10,180 research outputs found

    Supermarket Model on Graphs

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    We consider a variation of the supermarket model in which the servers can communicate with their neighbors and where the neighborhood relationships are described in terms of a suitable graph. Tasks with unit-exponential service time distributions arrive at each vertex as independent Poisson processes with rate Ξ»\lambda, and each task is irrevocably assigned to the shortest queue among the one it first appears and its dβˆ’1d-1 randomly selected neighbors. This model has been extensively studied when the underlying graph is a clique in which case it reduces to the well known power-of-dd scheme. In particular, results of Mitzenmacher (1996) and Vvedenskaya et al. (1996) show that as the size of the clique gets large, the occupancy process associated with the queue-lengths at the various servers converges to a deterministic limit described by an infinite system of ordinary differential equations (ODE). In this work, we consider settings where the underlying graph need not be a clique and is allowed to be suitably sparse. We show that if the minimum degree approaches infinity (however slowly) as the number of servers NN approaches infinity, and the ratio between the maximum degree and the minimum degree in each connected component approaches 1 uniformly, the occupancy process converges to the same system of ODE as the classical supermarket model. In particular, the asymptotic behavior of the occupancy process is insensitive to the precise network topology. We also study the case where the graph sequence is random, with the NN-th graph given as an Erd\H{o}s-R\'enyi random graph on NN vertices with average degree c(N)c(N). Annealed convergence of the occupancy process to the same deterministic limit is established under the condition c(N)β†’βˆžc(N)\to\infty, and under a stronger condition c(N)/ln⁑Nβ†’βˆžc(N)/\ln N\to\infty, convergence (in probability) is shown for almost every realization of the random graph.Comment: 32 page

    Randomized longest-queue-first scheduling for large-scale buffered systems

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    We develop diffusion approximations for parallel-queueing systems with the randomized longest-queue-first scheduling algorithm by establishing new mean-field limit theorems as the number of buffers nβ†’βˆžn\to\infty. We achieve this by allowing the number of sampled buffers d=d(n)d=d(n) to depend on the number of buffers nn, which yields an asymptotic `decoupling' of the queue length processes. We show through simulation experiments that the resulting approximation is accurate even for moderate values of nn and d(n)d(n). To our knowledge, we are the first to derive diffusion approximations for a queueing system in the large-buffer mean-field regime. Another noteworthy feature of our scaling idea is that the randomized longest-queue-first algorithm emulates the longest-queue-first algorithm, yet is computationally more attractive. The analysis of the system performance as a function of d(n)d(n) is facilitated by the multi-scale nature in our limit theorems: the various processes we study have different space scalings. This allows us to show the trade-off between performance and complexity of the randomized longest-queue-first scheduling algorithm

    Concentration of measure and mixing for Markov chains

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    We consider Markovian models on graphs with local dynamics. We show that, under suitable conditions, such Markov chains exhibit both rapid convergence to equilibrium and strong concentration of measure in the stationary distribution. We illustrate our results with applications to some known chains from computer science and statistical mechanics.Comment: 28 page
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