Analyzing state-dependent arrival in GI/BMSP/1/∞ queues

Performance Evaluation, 2013

In the past, many researchers have analysed queueing models with batch service. In such models, the server typically postpones service until the number of present customers reaches a service threshold, whereupon service is initiated of a batch consisting of several customers. In addition, correlation in the customer arrival process has been studied for many different queueing models. However, correlated arrivals in batch-service models has attracted only modest attention. In this paper, we analyse a discrete-time D-BMAP/G l,c /1 queue, whereby the service time of a batch is dependent on the number of customers within it. In addition, a timing mechanism is included, to avoid that customers suffer excessive waiting times because their service is postponed until the amount of customers reaches the service threshold. We deduce various useful performance measures related to the buffer content and we investigate the impact of the traffic parameters on the system performance through some numerical examples. We show that correlation merely has a small impact on the service threshold that minimizes the mean system content, and consequently, that the existing results of the corresponding independent system can be applied to determine a near-optimal service threshold policy, which is an important finding for practitioners. On the other hand, we demonstrate that for other purposes, such as performance evaluation and buffer management, correlation in the arrival process cannot be ignored, a conclusion that runs along the same lines as in queueing models without batch service.

Communications in statistics, 1991

The versatile Markovian point process was introduced by M. F. Neuts in 1979. This is a rich class of point processes which contains many familiar arrival process as very special cases. Recently, the Batch Markovian Arrival Process, a class of point processes which was subsequently shown tobe equivalent to Neuts' point process, has been studied using a more transparent notation. Recent results in the matrix-analytic approach to queueing theory have su.bstantially reduced the computational complexity of the algorithmic solution of single server queues with a general Markovian arrival process. We generalize these results to the single server queue with the batch arrival process and emphasize the resulting simplifications. Algorithms for the special cases of the PHIG/I and MMPPIGII queues are highlighted as these models are receiving renewed attention in the literature and the new algorithms proposed here are simpler than existing ones. In particular, the PHIG/I queue has additional structure which further enhances the efficiency of its algorithmic solution. Also, the two-state MMPPIGII queue, which has applications in communications modeling, has an extremely simple solution.

4OR, 2021

This paper analyzes a finite-buffer queueing system, where customers arrive in batches and the accepted customers are served in batches by a single server. The service is assumed to be dependent on the batch-size and follows a general bulk service rule. The inter-arrival times of batches are assumed to be correlated and they are represented through the batch Markovian arrival process (BM AP). Computation procedure of the queue-length distributions at the post-batch-service completion, an arbitrary, and the pre-batch-arrival epochs are discussed. Various performance measures along with the consecutive customer loss probabilities are studied considering batch-size-dependent renewal service time distributions. Further, the above finite-buffer bulk-service queueing model is also investigated considering correlated batch-service times which are presented through the Markovian service process (M SP). The phase-dependent consecutive loss probabilities for the correlated batch-service times are determined. In the form of tables and graphs, a variety of numerical results for different batch-service time distributions are presented in this paper. Keywords Finite-buffer queue • batch Markovian arrival process (BM AP) • Markovian service process (M SP) • batch-size-dependent bulk service • performance measures • consecutive customer loss (CCL)

Mathematics

In this paper, we discuss the waiting-time distribution for a finite-space, single-server queueing system, in which customers arrive singly following a Poisson process and the server operates under (a,b)-bulk service rule. The queueing system has a finite-buffer capacity ‘N’ excluding the batch in service. The service-time distribution of batches follows a general distribution, which is independent of the arrival process. We first develop an alternative approach of obtaining the probability distribution for the queue length at a post-departure epoch of a batch and, subsequently, the probability distribution for the queue length at a random epoch using an embedded Markov chain, Markov renewal theory and the semi-Markov process. The waiting-time distribution (in the queue) of a random customer is derived using the functional relation between the probability generating function (pgf) for the queue-length distribution and the Laplace–Stieltjes transform (LST) of the queueing-time distri...

Journal of Applied Mathematics and Stochastic Analysis, 2004

Vacation time queues with Markovian arrival process (MAP) are mainly useful in modeling and performance analysis of telecommunication networks based on asynchronous transfer mode (ATM) environment. This paper analyzes a single-server finite capacity queue wherein service is performed in batches of maximum size “b” with a minimum threshold “a” and arrivals are governed by MAP. The server takes a single vacation when he finds less than “a” customers after service completion. The distributions of buffer contents at various epochs (service completion, vacation termination, departure, arbitrary and pre-arrival) have been obtained. Finally, some performance measures such as loss probability and average queue length are discussed. Numerical results are also presented in some cases.

Journal of Probability and Statistics

This paper deals with a batch arrival infinite-buffer single server queue. The interbatch arrival times are generally distributed and arrivals are occurring in batches of random size. The service process is correlated and its structure is presented through a continuous-time Markovian service process (C-MSP). We obtain the probability density function (p.d.f.) of actual waiting time for the first and an arbitrary customer of an arrival batch. The proposed analysis is based on the roots of the characteristic equations involved in the Laplace-Stieltjes transform (LST) of waiting times in the system for the first, an arbitrary, and the last customer of an arrival batch. The corresponding mean sojourn times in the system may be obtained using these probability density functions or the above LSTs. Numerical results for some variants of the interbatch arrival distribution (Pareto and phase-type) have been presented to show the influence of model parameters on the waiting-time distribution....

Computers & Mathematics With Applications, 2009

This paper analyzes a discrete-time finite-buffer multi-server bulk-service queueing system in which the interarrival-and service-times are, respectively, arbitrarily and geometrically distributed. Using the supplementary variable and the imbedded Markovchain techniques, the queue is analyzed for the early arrival system. We obtain state probabilities at prearrival, arbitrary and outside observer's observation epochs. Some performance measures, waiting-time distribution in the queue along with some numerical results, and special cases of the model have also been discussed. Finally, it is shown that in the limiting case the results obtained in this paper tend to the continuous-time counterpart.

Computers & Industrial Engineering, 2006

The independence of processes in queueing systems is generally assumed when developing queueing models. However, real systems often involve several process dependencies, and failure to take these into consideration can lead to serious underestimation of the performance measures. We consider herein a single server queueing system with a Markov renewal process (MRP) for its arrival process and a general service time distribution, and derive the distribution function and correlation coefficient of the departure process. Since the departure process also often corresponds to an arrival process in downstream queues, the results obtained here can be used to derive a better approximation of the performance measures of a non-product form general queueing network.

2017

In many queueing systems the server processes several customers simultaneously. Although the capacity of a batch, that is the number of customers that can be processed simultaneously, is often variable in practice, nearly all batch-service queueing models in literature consider a constant capacity. In this paper, we extend previous work on a batch-service queueing model with variable server capacity, where customers of two classes are accommodated in a common first-come-first-served single-server queue. We include correlation between the classes of consecutive customers, and the service times are geometrically distributed. We establish the equations that govern the system behaviour, the stability condition, and an expression for the steady-state probability generating function of the system occupancy at random slot boundaries. In addition, some numerical results are shown to study the impact of the mean service times and of the customer-based correlation in the arrival process on the performance of the queueing system.

Journal Européen des Systèmes Automatisés

Computing and logistics management systems have a wide area of applications with compound Poisson process Markov system with a batch servicing facility where customers arrive either independently or batches for service into the multi-server queues. The service of the customers is processed either independently or batch-wise based on the requirement of various sizes. The order of service has been found to follow First Come First Service while customers arrive according to the exponential distribution. A mathematical model is proposed to process customers by using generalized spectral expansion method. The explicit type required to service the system is measured as buffer size. For accurate assessment of performance, numerical results have been depicted in graphical form.