Search

In this paper, we consider a single-server finite-capacity queue with general bulk service rule where customers arrive according to a discrete-time batch Markovian arrival process (D-BMAP). The mod...

A complete solution for the stationary queue-length distribution of a bulk-arrival, bulk-service (GIX/MY/1) queue is presented. Beginning with a known expression for the probability generating function of the stationary pre-arrival-epoch queue-length distribution, the roots method is used to invert it and determine all probabilities. Next, using level crossing arguments, theoretical relationships between pre-arrival and arbitrary-epoch probabilities are developed. These relationships are then used to directly determine a complete set of probabilities for the arbitrary-epoch queue-length distribution. Finally, selected examples are presented. These demonstrate how, given arbitrary arrival time, arrival group size and service batch size probability distributions, a complete solution for the stationary queue-length probabilities can be readily determined.

The high speed multi-access communication channels such as BISDN and ATM operate on a discrete-time basis rather than on a continuous-time basis. The usual Erlang loss formula developed for continuous-time is not valid for discrete-time queues, though models such as these can be derived from the corresponding discrete-time models. In this paper, we consider the discrete-time GI/Geom/m/m queue. Two variants of this model viz. late-arrival system with delayed access and early arrival system have been discussed and analyzed. Distributions of numbers of busy channels at various epochs have been obtained using the supplementary variable technique. The performance measures such as probability of loss, and average number of busy channels have been presented for interarrival distributions: geometric, deterministic, arbitrary, etc. It is hoped that the results obtained in this paper may provide useful information to designers of telecommunication systems, practitioners, and others.

In this paper, we propose an alternative method to obtain the limiting distribution of number in system for bulk-arrival queues of the type MX/G/1. The method discussed is based on easily obtainable roots of the associated characteristic equation of the model. It is also shown that the tail probabilities can be obtained much more easily. Numerical aspects have been tested for a variety of service-time distributions such as mixed generalized Erlang, generalized Erlang, generalized hyperexponential, etc., and only representative results have been included in the form of tables and graphs. As the model MX/G/1 has wide applications, it is hoped that the results presented here might be of use to queueing theorists, practitioners, and others.

This paper deals with a bulk-service queuing model denoted by It is assumed that the interarrival times are i.i,d.r,v,’s each having the A-phase Erlangian distribution, the service is in bulk, and there is a single exponential server of capacity b. The random variable X denotes the number of units already present with the server at a service epoch, and if X = m (≤ b), then the server takes (b - m) more units or the whole queue length, whichever is smaller. Under the assumption that at a service epoch the server does not wait for the units to arrive if he finds the queue empty, we derive limiting distributions of queue sizes at random, post-departure, and pre-arrival epochs. The first two moments of the distributions are listed. The results are specialized to suit the case of deterministic input. One of the main features of this study is that the results are expressed in terms of a unique real root of the characteristic equation and are thus easy to compute numerically. A sample listing of the root...

This paper analyses a transient solution of a bulk service queueing problem wherein the arrivals follow a Poisson distribution, but the server is available in alternate, independently and exponentially distributed intervals with different means; except that, when the system is empty, he is available immediately to the arriving customer or customers in the bulk service case.