Quantitative analysis of QPNs can be performed by analysis of the embedded Markov chain specified by the QPN's stochastic process. This procedure is analogous to the quantitative analysis of GSPNs [2]. In the following we assume that the QPN is bounded, implying a finite state space. The state space of a QPN is the disjunct union of tangible and vanishing states, where Let denote the probability of changing from to , be the rate transiting from state to and .

specifies the corresponding probability. Define four matrices .

Since the sojourn time in vanishing states is zero by definition, we are only interested in tangible states. Thus it is sufficient to solve the global balance equations of the reduced embedded Markov chain given by

provided exists. If (4) has a unique solution the steady state distribution of the QPN's stochastic process is given by

Performance measures can be computed from this steady state distribution as usual.

The description above shows that two important preconditions have to be satisfied for a successful quantitative analysis of QPNs:

- a)
- must exist and
- b)
- Equation (4) must have a unique solution.

Define and , i.e and contain all vanishing states for which the sum of elements of the corresponding row in matrix equals one or is smaller than one resp. Note that, being in a vanishing state, a tangible state can only be reached via some state of .

In QPN terminology this condition is specified by: .

Matrix characterizes the changes between vanishing states. Such states are determined by enabling of immediate transitions or scheduling in immediate queueing places according to the QPN description. Thus, if matrix contains a trap this indicates the possible occurrence of an infinite firing of immediate transitions or in pathological cases an infinite number of actions in an immediate queueing place. Therefore we will refer to this situation calling it a 'timeless trap'. Figure 6 is meant to illustrate a timeless trap caused by never ending firings of immediate transitions. The next theorem proves that in a live QPN the stochastic process can not be "caught" in such timeless traps.

implying that matrix has no trap. **Proof: **
Assume there is a trap, i.e.
.

Consider the case . Choose an arbitrary transition
and a colour . Liveness yields
and
and .
Thus is a tangible state which can only be reached from via some
state .

In the case of choose an arbitrary timed queueing place
,
and a transition .
Liveness implies that there exists an infinite firing sequence containing
, thus colour tokens are fired infinitely often onto place .
From (#>

So in bounded and live QPNs the existence of
is guaranteed.
The second requirement, a unique solution of (4),
is satisfied if the state space of the reduced embedded Markov chain
contains only one irreducible subset of states, provided the state space
is finite. There is only one such subset iff
the QPN has at least one home state.

Summarizing all arguments we realize that quantitative analysis of QPNs can be performed by steady state analysis of the reduced embedded Markov chain, if the QPN is bounded, live and has home states. With that, qualitative properties of the QPN are essential preconditions for a successful quantitative analysis.

The next section is dedicated to a more efficient determination of some of these properties by exploiting the PN part of the QPN.

Tue Jan 9 09:36:45 MET 1996