2015-07-01
Analysis of Markov-modulated infinite-server queues in the central-limit regime
Publication
Publication
Probability in the Engineering and Informational Sciences , Volume 29 - Issue 3 p. 433- 459
This paper focuses on an infinite-server queue modulated by an independently evolving finite-state Markovian background process, with transition rate matrix $Q\equiv(q_{ij})_{i,j=1}^d$. {Both arrival rates and service rates are depending on the state of the background process.} The main contribution concerns the derivation of central limit theorems for the number of customers in the system at time $t\ge 0$, in the asymptotic regime in which the arrival rates $\lambda_i$ are scaled by a factor $N$, and the transition rates $q_{ij}$ by a factor $N^\alpha$, with $\alpha \in \mathbb R^+$. The specific value of $\alpha$ has a crucial impact on the result: (i)~for $\alpha>1$ the system essentially behaves as an M/M/$\infty$ queue, and in the central limit theorem the centered process has to be normalized by $\sqrt{N}$; (ii)~for $\alpha<1$, the centered process has to be normalized by $N^{{1-}\alpha/2}$, with the deviation matrix appearing in the expression for the variance.
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Cambridge U.P. | |
doi.org/10.1017/S026996481500008X | |
Probability in the Engineering and Informational Sciences | |
Coarse grained stochastic methods for biochemical reactions | |
Organisation | Evolutionary Intelligence |
Blom, J., de Turck, K., & Mandjes, M. (2015). Analysis of Markov-modulated infinite-server queues in the central-limit regime. Probability in the Engineering and Informational Sciences, 29(3), 433–459. doi:10.1017/S026996481500008X |
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