Approximations for the Moments of Nonstationary and State Dependent Birth-Death Queues
Approximations for the Moments of Nonstationary and State Dependent Birth-Death Queues
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非平稳和状态相关的生死队列矩的近似
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Jamol Pender
中科院分区：
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作者：
Stefan Engblom;Jamol Pender
本文提出了一种逼近一维马尔可夫生灭过程非平稳矩动态的新方法。通过将马尔可夫过程的转移概率展开为Poisson-Charlier多项式，我们能够估计马尔可夫过程的任意矩，即使矩方程组可能不是闭合的。利用新的加权离散Sobolev空间，我们得到了转移概率的显式误差界和利用截断展开逼近马尔可夫过程矩的新的弱先验估计。利用我们的误差界和估计，我们能够证明当我们在展开式中增加更多的项并给出截断误差的显式界时，我们的近似收敛到真正的随机过程。因此，我们是排队文献中第一篇提供矩闭合近似的误差界和性能估计的论文。最后，我们对排队论文献中的一些重要模型进行了数值实验，结果表明我们的展开技术在估计这些马尔可夫过程的矩动态时是准确的，只需要很少的展开项。
In this paper we propose a new method for approximating the nonstationary moment dynamics of one dimensional Markovian birth-death processes. By expanding the transition probabilities of the Markov process in terms of Poisson-Charlier polynomials, we are able to estimate any moment of the Markov process even though the system of moment equations may not be closed. Using new weighted discrete Sobolev spaces, we derive explicit error bounds of the transition probabilities and new weak a priori estimates for approximating the moments of the Markov processs using a truncated form of the expansion. Using our error bounds and estimates, we are able to show that our approximations converge to the true stochastic process as we add more terms to the expansion and give explicit bounds on the truncation error. As a result, we are the first paper in the queueing literature to provide error bounds and estimates on the performance of a moment closure approximation. Lastly, we perform several numerical experiments for some important models in the queueing theory literature and show that our expansion techniques are accurate at estimating the moment dynamics of these Markov process with only a few terms of the expansion.