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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.

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Index Terms:
Stochastic Petri nets, fixed-point iteration, existence, sufficient conditions.
Citation:
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
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