On the sensitivity of transient solutions of Markov models

Abstract

We consider the sensitivity of transient solutions of Markov models to perturbations in their generator matrices. The perturbations can either be of a certain structure or can be very general. We consider two different measures of sensitivity and derive upper bounds on them. The derived bounds are sharper than previously reported bounds in the literature. Since the sensitivity analysis of transient solutions is intimately related to the condition of the exponential of the CTMC matrix, we derive an expression for the condition number of the CTMC matrix exponential which leads to some interesting implications. We compare the derived sensitivity bounds both numerically and analytically with those reported in the literature.

Component Importance Measures for Real-Time Computing Systems in the Presence of Common-Cause Failures

“…In fact, several kinds of reward functions have been discussed in the past literatures [29,46]. Since this paper deals with reliability and availability measures, we consider instantaneous reward functions in MRM.…”

“…∂R k (t) . (5.45) Essentially, these measures can be computed from AIB k and RIB k (t), i.e., 46) and…”

On the Sensitivity of a Dynamic Measure of Financial Inequality

“…The sensitivity of steady state solution of Markov chain has been investigated, among others, in (Schweitzer, 1968) and successively a study of the sensitivity bounds for stationary distribution of Markov chain has been presented in (Haviv and Van der Heyden, 1984). Ramesh and Trivedi (1993) the authors derive measure of sensitivity and its bounds for transient solutions under structured and unstructured perturbations for CTMC. The study of the effect of perturbations for a discrete-time Markov reward process on the total expected reward has, instead, been performed in (Yu et al, 2009).…”

“…To the best of our knowledge, a sensitivity analysis of the dynamic inequality with respect to the uncertainty in the rating dynamics has never been faced. Following (Ramesh and Trivedi, 1993), we introduce a structured perturbation and we perform several simulations by varying the perturbation parameters and the way they affect the generator of the CTMC. This is done to analyze the differences on the measure of inequality assessed in the nominal and the perturbed model (i.e., that one without perturbation and that one with the introduction of a perturbation, respectively).…”

Computing exponentials of essentially non-negative matrices entrywise to high relative accuracy

“…As they are at best normwise backward stable [19], the computed matrix exponentials have relative errors that are bounded in norm and dependent on a condition number for normwise perturbations. This condition number is well bounded for normal matrices and generator matrices of Markov chains, but in general, it may grow very quickly as the norm of the matrix increases [17,22,24]. Moreover, even for problems where this condition number is well bounded, the entrywise relative accuracy is not guaranteed by these two methods; namely smaller entries of the exponential matrix may be computed with lower relative accuracy.…”