Stochastic Comparison of Phase-type Distributions

Abstract

Abstract: In this paper, we analyze stochastic comparison of phase-type(PH)distributions by using stochastic dominance in applied probability, sufficient conditions and necessary and sufficient conditions for comparing two phase-type distributions under first and second degree stochastic dominance are provided. We prove that conditions to compare variabilities of two discrete phase-type distributions are different from those comparing variabilities of two continuous phase-type distributions. Conditions comparing two continuous phase-type distributions under second degree stochastic dominance are independent of their means, but conditions comparing two discrete phase-type distributions depend on their means. The results in this article can be applied to the problem for the minimization coefficient of variation and variabilities of phase-type distributions, and they can also be used to analyze the effect of variability of arrival process or service time on queue length or waiting time in queueing systems.

Key words: phase-type distributions, variability, stochastic dominance, uniformization method, service systems

摘要： 本文运用应用概率中的随机占优研究位相型(PH)分布的随机比较问题,具体给出在一阶、二阶随机占优下比较两个离散PH分布或两个连续PH分布的充分条件及充分必要条件。研究表明,比较两个离散PH分布可变性的条件与比较两个连续PH分布可变性的条件不同,在二阶随机占优意义下比较两个连续PH分布的条件与均值无关,而比较两个离散PH分布的条件与均值有关。本文的结果可用于研究PH分布的最小变异系数问题和可变性问题,也可用于研究带有PH到达间隔或PH服务的排队系统中到达过程或服务时间可变性对系统队长或等待时间的影响。

关键词: 位相型分布, 可变性, 随机占优, 一致化方法, 服务系统

CLC Number:

O226

YU Hai-bo. Stochastic Comparison of Phase-type Distributions[J]. Operations Research and Management Science, 2014, 23(2): 64-72.

禹海波. 服务系统中位相型分布的随机比较[J]. 运筹与管理, 2014, 23(2): 64-72.

References

[1] Neuts M F. Probability distributions of phase type[A]. In Liber Amicorum, Prof. Emeritus H. Florin, University of Louvain, Belgium, 1975. 173-206.
[2] Neuts M F. Matrix-geometric solutions in stochastic models: an algorithmic approach[M]. Baltimore: Johns Hopkins University Press, 1981.
[3] Neuts M F. Structured stochastic matrices of M/G/1 type and their applications[M]. New York: Marcel Dekker, 1989.
[4] Neuts M F. Algorithmic probability: a collection of problems[M]. New York: Chapman & Hall, 1995.
[5] O'Cinneide C A. Characterization of phase-type distributions[J]. Stochastic Models, 1990, 6(1): 1-57.
[6] Latouche G, Ramaswami V. Introduction to matrix analytic methods in stochastic models[J]. SIAM, Philadelphia, 1999.
[7] 田乃硕,李泉林.PH分布及其在随机模型中的应用[J].应用数学与计算数学学报,1995,9(1):1-15.
[8] 张忠君,周岩,田乃硕.M/G/1休假排队中的PH封闭性[J].燕山大学学报,1999,23(2):153-156.
[9] 徐秀丽,田乃硕.离散PH分布的一个封闭性质及其应用[J].运筹与管理,2001,10(1): 6-9.
[10] 禹海波,周家良,聂赞坎.具有位相型修理的离散时间可修排队系统[J].运筹学学报,2004,8(1):87-96.
[11] Assaf D, Lexikson B. Closure of phase type distributions under operations arising in reliabiling theory[J].Ann. Prob.,1982, 10(2): 265-269.
[12] Asmussen S. Ruin probabilities[M]. London: World Scientific, 2000.
[13] Asmussen S. Applied probability and queues[M]. 2nd edition.New York: Springer-Verlag, 2003.
[14] Aldous D, Shepp L. The least variable phase type distribution is erlang[J]. Stochastic Model,1987, (3): 467-473.
[15] O'Cinneide C A. Phase-type distributions and majorization[J]. Annals of Applied Probability,1991, 1(2): 219-227.
[16] O'Cinneide C A. Phase-type distributions: open problems and a few properties[J]. Stochastic Models, 1999, 15(4): 731-757.
[17] He Q M, Zhang H. PH-invariant polytopes and coxian representations of phase type distributions[J]. Stochastic Models, 2006a, 22(3): 383-409.
[18] He Q M, Zhang H. Spectral polynomial algorithms for computing bi-diagonal representations for phase-type distributions and matrix exponential distributions[J]. Stochastic Models, 2006b, 22: 289-317.
[19] He Q M, Zhang H. On matrix exponential distributions[J]. Advances in Applied Probability, 2007, 39(1): 271-292.
[20] He Q M, Zhang H. An algorithm for computing minimal coxian representations[J]. INFORMS Journal on Computing, 2008, 20(2): 179-190.
[21] He Q M, Zhang H, Xue J G. Algorithms for coxianization of phase-type generators[J]. INFORMS Journal on Computing Winter, 2011, 23(1): 153-164.
[22] Telek M. The minimal coefficient of variation of DPH distributions[C]. In Proceedings of the Third International Conference on Matrix-analytic Methods in Stochastic Models, Latouch G., Taylor P. G. eds. Leuven, Belgium: Notable Publications, 2000. 391-400.
[23] Bobbio A, Horvath A, Scarpa M, Telek M. Acyclic discrete phase type distributions: properties and a parameter estimation algorithm[J]. Performance Evaluation, 2003, 54: 1-32.
[24] Ross S M. Stochastic processes[M]. New York: Wiley, 1983.
[25] Shaked M, Shantikumar J G. Stochastic orders and their applications[M]. New York: Academic Press, 1994.
[26] Stoyan D. Comparison methods for queues and other stochastic models[M]. New York: Wiley, 1983.
[27] 禹海波著.供应链系统的随机比较[M].北京:科学出版社,2013.
[28] YU H B, Q M H E, Zhang H. Convexity of some functions associated with denumerable-state-space markov chains and applications to queuing systems[J]. Probability in the Engineering and Informational Sciences, 2006, 20: 67-86.
[29] Grassmann W K. Transient solutions in markovian queues: an algorithm for finding them and determining their waiting-time distributions[J]. European Journal of Operational Research, 1977, 1(6): 396-402.
[30] Ross S M. Introduction to probability models[M]. New York: Academic Press, Elsevier Inc, 2007.
[31] Cassandras C G, Lafortune S. Introduction to discrete event systems[M]. New York: Springer, 2008.
[32] Stewart W J. Probability, Markov chains, queues, and simulation: the mathematical basis of performance modeling[M]. Princeton: Princeton University Press, 2009. 361.