We find scaling limits for the sizes of the largest components at criticality for rank-1 inhomogeneous random graphs with power-law degrees with power-law exponent \tau. We investigate the case where $\tau\in(3,4)$, so that the degrees have finite variance but infinite third moment. The sizes of the largest clusters, rescaled by $n^{-(\tau-2)/(\tau-1)}$, converge to hitting times of a "thinned" L\'{e}vy process, a special case of the general multiplicative coalescents studied by Aldous [Ann. Probab. 25 (1997) 812-854] and Aldous and Limic [Electron. J. Probab. 3 (1998) 1-59]. Our results should be contrasted to the case \tau>4, so that the third moment is finite. There, instead, the sizes of the components rescaled by $n^{-2/3}$ converge to the excursion lengths of an inhomogeneous Brownian motion, as proved in Aldous [Ann. Probab. 25 (1997) 812-854] for the Erd\H{o}s-R\'{e}nyi random graph and extended to the present setting in Bhamidi, van der Hofstad and van Leeuwaarden [Electron. J. Probab. 15 (2010) 1682-1703] and Turova [(2009) Preprint].Comment: Published in at http://dx.doi.org/10.1214/11-AOP680 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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We investigate the component sizes of the critical configuration model, as well as the related problem of critical percolation on a supercritical configuration model. We show that, at criticality, the finite third moment assumption on the asymptotic degree distribution is enough to guarantee that the sizes of the largest connected components are of the order $n^{2/3}$ and the re-scaled component sizes (ordered in a decreasing manner) converge to the ordered excursion lengths of an inhomogeneous Brownian Motion with a parabolic drift. We use percolation to study the evolution of these component sizes while passing through the critical window and show that the vector of percolation cluster-sizes, considered as a process in the critical window, converge to the multiplicative coalescent process in the sense of finite dimensional distributions. This behavior was first observed for Erd\H{o}s-R\'enyi random graphs by Aldous (1997) and our results provide support for the empirical evidences that the nature of the phase transition for a wide array of random-graph models are universal in nature. Further, we show that the re-scaled component sizes and surplus edges converge jointly under a strong topology, at each fixed location of the scaling window.Comment: 33 pages. Minor improvement

We consider a system of N parallel single-server queues with unit exponential service rates and a single dispatcher where tasks arrive as a Poisson process of rate λ(N). When a task arrives, the dispatcher assigns it to a server with the shortest queue among d(N) randomly selected servers (1 d(N) N). This load balancing strategy is referred to as a JSQ(d(N)) scheme, marking that it subsumes the celebrated Join-the-Shortest Queue (JSQ) policy as a crucial special case for d(N) = N.We construct a stochastic coupling to bound the difference in the queue length processes between the JSQ policy and a JSQ(d(N)) scheme with an arbitrary value of d(N). We use the coupling to derive the fluid limit in the regime where λ(N)/N → λ < 1 as N → ∞ with d(N) → ∞, along with the associated fixed point. The fluid limit turns out not to depend on the exact growth rate of d(N), and in particular coincides with that for the JSQ policy. We further leverage the coupling to establish that the diffusion limit in the critical regime where (N − λ(N))/ √ N → β > 0 as N → ∞ with d(N)/( √ N log(N)) → ∞ corresponds to that for the JSQ policy. These results indicate that the optimality of the JSQ policy can be preserved at the fluid-level and diffusion-level while reducing the overhead by nearly a factor O(N) and O( √ N/ log(N)), respectively. * d.mukherjee@tue.nl

We consider the fixed-cycle traffic-light (FCTL) queue, where vehicles arrive at an intersection controlled by a traffic light and form a queue. The traffic light signal alternates between green and red periods, and delayed vehicles are assumed to depart during the green period at equal time intervals. Most of the research done on the FCTL queue assumes that the vehicles arrive at the intersection according to a Poisson process and focusses on deriving formulas for the mean queue length at the end of green periods and the mean delay. For a class of discrete arrival processes, including the Poisson process, we derive the probability generating function of both the queue length and delay, from which the whole queue length and delay distribution can be obtained. This allows for the evaluation of performance characteristics other than the mean, such as the variance and percentiles of the distribution. We discuss the numerical procedures that are required to obtain the performance characteristics, and give several numerical examples.