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Transient behavior of fractional queues and related processes

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Abstract

We propose a generalization of the classical M/M/1 queue process. The resulting model is derived by applying fractional derivative operators to a system of difference-differential equations. This generalization includes both non-Markovian and Markovian properties, which naturally provide greater flexibility in modeling real queue systems than its classical counterpart. Algorithms to simulate M/M/1 queue process and the related linear birth-death process are provided. Closed-form expressions of the point and interval estimators of the parameters of these fractional stochastic models are also presented. These methods are necessary to make these models usable in practice. The proposed fractional M/M/1 queue model and the statistical methods are illustrated using S&P data.

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Fractional Poisson process

(2003)
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Parameter estimation for fractional Poisson processes

(2010)
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On a fractional linear birth--death process

,   (2011)
In this paper, we introduce and examine a fractional linear birth--death process $$N_{\nu}(t)$$, $$t>0$$, whose fractionality is obtained by replacing the time derivative with a fractional derivative in the system of difference-differential equations governing the state probabilities $$p_k^{\nu}(t)$$, $$t>0$$, $$k\geq0$$. We present a subordination relationship connecting $$N_{\nu}(t)$$, $$t>0$$, with the classical birth--death process $$N(t)$$, $$t>0$$, by means of the time process $$T_{2\nu}(t)$$, $$t>0$$, whose distribution is related to a time-fractional diffusion equation. We obtain explicit formulas for the extinction probability $$p_0^{\nu}(t)$$ and the state probabilities $$p_k^{\nu}(t)$$, $$t>0$$, $$k\geq1$$, in the three relevant cases $$\lambda>\mu$$, $$\lambda<\mu$$, $$\lambda=\mu$$ (where $$\lambda$$ and $$\mu$$ are, respectively, the birth and death rates) and discuss their behaviour in specific situations. We highlight the connection of the fractional linear birth--death process with the fractional pure birth process. Finally, the mean values $$\mathbb{E}N_{\nu}(t)$$ and $$\operatorname {\mathbb{V}ar}N_{\nu}(t)$$ are derived and analyzed.
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Journal
26 March 2013
2013-08-22
1303.6695 10.1007/s11009-013-9391-2