Bounding the equilibrium distribution of Markov population models

Abstract

SUMMARYWe propose a bounding technique for the equilibrium probability distribution of continuous-time Markov chains with population structure and infinite state space. We use Lyapunov functions to determine a finite set of states that contains most of the equilibrium probability mass. Then we apply a refinement scheme based on stochastic complementation to derive lower and upper bounds on the equilibrium probability for each state within that set. To show the usefulness of our approach, we present experimental results for several examples from biology.

Infinite level-dependent QBD processes and matrix-analytic solutions for stochastic chemical kinetics

“…Recently a method that computes componentwise bounds on the stationary probability distribution of Markov population models was proposed [6]. The work in this paper is complementary to that in several ways.…”

“…The work in this paper is complementary to that in several ways. Although both techniques use a suitable Lyapunov function to locate a finite subset of states where a specified percentage of the stationary probability mass resides, the technique proposed in this paper for the LDQBD model requires a larger finite subset of states with which to work than the technique in [6] for the same percentage. This clearly implies larger memory requirements.…”

“…Note that the largest linear system solved has as many equations and unknowns as the number of states in level high. On the contrary, the computation of bounds with comparable accuracy with the bounding technique in [6] would require a finite subset of states with at least comparable size to those that lie between levels low and high to be explored and a system of linear equations with coefficient matrix incident on this relatively larger subset to be solved for multiple right-hand sides. Although there is no duplication of work when the coefficient matrix is relatively small to be factorized and a direct solution method employed, the nonzero entries of the same coefficient matrix will be used over and over again when the finite subset is relatively large to warrant the use of an iterative method in the solution procedure [26, pp.…”

A computational approach to steady-state convergence of fluid limits for Coxian queuing networks with abandonment

“…Thanks to Proposition 1, it holds that Z π N (Z)dg(Z) = 0 and we can use the idea from [11] to infer…”

“…In this section we extend our exponential model (11) and allow the abandonment distribution to depend on the fact whether a customer is waiting or being served. While this usually does not apply to computer systems where clients are jobs and abandonment is timeout, in the case of call centers, this can be explained by the changed mindset of a customer that entered service.…”