This Article 
 Bibliographic References 
 Add to: 
Sufficient Conditions for Existence of a Fixed Point in Stochastic Reward Net-Based Iterative Models
September 1996 (vol. 22 no. 9)
pp. 640-653

Abstract—Stochastic Petri net models of large systems that are solved by generating the underlying Markov chain pose the problem of largeness of the state-space of the Markov chain. Hierarchical and iterative models of systems have been used extensively to solve this problem. A problem with models which use fixed-point iteration is the theoretical proof of existence, uniqueness, and convergence of the fixed-point equations, which still remains an "art." In this paper, we establish conditions, in terms of the net structure and the characteristics of the iterated variables, under which existence of a solution is guaranteed when fixed-point iteration is used in stochastic Petri nets. We use these conditions to establish the existence of a fixed point for a model of a priority scheduling system, at which tasks may arrive according to a Poisson process or due to spawning or conditional branching of other tasks in the system.

[1] M.Ajmone Marsan,G. Balbo,, and G. Conte,“A class of generalized stochastic Petri nets for the performance evaluation of multiprocessor systems,” ACM Trans. Computer Systems, pp. 93-122, vol. 2, no. 2, May 1984.
[2] J.T. Blake and K.S. Trivedi, "Reliability Analysis of Interconnection Networks Using Hierarchical Composition," IEEE Trans. Reliability, vol. 38, no. 1, pp. 111-120, Apr. 1989.
[3] A. Chesnais, E. Gelenbe, and I. Mitrani, "On the Modeling of Parallel Access to Shared Data," Comm. ACM, vol. 26, no. 3, pp. 196-202, Mar. 1983.
[4] H. Choi and K. Trivedi, "Approximated Performance Models of Polling Systems Using Stochastic Petri Nets," Proc. IEEE INFORCOM '92, pp. 2,306-2,314,Florence, Italy, May 1992.
[5] G. Ciardo, "Analysis of Large Stochastic Petri net Models," PhD thesis, Duke Univ., Durham, N.C., 1989.
[6] G. Ciardo, A. Blakemore, P.F. Chimento, and K.S. Trivedi, "Automated Generation and Analysis of Markov Reward Models Using Stochastic Reward Nets, "Linear Algebra, Markov Chains, and Queueing Models, IMA Volumes in Math. and its Applications, A. Friedman and J.W. Miller, eds., vol. 48, pp. 145-191.Heidelberg, Germany: Springer-Verlag, 1993.
[7] G. Ciardo, J. Muppala, and K. Trivedi, SPNP: Stochastic Petri Net Package Proc. Third Int'l Workshop Petri Nets and Performance Models, pp. 142-151, 1989.
[8] G. Ciardo and K. Trivedi, “A Decomposition Approach for Stochastic Reward Net Models,” Performance Evaluation, vol. 18, no. 1, pp. 37-59, 1993.
[9] J. Daigle and C.E. Houstis, "Analysis of a Task-Oriented Multipriority Queueing System," IEEE Trans. Comm., vol. 29, no. 11, pp. 1,669-1,677, Nov. 1981.
[10] W. Fleming, Functions of Several Variables, second edition. New York: Springer-Verlag, 1977.
[11] B.R. Haverkort, "Approximate Performability and Dependability Analysis Using Generalized Stochastic Petri Nets," Performance Evaluation, vol. 18, no. 1, pp. 61-78, July 1993.
[12] P. Heidelberger and K. Trivedi, "Queueing Network Models for Parallel Processing with Asynchronous Tasks," IEEE Trans. Computers, vol. 31, no. 11, pp. 1,099-1,108, Nov. 1982.
[13] P. Heidelberger and K.S. Trivedi, "Analytic Queueing Models for Programs with Internal Concurrency," IEEE Trans. Computers, vol. 32, pp. 73-82, 1983.
[14] J.-B. Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms.Berlin: Springer-Verlag, 1993.
[15] V.G. Kulkarni, Modeling and Analysis of Stochastic Systems. Chapman&Hall, 1995.
[16] P. Lancaster and M. Tismenetsky, The Theory of Matices, second edition. San Diego: Academic Press, 1985.
[17] V. Mainkar and K.S. Trivedi, "Approximate Analysis of Priority Scheduling Systems Using Stochastic Reward Nets," Proc. 13th Int'l Conf. Distributed Computing Systems, pp. 466-473,Pittsburgh, Pa., May 1993.
[18] M.K. Molloy, "Performance Analysis Using Stochastic Petri Nets, IEEE Trans. Computers, vol. 31, no. 9, pp. 913-917, Sept. 1982.
[19] J.K. Muppala and K.S. Trivedi, "Composite Performance and Availability Analysis Using a Hierarchy of Stochastic Reward Nets," Proc. Fifth Int'l Conf. Modeling Techniques and Tools for Computer Performance Evaluation, G. Balbo, ed., Torino, Italy, Feb. 1991.
[20] T. Nishida, "Approximate Analysis for Heterogeneous Multiprocessor Systems with Priority Jobs," Performance Evaluation, vol. 15, pp. 77-88, June 1992.
[21] J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, Inc., San Diego, CA, 1970.
[22] J.L. Peterson, Petri Net Theory and the Modeling of Systems.Englewood Cliffs, N.J.: Prentice Hall, 1981.
[23] S. Sahner and K. Trivedi,“A hierarchical, combinatorial-markov method of solving complex reliability models,”inProc. 1986 Fall Joint Comput. Conf.,pp. 817–825, Nov. 1986.
[24] B. Simon, "Priority Queues with Feedback," J. ACM, vol. 31, no. 1, pp. 134-149, 1984.
[25] A. Thomasian and P.F. Bay, "Analytic Queueing Network Models for Parallel Processing of Task Systems," IEEE Trans. Computers, vol. 35, no. 12, pp. 1,045-1,054, Dec. 1986.
[26] L.A. Tomek and K.S. Trivedi, "Fixed-Point Iteration in Availability Modeling," Informatik-Fachberichte, M. Dal Cin, ed., vol. 91, Fehlertolerierende Rechensysteme, pp. 229-240, Springer-Verlag, Berlin, 1991.
[27] K.S. Trivedi, Probability and Statistics with Reliability, Queuing, and Computer Science Applications. Prentice Hall, 1982.
[28] K.S. Trivedi and R. Geist, "Decomposition in Reliability Analysis of Fault-Tolerant Systems," IEEE Trans. Reliability, vol. 32, no. 5, pp. 463-468, Dec. 1983.
[29] M. Veeraraghavan and K.S. Trivedi, "Hierarchical Modeling for Reliability and Performance Measures," Concurrent Computations: Algorithms, Architecture, and Technology, S.K. Tewksbury, B.W. Dickson, and S.C. Schwartz, eds. New York: Plenum Press, 1987.

Index Terms:
Stochastic Petri nets, fixed-point iteration, existence, sufficient conditions.
Varsha Mainkar, Kishor S. Trivedi, "Sufficient Conditions for Existence of a Fixed Point in Stochastic Reward Net-Based Iterative Models," IEEE Transactions on Software Engineering, vol. 22, no. 9, pp. 640-653, Sept. 1996, doi:10.1109/32.541435
Usage of this product signifies your acceptance of the Terms of Use.