Perturbation Theory for the Asymptotic Decay Rates in the Queues with Markovian Arrival Process and/or Markovian Service Process

Three kinds of queues with Markovian arrival process and/or Markovian service process, are considered in this paper. In great generality, their basic steady-state distributions have asymptotically exponential tails. We investigate the sensitivity of these asymptotic decay rates to the small entrywise perturbations in the parameter matrices of the Markovian arrival process.     

Diffusion Approximations for Queues with Markovian Bases

Consider a base family of state-dependent queues whose queue-length process can be formulated by a continuous-time Markov process. In this paper, we develop a piecewise-constant diffusion model for an enlarged family of queues, each of whose members has arrival and service distributions generalized from those of the associated queue in the base. The enlarged family covers many standard queueing systems with finite waiting spaces, finite sources and so on. We provide a unifying explicit expression for the steady-state distribution, which is consistent with the exact result when the arrival and service distributions are those of the base. The model is an extension as well as a refinement of the M/M/s-consistent diffusion model for the GI/G/s queue developed by Kimura [13] where the base was a birth-and-death process. As a typical base, we still focus on birth-and-death processes, but we also consider a class of continuous-time Markov processes with lower-triangular infinitesimal generators.     

Distributional Form of Little's Law for FIFO Queues with Multiple Markovian Arrival Streams and Its Application to Queues with Vacations

This paper considers stationary queues with multiple arrival streams governed by an irreducible Markov chain. In a very general setting, we first show an invariance relationship between the time-average joint queue length distribution and the customer-average joint queue length distribution at departures. Based on this invariance relationship, we provide a distributional form of Little's law for FIFO queues with simple arrivals (i.e., the superposed arrival process has the orderliness property). Note that this law relates the time-average joint queue length distribution with the stationary sojourn time distributions of customers from respective arrival streams. As an application of the law, we consider two variants of FIFO queues with vacations, where the service time distribution of customers from each arrival stream is assumed to be general and service time distributions of customers may be different for different arrival streams. For each queue, the stationary waiting time distribution of customers from each arrival stream is first examined, and then applying the Little's law, we obtain an equation which the probability generating function of the joint queue length distribution satisfies. Further, based on this equation, we provide a way to construct a numerically feasible recursion to compute the joint queue length distribution.     

具有时滞和参数不确定性的网络系统的滑模控制

研究了一类具有时滞的不确定网络控制系统(NCS)的滑模控制.首先,用一种特殊的变换将不确定时滞系统化为非延迟不确定系统形式.然后基于Lyapunov稳定性理论及线性矩阵不等式(LMI)技术,以线性矩阵不等式的形式给出了该系统的一些新的稳定性判据.所提出的控制器能够使具有不确定性的时滞系统渐近稳定性,而且控制器结构简单,计算方便.最后通过数值模拟,验证了所设计方法的有效性.     

带有阈值转换和启动时间的优先权排队

在诸如ＩＳＤＮ的通信网络中,多种信息共用一条线路,为了满足不同类型信息的服务质量要求,带有阈值转换的优先权排队系统应是一种合适的模型。本文研究单服务员、两类顾客的带有阈值转换和启动时间的优先权排队系统,首先,分别就抢占和非抢占情形讨论了具有泊松到达、服务时间和启动时间均有指数贩系统,然后就非抢占情况进上步考虑了服务时间和启动时间有一般分布的系统,求出了系统中两类顾客队长的稳态联合概率母函数,藉助这     

Decay properties of Markovian bulk-arrival and bulk-service queues with state-independent control

We consider decay properties regarding decay parameter and invariant measures of Markovian bulk-arrival and bulk-service queues with state-independent control. The exact value of the decay parameter, denoted by λz, is firstly revealed. A criterion regarding ）λz-recurrence and λz-positive is obtained. The corresponding λz-subinvariant/invariant measures and λz-subinvariant/invariant vectors are then presented.     

Indefinite Stochastic Linear Quadratic Control with Markovian Jumps in Infinite Time Horizon

This paper studies a stochastic linear quadratic (LQ) control problem in the infinite time horizon with Markovian jumps in parameter values. In contrast to the deterministic case, the cost weighting matrices of the state and control are allowed to be indinifite here. When the generator matrix of the jump process – which is assumed to be a Markov chain – is known and time-invariant, the well-posedness of the indefinite stochastic LQ problem is shown to be equivalent to the solvability of a system of coupled generalized algebraic Riccati equations (CGAREs) that involves equality and inequality constraints. To analyze the CGAREs, linear matrix inequalities (LMIs) are utilized, and the equivalence between the feasibility of the LMIs and the solvability of the CGAREs is established. Finally, an LMI-based algorithm is devised to slove the CGAREs via a semidefinite programming, and numerical results are presented to illustrate the proposed algorithm.     

Queues with Equally Heavy Sojourn Time and Service Requirement Distributions

For the G/G/1 queue with First-Come First-Served, it is well known that the tail of the sojourn time distribution is heavier than the tail of the service requirement distribution when the latter has a regularly varying tail. In contrast, for the M/G/1 queue with Processor Sharing, Zwart and Boxma [26] showed that under the same assumptions on the service requirement distribution, the two tails are equally heavy. By means of a probabilistic analysis we provide a new insightful proof of this result, allowing for the slightly weaker assumption of service requirement distributions with a tail of intermediate regular variation. The new approach allows us to also establish the tail equivalence for two other service disciplines: Foreground–Background Processor Sharing and Shortest Remaining Processing Time. The method can also be applied to more complicated models, for which no explicit formulas exist for (transforms of) the sojourn time distribution. One such model is the M/G/1 Processor Sharing queue with service that is subject to random interruptions. The latter model is of particular interest for the performance analysis of communication networks.     

Invariant Subsets and Invariant Measures for Irreducible Actions on Zero-Dimensional Groups

Algebraic higher-rank actions on connected groups are oftenremarkably rigid in their topological and measurable structure.In contrast to this, the author of this paper constructs uncountablymany closed invariant sets and uncountably many invariant measureswith positive entropy for irreducible algebraic Z^d-actions onzero-dimensional groups. 2000 Mathematics Subject Classification37A45 (primary), 37B50, 37A35, 37B40 (secondary).     

带小扰动的随机发展方程的不变测度

本文考虑带小扰动的随机发展方程,证明如何建立此方程的耦合解.作为应用,我们证明解的Feller连续性和不变测度的存在唯一性.还进一步建立了当扰动趋于零时,关于这族不变测度的大偏差原理.     

Invariant Measures for Monotone SPDEs with Multiplicative Noise Term

We study diffusion processes corresponding to infinite dimensional semilinear stochastic differential equations with local Lipschitz drift term and an arbitrary Lipschitz diffusion coefficient. We prove tightness and the Feller property of the solution to show existence of an invariant measure. As an application we discuss stochastic reaction diffusion equations.     

Invariant Measures for a Random Evolution Equation with Small Perturbations

In this paper we consider a random evolution equation with small perturbations, and show how to construct coupled solutions to the equations. As applications, we prove the Feller continuity of the solutions and the existence and uniqueness of invariant measures. Furthermore, we establish a large deviations principle for the family of invariant measures as the perturbations tend to zero. Received March 20,1998, Accepted June 1, 2000     

The Decay Rate of Patterson-Sullivan Measures with Potential Functions and Critical Exponents

Basing upon the recent development of the Patterson-Sullivan measures with a Hölder continuous nonzero potential function, we use tools of both dynamics of geodesic flows and geometric properties of negatively curved manifolds to present a new formula illustrating the relation between the exponential decay rate of Patterson-Sullivan measures with a Hölder continuous potential function and the corresponding critical exponent.     

Long‐Time Behavior,Invariant Measures,and Regularizing Effects for Stochastic Scalar Conservation Laws

We study the long‐time behavior and regularity of the pathwise entropy solutions to stochastic scalar conservation laws with random‐in‐time spatially homogeneous fluxes and periodic initial data. We prove that the solutions converge to their spatial average, which is the unique invariant measure of the associated random dynamical system, and provide a rate of convergence, the latter being new even in the deterministic case for dimensions higher than 2. The main tool is a new regularization result in the spirit of averaging lemmata for scalar conservation laws, which, in particular, implies a regularization by noise‐type result for pathwise quasi‐solutions.© 2016 Wiley Periodicals, Inc.     

Existence and Uniqueness of Invariant Measures for SPDEs with Two Reflecting Walls

In this article, we study stochastic partial differential equations with two reflecting walls h ¹ and h ², driven by space-time white noise with non-constant diffusion coefficients under periodic boundary conditions. The existence and uniqueness of invariant measures is established under appropriate conditions. The strong Feller property is also obtained.     

Nonfragile Robust Controller for Linear Markovian Jumping Parameter Systems with Multiplicative Brownian Disturbance

This paper deals with the uncertain class of continuous-time linear systems with Markovian jumping parameters and multiplicative Brownian disturbance. A design method for a nonfragile robust controller for this class of systems is proposed when the uncertainties are of the norm-bounded type. An LMI based sufficient condition is developed. The methodology used is based mainly on the Lyapunov approach. A numerical example is presented to show the usefulness of the proposed results.     

On the Existence of Invariant Measures for Networks with String Transitions

Recently a new class of Markov network processes was introduced, characterized by so-called string transitions. These are continuous-time Markov processes on a discrete state space. It is known that they possess an invariant measure of a special form, called a product-form, provided that a certain system of so-called traffic equations possesses a solution. Little is known about the existence of solutions of the traffic equations. The present paper deals with this question, focussing on the most important special case of unit vector string transitions. It is shown for open networks with unit vector string transitions of bounded lengths that the traffic equations possess a solution. Furthermore, it is shown for a prominent example of a network featuring signals and batch services that the traffic equations possess a solution.     

跳过程μ正则性和不变测度存在性

本文给出了一般状态跳过程μ正则的充分条件,作为其推论得到跳跃链常返的跳过程是μ正则的,证明了跳跃链常返的跳过程,其q对的不变测度是跳过程的不变测度.还证明了跳跃链常返的跳过程存在唯一的不变测度.     

一阶格点系统的不变测度与Liouville型方程

该文讨论一阶格点系统的解在相空间中的概率分布问题.作者先证明该格点系统的解算子生成的过程存在拉回吸引子,然后证明拉回吸引子上存在唯一的Borel不变概率测度,且该不变测度满足Liouville型方程.