05. Stochastic Petri Nets - CS4 Modelling and Simulation...

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CS4 Modelling and Simulation LN-5 5 Stochastic Petri Nets In this lecture note we consider an important class of high level performance modelling paradigms—stochastic extensions of Petri nets . These are Petri net formalisms into which random variables have been added to represent the duration of activities, or the delay until events. The basic extension, Stochastic Petri Nets (SPN), present a very straightforward mapping between events in the SPN model and events in the underlying Markov process. As in the Markov process, a delay, represented by a random variable, is associated with every event in the model. This straightforward mapping has the advantage that gener- ating the Markov process from any SPN model is simple and easy to implement. The disadvantage is that models developed in this way tend to result in Markov processes which have a large number of states. We will go on to consider Generalised Stochastic Petri Nets (GSPN). The generalisation involves adding some simple additional features to the set of modelling primitives provided for the modeller. As a result the mapping between events in the model and events in the underlying Markov process is more sophisticated. However the resulting state space is usually more compact, and often a closer representation of the behaviour of the system can be achieved. Although Petri nets have been used for qualitative modelling of computer and commu- nication systems since the 1960s, their use as a performance modelling paradigm started about twenty years ago. In particular they were found to be especially useful for mod- elling distributed and parallel systems. Such systems were difficult to model with queueing networks, which were the prevailing performance modelling paradigm at the time. 5.1 Petri Nets Petri nets provide a graphical notation for the formal description of the dynamic behaviour of systems. They are particularly well suited to systems which exhibit concurrency, syn- chronisation, mutual exclusion and conflict. The primitives of the notation are the following: PLACES Places are used to represent conditions or local system states, e.g. a place may relate to one phase in the behaviour of a particular component. TRANSITIONS Transitions are used to describe events that occur in the system; these will usually result in a modiFcation to the system state. The occurrence of the event in the system is represented by the fring of the corresponding transition in the Petri net. TOKENS Tokens are identity-less markers that reside in places. The pres- ence of a token in a place indicates that the corresponding con- dition or local state holds. 31
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CS4 Modelling and Simulation LN-5 - ARCS Arcs specify the relationships between local states or conditions (places) and events (transitions). An arc from ap lace to a tran- sition is termed an input arc . This indicates the local state in which the event can occur. An arc to from a transition is termed an output arc . This indicates the local transformations which will be induced by the event.
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05. Stochastic Petri Nets - CS4 Modelling and Simulation...

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