The Law of the Iterated Logarithm for Extreme Waiting Time in Multiphase Queueing Systems

The objective of this research in the queueing theory is the law of the iterated logarithm (LIL) under the conditions of heavy traffic in multiphase queueing systems (MQS). In this paper, the LIL is proved for the extreme values of some important probabilistic characteristics of the MQS, namely, maxima and minima of the summary waiting time of a customer, and maxima and minima of the waiting time of a customer.     

On the Analysis of the Virtual Waiting Time in Open Queueing Networks

The model of an open queueing network in heavy traffic has been developed. These models are mathematical models of computer networks in heavy traffic. A limit theorem has been presented for the virtual waiting time of a customer in heavy traffic in open queueing networks. Finally, we present an application of the theorem—a reliability model from computer network practice.     

批到达离散时间轮询系统中顾客等待时间及队长

本对批到达离散时间轮询系统进行研究,在门限服务原则下,推出了原客等待时间和轮询周期的概率母函数,利用Markov链理论,得出了队列队长均值。     

批量到达的、具有两种服务阶段和服务中断的重试排队系统的适定性

运用Hille-Yosida定理,Phillips定理与Fattorini定理证明批量到达、具有两种服务阶段和服务中断的重试排队系统存在唯一的、非负的、满足概率性质的时间依赖解.     

启动时间的GI／G／1排队系统的非平衡理论

本文利用侯振挺等提出的马尔可夫骨架过程理论讨论了启动时间的GI/G/I排队系统，得到了此系统到达过程，队长，及等待时间的概率分布/     

Application of the Superposition of Renewal Processes to the Study of Batch Arrival Queues

The G /G/1-type batch arrival system is considered. We deal with non-steady-state characteristics of the system like the first busy period and the first idle time, the number of customers served on the first busy period. The study is based on a generalization of Korolyuk's method which he developed for semi-Markov random walks.     

Some New Results for Departure Process in the M X /G/1 Queueing System with a Single Vacation and Exhaustive Service

A batch arrival queueing system with a single vacation between two successive busy periods and with exhaustive service is considered.

The departure process h(t) is studied first on a single vacation cycle. The approach based on renewal theory is applied to obtain results in the general case. In particular, the explicit representation for the generating function of Laplace transform of the probability function of h(t) is derived. All formulae are written in terms of input parameters of the system and factors of a certain canonical factorization of Wiener–Hopf type. A numerical approach to results is discussed as well.     

The Virtual Waiting Time of the M/G/1 Queue with Impatient Customers

The M/G/1 queue with impatient customers is studied. The complete formula of the limiting distribution of the virtual waiting time is derived explicitly. The expected busy period of the queue is also obtained by using a martingale argument.     

Analytical Distribution of Waiting Time in the M/{iD}/1 Queue

We give an analytical formula for the steady-state distribution of queue-wait in the M/G/1 queue, where the service time for each customer is a positive integer multiple of a constant D > 0. We call this an M/{iD}/1 queue. We give numerical algorithms to calculate the distribution. In addition, in the case that the service distribution is sparse, we give revised algorithms that can compute the distribution more quickly.AMS subject classification: 60K25, 90B22     

Combined Elapsed Time and Matrix-Analytic Method for the Discrete Time GI/G/1 and GI X /G/1 Systems

In this paper, we show that the discrete GI/G/1 system can be easily analysed as a QBD process with infinite blocks by using the elapsed time approach in conjunction with the Matrix-geometric approach. The positive recurrence of the resulting Markov chain is more easily established when compared with the remaining time approach. The G-measure associated with this Markov chain has a special structure which is usefully exploited. Most importantly, we show that this approach can be extended to the analysis of the GI ^X /G/1 system. We also obtain the distributions of the queue length, busy period and waiting times under the FIFO rule. Exact results, based on computational approach, are obtained for the cases of input parameters with finite support – these situations are more commonly encountered in practical problems.     

每个忙期中第一个顾客被拒绝服务的M／M／1排队模型的另一个特征值

研究每个忙期中第一个顾客被拒绝服务的M／M／1排队模型主算子在左半复平面中的特征值,证明2√λμ-λ-μ是该主算子的几何重数为1的特征值。     

每个忙期中第一个顾客被拒绝服务的M／M/1排队模型的其他特征值

研究每个忙期中第一个顾客被拒绝服务的M／M／1排队模型的主算子在左半复平面中的特征值,证明对一切θ∈（0,1）,（2√λμ-λ—μ）θ是该主算子的几何重数为1的特征值．     

Analysis of the M x /G/1 Queue with a Random Setup Time

Abstract

We consider an M ^x /G/1 queueing system with a random setup time, where the service of the first unit at the commencement of each busy period is preceded by a random setup time, on completion of which service starts. For this model, the queue size distributions at a random point of time as well as at a departure epoch and some important performance measures are known [see Choudhury, G. An M ^x /G/1 queueing system with setup period and a vacation period. Queueing Sys. 2000, 36, 23–38]. In this paper, we derive the busy period distribution and the distribution of unfinished work at a random point of time. Further, we obtain the queue size distribution at a departure epoch as a simple alternative approach to Choudhury4 Choudhury, G. 2000. An M^x/G/1 queueing system with setup period and a vacation period. Queueing Syst., 36: 23–38. [CROSSREF][Crossref], [Web of Science ®] , [Google Scholar]. Finally, we present a transform free method to obtain the mean waiting time of this model.     

带随机启动时间与双阈值(m,N)-策略的M/G/1可修排队系统的最优控制策略

本文研究带随机启动时间与双阈值(m,N)-策略的M/G/1可修排队系统,首先讨论系统有关的排队指标,接着研究因为故障而产生的系统的下列可靠性指标,如:服务台首次失效前的寿命分布、不可用度和(0,t]时间内的平均故障次数。最后,在建立费用模型的基础上,结合实际中检测公司检测样品的这一现实情况,研究了双阈值最优控制策略(m^*,N^*),并在同一组参数下与服务台不发生故障时系统的双阈值最优控制策略进行了比较。     

Some aspects of an<Emphasis Type="Italic">M/G/1</Emphasis> queueing system with optional second service

This paper examines the steady state behaviour of anM/G/1 queue with a second optional service in which the server may provide two phases of heterogeneous service to incoming units. We derive the queue size distribution at stationary point of time and waiting time distribution. Moreover we derive the queue size distribution at the departure point of time as a classical generalization of the well knownPollaczek Khinchin formula. This is a generalization of the result obtained by Madan (2000). This work is supported by Department of Atomic Energy, Govt. of India, NBHM Project No. 88/2/2001/R&D II/2001.     

Operating characteristics of MX/G/1 queue with N-policy

We consider aM ^X/G/1 queueing system withN-policy. The server is turned off as soon as the system empties. When the queue length reaches or exceeds a predetermined valueN (threshold), the server is turned on and begins to serve the customers. We place our emphasis on understanding the operational characteristics of the queueing system. One of our findings is that the system size is the sum of two independent random variables: one has thePGF of the stationary system size of theM ^X/G/1 queueing system withoutN-policy and the other one has the probability generating function _j=0 ^N=1 _j z ^j/ _j=0 ^N=1 _j , in which _j is the probability that the system state stays atj before reaching or exceedingN during an idle period. Using this interpretation of the system size distribution, we determine the optimal thresholdN under a linear cost structure.     

A Polynomial Factorization Approach for the Discrete Time GIX/>G/1/K Queue

This paper proposes a polynomial factorization approach for queue length distribution of discrete time GI ^X /G/1 and GI _X /G/1/K queues. They are analyzed by using a two-component state model at the arrival and departure instants of customers. The equilibrium state-transition equations of state probabilities are solved by a polynomial factorization method. Finally, the queue length distributions are then obtained as linear combinations of geometric series, whose parameters are evaluated from roots of a characteristic polynomial.     

Reliability Approximation for Markov Chain Imbeddable Systems

In the present article, a simple method is developed for approximating the reliability of Markov chain imbeddable systems. The approximating formula reduces the problem to the reliability assessment of smaller systems with structure similar to the original systems. Two specific reliability structures which have attracted considerable research interest recently (r-within-consecutive-k-out-of-n system and two dimensional r-within-k₁ × k₂-out-of-n₁ × n₂ system) are studied by the new approach and numerical calculations are carried out, which reveal the high quality of our approximations. Several possible extensions and generalizations are also presented in brief.     

Some structural properties of a Markovian storage/production system

We consider a storage/production system with state-dependent production rate and state-dependent demand arrival rate. Every arriving demand gives rise to a 'peak' in the trajectory of the content process. We characterize the processes N ₀(x)and N₁(x), defined as the number of peaks and the number of record peaks, respectively, before the content reaches the level x. The results are applied to the virtual waiting time process W(t) of a M/G/1 queue. Assuming that W(0)= x₀, M(x) is defined to be the number of arrivals before the virtual waiting time drops from x₀ to x₀-x(0⩽ x ⩽ x₀) ; in particular, Mx₀)is the number of customers arriving during the first busy period. It is shown that (M(x))(0⩽ x ⩽ x₀)is a compound Poisson process, and its jump size distribution is derived in closed form. This revised version was published online in June 2006 with corrections to the Cover Date.