This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Computing Bounds for the Performance Indices of Quasi-Lumpable Stochastic Well-Formed Nets
July 1994 (vol. 20 no. 7)
pp. 516-525

Structural symmetries in stochastic well-formed colored Petri nets (SWN's) lead to behavioral symmetries that can be exploited by using the symbolic reachability graph (SRG) construction algorithm. The SRC allows one to compute an aggregated reachability graph (RG) and a "lumped" continuous time Markov chain (CTMC) that contain all the information needed to study the qualitative properties and the performance of the modeled system, respectively. Some models exhibit qualitative behavioral symmetries that are not completely reflected at the CTMC level. We call them quasi-lumpable SWN models. In these cases, exact performance indices can be obtained by avoiding the aggregation of those markings that are qualitatively, but not quantitatively, equivalent. An alternative approach consists of aggregating all the qualitatively equivalent states and computing approximated performance indices. In this paper, a technique is proposed to compute bounds on the performance of SWN models of this kind, using the results we have presented elsewhere. The technique is based on the Courtois and Semal bounded aggregation method.

[1] G. Chiola, C. Dutheillet, G. Franceschinis, and S. Haddad, "Stochastic well-formed colored nets and symmetric modeling,"IEEE Trans. Comput., vol. 12, no. 11, pp. 1343-1360, Nov., 1993.
[2] P. J. Courtois and P. Semal, "Computable bounds on conditional steady-state probabilities in large markov chains and queueing models,"IEEE J. Select. Areas Commun., vol. 4, no. 6, pp. 926-937, Sept. 1986.
[3] C. Dutheillet and S. Haddad, "Regular stochastic Petri nets," inProc. 10th Int. Conf. on Application and Theory of Petri Nets(Bonn, Germany), June 1989.
[4] G. Franceschinis and R. R. Muntz, "Bounds for quasi-lumpable Markov chains," inProc. Performance '93, Rome, Italy, Sept. 1993.
[5] S. Haddad, "Une categorie regulier de reseau de Petri de haut niveau: Definition, proprietes et reductions" (in French), Ph.D. dissertation, These de Doctorat, RR87/197, Lab. MASI, Universite P. et M. Curie, Paris, France, 1987.
[6] P. Huber, A. M. Jensen, L. O. Jepsen, and K. Jensen, "Towards reachability trees for high-level Petri nets,"Lecture Notes Comput. Sci., vol. 188, pp. 215-233, 1985.
[7] K. Jensen, "Colored Petri nets and the invariant method,"Theoretical Comput. Sci., vol. 14, pp. 317-336, 1981.
[8] J. G. Kemeny and J. L. Snell,Finite Markov Chains. Princeton, NJ: Van Nostrand, 1960.
[9] C. Lin and C. Marinescu, "On stochastic high-level Petri nets," inProc. Int. Workshop on Petri Nets and Performance Models(Madison, WI), Aug. 1987.

Index Terms:
Petri nets; stochastic processes; Markov processes; performance evaluation; performance indices; quasilumpable stochastic well-formed Nets; stochastic well-formed colored Petri nets; symbolic reachability graph; aggregated reachability graph; continuous time Markov chain; qualitative properties; quasilumpable SWN models; approximated performance indices; SWN models; bounded aggregation method
Citation:
G. Franceschinis, R.R. Muntz, "Computing Bounds for the Performance Indices of Quasi-Lumpable Stochastic Well-Formed Nets," IEEE Transactions on Software Engineering, vol. 20, no. 7, pp. 516-525, July 1994, doi:10.1109/32.297940
Usage of this product signifies your acceptance of the Terms of Use.