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Abstract: A Queue with Correlated Arrivals

This paper deals with a queue with a Markov renewal arrival process (MRP) which is autocorrelated. Our choice of models for the arrival process has been motivated by the need to keep the marginal distributions the same as far as possible. By doing so, it is possible to better expose the pure effects of the parameters in the arrival process on the correlation coefficient and thence on the mean queue length. We consider the various effects of 4 parameters: the ``stickiness'' of the underlying Markov chain ($p$), differences in the mean interarrival times of each type ($m_i-m_j$), the variance of the arrival times of each type ($v _j $), and the number of states ($n$). We show that $p$ and $m_ i-m_j$ interact in such a way that the rate of convergence of mean queue length to infinity is faster for large $m _i-m_j$ as a function of $p$. It is possible for the queue length process to be in steady state but the mean queue length to be arbitrary large solely due to correlations. We also show that decreases in $v_j$ increase correlations but can decrease the mean queue length. Also, the number of states, acting through the correlation coefficient can have additional effects on the mean queue length especially in the case of ``sticky'' MRP's. It would appear that more attention should be paid to the correlations especially in situations where the traffic intensity is high and where correlations can be present and can be large.