Discrete-time bulk-service queues with accessible and non-accessible batches

Steyaert

Computers & Operations Research

et al. 2013

We deduce approximations for the tail probabilities of the customer delay in a discrete-time queueing model with batch arrivals and batch service. As in telecommunications systems transmission times are dependent on packet sizes, we consider a general dependency between the service time of a batch and the number of customers within it. The model also incorporates a timer mechanism to avoid excessive delays stemming from the requirement that a service can only be initiated when the number of present customers reaches or exceeds a service threshold. The service discipline is first-come, first-served (FCFS). We demonstrate in detail that our approximations are very useful for the purpose of assessing the order of magnitude of the tail probabilities of the customer delay, except in some special cases that we discuss extensively. We also illustrate that neglecting batch-size dependent service times or a timer mechanism can lead to a devastating assessment of the tail probabilities of the customer delay, which highlights the necessity to include these features in the model. The results from this paper can, for instance, be applied to assess the quality of service (QoS) of Voice over IP (VoIP) conversations, which is typically expressed in terms of the order of magnitude of the probability of Computers & Operations Research May 1, 2013 packet loss due to excessive delays. Preprint submitted to

Section: Pr ŵ 2 > Wmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Tail probabilities of the delay in a batch-service queueing model with batch-size dependent service times and a timer mechanism

Steyaert

Computers & Operations Research

et al. 2013

“…Next, approximation (18) and exact formula (12) for Pr [W 2 > w] are depicted versus w in Figures 5-7 for various settings of the system parameters. The figures highlight that there might be a discernable error for small values of w, whereas it becomes small for larger values of w. We also observe that although the relative error might sometimes be quite large, the order of magnitude of Pr [W 2 > w] is well approximated, which is sufficient for e.g.…”

Section: Accuracy Of the Formulasmentioning

confidence: 99%

“…The emphasis was laid on the amount of packets in the system (e.g. [1], [5], [6], [8], [17], [18], [19], [21], [24], [28], [29], [30], [32], [33]). The packet delay, however, has only attracted attention in [7], [13], [14], [16], [22], [23], [26] and [27].…”

Section: Introductionmentioning

confidence: 99%

Tail distribution of the delay in a general batch-service queueing model

Steyaert

Computers & Operations Research

et al. 2012

Batch servers are capable of processing batches of packets instead of individual packets. Although batch-service queueing models have been studied extensively during the past decades, the focus was mainly put on calculating performance measures related to the buffer content, whereas less attention has been paid to the packet delay. In this paper, we focus on the tail probabilities of the delay that a random packet experiences in a general batch-service queueing model. More specifically, we establish approximations for these probabilities, which are highly accurate and easy to calculate. These results, for instance, allow to accurately assess the probability that real-time packets experience an excessive delay in practical telecommunication systems.

“…Batch-service queueing models have been studied extensively during the past decades [2], [6]- [9], [11]- [12], [16]- [17], [19], [21], [24], [26], [28]- [29], [31]. However, these surveys share the common feature that only models with an uncorrelated arrival process are considered, which is unrealistic in several real-life situations.…”

Section: Introductionmentioning

confidence: 99%

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

Analytical and Stochastic Modeling Techniques and Applications

Laevens

et al. 2010

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.