Every QPN describes a stochastic process. Similar to the stochastic process specified by a GSPN , the states of this process might be vanishing or tangible. In vanishing states only immediate transitions are enabled and/or scheduling in immediate queueing places might occur. In tangible states a change to another state occurs due to the firing of a timed transition or some action concerning the service of a token in a queue of a timed queueing place.
For simplification of the formal description, we assume that timed queueing places are numbered from to and immediate queueing places from to . Define and Each state of a QPN is an element of The state space, denoted by , of the QPN's corresponding stochastic process can be constructed like the reachability set of PNs. We assume that the state space is finite. The initial state is given by where denotes the appropriate element of .
In order to define the successor states of a given state we need the following definitions.
maps a state of the QPN onto the corresponding marking of the underlying CPN taking also tokens in queues into consideration. neglects these tokens.
Furthermore in contrast to ordinary QNs we have to consider the case of bulk arrivals at the queue of a queueing place. There are several possibilities of defining the queue's state after acceptance of the new arriving tokens, e.g. apply the AC-function successively to randomly chosen elements of the bulk or define an order on arriving tokens. We choose the last possibility, assuming that the colours of all places are indexed by natural numbers and the smallest index specifies the colour with the highest rank. In case of bulk arrivals the AC-function can recursively be applied to the arriving tokens, which is described by a function R-AC defined as follows.
The definition of pays attention to the priority of immediate transitions over timed transitions, thus if is non-empty, either or holds. The successor states of a given state can be calculated as follows.
The relation between and one of its successor states will be denoted by and is defined as the reflexive and transitive closure of this relation.
In the next subsections we will discuss qualitative and quantitative analysis of QPNs.