A fork-join queueing model: Diffusion approximation,integral representations and asymptotics

We consider two parallel M/M/1 queues which are fed by a single Poisson arrival stream. An arrival splits into two parts, with each part joining a different queue. This is the simplest example of a fork-join model. After the individual parts receive service, they may be joined back together, though we do not consider the join part here. We study this model in the heavy traffic limit, where the service rate in either queue is only slightly larger than the arrival rate. In this limit we obtain asymptotically the joint steady-state queue length distribution. In the symmetric case, where the two servers are identical, this distribution has a very simple form. In the non-symmetric case we derive several integral representations for the distribution. We then evaluate these integrals asymptotically, which leads to simple formulas which show the basic qualitative structure of the joint distribution function.     

On the stability of a partially accessible multi‐station queue with state‐dependent routing

We consider a multi‐station queue with a multi‐class input process when any station is available for the service of only some (not all) customer classes. Upon arrival, any customer may choose one of its accessible stations according to some state‐dependent policy. We obtain simple stability criteria for this model in two particular cases when service rates are either station‐ or class‐independent. Then, we study a two‐station queue under general assumptions on service rates. Our proofs are based on the fluid approximation approach. This revised version was published online in June 2006 with corrections to the Cover Date.     

Cost evaluation in M/G/1 queue with T-policy revisited, technical note

This note clarifies the concept of regeneration cycle used in evaluating the average operating cost of the M/G/1 queue with T-policy studied in 70s. Two ways of defining the regeneration cycle are compared and advantages and disadvantages of each way are pointed out. In addition, we establish the convexity of the cost function based on the service cycle.     

Block-Structured Fluid Queues Driven by QBD Processes

Abstract

In this paper, a unified approach for studying block-structured fluid models is proposed by means of the RG-factorization. When the stochastic environment (or background) is assumed to be a quasi-birth-and death (QBD) process, with either infinitely many levels or finitely many levels, the Laplace transform for the stationary probability distribution of the buffer content is expressed in terms of the R-measure. At the same time, the Laplace-Stieltjes transforms for both the conditional distribution and the conditional mean of a first passage time in such a fluid queue are derived by the same approach.     

网络结构优化设计模型及遗传算法

本文以网型通信网问题为列建立了优化问题的数学模型,并用遗传算法并行求解该问题,该方法较好地解决了既考虑经济效益又考虑需求的网络结构优化问题,该问题的研究对实际应用及动态网络的研究有十分重要的意义。     

Features of some discrete-time cyclic queueing networks

A class of discrete-time closed cyclic networks is analyzed, where queues at each node have ample waiting room and have independent geometric service times with possibly unequal means. If each node has a single server or if there are sufficiently many parallel servers at each node to accommodate all jobs, equilibrium vectors of product form are obtained. For some other cases, equilibrium vectors of product form need not exist. For the single-server model, a normalization constant is computed and used to determine the queue-length distribution at a node.     

The equivalence of the transient behaviour formulae for the single server queue

It is indeed obvious to expect that the different results obtained for some problem are equal, but it needs to established. For the M/M/1/N queue, using a simple algebraic approach we will prove that the results obtained by Takâcs [17] Takâcs, L. 1960. Introduction to the Theory of Queues Oxford University Press.  [Google Scholar] and Sharma and Gupta [13] Sharma, O.P. and Gupta, U.C. 1982. Transient Behaviour of an M/M/l/N Queue. Stoch. Process Appl., 13: 327–331. [Crossref] , [Google Scholar] are equal. Furthermore, a direct proof to the equivalence between all formulae of the M/M/l/∞ queue is established. At the end of this paper, we will show that the time-dependent state probabilities for M/M/l/N queue can be written in series form; its coefficients satisfy simple recurrence relations which would allow for the rapid and efficient evaluation of the state probabilities. Moreover, a brief comparison of our technique, Sharma and Gupta's formula and Takâcs result is also given, for the CPU time computing the state probabilities.     

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.     

Corrigendum to “Tail behaviour of the busy period of a GI/GI/1 queue with subexponential service times” [Stochastic Process. Appl. 111 (2004) 237-258]

The purpose of this note is to correct an error in Baltrunas et al. (2004) [1], and to give a more detailed argument to a formula whose validity has been questioned over the years. These details close a gap in the proof of Theorem 4.1 as originally stated, the validity of which is hereby strengthened.     

On the rod placement theorem of Rybko and Shlosman

Given n−1 points on the real line and a set of n rods of strictly positive lengths , we get to choose an n-th point x_n anywhere on the real line and to assign the rods to the points according to an arbitrary permutation π. The rod is thought of as the workload brought in by a customer arriving at time x_k into a first in -first out queue which starts empty at − ∞. If any x_i equals x_j for i < j, service is provided to the rod assigned to x_i before the rod assigned to x_j. Let denote the set of departure times of the customers (rods). Let denote the number of choices for the location of x_n for which . Rybko and Shlosman proved that