ON REPRESENTING MULTICLASS M/M/k QUEUES BY GENERALIZED STOCHASTIC PETRI NETS

In this paper we study the relations between multiclass BCMP-like service stations and generalized stochastic Petri nets (GSPN). Representing queuing discipline with GSPN models is not easy. We focus on representing multi-class queuing systems with different queuing disciplines by defining appropriate finite GSPN models. Note that queuing 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 structurally finite GSPNs equivalent to the multi-class M/M/k queuing system with FCFS, LCFSPR, Processor Sharing (PS) and Infinite Servers (IS). Equivalence holds in terms of steady state probability function and average performance measure. The main idea is to define a finite GSPN model that simulates the behavior of a given queue discipline with some appropriate random choice. Moreover, we prove that the combination of the introduced equivalent models has a closed-form steady state probability by the M ⇒ M property. We consider queuing systems with both a single server with load dependent service rate, and multiple servers with constant service rate.