One-place unbounded stochastic petri nets: Ergodic criteria and steady-state solutions
This paper presents the analysis of Markov stochastic Petri nets having one unbounded place. These nets represent queues the arrival and departure of customers of which can be grouped and depend on a finite homogeneous Markov process. In the first section we give a formal definition of these nets, called C-Q nets (complex-queue nets), and we state some qualitative properties of such nets (particularly we state that the reachability graph is composed by an infinite sequence of isomorphic subgraphs). In the second section we give a complete classification of ergodic properties of C-Q nets. The Markov process associated with the marking are transient, null recurrent, and positive recurrent, according to the sign of the scalar quantity C"1, N"m"a"x (where C"1 is the incidence vector of the unbounded place p"1, N"m"a"x is the maximal expected firing rate of transitions of the net). The third section is devoted to the computation of the steady-state probabilities of C-Q nets. Because the generator of the marking process is a quasi-birth-and-death matrix, we recall the results for such processes (matrix geometric solution). We present the solution based on the generalized eigenvalues. The particular structure of the generator associated to a C-Q net induces a third method to compute the steady-state probabilities that are more efficient for grouped arrival and departure.