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Long time behavior of a stochastically modulated infinite server queue

Abhishek Pal Majumder

Abstract

We consider an infinite server queue where the arrival and the service rates are both modulated by a stochastic environment governed by an $S$-valued stochastic process $X$ that is ergodic with a limiting measure $π\in \mathcal{P}(S)$. Under certain conditions when $X$ is semi-Markovian and satisfies the renewal regenerative property, long-term behavior of the total counts of people in the queue (denoted by $Y:=(Y_{t}:t\ge 0)$) becomes explicit and the limiting measure of $Y$ can be described through a well-studied affine stochastic recurrence equation (SRE) $X\stackrel{d}{=}CX+D,\,\, X\perp\!\!\!\perp (C, D)$. We propose a sampling scheme from that limiting measure with explicit convergence diagnostics. Additionally, one example is presented where the stochastic environment makes the system transient, in absence of a `no-feedback' assumption.

Long time behavior of a stochastically modulated infinite server queue

Abstract

We consider an infinite server queue where the arrival and the service rates are both modulated by a stochastic environment governed by an -valued stochastic process that is ergodic with a limiting measure . Under certain conditions when is semi-Markovian and satisfies the renewal regenerative property, long-term behavior of the total counts of people in the queue (denoted by ) becomes explicit and the limiting measure of can be described through a well-studied affine stochastic recurrence equation (SRE) . We propose a sampling scheme from that limiting measure with explicit convergence diagnostics. Additionally, one example is presented where the stochastic environment makes the system transient, in absence of a `no-feedback' assumption.

Paper Structure

This paper contains 20 sections, 7 theorems, 133 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and description of the semi-Markovian $X$
  3. Main results
  4. Sampling scheme
  5. Exceedance event computation
  6. Example $1$:
  7. Example $2$:
  8. Sharpness of the Assumption \ref{['A0']}
  9. Conclusion and future directions
  10. Proof of Theorem \ref{['T1']}
  11. Set-up for the proof of Lemma \ref{['L1']}
  12. Proof of Lemma \ref{['L1']}
  13. Proof of Step $1$
  14. Proof of Step $2$
  15. Proof of Claim $1$
  16. ...and 5 more sections

Key Result

Proposition 2.2

Suppose Assumption As0 holds. For any $t>\tau_{0}^{j},$

Figures (4)

  • Figure 1: A sample trajectory of \ref{['inftysq']} where $X$ is a $\{0,1\}$-valued CTMC with switching rates $\lambda_{01},\lambda_{10}$ where $\lambda_{10}=1000\lambda_{01}$, and $\lambda(\cdot)=1, \mu(X_{s})=X_{s}$ for any $s\ge 0.$
  • Figure 2: In Example $1$ we generated samples from mixture measure $\mathcal{L}(W_{{\mathbf j}})$ applying the sampling scheme for $n=20,$ for ${\mathbf j}=(2,8)$ and ${\mathbf j}=(6,4)$ and the histograms of sampled Poisson counts with the mixture $W_{{\mathbf j}}$ are also presented
  • Figure 3: In Example $2$ for $\mathcal{G}_{\text{Exp}}$ we generated samples from mixture measure $\mathcal{L}(W_j)$ applying the sampling scheme (in Section \ref{['SS']}) for $n=20,$ for $j=3$ and $j=0$ and the histograms of sampled Poisson counts with the mixture $W_{j}$ are also presented
  • Figure 4: In Example 2 for $\mathcal{G}_{\text{Pareto}}$, samples were generated from the mixing measure $\mathcal{L}(W_j)$ using the sampling scheme for $n=20$, for $j=3$ and $j=0$. The histograms of the sampled Poisson counts with the mixture $W_j$ are also shown.

Theorems & Definitions (18)

  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Theorem 1
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Proposition 4.1
  • Remark 4.2
  • proof
  • ...and 8 more