BMAP/G/1 queue with correlated arrivals of customers and disasters

We consider a single server queue with disasters where the arrivals of customers and disasters are correlated. When a disaster occurs, it removes all the customers in the system and there requires repair time for the system to be operated normally. The stationary queue length distribution at the embedded points and at an arbitrary time are presented.     

Rate conservation laws: A survey

We survey the rate conservation law, RCL for short, arising in queues and related stochastic models. RCL was recognized as one of the fundamental principles to get relationships between time and embedded averages such as the extended Little's formulaH=G, but we show that it has other applications. For example, RCL is one of the important techniques for deriving equilibrium equations for stochastic processes. It is shown that the various techniques, including Mecke's formula for a stationary random measure, can be formulated as RCL. For this purpose, we start with a new definition of the rate with respect to a random measure, and generalize RCL by using it. We further introduce the notion of quasi-expectation, which is a certain extension of the ordinary expectation, and derive RCL applicable to the sample average results. It means that the sample average formulas such asH=G can be obtained as the stationary RCL in the quasi-expectation framework. We also survey several extensions of RCL and discuss examples. Throughout the paper, we would like to emphasize how results can be easily obtained by using a simple principle, RCL.     

A matrix-geometric approximation for tandem queues with blocking and repeated attempts

Our interest is in the study of the MAP/PH/1/1→·/PH/1/K+1 queue with blocking and repeated attempts. The main feature of its infinitesimal generator is the spatial heterogeneity caused by the transitions due to successful repeated attempts. We develop an algorithmic solution by making a simplifying approximation which yields an infinitesimal generator which is spatially homogeneous and has a modified matrix-geometric stationary vector. The essential tool in our analysis is the general theory on quasi-birth-and-death processes.     

Admission control of a service station when the arrivals are internal

All studies in the admission control of a service station make decisions at arrival epochs. When arrivals are internal and are rejected from a queue, the rejected jobs have to be routed to other stations in the system. However the system will not know whether a job will be admitted to a queue or not until its arrival epoch to that queue. Thus, the system has to react dynamically and agilely to the decisions made at a specific queue and may try several queues before finding a queue that admits the job. This paper remedies these difficulties by changing the decision epochs of the admission control from arrival epochs to departure epochs with the actions of switching (keeping) the arrival stream on or off. Thus upstream stations will have information on the admission status of their downstream stations all the time. It is proved that the optimal policy for this revised admission control system is of control limit type for an M/G/1 queue. Comparisons of the optimal values and optimal policies for the admission controls made at arrival epochs and at departure epochs are included in the paper.     

N-策略单重休假M/G/1排队系统的队长分布

本文考虑N-策略单重休假M/G/1排队系统,通过引进"服务员忙期"和使用全概率分解技术,从任意初始状态出发,研究了队长的瞬态分布和稳态分布,首次导出了在任意时刻t瞬态队长分布的L变换的递推表达式和稳态队长分布的递推表达式,以及平稳队长的随机分解.特别地,通过本文可直接获得一些特殊排队系统相应的结果.     

Logarithmic asymptotics for a single-server processing distinguishable sources

We consider a single-server first-in-first-out queue fed by a finite number of distinct sources of jobs. For a large class of short-range dependent and light-tailed distributed job processes, using functional large deviation techniques we prove a large deviation principle and logarithmic asymptotics for the joint waiting time and queue lengths distribution. We identify the paths that are most likely to lead to the rare events of large waiting times and long queue lengths. A number of examples are presented to illustrate salient features of the results.      

GI/G/1排队系统队长的强大数定律和中心极限定理

本文首先证明当服务强度小于1时,GI/G/1排队系统的队长是一个特殊的马尔可夫骨架过程——正常返的Doob骨架过程,然后运用马尔可夫骨架过程的强大数定律和中心极限定理等重要结果,给出了队长的累积过程的期望和方差,并给出了该累积过程满足强大数定律和中心极限定理的充分条件。     

A new upper bound on the queuenumber of hypercubes

A queue layout of a graph consists of a linear order of its vertices, and a partition of its edges into queues, such that no two edges in the same queue are nested. In this paper, we show that the n-dimensional hypercube Q_n can be laid out using n−3 queues for n?8. Our result improves the previously known result for the case n?8.