System Occupancy of a Two-Class Batch-Service Queue with Class-Dependent Variable Server Capacity

Abstract

Abstract. 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.

System Performance Of A Variable-Capacity Batch-Service Queue With Geometric Service Times And Customer-Based Correlation

“…With these results we can also obtain the result of our previous paper, see Baetens et al 2016, by using a mean service time µ = 1, which corresponds to single-slot service times and β = 1 − α. Substituting these assumptions in Eq.…”

“…In many of these papers, the service capacity is generically distributed and does not depend on any parameter of the system, like the number of customers waiting in the queue, see the papers of Chaudhry and Chang 2004;Germs and Foreest 2010;Pradhan et al 2015;Sikdar and Samanta 2016. A more detailed description of these papers can be found in our paper (Baetens et al 2016). Also, Germs and Foreest (2013) have developed an algorithmic method to analyse the performance of continuous-time batch service queueing systems with arrival process, service time distribution and variable service capacity that depend on the number of customers in the queue.…”

“…This kind of queueing systems can be found in many telecommunication technologies like optical burst switching networks (Chen et al 2004) and wireless local area networks (Lu et al 2005). To the best of our knowledge, the combination of batch service with variable capacity and multiple customer classes has only appeared in our previous paper Baetens et al (2016). The difference with that paper is that we relax the assumption of fixed single-slot service times by considering geometric service times instead.…”

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

“…In order to find the pgf of the delay of a random customer, we must first find the partial pgfs of the number of customers in the queue at customer arrival and the first customer in the queue is of class A or B after arrival. For this, we will use the results from Baetens et al (2016) for the idle probability and the partial pgf's of the system occupancy at random slot boundaries if the server is processing a class A or B batch. The probability u I that the server is idle in a random slot is given by…”

“…This, for instance, means that if the first customer is of class A, all of the following class A customers are also grouped in the batch that will be taken into service, until the next customer is of class B or no more customers are present, whichever occurs first. The system occupancy at random slot boundaries has been studied previously in our conference paper Baetens et al (2016). To the best of our knowledge, no other papers in the literature have studied the delay in a system that combines variable capacity batch service and multiple customer classes.…”