Long time behavior of a stochastically modulated infinite server queue
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929566809849856 |
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| author | Majumder, Abhishek Pal |
| author_facet | Majumder, Abhishek Pal |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21910 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long time behavior of a stochastically modulated infinite server queue Majumder, Abhishek Pal Probability Computation 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. |
| title | Long time behavior of a stochastically modulated infinite server queue |
| topic | Probability Computation |
| url | https://arxiv.org/abs/2410.21910 |