Abstract
The combinatorial explosion of the state space of Stochastic Petri Nets (SPNs) is a well known problem that inhibits the exact solution of large SPNs, and therefore a broad use of this kind of Petri Nets as a modelling tool. The same problem exists also for other modelling formalisms like for example Queueing Networks (QNs). In [13, 3] a class of QNs whose solution can be computed in an easy way was defined. For this class of models the solution can be factorized into terms that refer to each single queue of the network. This solution is known as Product Form Solution (PFS).
In this paper we compare two different approaches to PFS for SPNs. In both proposals the solution is obtained as a product form of terms, each term corresponding to a place in the SPN.
The first approach (by Lazar and Robertazzi) allows the PFS to be detected at state space level by inspecting the structure of the reachability graph. The second one (by Henderson, Lucic and Taylor) allows the PFS to be detected at structural level, that is to say without inspection of the reachability graph. In this paper we try to put the two approaches into a common framework and to show the important role played by T-invariants.
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References
I.F. Akyildiz. Exact product form solution for queueing networks with blocking. IEEE Transactions on Computers, 36(1):122–125, January 1987.
I.F. Akyildiz. On the exact and approximate throughput analysis of closed queueing networks with blocking. IEEE Transactions on Software Engineering, 14(1):62–70, January 1988.
F. Baskett, K. M. Chandy, R. R. Muntz, and F. Palacios. Open, closed and mixed networks of queues with different classes of customers. Journal of the ACM, 22(2):248–260, April 1975.
A. Bertoni and M. Torelli. Probabilistic Petri nets and semi-markov processes. In Proc. 2 nd European workshop on Petri nets, Bad Honnef, West Germany, September 1981.
J. P. Buzen. Computational algorithms for closed queueing networks with exponential servers. Communications of the ACM, 16(9):527–531, September 1973.
G. Florin and S. Natkin. Les reseaux de Petri stochastiques. Technique et Science Informatiques, 4(1), February 1985.
G. Florin and S. Natkin. Matrix product form solution for closed synchronized queueing networks. In Proc. 3rd Intern. Workshop on Petri Nets and Performance Models, pages 29–39, Kyoto, Japan, December 1989. IEEE-CS Press.
W. Henderson and D. Lucic. Exact results in the aggregation and disaggregation of stochastic Petri nets. In Proc. 4th Intern. Workshop on Petri Nets and Performance Models, pages 166–175, Melbourne, Australia, December 1991. IEEE-CS Press.
W. Henderson, D. Lucie, and P.G. Taylor. A net level performance analysis of stochastic petri nets. Journal of Astraulian Mathematical Society, Ser. B, 31:176–187, 1989.
W. Henderson and P.G. Taylor. Aggregation methods in exact performance analysis of stochastic Petri nets. In Proc. 3rd Intern. Workshop on Petri Nets and Performance Models, pages 12–18, Kyoto, Japan, December 1989. IEEE-CS Press.
W. Henderson and P.G. Taylor. Embedded processes in stochastic petri nets. IEEE Transactions on Software Engineering, 17(2), February 1991.
A. Hordijk and N. van Dijk. Networks of queues with blocking. In Proc. PERFORMANCE '81, pages 51–65, 1981.
J. R. Jackson. Jobshop-like queueing systems. Management Science, 10(1):131–142, October 1963.
A.A. Lazar. An algebraic topological approach to markovian queueing networks. In Proc. of the 1984 Information Sciences and Systems Conference, Princeton, NJ, USA, March 1984. pp.437–442.
A.A. Lazar and T.G. Robertazzi. The lattice structure of a bus oriented multiprocessor system. In Proc. of 25 th Allerton Conference on Communication, Control, & Computing, Urbana-Champaigne, IL, USA, 1987.
A.A. Lazar and T.G. Robertazzi. Markovian Petri net protocols with product form solution. In Proc. of IEEE INFOCOM'87, San Francisco, CA, USA, March 1987. Also in Performance Evaluation, Vol.12, 1991, pp.67–77.
M. K. Molloy. Performance analysis using stochastic Petri nets. IEEE Transaction on Computers, 31(9):913–917, September 1982.
T. Murata. Petri nets: properties, analysis, and applications. Proceedings of the IEEE, 77(4):541–580, April 1989.
M. Reiser and S. S. Lavenberg. Mean value analysis of closed multichain queueing networks. Journal of the ACM, 27(2):313–322, April 1980.
T. G. Robertazzi. Why most stochastic Petri nets are non-product form networks. Technical report, University of New York, 1991.
T.G. Robertazzi. Computer Networks and Systems: Queueing Theory and Performance Evaluation. Springer Verlag, 1990.
M. Silva. Las Redes de Petri en la Automatica y la Informatica. Editorial AC, Madrid, Spain, 1985. (in Spanish).
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© 1992 Springer-Verlag Berlin Heidelberg
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Donatelli, S., Sereno, M. (1992). On the product form solution for Stochastic Petri Nets. In: Jensen, K. (eds) Application and Theory of Petri Nets 1992. ICATPN 1992. Lecture Notes in Computer Science, vol 616. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55676-1_9
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DOI: https://doi.org/10.1007/3-540-55676-1_9
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