Sufficient Conditions for Existence of a Fixed Point in Stochastic Reward Net-Based Iterative Models
Issue No. 09 - September (1996 vol. 22)
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/32.541435
<p><b>Abstract</b>—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 <it>fixed-point iteration</it> 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 <it>existence</it> 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.</p>
Stochastic Petri nets, fixed-point iteration, existence, sufficient conditions.
K. S. Trivedi and V. Mainkar, "Sufficient Conditions for Existence of a Fixed Point in Stochastic Reward Net-Based Iterative Models," in IEEE Transactions on Software Engineering, vol. 22, no. , pp. 640-653, 1996.