Abstract
A class of synchronized queueing networks with deterministic routing
is identified to be equivalent to a subclass of timed Petri nets
called marked graphs. First some structural and behavioral properties
of marked graphs are recalled and used to show interesting properties
of this class of performance models. In particular, ergodicity is
derived from boundedness and liveness of the underlying Petri net
representation, which can be efficiently computed in polynomial time
on the net structure. In case of unbounded (i.e., non-strongly-connected)
marked graphs, ergodicity is computed as a function of the average
transition firing delays. Then the problem of computing both upper
and lower bounds for the steady-state performance of timed and stochastic
marked graphs is studied. In particular, linear programming problems
defined on the incidence matrix of the underlying Petri nets are
used to compute tight (i.e., attainable) bounds for the throughput
of transitions for marked graphs with deterministic or stochastic
time associated with transitions. These bounds depend on the initial
marking and the mean values of the delays but not on the probability
distribution functions (thus including both the deterministic and
the stochastic cases). The benefits of interleaving qualitative and
quantitative analysis of marked graph models are shown.
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