Deficiency Zero Petri Nets and Product Form

TL;DR

This paper explores conditions under which Markovian Petri nets exhibit a product form stationary distribution, linking deficiency zero conditions to free-choice nets and Jackson networks.

Contribution

It establishes that deficiency zero Petri nets have a product form and characterizes free-choice nets with this property as Jackson networks.

Findings

Deficiency zero Petri nets admit a product form stationary distribution.

Only state machine free-choice nets have a product form among such Petri nets.

These nets can be viewed as Jackson networks.

Abstract

Consider a Markovian Petri net with race policy. The marking process has a "product form" stationary distribution if the probability of viewing a given marking can be decomposed as the product over places of terms depending only on the local marking. First we observe that the Deficiency Zero Theorem of Feinberg, developped for chemical reaction networks, provides a structural and simple sufficient condition for the existence of a product form. In view of this, we study the classical subclass of free-choice nets. Roughly, we show that the only such Petri nets having a product form are the state machines which can alternatively be viewed as Jackson networks.