Simulation of Generalised Semi-Markov Processes based on Graph Transformation Systems

Abstract

Stochastic Graph Transformation combines graphical modelling of various software artefacts with stochastic analysis techniques. Existing approaches are restricted to processes with exponential time distribution. Such processes are sufficient for modelling a significant class of stochastic systems, however there are interesting systems which cannot be specified appropriately in such a framework. In several cases one needs to consider non-exponential time distributions. This paper proposes a stochastic model based on graph transformation with general probability distributions. This model is well suited to represent concurrency and performance aspects of architecture reconfiguration. It is also possible to apply Monte Carlo simulation techniques in order to analyse behaviour of complex stochastic systems. The new model is implemented and used to simulate simple networks.

SimSG: Rule-based Simulation using Stochastic Graph Transformation.

“…This is useful whenever a stochastic model is required to model waiting times, i.e., transitions becoming more likely the longer they have to wait. When transition delays are not described by the memoryless exponential distribution, we speak of semi-Markov processes [KL07]. When we think about graph transformations in the context of CTMCs, we can describe a state by a graph G and the set of states by a set of graphs G. Hence, the transition from a state G to G corresponds to the application of rule r :…”

“…A well-known approach from biochemistry that is used in various simulation tools is Gillespie's algorithm [TG77]. Another approach that was used for the simulation of peer-to-peer networks is the generalised semi-Markov scheme [KL07]. Both works describe how to determine the next state transition as well as the transition delay in their respective domains.…”

“…Generalised semi-Markov scheme (GSMS) devised by Kosiuczenko et al [KL07] represents a different approach for determining transitions and the corresponding delay compared to Gillespie's algorithm. A GSM S = (G, E) has a set of graphs G ∈ G representing states and a set of events e ∈ E. An event e(m, r) contains a rule-match pair, with m ∈ m(G, r) and r ∈ R. A GSMS tracks individual matches to preconditions of rules as distinct events, where each event is annotated with its own continuous distribution function.…”

Preface

“…Andrea Corradini gave an invited talk about "Verification of Graph Transformation Systems (using Concurrency)". The accepted papers were seperated into sections "Theoretical aspects," which included [3,8,1,5,2], covering mostly areas (3) and (4) above, and "Quantitave aspects", which included [6,4,7], covering areas (1) and (3). Thus, we see that areas (2) and (5) be sought in the fact that a number of related workshops such as GraMoT and GraBaTs, as well as the ICGT conference, took place shortly after GT-VC (within six weeks).…”