Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

Abstract

The paper is largely of a review nature. It considers two main methods used to study stability and obtain appropriate quantitative estimates of perturbations of (inhomogeneous) Markov chains with continuous time and a finite or countable state space. An approach is described to the construction of perturbation estimates for the main five classes of such chains associated with queuing models. Several specific models are considered for which the limit characteristics and perturbation bounds for admissible "perturbed"processes are calculated.

Ergodicity and perturbation bounds forMt/Mt/1 queue with balking, catastrophes, server failures and repairs

“…The next statement follows immediately from Theorem 1 [21] (see also the first corresponding homogeneous result in [12] and for inhomogeneous situation in [15]). for some positive N, γ 0 .…”

“…Consider here the application of general perturbation bounding (see the recent review in [21]) for the models under study. Consider a "perturbed" queue-length processX(t), t ≥ 0 with the corresponding transposed intensity matrixĀ(t), where the "perturbation" matrix A(t) = A(t) −Ā(t) is small in a sense.…”

“…Remark. Applying estimates (21), ( 22) from [21], one can also obtain estimates for timedependent perturbations p(t) −p(t) 1D , in the case of coinciding the corresponding initial conditions X(0) =X(0). In addition, using such estimates, we can make the necessary corrections to the inaccurately known intensities values by changing them accordingly.…”

Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm Method

“…Moreover, whenever X(t) is weakly ergodic, the analysis can be carried on beyond what is stated in the Theorem 1. For example, one can obtain the perturbation bounds (see e.g., [40]) and study different state space truncation options: one-sided or two sided (see e.g., [29,41,42]).…”

Ergodicity and Perturbation Bounds for $M_t/M_t/1$ Queue with Balking, Catastrophes, Server Failures and Repairs

“…Consider here the application of general perturbation bounding (see the recent review in [15]) for the models under study. Consider a "perturbed" queue-length process X(t), t ≥ 0 with the corresponding transposed intensity matrix Ā(t), where the "perturbation" matrix Â(t) = A(t) − Ā(t) is small in a sense.…”