Queues with Galton-Watson-type arrivals

Input traffic at various nodes in packet switched telecommunication networks typically exhibits various levels of correlation. It is well known that input correlation significantly affects queueing performance and hence there is a continuing interest in analytically tractable queueing models which can accurately capture arrival correlation. There is a particular interest in Markovian arrival models, including models with a finite state space such as the discrete-time batch-Markovian arrival model [1, 2], or with a structured infinite state space such as the discrete autoregressive arrival models [3, 4] and the train and session arrival models [5, 6]. Queueing metrics for unstructured finite state space arrival models are not available in closed form. However, efficient algorithms are devised which yield the various performance measures in no time. In contrast, by imposing a structure on the state space of the arrival process, closed-form expressions for the various performance measures can be obtained. In this paper, we propose a Markovian arrival process with a structured infinite statespace, the Galton-Watson arrival process. A discrete-time queueing system is analysed where the arrivals during the consecutive slots stem from a multi-type Galton-Watson branching processes with migration [7]. It is shown that such an arrival process exhibits intricate arrival correlation while closed form expressions for the probability generating functions of queue content and packet delay are obtained. The remainder of the paper is organised as follows. In the next section, the arrival process and corresponding queueing model are introduced first. By a probability generating functions approach, we then obtain expressions for various performance measures. Our results are illustrated by some numerical examples in section 3. Finally, conclusions are drawn in section 4.