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## Properties and Performance Bounds for Timed Marked Graphs.

Campos, J.;
Chiola, G.;
Colom, J.M.;
Silva, M.
In:
*IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Vol. 39, No. 5*, pages 386-401.
May 1992.

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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