Skip to main content Accessibility help

On closed support T-Invariants and the traffic equations

  • Richard J. Boucherie (a1) and Matteo Sereno (a2)


The traffic equations 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 traffic equations are therefore of great importance. This note shows that the new condition stating that each transition is covered by a minimal closed support T-invariant, is necessary and sufficient for the existence of a solution for the traffic equations for batch routing queueing networks and stochastic Petri nets.


Corresponding author

Postal address: Department of Econometrics, Universiteit van Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands. E-mail address:
∗∗ Postal address: Dipartimento di Informatica, Università di Torino, Corso Svizzera 185, 10149 Torino, Italy.


Hide All
[1] Ajmone Marsan, M., Balbo, G., Bobbio, A., Chiola, G., Conte, G., and Cumani, A. (1989). The effect of execution policies on the semantics and analysis of stochastic Petri nets. IEEE Trans. Software Eng. 15, 832846.
[2] Baskett, F., Chandy, K.M., Muntz, R.R., and Palacios, F.G. (1975). Open, closed and mixed networks of queues with different classes of customers. J. ACM 22, 248260.
[3] Boucherie, R.J., and Van Dijk, N.M. (1991). Product forms for queueing networks with state dependent multiple job transitions. Adv. Appl. Prob. 23, 152187.
[4] Donatelli, S., and Sereno, M. (1992). On the product form solution for stochastic Petri nets. In Proc. 13th International Conference on Application and Theory of Petri Nets, Sheffield, UK, pp. 154172.
[5] Frosch, D., and Natarajan, K. (1992). Product form solutions for closed synchronized systems of stochastic sequential processes. In Proc. 1992 International Computer Symposium, Taichung, Taiwan, pp. 392402.
[6] Henderson, W., Lucic, D., and Taylor, P.G. (1989). A net level performance analysis of stochastic Petri nets. J. Austral. Math. Soc. Series B 31, 176187.
[7] Henderson, W., Pearce, C.E. M., Taylor, P.G., and Van Dijk, N.M. (1990). Closed queueing networks with batch services. Queueing Systems 6, 5970.
[8] Henderson, W., and Taylor, P.G. (1991). Some new results on queueing networks with batch movement. J. Appl. Prob. 28, 409421.
[9] Lazar, A.A., and Robertazzi, T.G. (1991). Markovian Petri net protocols with product form solution. Performance Evaluation 12, 6777.
[10] Martinez, J., and Silva, M. (1982). A simple and fast algorithm to obtain all invariants of a generalized Petri net. In Application and theory of Petri nets – selected papers from the 1st and 2nd European Workshop on Application and Theory of Petri Nets, ed. Girault, C. and Resig, W. (Informatik–Fachberichte 52.) Springer-Verlag, pp. 301310.
[11] Memmi, G., and Roucairol, G. (1979). Linear algebra in net theory. In Net Theory and Applications, – Proc. Advanced Course on General Net Theory of Processes and Systems, Hamburg. (Lecture Notes in Computer Science 84,.) Pp. 213223.
[12] Molloy, M.K. (1982). Performance analysis using stochastic Petri nets. IEEE Trans. Computers 31, 913917.
[13] Murata, T. (1989). Petri nets: properties, analysis and applications. In Proc. IEEE 77, 541580.
[14] Silva, M. (1985). Las Redes de Petri en la Automatica y la Informatica. Editorial AC, Madrid, Spain. (In Spanish.)


MSC classification

On closed support T-Invariants and the traffic equations

  • Richard J. Boucherie (a1) and Matteo Sereno (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed