Operations Research Letters 36 (2008) 133 – 139 Operations Research Letters www.elsevier.com/locate/orl A new renewal approximation for certain autocorrelated processes Mojtaba Araghi, Barı¸ s Balcıo ˜ glu ∗ Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ont., Canada M5S 3G8 Received 1 March 2007; accepted 29 March 2007 Available online 4 July 2007 Abstract We propose a new renewal approximation for autocorrelated streams with limiting index of dispersion less than unity. The superposition of independent processes and the splitting of an autocorrelated process are studied. The proposed approximation performs well on predicting the mean delay in a single server queueing system receiving autocorrelated processes. Crown Copyright © 2007 Published by Elsevier B.V. All rights reserved. Keywords: Autocorrelation; Indices of Dispersion; Superposition; Splitting; Queueing networks; Mean waiting time 1. Introduction Statistical analysis of data from underlying stochastic pro- cesses in delay systems can reveal temporal dependence. When this is the case, incorporating such processes in analytical mod- els can become quite difficult. A viable approach would be to summarize the important statistical characteristics of the orig- inal data so that using them an approximating renewal process could be fit. Once this is done, existing theory operating on renewal stochastic processes assumption can be exploited to predict the performance of the original system. Our interest in this topic is triggered by the promising results obtained via the three-parameter exponential residual (ER) re- newal approximation proposed in [11] in predicting the mean waiting time in the G/G/1 queue. The ER approximation is in agreement with the two-parameter renewal approximation pro- posed in [14] in equating the arrival rate of the original stream with that of the approximating renewal process. Moreover, both approximations in [14,11] estimate the limiting index of dis- persion of the autocorrelated stream, which is a measure of its variability and autocorrelation information, and equate that to the squared-coefficient of variation of the renewal process. The additional third parameter in the ER approximation not only improves the accuracy of predictions but due to the ∗ Corresponding author. E-mail addresses: mojtaba@mie.utoronto.ca (M. Araghi), baris@mie.utoronto.ca (B. Balcıo˜ glu). 0167-6377/$ - see front matter Crown Copyright © 2007 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2007.03.007 possibility of obtaining the density function Laplace transform (LT) of the approximating r.v. of interest, it can be also used to analyze autocorrelated processes other than the job arrivals in a delay system (such as [6] studying the autocorrelated times- to-failure). However, as shown in [5], the ER approximation can approximate an autocorrelated stream only if its limiting index of dispersion is greater than or equal to 0.5. Hence, in this paper, we will propose a new three-parameter generalized Erlang (GE) renewal approximation for streams that cannot be studied by the ER approximation. We will first discuss how the parameters of the GE approximation can be estimated. Next, we will show how it can be employed to predict mean delay in a single server queueing system receiving the superposition of independent processes or a thinned/split autocorrelated process as the arrival process. In the literature, the flow processes are approximated by re- newal processes, although they may have significant temporal dependence that worsens queueing performance dramatically. Researchers mostly pay attention to arrival streams with pos- itive autocorrelation since high levels of positive autocorrela- tion significantly increase the delay experienced by customers (see [4,11] and the references therein). However, similar to the numerical examples we study in Section 4 involving the super- position of independent Erlangian processes, one can observe negative autocorrelation in arrival streams. Predicting the mean delay in systems receiving such job arrival processes is not straightforward, either.