This paper surveys the theoretical developments in the ÿeld of stochastic process algebras, process algebras where action occurrences may be subject to a delay that is determined by a random variable. A huge class of resource-sharing systems -like large-scale computers, clientserver architectures, networks -can accurately be described using such stochastic speciÿcation formalisms. The main emphasis of this paper is the treatment of operational semantics, notions of equivalence, and (sound and complete) axiomatisations of these equivalences for di erent types of Markovian process algebras, where delays are governed by exponential distributions. Starting from a simple actionless algebra for describing time-homogeneous continuous-time Markov chains, we consider the integration of actions and random delays both as a single entity (like in known Markovian process algebras like TIPP, PEPA and EMPA) and as separate entities (like in the timed process algebras timed CSP and TCCS). In total we consider four related calculi and investigate their relationship to existing Markovian process algebras. We also brie y indicate how one can proÿt from the separation of time and actions when incorporating more general, non-Markovian distributions.
We consider a queuing network with M exponential service stations and with N customers. We study the behavior of a subsystem u, which has a single node as input and a single node as output, when the subsystem parameters are varied. An "equivalent" network is constructed in which all queues except those in subsystem u are replaced by a single composite queue. We show that for certain classes of system parameters, the behavior of subsystem u in the equivalent network is the same as in the given network. The analogy to Norton's theorem in electrical circuit theory is demonstrated. In addition, the equivalent network analysis can be applied to open exponential networks.
We introduce Stochastic Process Algebras as a novel approach for the structured design and analysis of both the functional behaviour and performance characteristics of parallel and distributed systems. This is achieved by integrating performance modelling and analysis into the powerful and well investigated formal description technique of process algebras. After advocating the use of stochastic process algebras as a modelling technique we recapitulate the foundations of classical process algebras. Then we present extensions of process algebras such that the requirements of performance analysis are taken into account. Examples illustrate the methodological advantages that are gained. 123 1.2 Precise and Modular Description of System Behaviour Classical queueing and loss models are the best known and most often used performance evaluation techniques. They mainly use a mixture of (almost) standardized box symbols, precisely defined stochastic parameters and colloquially formulated scheduling strategies. These semi-formal techniques have been most successfully applied to describe and analyse the behaviour of many dynamic systems. Modelling, however, the complex interdependencies within distributed and parallel processor systems we need a more precise, expressive and unambiguous description. Formal languages and particularly process algebras (two of them, CSP [Hoa85] and CCS [Mi189], are the basis of the quite remarkable programming languages OCCAM and LOTOS) are best suited and generally accepted as appropriate means. Stochastic Process Algebras areas we will elaborate during this tutorial-an extension of these classical abstract languages including random time variables and operators for the description of probabilistic behaviour. Some examples are shown below to illustrate intuitively their expressive power. (Here we assume exponentially distributed time intervals, characterized by their rates; in general these rates just have to be replaced by an appropriate parameter set.) 9 The sequential arrival of three different jobs is specified by a process Jobstream describing explicitly each arrival point before halting:
An approximate iterative technique for the analysis of complex queuing networks with general service times is presented. The technique is based on an application of Norton's theorem from electrical circuit theory to queuing networks which obey local balance. The technique determines approximations of the queue length and waiting time distributions for each queue in the network. Comparison of results obtained by the approximate method with simulated and exact results shows that the approximate method has reasonable accuracy.
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