinism, which are properties that enable us to merge subsequent timed delays and to impose their synchronous expiration. Stochastic delays, however, interact with respect to a so-called race condition that determines the set of delays that expire first, which is guided by an (implicit) probabilistic choice. The race condition precludes the property of time additivity as the merger of stochastic delays alters this probabilistic behavior. To this end, we resolve the race condition using conditionally-distributed unit delays. We give a sound and ground-complete axiomatization of the process theory comprising the standard set of ACP-style operators. In this generalized setting, the alternative composition is no longer associative, so we have to resort to special normal forms that explicitly resolve the underlying race condition. Our treatment succeeds in the initial challenge to conservatively extend standard time with stochastic time. However, the ‘dissection’ of the stochastic delays to conditionally-distributed unit delays comes at a price, as we can no longer relate the resolved race condition to the original stochastic delays. We seek a solution in the field of probabilistic refinements that enable the interchange of probabilistic and nondeterministic choices.

Highlights

Like other approaches in concurrency theory, process algebra originally focused on qualitative aspects of systems, and gradually included quantitative aspects like time, probabilities, Markovian, and generally-distributed stochastic delays [Bae05]
Embedding of timed delays in stochastic process algebras was attempted in [MV08], where we introduced the notion of context-sensitive time interpolation, which is supposed to mimic the trivial race condition for timed delays
To conservatively extend timed process algebras with stochastic delays, we introduced the concept of racing timed delays, which represent conditionally-distributed unit timed delays that are employed to explicitly resolve the race condition

Summary

Introduction

Like other approaches in concurrency theory, process algebra originally focused on qualitative aspects of systems, and gradually included quantitative aspects like time, probabilities, Markovian (exponential), and generally-distributed stochastic delays [Bae05]. The semantics of stochastic process algebras is given using clocks that represent the stochastic delays at a symbolic level Such a symbolic representation allows for the manipulation of finite structures, e.g., stochastic automata [DK05a] that support SPADES or extensions of generalized semi-Markov processes [Bra02] for IGSMPs. The concrete execution model is subsequently obtained by sampling the clocks, frequently yielding infinite probabilistic timed transition systems. To conservatively extend timed process algebras with stochastic delays, we introduced the concept of racing timed delays, which represent conditionally-distributed unit timed delays that are employed to explicitly resolve the race condition. 5, we develop the process theory DTCPdresct that comprises derived stochastic delays and delayable actions and we illustrate how to apply it to resolve the race condition by revisiting a discrete-time variant of the G/G/1/∞ queue. We refer to [Mar08] for technical details

Race condition

Process theory TCPdrst

Equational theory

Derived delayable action and stochastic delay prefixes

Towards reconciling real and stochastic time

Concluding remarks