Mean queue size in a queue with discrete autoregressive arrivals of order <Emphasis Type="Italic">p</Emphasis>

We consider a discrete time single server queueing system where the arrival process is governed by a discrete autoregressive process of order p (DAR(p)), and the service time of a customer is one slot. For this queueing system, we give an expression for the mean queue size, which yields upper and lower bounds for the mean queue size. Further we propose two approximation methods for the mean queue size. One is based on the matrix analytic method and the other is based on simulation. We show, by illustrations, that the proposed approximations are very accurate and computationally efficient.     

On the asymptotic analysis of statistical multiplexers with hyper-bursty arrivals

We consider multiplexers in discrete time fed by the superposition of Ternary Markov Sources. Such sources are the natural extension of the Binary Markov Sources (BMS) recently used to model bursty arrivals in a high speed environment. Unlike BMS, we allow sources to have arbitrary (large) variance in the duration of their OFF (silence) or ON (burst) periods.This paper focuses mainly on the impact of large variability either in the ON or OFF period on the performance. Following some asymptotic analysis, simple results on the tail behavior of the number of cells queued in the multiplexer are given.Our results indicate that ignoring the variability in the ON period may grossly underestimate the cell buildup in the multiplexer queue for all levels of the utilization. Furthermore, the impact of large variability of the OFF period depends very much on the utilization of the system. For a lightly-loaded multiplexer (utilization below a given threshold), the impact of large variability of the OFF period is minimal. However, for a heavy-loaded multiplexer (utilization above the threshold) the impact of the large variability in the OFF period is similar to that of the ON period.     

On Matrix-Geometric Solution of Nested QBD Chains

In this paper, a generalization of the level dependent Quasi-Birth-and-Death (QBD) chains is presented. We analyze nested level dependent QBD chains and provide the complete characterization of their fundamental matrices in terms of minimal non-negative solutions of a number of matrix quadratic equations. Our results provide mixed matrix-geometric solution for the stationary distribution of nested QBD chains. Applications in overload control in communication networks are also discussed.     

On the Exact Analysis of a Discrete-Time Queueing System with Autoregressive Inputs

In this paper, we provide an exact analysis of a discrete-time queueing system driven by a discrete autoregressive model of order 1 (DAR(1)) characterized by an arbitrary marginal batch size distribution and a correlation coefficient. Closed-form expressions for the probability generating function and mean queue length are derived. It is shown that the system performance is quite sensitive to the correlation of the arrival process. In addition, a comparison with traditional Markovian processes shows that arrival processes of DAR(1) type exhibit larger queue length as compared with the traditional Markovian processes when the marginal densities and correlation coefficients are matched.