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Estimating queue length distributions for queues with random arrivals

Daniel S. Myers, Mary K. Vernon

2012
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Performance Evaluation Review
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This work develops an accurate and efficient two-moment approximation for the queue length distribution in the M/G/1 queue. Queue length distributions can provide insight into the impact of system design changes that go beyond simple averages, but conventional queueing theory lacks efficient techniques for estimating the long-run queue length distribution when service times are not exponential. The approximate queue lengths depend on only the first and second moments of the service time rather
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... han the full service time distribution, resulting in a model that is applicable to a wide variety of systems. Validation results show that the new approximation is highly accurate for light-tailed service time distributions. Work in progress includes developing accurate approximations for multi-server queues and heavytailed service distributions. INTRODUCTION Modern computer systems can be viewed as highly complex collections of shared resources. Customized simulations and analytic models quantify the behavior of these systems and provide insight into system bottlenecks, leading to improved designs. These models typically provide accurate approximations in cases where the exact solution is intractable. One limitation in the queueing theory that customized analytic models draw on, to date, is the lack of efficient, accurate calculations of the queue length distribution in the M/G/1 queue. Queue length distributions can give further insight into the impact of system design changes than mean queue length and mean residence time alone. For example, the queue length distributions at disk queues provides valuable design insight for large-scale storage systems. Closed form expressions for the queue length distribution are available only for service time distributions for which the Pollaczek-Khinchin z-transform of the queue length distribution can be inverted to create such solutions, notably including systems with exponential or hyperexponential service time distributions [4] . In general, inversion of the transform is computationally infeasible [3] . This paper develops a simple and efficient two-moment approximation for the M/G/1 queue length distribution that has three key features. First, its derivation uses only Little's result, the random arrival property of the queue, and the well-known formula for mean residence time in the M/G/1 queue. Second, the new approximate queue length distribu-Copyright is held by author/owner(s). tion depends only on the first and second moments of the service time distribution, which can easily be obtained for most real-world systems. Finally, the approximation is accurate for a wide range of light-tailed service time distributions with high and low coefficients of variation. DERIVATION Consider an M/G/1 queue with average service time x, Poisson arrival rate λ, and a squared coefficient of variation of service time c 2 x . Number the positions of the queue, beginning with number one for the server, two for the first position in the waiting line, and so forth. Let U k be the utilization of position k, which is the fraction of time that position k is occupied. Let N denote the number of customers in the queue at a random point in time after the queue has reached equilibrium. Note that and For position one, U1 is the server utilization, U = λx. More generally, for position k + 1 > 1, we let x k+1 denote the expected time a customer spends in position k + 1 and use Little's result applied to position k + 1 to obtain Given x k+1 for k + 1 > 1, we can estimate P [N > k] for k > 1. To derive the average time at position k + 1 > 1, consider an arbitrary "tagged customer" that arrives to the queue in equilibrium. Due to the PASTA property, the queue length probability distribution at the arrival instant is equal to the equilibrium queue length distribution at a random point in time. There are three cases to consider: 1. The tagged customer arrives when there are fewer than k customers in the queue. In this case, the customer spends no time at position k + 1 and x k+1 = 0. 2. With probability P [N = k], the tagged customer arrives when there are exactly k customers in the queue, in which case the customer arrives directly to position k + 1 and x k+1 = r k+1 , where r k+1 is the average remaining service time for the customer in service conditioned on the knowledge that there are k customers in the queue at the arrival instant, discussed further below.

doi:10.1145/2425248.2425268
fatcat:kppon3yw75hv5fq3r6j7trktvy