K. Sikdar

Analytic and numerical aspects of batch service queues with single vacation

This paper deals with an M/G/1 batch service queue where customers are served in batches of maximum size b with a minimum threshold value a. The server takes a single vacation when he finds less than a customers after the service completion. The vacation time of the server is arbitrarily distributed. Using the supplementary variable method we obtain the probability generating functions of the queue length distributions at various epochs. We also obtain relations among queue length distributions at arbitrary, service (vacation) termination epochs. Further their evaluation is also discussed. Finally, some numerical results and graphs are presented.

Computing queue length distributions in MAP/G/1/N queue under single and multiple vacation

This paper studies a single server queue with finite waiting room in which the server takes vacation(s) whenever the system becomes empty and we consider both single and multiple vacation(s). Whereas the input process is a Markovian Arrival Process (MAP), the service and vacation times are arbitrarily distributed. The distributions of number of customers in the queue at service completion, vacation termination, departure, arbitrary and pre-arrival epochs have been obtained. Computational procedure has been given when the service- and vacation-time distributions are of phase type (PH-distribution).

On the batch arrival batch service queue with finite buffer under server's vacation: MX/GY/1/N queue

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.

The queue length distributions in the finite buffer bulk-service MAP/G/1 queue with multiple vacations

We consider a finite buffer batch service queueing system with multiple vacations wherein the input process is Markovian arrival process (MAP). The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. The service- and vacation- times are arbitrarily distributed. We obtain the queue length distributions at service completion, vacation termination, departure, arbitrary and pre-arrival epochs. Finally, some performance measures such as loss probability, average queue lengths are discussed. Computational procedure has been given when the service- and vacation- time distributions are of phase type (PH-distribution).

The analysis of a finite-buffer general input queue with batch arrival and exponential multiple vacations

Vacation queueing models have wide range of application in several areas including computer-communication, and manufacturing systems. A finite-buffer single-server queue with renewal input and multiple exponential vacations has been analysed by Karaesmen and Gupta (1996). In this paper we extend the analysis to cover the batch arrivals, i.e. we consider a batch arrival single-server queue with renewal input and multiple exponential vacations. Using the imbedded Markov chain and supplementary variable techniques we obtain steady-state distribution of number of customers in the system at pre-arrival and arbitrary epochs. The Laplace-Stieltjes transforms of the actual waiting-time distribution of the first-, arbitrary- and last-customer of a batch under First-Come-First-Serve discipline have been derived. Finally, we present useful performance measures of interest such as probability of blocking, average queue (system) length. Some tables and graphs showing the effect of model parameters on key performance measures are presented.

Analysis of finite-buffer bulk-arrival bulk-service queue with variable service capacity and batch-size-dependent service: M X /G Y r /1/N

Over the past few decades, bulk-arrival bulk-service queues have found wide application in several areas including computer-communication and telecommunication systems. In this paper, we consider a single server finite-buffer queue where customers arrive in batches according to the compound Poisson process and are served in batches of variable service capacity. The service times of the batches are arbitrarily distributed and depend on the size of the batch taken into for service. We obtain the joint distribution of the number of customers in the queue and number with the server, and other distributions such as number of customers in the queue, in the system, and number with the server. Various performance measures such as average number of customers in the system (queue), with the server, blocking probabilities, etc. are obtained. Several numerical results are presented and comparative studies of batch-size-dependent service with the one when service time of the batches are independent of the size of the batch have been carried out.

Computing system length distribution of a finite-buffer bulk-arrival bulk-service queue with variable server capacity

In a recent paper, Chang et al. (2004) analysed finite-buffer bulk-arrival bulk-service queue with variable server capacity: MX/GY/1/N, and obtained queue length distributions at departure-, arbitrary- and arrival-epochs. They only obtained the distribution of the number of customers in the queue when server is idle/busy. From their analysis one cannot obtain the system length distribution or distribution of the number of customers in the batch undergoing service with the server when the server is busy. In this paper, we reinvestigate the model and first obtain the joint distribution of the number of customers in the queue and number with the departing batch at departure-epoch and then using it we derive the joint distribution of the number of customers in the queue and number with the server at arbitrary- and arrival-epoch. Besides obtaining system length distribution, we also obtain distribution of number of customers in the queue when the server is idle/busy and several performance measures viz. blocking probability, average number of customers in the system (queue) and average number of customers with the server.

A finite capacity bulk service queue with single vacation and Markovian arrival process

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.

The N Threshold Policy in the Finite Buffer GI/MSP/1 Queues

Abstract This article deals with a finite buffer GI/MSP1 queue with an N threshold policy. In this system, the server is turned off whenever the system is empty and inspects the queue length every time a customer arrives. When the queue length reaches a pre-specified value N(N > 1), the server turns on and serves customers continuously until the system becomes empty. The distributions of the number of customers in the system at pre-arrival and arbitrary epochs, as well as the distributions of the waiting time (in the system) are established. Finally, some tables and graphs showing the effect of model parameters on key performance measures are presented. The model has potential applications in the area of computer networks, telecommunication systems, and manufacturing systems.

GI/M/1 Queues with Server Vacations and Dependent Interarrival and Service Times

Abstract We derive the equilibrium distribution at pre-arrival and arbitrary epochs and the waiting time distribution in a GI/M/1 queueing system with dependence between the service time of each customer and the subsequent interarrival times. In addition, the server takes exponentially distributed vacations when there are no customers left to serve in the queue.