Stochastic Petri nets with Low variation matrix exponentially distributed firing time
Peter Buchholz, András Horváth, Miklós Telek · 2011
The set of order n matrix exponential (ME) distributions is strictly larger than the set of phase type (PH) distributions for n> 2 and contains elements with squared coefficient of variation (scv) significantly lower than 1/n. ME distributions with very low scv are such that the density function becomes zero at some points in (0,∞). For such distributions there is no equivalent finite dimensional PH representation, which inhibits the application of existing methodologies for the numerical analysis of stochastic Petri nets (SPNs) with this kind of ME distributed firing time. To overcome the limitations of existing methodologies we apply the flow interpretation of ME distributions introducedbyBladt and Neuts andstudy thetransient andthe stationary behaviour of stochastic Petri nets with low variation matrix exponentially distributed firing times via extended differential and linear equations, respectively. Since the proof we present in the paper is general, it applies to all kind of ME distributions and therefore shows that ME distributions can be used like PH distributions in stochastic Petri nets and a numerical computation of transient or stationary measures like token populations at places or transition throughputs is possible with methods similar to the methods used for Markov models.