Analysis of queues under correlated arrivals with applications to web server performance

Abstract

The complexity of many high-volume Web sites often makes it difficult to mathematically analyze various performance measures. Since these complex behaviors can have a significant impact on performance, it is important to capture them in sufficient detail in the analysis of the corresponding queueing systems.We consider the access logs from a particular class of high-volume Web sites serving dynamic content to obtain a better understanding of the complexities of user request patterns in such environments. Our analysis demonstrates that these arrival patterns exhibit strong dependence structures which can be accurately represented by an arrival process with strong (short-range) correlations, at least for the class of Web sites motivating our study [2]. Based on these results, we develop a methodology for approximating this class of dependent arrival processes by a set of phase-type distributions. Our approach consists of formulating and solving a nonlinear optimization problem that fits a set of dependent stochastic models to approximate the interarrival time patterns from the data, which includes matching the autocorrelation function. To evaluate the effectiveness of our approach, we conduct a large number of statistical tests and experiments showing that our methodology provides an excellent match between the real user request data and the fitted approximate arrival process.Given this dependent arrival process as input, we then derive an exact matrix-analytic analysis of a general multi-server queue under two server queueing disciplines. This analysis yields results that provide significant reductions in the numerical computation required to solve the queueing models. To demonstrate the accuracy of the performance measures obtained under these methods, a large number of experiments were performed and detailed comparisons were made between the sojourn time measures from our analysis and the corresponding measures obtained from simulation of the queueing system under the actual user request data. These results show both sets of performance measures to be in excellent agreement, with relative errors consistently less than 5%, and further demonstrate the robustness of our approach. We also conduct a set of numerical experiments that exploit our matrix-analytic analysis and its computational efficiency, which are then used to establish some important results for multi-server queues under dependent arrival processes. This includes the notion of effective stability where the point at which the mean sojourn time of the queue exceeds a large constant (e.g., 1000) multiplied by the mean service time occurs well before the theoretical stability condition for the queue.Due to space limitations, we simply summarize a subset of our results in this extended abstract. We refer the interested reader to [1] for additional details, references and results.