First Passage Time Computation in Tagged GSPNs with Queue Places

Abstract

This paper presents an extension of the Generalized Stochastic Petri Net (GSPN) formalism that enables the computation of first passage time distributions. The tagged customer technique typical of queuing networks is adapted to the GSPN context by providing a formal definition and an automatic computation of the groups of tokens that can be identified as customers, i.e. classes of homogeneous entities behaving in a similar manner. Passage times are identified through the concept of events that correspond to the firing of transitions placed at the boundaries of a subnet. The extended model obtained with this specifications is translated into an ordinary GSPN by isolating a customer from the group and highlighting its path through the net thus obtaining a representation suited for the passage time analysis. Proofs are provided to show the equivalence between these models with respect to their steady state distributions. An important and original aspect treated in this paper is the possibility of specifying several scheduling policies of tokens at places, an information not present in ordinary GSPN models, but that is vital for the precise computation of first passage time distributions as shown by a few results computed for a simple Flexible Manufacturing application.

Expressing and Computing Passage Time Measures of GSPN Models with HASL

“…Since Cosmos also allows for the parallelisation of trajectory simulation we also considered the level of parallelisation (expressed in terms of chosen number of cores over which an experiment is distributed) as a parameter of each experiment. Observe that, quite sensibly, the runtime gain is less than linear wrt the level of parallelisation ¿¿¿¿¿¿¿ .r673 and in particular the gain drastically decreases when the chosen number of cores over which the computation is distributed is beyond the actual cores the CPU consists of (in our experiments two) 6 . Finally observe that Cosmos adopts a model-driven codegeneration scheme, i.e.…”

“…The work in [6] proposes to exploit invariant properties (i.e. p-invariant) to identify places where tokens with indistinguishable behavior are preserved so that such tokens can be treated similarly to the customers of Queuing Network models and can thus be tagged by the modeler to compute the distribution of the time required by one specific token to travel between points of the net.…”

“…This is the model of a Flexible Manufacturing System (FMS), a model that was already used in [6] to explain tagged GSPN, and that we report in Fig. 10 for ease of reference.…”

Simulation-based verification of hybrid automata stochastic logic formulas for stochastic symmetric nets

“…Classical performance indices includes the distribution (or mean value) of tokens in a place and the throughput of a transition. More elaborate performance indices, like the distribution of the passage time over a subnet, may not be directly computed upon the basic performance indicators [2] hence more elaborate ways of defining performance indices for a GSPN are needed. In this paper we concentrate on the computation of performance indices based on timed paths, where a path is a net execution (an (in)finite sequence of transitions and associated delays), in other words, a simulation trace of the GSPN.…”

Stochastic Petri Nets Sensitivity to Token Scheduling Policies

“…When timed transitions are interpreted as servers, firing delays as activity durations, and tokens as customers that are moved from input to output places upon transition firings, response time distributions may become important and the need arises of specifying queueing policies for input places (an issue that has been studied in [2]) and of accounting for the relevant results of queueing theory that apply within this context. The impact that service policies have on the behavior of queueing models is well understood and average performance indices such as throughputs, utilizations, mean queue lengths, and mean response times are shown to be insensi-tive to queueing policies, as long as they are work-conservative and the service time distribution is negative exponential [6].…”