Heavy-traffic single-server queues and the transform method

Heavy-traffic limit theory is concerned with queues that operate close to criticality and face severe queueing times. Let W denote the steady-state waiting time in the GI/G/1 queue. Kingman (1961) showed that W, when appropriately scaled, converges in distribution to an exponential random variable as the system's load approaches 1. The original proof of this famous result uses the transform method. Starting from the Laplace transform of the pdf of W (Pollaczek's contour integral representation), Kingman showed convergence of transforms and hence weak convergence of the involved random variables. We apply and extend this transform method to obtain convergence of moments with error assessment. We also demonstrate how the transform method can be applied to so-called nearly deterministic queues in a Kingman-type and a Gaussian heavy-traffic regime. We demonstrate numerically the accuracy of the various heavy-traffic approximations.</p

Roots, symmetry and contour integrals in queueing systems

Many queueing systems are analysed using the probability-generating- function (pgf) technique. This approach often leads to expressions in terms of the (complex) roots of a certain equation. In this paper, we show that it is not necessary to compute the roots in order to evaluate these expressions. We focus on a certain class of pgfs with a rational form and represent it explicitly using symmetric functions of the roots. These functions can be computed using contour integrals. We also study when the mean of the random variable corresponding to the considered pgf is an additive function of the roots. In this case, it may be found using one contour integral, which is more reliable than the root-finding approach. We give a necessary and sufficient condition for an additive mean. For example, the mean is an additive function when the numerator of the pgf has a polynomial-like structure of a certain degree, which means that the pgf can be represented in a special product form. We also give a necessary and sufficient condition for the mean to be independent of the roots

Novel heavy-traffic regimes for large-scale service systems

We introduce a family of heavy-traffic regimes for large scale service systems, presenting a range of scalings that include both moderate and extreme heavy traffic, as compared to classical heavy traffic. The heavy-traffic regimes can be translated into capacity sizing rules that lead to Economies-of-Scales, so that the system utilization approaches 100% while congestion remains limited. We obtain heavy-traffic approximations for stationary performance measures in terms of asymptotic expansions, using a non-standard saddle point method, tailored to the specific form of integral expressions for the performance measures, in combination with the heavy-traffic regimes

Optimal capacity allocation for heavy-traffic fixed-cycle traffic-light queues and intersections

Setting traffic light signals is a classical topic in traffic engineering, and important in heavy-traffic conditions when green times become scarce and longer queues are inevitably formed. For the fixed-cycle traffic-light queue, an elementary queueing model for one traffic light with cyclic signaling, we obtain heavy-traffic limits that capture the long-term queue behavior. We leverage the limit theorems to obtain sharp performance approximations for one queue in heavy traffic. We also consider optimization problems that aim for optimal division of green times among multiple conflicting traffic streams. We show that inserting heavy-traffic approximations leads to tractable optimization problems and close-to-optimal signal prescriptions. The same type of limiting result can be established for several vehicle-actuated strategies which adds to the general applicability of the framework presented in this paper

Queueing models for cable access networks

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