Stationary Mode and Binomial Moments for Networks of the Type [SM|GI|∞] r

By using methods of the Markov renewal theory, we find conditions for the existence of a stationary mode for multichannel networks with semi-Markov input stream. As a tool of stationary distribution analysis, we introduce multivariate binomial moments and investigate their asymptotic properties. For a multichannel queuing system with periodic input stream, we construct the generating function of the stationary distribution in explicit form in terms of the parameters of the system under consideration.     

Sojourn time problems in feedback queues

A brief survey of the literature on sojourn time problems in single node feedback queueing systems is presented. The derivation of the distribution and moments of the sojourn time of a typical customer in a Markov renewal queue with state dependent feedback is considered in depth. The techniques used relate to the derivation of a first passage time distribution in a particular Markov renewal process. These results are applied to birth-death queues with state dependent feedback. For such models an alternative approach using the theory of Markov chains in continuous time is also examined.     

The Geometric Markov Renewal Processes with Application to Finance

We introduce the geometric Markov renewal processes as a model for a security market and study this processes in a series scheme. We consider its approximations in the form of averaged, merged and double averaged geometric Markov renewal processes. Weak convergence analysis and rates of convergence of ergodic geometric Markov renewal processes are presented. Martingale properties, infinitesimal operators of geometric Markov renewal processes are presented and a Markov renewal equation for expectation is derived. As an application, we consider the case of two ergodic classes. Moreover, we consider a generalized binomial model for a security market induced by a position dependent random map as a special case of a geometric Markov renewal process.     

Probabilities for Queueing Systems with Embedded Markov Chains

Abstract

Transition probabilities of embedded Markov chain for single-server queues are considered when the distribution of the inter-arrival time or that of the service time is specified. A comprehensive collection of formulas is derived for the transition probabilities, covering some seventeen flexible families. The corresponding estimation procedures are also derived by the method of moments. It is expected that this work could serve as a useful reference for the modeling of queuing systems with embedded Markov chains.     

TheGr X n/G n/∞ system: System size

An important property of most infinite server systems is that customers are independent of each other once they enter the system. Though this non-interacting property (NIP) has been instrumental in facilitating excellent results for infinite server systems in the past, the utility of this property has not been fully exploited or even fully recognized. This paper exploits theNIP by investigating a general infinite server system with batch arrivals following a Markov renewal input process. The batch sizes and service times depend on the customer types which are regulated by the Markov renewal process. By conditional approaches, analytical results are obtained for the generating functions and binomial moments of both the continuous time system size and pre-arrival system size. These results extend the previous results on infinite server queues significantly.     

Queue tests for renewal processes

Queuing models can be used to test whether a stochastic point process can be represented as a renewal process. The test queuing model is analyzed, perhaps by simulation, using the prrocess of interest to generate arrivals or service times. Various congestion measures indicate departure from the renewal property.     

Determination of the Spectral Index of Ergodicity of a Birth-and-Death Process

We obtain a new explicit relation for the calculation of the spectral index of ergodicity of a birth-and-death process with continuous time. The calculation of the index is reduced to the solution of an optimization problem of nonlinear programming that contains the infinitesimal matrix of the process. As an example, we use the proposed method for finding the exact values of the indices of exponential ergodicity for certain Markov queuing systems.     

关于红利-惩罚等式的相关结果

本文研究了在一类马氏相关更新风险模型中的红利-惩罚等式的问题.推导了在常数红利边界下,折扣惩罚函数满足的方程,利用解微分-积分方程的方法,更简洁的推出了红利-惩罚等式相关的结果,推广了文献[1]的结论.     

马尔可夫排队过程的数学模型研究

本文用研究了一个广义（Ａ｜Ｂ｜ｃ｜ｎ｜ｍ）的排队系统模型，把其它的各种排队系统看成是该模型的一个特例，使得对排队系统的研究既有较强系统性而又简明扼要。     

On Numerical Solution of the Markov Renewal Equation: Tight Upper and Lower Kernel Bounds

We develop tight bounds and a fast parallel algorithm to compute the Markov renewal kernel. Knowledge of the kernel allows us to solve Markov renewal equations numerically to study non-steady state behavior in a finite state Markov renewal process. Computational error and numerical stability for computing the bounds in parallel are discussed using well-known results from numerical analysis. We use our algorithm and computed bounds to study the expected number of departures as a function of time for a two node overflow queueing network.