A Batch-Service Queueing Model with a Discrete Batch Markovian Arrival Process

Abstract

Abstract. Queueing systems with batch service have been investigated extensively during the past decades. However, nearly all the studied models share the common feature that an uncorrelated arrival process is considered, which is unrealistic in several real-life situations. In this paper, we study a discrete-time queueing model, with a server that only initiates service when the amount of customers in system (system content) reaches or exceeds a threshold. Correlation is taken into account by assuming a discrete batch Markovian arrival process (D-BMAP), i.e. the distribution of the number of customer arrivals per slot depends on a background state which is determined by a first-order Markov chain. We deduce the probability generating function of the system content at random slot marks and we examine the influence of correlation in the arrival process on the behavior of the system. We show that correlation merely has a small impact on the threshold that minimizes the mean system content. In addition, we demonstrate that correlation might have a significant influence on the system content and therefore has to be included in the model.

Analysis of a versatile batch-service queueing model with correlation in the arrival process

“…Before closing this section, we evaluate the impact of the probabilities β n on the mean customer delay (equal to E [U ] /λ due to Little's law). In Fig.7, the mean delay is depicted versus the load both for the case β n = n/l and β n = 0 (the latter case was considered in our paper [27]). The left pane of the figure corresponds to γ = 0.9 and the right pane represents γ = −0.9.…”

“…Our conference paper [27] served as a starting point for this research. In [27], we have studied the system content in a batch-service queueing model with a service threshold, with geometrically distributed service times that are independent of the number of served customers, and with a customer arrival process modelled by a D-BMAP.…”

“…In [27], we have studied the system content in a batch-service queueing model with a service threshold, with geometrically distributed service times that are independent of the number of served customers, and with a customer arrival process modelled by a D-BMAP. This paper is an extension of [27], in the sense that we consider a more versatile model with service times that are generally distributed and dependent on the number of items in a served batch. In addition, a timing mechanism is included, to avoid excessive delays due to postponing service until the service threshold is reached.…”

“…In addition, a timing mechanism is included, to avoid excessive delays due to postponing service until the service threshold is reached. Furthermore, we establish various quantities related to the number of customers in the queue and server at specific time instants, instead of only the system content at random slot bounds as in [27]. We also elaborate upon the influence of correlation on the behaviour of the system.…”