A subgeometric convergence formula for finite-level M/G/1-type Markov chains: via a block-decomposition-friendly solution for the Poisson equation of deviation matrix

Abstract

This paper considers finite-level M/G/1-type Markov chains. We introduce the fundamental deviation matrix of the infinite-level M/G/1-type Markov chain, which is a solution of the Poisson equation that the deviation matrix satisfies. With the fundamental deviation matrix, we describe a difference formula for the respective stationary distributions of the finite-level chain and its infinite-level limit. From the difference formula, we derive a subgeometric convergence formula for the stationary distribution of the finite-level chain as its maximum level goes to infinity. Using the obtained formula, we show an asymptotic formula for the loss probability in the MAP/G/1/N + 1 queue.

A sequential update algorithm for computing the stationary distribution vector in upper block-Hessenberg Markov chains

“…On the other hand, α * n of the MIP form (3. 19) is an optimal solution of Problem 3.1, which does not need any input parameter set (v, b, C) satisfying Conditions 1 and 2. The objective function of Problem 3.1 requires u * n,K given in (3.1), instead of y n .…”

“…Recall here that P (θ 0 ) is an M/G/1-type stochastic matrix. Thus, there exists some ε > 0 such that (see [19,Lemma 3.3]) ). We first note that (I − A − β A g) −1 exists due to Assumption B.1 (see, e.g., the proof of [21, Theorem 3.1.1]).…”

Level-Wise Subgeometric Convergence of the Level-Increment Truncation Approximation of M/G/1-Type Markov Chains

“…In fact, there is a study [11] on the above-mentioned unfavorable cases for the lastcolumn-block-augmented (LCBA) truncation approximation of M/G/1-type Markov chains. The LCBA truncation approximation to an M/G/1-type Markov chain can be considered a finite-level M/G/1-type Markov chain (see [11,Remark 2.2]).…”

“…In fact, there is a study [11] on the above-mentioned unfavorable cases for the lastcolumn-block-augmented (LCBA) truncation approximation of M/G/1-type Markov chains. The LCBA truncation approximation to an M/G/1-type Markov chain can be considered a finite-level M/G/1-type Markov chain (see [11,Remark 2.2]). The study [11] presents a subgeometric convergence formula for the level-wise difference between the original stationary distribution and its LCBA truncation approximation.…”

“…The LCBA truncation approximation to an M/G/1-type Markov chain can be considered a finite-level M/G/1-type Markov chain (see [11,Remark 2.2]). The study [11] presents a subgeometric convergence formula for the level-wise difference between the original stationary distribution and its LCBA truncation approximation. The subgeometric convergence formula requires two conditions: (i) the finiteness of the second moment of level increments; and (ii) the subexponentiality of the integrated tail distribution of nonnegative level increments in steady state.…”