马尔可夫骨架过程理论在两个数学模型中的应用
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摘要
本文主要是利用马尔可夫骨架过程理论来研究商店出售易腐烂物品模型和两部件热储备可修复模型。马尔可夫骨架过程是侯振挺教授等人于1997年首次提出的一类较为综合的随机过程，它包含了许多已有的随机过程模型，如马尔可夫过程、半马尔可夫过程、逐段决定马尔可夫过程等一系列经典的随机过程，具有重要的理论和应用价值．侯振挺等利用马尔可夫骨架过程理论成功地解决了排队论的瞬时分布、平稳分布、遍历性等一系列经典难题，并提出许多新问题和新思想．最近，此理论又得到了进一步的补充完善。
对于商店出售易腐烂物品模型，它是在库存模型的基础上，通过引进商店的盈利额过程而得到的一个模型。在原来的库存模型中主要考虑的是仓库的存货量，而在这个新的模型中我们不但考虑存货量，还考虑商店的盈利额。本文应用马尔可夫骨架过程理论给出了模型状态的瞬时分布和极限性态，
对于两部件热储备可修复模型，前人主要考虑的是工作部件和热储备部件的寿命以及故障部件的修理时间均为指数分布的情形。本文是利用马尔可夫骨架过程方法研究工作部件的寿命和故障部件的修理时间均为一般分布，仅热储备部件的寿命为指数分布的情形。
本文的主要结果有：
第一，分别利用马尔可夫骨架过程法和密度演化法列出了商店出售易腐烂物品模型的状态{X(t)，S(t)，θ(t)，(?)(t)；t≥0}的瞬时分布所满足的方程组。对于马尔可夫骨架过程法，我们证明了盈利额的概率分布是某一方程的最小非负解，又讨论了盈利额的极限分布。进一步又找出了该模型的Doob骨架过程，利用Doob骨架过程理论和极限理论给出了盈利额的广义极限分布、极限分布以及不变概率测度的存在性条件和表达式。对于密度演化法，我们列出了模型状态的概率密度P_t(t…)所满足的偏微积分方程组、边界条件和初始条件，并给予了详细证明。
第二，利用密度演化法给出了两部件热储备可修复模型的状态{L(t)；X(t)，Y(t)；t≥0)的概率密度所满足的偏微积分方程组、边界条件、初始条件和正则条件，并给予证明。进一步用马尔可夫骨架过程法给出了该模型状态的瞬时分布所满足的积分方程组，并给予证明。最后分析了其极限性态。
In this thesis, we discuss the solding of perishable goods model and the reliability of repairable hot reserve model with two-part applying the Markov skeleton process theory. Markov skeleton process is a kind of comprehensive stochastic process, which is firstly put forward by Prof. Hou zhenting and his colleagues in 1997. The process contains many extant classical stochastic process models,such as Markov process,semi-Markov process, piecewise deterministic Markov process etc. They have important value in theory and application. Prof. Hou zhenting and his colleagues have successfully solved a series of classical difficult problems of transient distribution, steady-state distribution, ergodicity ergodicity in queueing system, meanwhile posed many new problems and new thoughts.Recently, the theory has got complemented and perfected.
As for the solding of perishable goods model, we introduce the payoff to the inventory model, and get this model. In the inventory model, the predecessors mainly studied the property of the stocks. In the solding of perishable goods model ,we use Markov skeleton process theory not only study the property of the stocks ,but also study the property of the payoff. We also get the limit distribution and instantaneous distribution of the state of the model.
As for the repairable hot reserve system with two-part model, the predecessors mainly studied this model when both the distribution of the age of the work part and hot reserve part and the distribution of the repair time of the fault part are exponential distribution. We use Markovskeleton process theory study the model when both the distribution of the age of the work part and the distribution of the repair time of the fault part are general distribution. Only the distribution of the the age of the hot reserve part be exponential distribution.
In this dissertation, we drew the following conclusions:
Firstly, in the solding of perishable goods model, applying the Markov skeleton process method and the density evolution method respectively, we present the equations which satisfy the transient distribution of the payoff level {X(t), S(t),θ(t),(?)(t); t≥0} for the solding of perishable goods model. To Markov skeleton process method, we prove that the probability distribution of the payoff level is the minimal nonnegative solution of some equation. And we study the limit distribution of payoff level. Furthermore,we find out a Doob skeleton process of the model and give out its limit distribution, generalized limit distribution, the existence condition and its expression of invariant probability measure. To the density evolution method, we list the partial differentio-integral equations system ,boundary conditions and initial conditions that the state probability density P_i(t…) satisfies,and give out the detailed proof.
Secondly, in the repairable hot reserve system with two-part model, applying the density evolution method, we list the partial differentio -integral equations system ,boundary conditions,the initial conditions and regular conditions that the state probability density P_i (t…) satisfies,and give out the detailed proof. Furthermore, we use Markov skeleton process theory list the partial integral equations system that the state {L(t);X(t),Y(t);t≥0} satisfies, and give out the detailed proof. In the end, we discuss its limit distribution.
对于商店出售易腐烂物品模型，它是在库存模型的基础上，通过引进商店的盈利额过程而得到的一个模型。在原来的库存模型中主要考虑的是仓库的存货量，而在这个新的模型中我们不但考虑存货量，还考虑商店的盈利额。本文应用马尔可夫骨架过程理论给出了模型状态的瞬时分布和极限性态，
对于两部件热储备可修复模型，前人主要考虑的是工作部件和热储备部件的寿命以及故障部件的修理时间均为指数分布的情形。本文是利用马尔可夫骨架过程方法研究工作部件的寿命和故障部件的修理时间均为一般分布，仅热储备部件的寿命为指数分布的情形。
本文的主要结果有：
第一，分别利用马尔可夫骨架过程法和密度演化法列出了商店出售易腐烂物品模型的状态{X(t)，S(t)，θ(t)，(?)(t)；t≥0}的瞬时分布所满足的方程组。对于马尔可夫骨架过程法，我们证明了盈利额的概率分布是某一方程的最小非负解，又讨论了盈利额的极限分布。进一步又找出了该模型的Doob骨架过程，利用Doob骨架过程理论和极限理论给出了盈利额的广义极限分布、极限分布以及不变概率测度的存在性条件和表达式。对于密度演化法，我们列出了模型状态的概率密度P_t(t…)所满足的偏微积分方程组、边界条件和初始条件，并给予了详细证明。
第二，利用密度演化法给出了两部件热储备可修复模型的状态{L(t)；X(t)，Y(t)；t≥0)的概率密度所满足的偏微积分方程组、边界条件、初始条件和正则条件，并给予证明。进一步用马尔可夫骨架过程法给出了该模型状态的瞬时分布所满足的积分方程组，并给予证明。最后分析了其极限性态。
In this thesis, we discuss the solding of perishable goods model and the reliability of repairable hot reserve model with two-part applying the Markov skeleton process theory. Markov skeleton process is a kind of comprehensive stochastic process, which is firstly put forward by Prof. Hou zhenting and his colleagues in 1997. The process contains many extant classical stochastic process models,such as Markov process,semi-Markov process, piecewise deterministic Markov process etc. They have important value in theory and application. Prof. Hou zhenting and his colleagues have successfully solved a series of classical difficult problems of transient distribution, steady-state distribution, ergodicity ergodicity in queueing system, meanwhile posed many new problems and new thoughts.Recently, the theory has got complemented and perfected.
As for the solding of perishable goods model, we introduce the payoff to the inventory model, and get this model. In the inventory model, the predecessors mainly studied the property of the stocks. In the solding of perishable goods model ,we use Markov skeleton process theory not only study the property of the stocks ,but also study the property of the payoff. We also get the limit distribution and instantaneous distribution of the state of the model.
As for the repairable hot reserve system with two-part model, the predecessors mainly studied this model when both the distribution of the age of the work part and hot reserve part and the distribution of the repair time of the fault part are exponential distribution. We use Markovskeleton process theory study the model when both the distribution of the age of the work part and the distribution of the repair time of the fault part are general distribution. Only the distribution of the the age of the hot reserve part be exponential distribution.
In this dissertation, we drew the following conclusions:
Firstly, in the solding of perishable goods model, applying the Markov skeleton process method and the density evolution method respectively, we present the equations which satisfy the transient distribution of the payoff level {X(t), S(t),θ(t),(?)(t); t≥0} for the solding of perishable goods model. To Markov skeleton process method, we prove that the probability distribution of the payoff level is the minimal nonnegative solution of some equation. And we study the limit distribution of payoff level. Furthermore,we find out a Doob skeleton process of the model and give out its limit distribution, generalized limit distribution, the existence condition and its expression of invariant probability measure. To the density evolution method, we list the partial differentio-integral equations system ,boundary conditions and initial conditions that the state probability density P_i(t…) satisfies,and give out the detailed proof.
Secondly, in the repairable hot reserve system with two-part model, applying the density evolution method, we list the partial differentio -integral equations system ,boundary conditions,the initial conditions and regular conditions that the state probability density P_i (t…) satisfies,and give out the detailed proof. Furthermore, we use Markov skeleton process theory list the partial integral equations system that the state {L(t);X(t),Y(t);t≥0} satisfies, and give out the detailed proof. In the end, we discuss its limit distribution.
引文
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[2] Hagel J．Ⅲ，Armstrong A．G．(王国瑞译)网络利益—通过虚拟社会扩大市场．北京：新华出版社，1998．
[3] 胡则成，罗荣桂，宋德昌，喻小军．随机存贮理论与应用．华中理工大学出版社，1994．
[4] RahimM. A., Kabadi S. N., Barnerjee P. K. A single-period perishable inventory model where deterioration begins at a random point in time. International Journal of Systems Science, 2000, 31(1):131-136.
[5] Nahmias S. Optimal ordering policy for perishable inventory-the second. Operations Research, 1975, 23:735-749.
[6] Ravichandran N. Stochastic analysis of a continuous review perishable inventory system with positive lead time and Poisson demand. European Journal of Operational Research, 1995, 84(2):444-457.
[7] Goswami A., Chaudhuri K. S. An EQQ model for deteriorating items with shortages and a linear trend in demand. Journal of the Operational Research Society, 1991, 42(12):1105-1110.
[8] Nahmias S. Perishable inventory theory:A review. Operations Research, 1982, 30:680-708.
[9] Abad P. L. Optimal Pricing and Lot-sizing under conditions of perishability and partial backordering. Management Science, 1996, 42(8): 1093-1104.
[10] LiuL. M., Shi D. H. (s, S) continuous review models for inventory with exponential lifetime and renewal demands, Naval Research Logistics, 1999, 46:39-56.
[11] Wee H. M., Yu J. A deteriorating inventory model with a temporary price discount. International Journal of Production Economics, 1997, 53: 81-90.
[12] Moinzadeh K., Aggarwal P. Analysis of a production/inventory system subject to Randomdisruptions. Management Science, 1997, 43(11):1577-1588.
[13] Makis A. & Fung J. An EMQ model with inspections and random machine failures. J. Opl. Res. Soc., 1998, 49:66-76.
[14] 徐光启，《农政全书》卷四：《玄扈先生井田考》．
[15] J. Graunt. Natural and political observations mentioned in a following index, and made upon the bills by mortality of John Graunt Citizen of London(1662)(reprinted in part in smith and keyfitz, Mathematical Demography).
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