Representing LCFSPR BCMP service center with Coxian service time by GSPN

In this paper we study the relations between multi-class BCMP-like service stations with Coxian service time and generalized stochastic Petri nets (GSPN). We consider multi-class product-form service center types of QN and their representation with GSPN, in order to investigate the relation between the product-form solutions of the two models. Representing queueing discipline with GSPN models is not easy. We focus on representing multi-class queueing systems with LCFSPR scheduling discipline and Coxian distributed service time. Note that the queueing discipline in general affects performance measures in multi-class systems. For example, BCMP-like service centers with First Come First Served (FCFS) and with Last Come First Served with Preemptive Resume (LCFSPR) have a (different) product-form solution under different hypotheses. We define a structurally finite GSPN model with product- form that does not belong to the class of known product-form GSPN. Then we show that this model is equivalent to the multi-class M/COX/k/LCFSPR queueing system, in terms of steady state probability and average performance measures. The main idea is to define a finite GSPN model that simulates the behavior of the given queue discipline by defining some appropriate random choices. Moreover, we prove that the combination of the introduced equivalent GSPN model has a closed-form steady state probability by the M ⇒ M property.