Queue Length and Server Content Distribution in an Infinite-Buffer Batch-Service Queue with Batch-Size-Dependent Service (original) (raw)
Related papers
Communications in Statistics - Theory and Methods, 2020
Queueing systems with batch Markovian arrival process (BMAP) have paramount applications in the domain of wireless communication. The BMAP has been used to model the superposition of video sources and to approximate the super-position of data, voice and video traffic. This paper analyzes an infinitebuffer generally distributed batch-service queue with BMAP, general bulk service (a, b) rule and batch-size-dependent service time. In this proposed analysis, we mainly focus on deriving the bivariate vector generating function of queue and server content distribution together at departure epoch using supplementary variable technique. The mathematical procedure for the complete extraction of distribution at departure epoch has been discussed and using those extracted probabilities, we achieve the queue and server content distribution at arbitrary epoch. Finally, numerical illustrations have been carried out in order to make a deep insight to the readers which contains deterministic as well as phase-type service time distributions.
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...
RAIRO - Operations Research, 2016
We consider an infinite-buffer single-server queue with renewal input and Markovian service process where customers are served in batches according to a general bulk service rule. Queue-length distributions at epochs of pre-arrival, arbitrary and post-departure have been obtained along with some important performance measures such as mean queue lengths and mean waiting times in both the system as well as the queue. We also obtain the steady-state service batch size distributions as well as system-length distributions. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of queue-length distribution at a pre-arrival epoch. Also, we provide analytical and numerical comparison between the roots method used in this paper and the matrix geometric method in terms of computational complexities and required computation time to evaluate pre-arrival epoch probabilities for both the methods. Later, we have established heavy-and light-traffic approximations as well as an approximation for the tail probabilities at pre-arrival epoch based on one root of the characteristic equation. Numerical results for some cases have been presented to show the effect of model parameters on the performance measures.
Quality Technology & Quantitative Management, 2021
Due to the widespread applicability of discrete-time queues in wireless networks or telecommunication systems, this paper analyzes an infinite-buffer batch-service queue with single and multiple vacation where customers/messages arrive according to the Bernoulii process and service time varies with the batch-size. The foremost focal point of this analysis is to get the complete joint distribution of queue length and server content at service completion epoch, for which first the bivariate probability generating function has been derived. We also acquire the joint distribution at arbitrary slot. We also provide several marginal distributions and performance measures for the utilization of the vendor. Transmission of data through a particular channel is skipped due to the high transmission error. As the discrete phase type distribution plays a noteworthy role to control this error, we include numerical example where service time distribution follows discrete phase type distribution. A comparison between batch-size dependent and independent service has been drawn through the graphical representation of some performance measures and total system cost.
Analysis of a versatile batch-service queueing model with correlation in the arrival process
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.
A Note on the Waiting-Time Distribution in an Infinite-Buffer GI[X]/C-MSP/1 Queueing System
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....
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)
Delay analysis of two batch-service queueing models with batch arrivals: Geo X /Geo c /1
4OR, 2010
In this paper, we compute the probability generating functions (PGF's) of the customer delay for two batch-service queueing models with batch arrivals. In the first model, the available server starts a new service whenever the system is not empty (without waiting to fill the capacity), while the server waits until he can serve at full capacity in the second model. Moments can then be obtained from these PGF's, through which we study and compare both systems. We pay special attention to the influence of the distribution of the arrival batch sizes. The main observation is that the difference between the two policies depends highly on this distribution. Another conclusion is that the results are considerably different as compared to Bernoulli (single) arrivals, which are frequently considered in the literature. This demonstrates the necessity of modeling the arrivals as batches.
A Model for Compound Poisson Process Queuing System with Batch Arrivals and Services
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.
On the batch arrival batch service queue with finite buffer under server’s vacation: queue
Computers & Mathematics with Applications, 2008
This paper considers a finite-buffer batch arrival and batch service queue with single and multiple vacations. The steady-state distributions of the number of customers in the queue at service completion, vacation termination, departure, arbitrary and pre-arrival epochs have been obtained. Finally, various performance measures such as average queue length, average waiting time, probability that the server is busy, blocking probabilities, etc. are discussed along with some numerical results. The effect of certain model parameters on the key performance measures have also been investigated. The model has potential application in several areas including manufacturing, internet web-server and telecommunication systems.