On the tra c equations for batch routing queueing networks and stochastic Petri nets

The tra c equations are a set of linear equations, which are the basis for the exact analysis of product form queueing networks, and the approximate analysis of non-product form queueing networks. Conditions characterising the structure of the network that guarantees the existence of a solution for the tra c equations are therefore of great importance. This note provides a necessary and su cient condition on the structure of the network for a solution of the tra c equations to exist. The basis of this structural characterisation is the equivalence between batch routing queueing networks and stochastic Petri nets at the level of the underlying stochastic process. Based on new and known results for stochastic Petri nets, this note shows that the new condition stating that each transition is covered by a minimal closed support T -invariant is necessary and su cient for the existence of a solution for the tra c equations for batch routing queueing networks and stochastic Petri nets. 1991 AMS Subject Classi cation: Primary: 60K25, Secondary: 90B22

[1]  Thomas G. Robertazzi,et al.  Markovian Petri Net Protocols with Product Form Solution , 1991, Perform. Evaluation.

[2]  Richard J. Boucherie,et al.  Spatial birth-death processes with multiple changes and applications to batch service networks and c , 1990 .

[3]  C. E. M. Pearce,et al.  Closed queueing networks with batch services , 1990, Queueing Syst. Theory Appl..

[4]  Peter Whittle,et al.  Systems in stochastic equilibrium , 1986 .

[5]  James L. Coleman.  Stochastic Petri nets with product form equilibrium distributions. , 1993 .

[6]  K. Mani Chandy,et al.  Open, Closed, and Mixed Networks of Queues with Different Classes of Customers , 1975, JACM.

[7]  Richard F. Serfozo,et al.  Markovian network processes: Congestion-dependent routing and processing , 1989, Queueing Syst. Theory Appl..

[8]  Tadao Murata,et al.  Petri nets: Properties, analysis and applications , 1989, Proc. IEEE.

[9]  Richard J. Boucherie,et al.  Product forms for queueing networks with state-dependent multiple job transitions , 1991, Advances in Applied Probability.

[10]  Anthony Unwin,et al.  Reversibility and Stochastic Networks , 1980 .

[11]  Dirk Frosch.  Product Form Solutions for Closed Synchronized Systems of Stochastic Sequential Processes , 1992, Universität Trier, Mathematik/Informatik, Forschungsbericht.

[12]  Michael K. Molloy.  Performance Analysis Using Stochastic Petri Nets , 1982, IEEE Transactions on Computers.

[13]  Matteo Sereno,et al.  On the Product Form Solution for Stochastic Petri Nets , 1992, Application and Theory of Petri Nets.

[14]  J. R. Jackson.  Networks of Waiting Lines , 1957 .

[15]  Peter G. Taylor,et al.  A net level performance analysis of stochastic Petri nets , 1989, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.