A batch-service queueing model with a discrete batch Markovian arrival process

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

Backpressure-based control protocols: design and computational aspects

Congestion control in packet-based networks is often realized by feedback protocols. In this paper we assess their performance under a back-pressure mechanism that has been proposed and standardized for Ethernet metropolitan networks. In such a mechanism the service rate of an upstream queue is reduced when the downstream queue is congested, in order to protect the downstream queue. We study a Markovian model that captures the essentials of the protocol, but at the same time allows for numerical analysis. We first derive explicit results for the stability condition of the model (which turns out to be nontrivial). Then we present logarithmic estimates of the probability of buffer overflow in the second queue, which are subsequentially used when devising an efficient simulation procedure based on importance sampling. We conclude the paper by presenting a number of numerical results, and some general design guidelines