带单重指数工作休假和休假中断的GI／M／1的排队系统

本文主要研究带有单重指数工作休假和休假中断策略的GI／M／1排队模型。利用分块矩阵表示出嵌入马尔可夫链的转移矩阵,并运用矩阵几何解的方法求得到达时刻队长的稳态分布,而且证明了其可以分解为三个独立随机变量的分布的和。     

绕积马氏链的状态

主要研究了绕积马氏链的各种状态,得到了相应的一些充要条件．同时利用Foguel的L1-理论对单链X常返性与瞬时性进行了讨论,回答了单链X是π-不可约链的本质,即绕积马氏链的相空间是最小闭集．     

The performance evaluation of a multi-stage JIT production system with stochastic demand and production capacities

This paper discusses a single-item, multi-stage, serial Just-in-Time (JIT) production system with stochastic demand and production capacities. The JIT production system is modeled as a discrete-time, M/G/1-type Markov chain. A necessary and sufficient condition, or a stability condition, under which the system has a steady-state distribution is derived. A performance evaluation algorithm is then developed using the matrix analytic methods. In numerical examples, the optimal numbers of kanbans are determined by the proposed algorithm. The optimal numbers of kanbans are robust for the variations in production capacity distribution and demand distribution.     

On strong ergodicity for nonhomogeneous continuous-time Markov chains

Let X(t) be a nonhomogeneous continuous-time Markov chain. Suppose that the intensity matrices of X(t) and some weakly or strongly ergodic Markov chain (t) are close. Some sufficient conditions for weak and strong ergodicity of X(t) are given and estimates of the rate of convergence are proved. Queue-length for a birth and death process in the case of asymptotically proportional intensities is considered as an example.     

The GI/M/1 queue with exponential vacations

In this paper, we give a detailed analysis of the GI/M/1 queue with exhaustive service and multiple exponential vacation. We express the transition matrix of the imbedded Markov chain as a block-Jacobi form and give a matrix-geometric solution. The probability distribution of the queue length at arrival epochs is derived and is shown to decompose into the distribution of the sum of two independent random variables. In addition, we discuss the limiting behavior of the continuous time queue length processes and obtain the probability distributions for the waiting time and the busy period.     

On Finite Markov Chain Imbedding and Its Applications

The finite Markov Chain Imbedding technique has been successfully applied in various fields for finding the exact or approximate distributions of runs and patterns under independent and identically distributed or Markov dependent trials. In this paper, we derive a new recursive equation for distribution of scan statistic using the finite Markov chain imbedding technique. We also address the problem of obtaining transition probabilities of the imbedded Markov chain by introducing a notion termed Double Finite Markov Chain Imbedding where transition probabilities are obtained by using the finite Markov chain imbedding technique again. Applications for random permutation model in chemistry and coupon collector’s problem are given to illustrate our idea.     

Strong stability of the embedded Markov chain in an GI/M/1 queue with negative customers

This paper is devoted to the investigation for sufficient conditions of the strong stability of the embedded Markov chain in GI/M/1 queueing system with negative customers. After perturbing the occurrence rate of the negative customers, we prove the strong stability of the considered Markov chain with respect to a convenient weight variation norm. Furthermore, we estimate the deviation of its transition operators and provide an upper bound to the approximation error. This results allow us to understand how the negative customers will affect the system’s level of performance.     

Performance and reliability analysis of an M/G/1-G retrial queue with orbital search and non-persistent customers

This paper treats an M/G/1 retrial queue with non-persistent customers, where the server is subject to failure due to the negative arrivals. After a completion of a service or a repair, the server searches for the customers in the orbit or remains idle. By using embedded Markov chain technique and the supplementary variable method, we present the necessary and sufficient condition for the system to be stable and the joint queue length distribution in steady state. The waiting process is also given. Some main reliability measures, such as the availability, failure frequency, and the reliability function of the server, are obtained. Finally, some numerical examples and cost optimization analysis are presented.     

多重休假的Bernoulli反馈M/G/1排队系统的稳态队长

系统地研究了多重休假的Bernoulli反馈排队模型,采用嵌入Markov链的方法,给出系统的一步转移概率矩阵进而求得稳态队长分布,并对排队顾客数进行随机分解,得到附加队长的母函数.     

The Limit Behaviour of Imprecise Continuous-Time Markov Chains

We study the limit behaviour of a nonlinear differential equation whose solution is a superadditive generalisation of a stochastic matrix, prove convergence, and provide necessary and sufficient conditions for ergodicity. In the linear case, the solution of our differential equation is equal to the matrix exponential of an intensity matrix and can then be interpreted as the transition operator of a homogeneous continuous-time Markov chain. Similarly, in the generalised nonlinear case that we consider, the solution can be interpreted as the lower transition operator of a specific set of non-homogeneous continuous-time Markov chains, called an imprecise continuous-time Markov chain. In this context, our convergence result shows that for a fixed initial state, an imprecise continuous-time Markov chain always converges to a limiting distribution, and our ergodicity result provides a necessary and sufficient condition for this limiting distribution to be independent of the initial state.     

Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching

In this paper, we prove that a stochastic logistic population under regime switching controlled by a Markov chain is either stochastically permanent or extinctive, and we obtain the sufficient and necessary conditions for stochastic permanence and extinction under some assumptions. In the case of stochastic permanence we estimate the limit of the average in time of the sample path of the solution by two constants related to the stationary probability distribution of the Markov chain and the parameters of the subsystems of the population model. Finally, we illustrate our conclusions through two examples.     

Ergodicity of a polling network

The polling network considered here consists of a finite collection of stations visited successively by a single server who is following a Markovian routing scheme. At every visit of a station a positive random number of the customers present at the start of the visit are served, whereupon the server takes a positive random time to walk to the station to be visited next. The network receives arrivals of customer groups at Poisson instants, and all customers wait until served, whereupon they depart from the network. Necessary and sufficient conditions are derived for the server to be able to cope with the traffic. For the proof a multidimensional imbedded Markov chain is studied.     

有启动失败和可选服务的M/G/1重试排队系统

考虑具有可选服务的M/G/1重试排队模型,其中服务台有可能启动失败.系统外新到达的顾客服从参数为λ的泊松过程.重试区域只允许队首顾客重试,重试时间服务一般分布.所有的顾客都必须接受必选服务,然而只有其中部分接受可选服务.通过嵌入马尔可夫链法证明了系统稳态的充要条件.利用补充变量的方法得到了稳态时系统和重试区域中队长分布.我们还得到重试期间服务台处于空闲的概率,重试区域为空的概率以及其他各种指标.并证出在把系统中服务台空闲和修理的时间定义为广义休假情况下也具有随机分解特征.     

Subexponential asymptotics of the stationary distributions of M/G/1-type Markov chains

This paper studies the subexponential asymptotics of the stationary distribution of an M/G/1-type Markov chain. We provide a sufficient condition for the subexponentiality of the stationary distribution. The sufficient condition requires only the subexponential integrated tail of level increments. On the other hand, the previous studies assume the subexponentiality of level increments themselves and/or the aperiodicity of the G-matrix. Therefore, our sufficient condition is weaker than the existing ones. We also mention some errors in the literature.     

On the class of Markov chains with finite convergence time

We study the necessary and sufficient conditions for a finite ergodic Markov chain to converge in a finite number of transitions to its stationary distribution. Using this result, we describe the class of Markov chains which attain the stationary distribution in a finite number of steps, independent of the initial distribution. We then exhibit a queueing model that has a Markov chain embedded at the points of regeneration that falls within this class. Finally, we examine the class of continuous time Markov processes whose embedded Markov chain possesses the property of rapid convergence, and find that, in the case where the distribution of sojourn times is independent of the state, we can compute the distribution of the system at time t in the form of a simple closed expression.     

Markov chain analysis of genetic algorithms applied to?fitness functions perturbed concurrently by additive and multiplicative noise

We analyze the transition and convergence properties of genetic algorithms (GAs) applied to fitness functions perturbed concurrently by additive and multiplicative noise. Both additive noise and multiplicative noise are assumed to take on finitely many values. We explicitly construct a Markov chain that models the evolution of GAs in this noisy environment and analyze it to investigate the algorithms. Our analysis shows that this Markov chain is indecomposable; it has only one positive recurrent communication class. Using this property, we establish a condition that is both necessary and sufficient for GAs to eventually (i.e., as the number of iterations goes to infinity) find a globally optimal solution with probability 1. Similarly, we identify a condition that is both necessary and sufficient for the algorithms to eventually with probability 1 fail to find any globally optimal solution. Our analysis also shows that the chain has a stationary distribution that is also its steady-state distribution. Based on this property and the transition probabilities of the chain, we compute the exact probability that a GA is guaranteed to select a globally optimal solution upon completion of each iteration.     

带启动-关闭期和N策略的多重休假M/G/1排队

研究多重休假带启动-关闭期和N策略的M/G/1排队系统,根据嵌入Markov链的方法推导出状态转移概率矩阵,利用M/G/1型排队系统结构矩阵解析法,得出顾客服务完离去后系统稳态队长分布及其母函数的表达式;从而由经典随机分解原理,给出稳态队长的随机分解结果.此外,利用LST变换处理卷积,得到忙期的母函数及数学期望的表达式;进而得到忙期、启动期和关闭期的母函数及在稳态下服务员处于各状态的概率.最后提出一些数值例子以验证结论.     

Studying an overload system using rotation

1. IntroductionThe motivation of writing this paper was from calculating the blocking probability foran overloaded finite system. Our numerical experiments suggested that this probability canbe approximated efficiently by rotating the transition matrix by 180". Some preliminaryresults were obtained and can be found in [11 and [2]. Rotating the transition matrix definesa new Markov chain, which is often called the dual process in the literature, for example,[3--7]. For a finite Markov chain, …     

Use of discrete transforms for the study of aGI/M/S queue with impatient customer phenomena

The problem is to determine an explicit form for the stationary distribution of an imbedded Markov Chain in aGI/M/S queue with balking and reneging.We first construct the discrete transforms — introduced bySubba Rao/Jaiswal [1969] — for this queueing system. Then, using this concept, we resolve the problem, obtaining a new way for determining the transition probabilities of the imbedded Markov chain.

---

Zusammenfassung Thema dieses Aufsatzes ist die Bestimmung der stationären Verteilung einer eingebetteten Markov-Kette in einemGI/M/S Wartesystem mit ungeduldigen Kunden. Anhand der vonSubba Rao/Jaiswal [1969] eingeführten diskreten Transformation gelingt es, die Übergangswahrscheinlichkeiten der eingebetteten Markov-Kette explizit zu bestimmen.