Numerical Calculation of the Stationary Distribution of the Main Multiserver Retrial Queue

We are concerned with the main multiserver retrial queue of M/M/c type with exponential repeated attempts. It is known that an analytical solution of this queueing model is difficult and does not lead to numerical implementation. Based on appropriate understanding of the physical behavior, an efficient and numerically stable algorithm for computing the stationary distribution of the system state is developed. Numerical calculations are done to compare our approach with the existing approximations.     

多重休假的带启动期Geom/G/1排队

本文研究多重休假的带启动期的Geom/G/1离散时间排队。给出稳态队长，等待时间分布的母函数及其随机分解结果，推导出忙期，假期和启动期的母函数等。     

A sequential technique for the control of traffic intensity in Markovian queues

A sequential parameter control technique previously introduced by the author is modified in this paper so as to make it simple in practice. The detailed procedure involving two phases, a warning phase with control limits and a testing phase using an appropriate test is illustrated for a queueing system with an embedded Markov chain. Operating characteristics of the procedure are also examined.     

A BMAP/G/1 Retrial Queue with a Server Subject to Breakdowns and Repairs

In this paper, we consider a BMAP/G/1 retrial queue with a server subject to breakdowns and repairs, where the life time of the server is exponential and the repair time is general. We use the supplementary variable method, which combines with the matrix-analytic method and the censoring technique, to study the system. We apply the RG-factorization of a level-dependent continuous-time Markov chain of M/G/1 type to provide the stationary performance measures of the system, for example, the stationary availability, failure frequency and queue length. Furthermore, we use the RG-factorization of a level-dependent Markov renewal process of M/G/1 type to express the Laplace transform of the distribution of a first passage time such as the reliability function and the busy period.     

Partial Pooling in Tandem Lines with Cooperation and Blocking

For a tandem line of finite, single-server queues operating under the production blocking mechanism, we study the effects of pooling several adjacent stations and the associated servers into a single station with a single team of servers. We assume that the servers are cross-trained (so that they can work at several different stations) and that two or more servers can cooperate on the same job. For such a system, we provide sufficient conditions on the service times and sizes of the input and output buffers at the pooled station under which pooling will decrease the departure time of each job from the system (and hence increase the system throughput). We also show that pooling decreases the total number of jobs in the system at any given time and the sojourn time of each job in the system if the departure time of each job from the system is decreased by pooling and there is an arrival stream at the first station. Moreover, we provide sufficient conditions under which pooling will improve the holding cost of each job in the system incurred before any given time, and extend our results to closed tandem lines and to queueing networks with either a more general blocking mechanism or probabilistic routing. Finally, we present a numerical study aimed at quantifying the improvements in system performance obtained through pooling and at understanding which stations should be pooled to achieve the maximum benefit. Our results suggest that the improvements gained by pooling may be substantial and that the bottleneck station should be among the pooled stations in order to obtain the greatest benefit. AMS subject classification: 90B22     

A Diffusion Approximation for a Markovian Queue with Reneging

Consider a single-server queue with a Poisson arrival process and exponential processing times in which each customer independently reneges after an exponentially distributed amount of time. We establish that this system can be approximated by either a reflected Ornstein–Uhlenbeck process or a reflected affine diffusion when the arrival rate exceeds or is close to the processing rate and the reneging rate is close to 0. We further compare the quality of the steady-state distribution approximations suggested by each diffusion.     

Large deviations of sojourn times in processor sharing queues

This paper presents a large deviation analysis of the steady-state sojourn time distribution in the GI/G/1 PS queue. Logarithmic estimates are obtained under the assumption of the service time distribution having a light tail, thus supplementing recent results for the heavy-tailed setting. Our proof gives insight into the way a large sojourn time occurs, enabling the construction of an (asymptotically efficient) importance sampling algorithm. Finally our results for PS are compared to a number of other service disciplines, such as FCFS, LCFS, and SRPT. 2000 mathematics subject classification: 60K25.     

Waiting time distributions for closed M/M/N processor sharing queues

We consider a multiple server processor sharing model with a finite number of terminals (customers). Each terminal can submit at most one job for service at any time. The think times of the terminals and the service time demands are independently exponentially distributed. We focus our attention on the exact detailed analysis of the waiting time distribution of a tagged job. We give the Laplace-Stieltjes transform of the waiting time distribution conditioned on the job's service time demand and the state of the other terminals and show that these transforms can be efficiently evaluated and inverted. Further results include the representation of conditioned waiting times as mixtures of a constant and several exponentially distributed components. The numerical precision of our results is being compared with results from a discrete approximation of the waiting time distributions.The main part of this research was carried out at the Institut für Mathematische Stochastik of the Technische UniversitÄt Braunschweig.     

A decomposition-related throughput property of tandem queueing networks with blocking

A general throughput property of tandem queueing networks with blocking that relates existing decomposition methods to throughput bounds is discussed using the sample path approach.     

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.