System occupancy of a two-class batch-service queue with class-dependent variable server capacity

Due to their wide area of applications, queueing models with batch service, where the server can process several customers simultaneously, have been studied frequently. An important characteristic of such batch-service systems is the size of a batch, that is the number of customers that are processed simultaneously. In this paper, we analyse a two-class batch-service queueing model with variable server capacity, where all customers are accommodated in a common first-come-first served single-server queue. The server can only process customers that belong to the same class, so that the size of a batch is determined by the number of consecutive same-class customers. After establishing the system equations that govern the system behaviour, we deduce an expression for the steady-state probability generating function of the system occupancy at random slot boundaries. Also, some numerical examples are given that provide further insight in the impact of the different parameters on the system performance

Delay analysis of a two-class batch-service queue with class-dependent variable server capacity

In this paper, we analyse the delay of a random customer in a two-class batch-service queueing model with variable server capacity, where all customers are accommodated in a common single-server first-come-first-served queue. The server can only process customers that belong to the same class, so that the size of a batch is determined by the length of a sequence of same-class customers. This type of batch server can be found in telecommunications systems and production environments. We first determine the steady state partial probability generating function of the queue occupancy at customer arrival epochs. Using a spectral decomposition technique, we obtain the steady state probability generating function of the delay of a random customer. We also show that the distribution of the delay of a random customer corresponds to a phase-type distribution. Finally, some numerical examples are given that provide further insight in the impact of asymmetry and variance in the arrival process on the number of customers in the system and the delay of a random customer

Queue Length and Server Content Distribution in an Infinite-Buffer Batch-Service Queue with Batch-Size-Dependent Service

We analyze an infinite-buffer batch-size-dependent batch-service queue with Poisson arrival and arbitrarily distributed service time. Using supplementary variable technique, we derive a bivariate probability generating function from which the joint distribution of queue and server content at departure epoch of a batch is extracted and presented in terms of roots of the characteristic equation. We also obtain the joint distribution of queue and server content at arbitrary epoch. Finally, the utility of analytical results is demonstrated by the inclusion of some numerical examples which also includes the investigation of multiple zeros

Analysis of a batch-service queue with variable service capacity, correlated customer types and generally distributed class-dependent service times

Queueing models with batch service have been studied frequently, for instance in the domain of telecommunications or manufacturing. Although the batch server's capacity may be variable in practice, only a few authors have included variable capacity in their models. We analyse a batch server with multiple customer classes and a variable service capacity that depends on both the number of waiting customers and their classes. The service times are generally distributed and class-dependent. These features complicate the analysis in a non-trivial way. We tackle it by examining the system state at embedded points, and studying the resulting Markov Chain. We first establish the joint probability generating function (pgf) of the service capacity and the number of customers left behind in the queue immediately after service initiation epochs. From this joint pgf, we extract the pgf for the number of customers in the queue and in the system respectively at service initiation epochs and departure epochs, and the pgf of the actual server capacity. Combined with additional techniques, we also obtain the pgf of the queue and system content at customer arrival epochs and random slot boundaries, and the pgf of the delay of a random customer. In the numerical experiments, we focus on the impact of correlation between the classes of consecutive customers, and on the influence of different service time distributions on the system performance. (C) 2019 Elsevier B.V. All rights reserved

System occupancy in a multiclass batch-service queueing system with limited variable service capacity

In multi-class telecommunications or manufacturing systems, customers belonging to the same class can often be processed together. This results in a service capacity that depends on the classes of the customers in the queue. In this paper, we analyse a discrete-time batch- service queue with two customer classes. The single batch server can group all same-class customers at the head of the queue up to a constant class-dependent maximum service capac- ity. We focus on the analysis of the system occupancy at service initiation opportunities, and also compute both a light- and heavy traffic approximation in order to reduce the numeri- cal complexity introduced by the maximum service capacities. Additionally, we propose a method for interpolating between these approximations in order to study the behaviour in the intermediate region. We also deduce the system occupancy and its approximations at random slot boundaries. In the numerical experiments, we examine the conditions under which these proposed approximations are accurate