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Renewal theory
About: Renewal theory is a research topic. Over the lifetime, 2381 publications have been published within this topic receiving 54908 citations.
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TL;DR: In this article, the authors extend some results of Hammersley and Welsh concerning first-passage percolation on the two-dimensional integer lattice and give an L 1-ergodic theorem for the unrestricted first passage time from (0, 0) to the line X = n.
Abstract: We extend some results of Hammersley and Welsh concerning first-passage percolation on the two-dimensional integer lattice. Our results include: (i) weak renewal theorems for the unrestricted reach processes; (ii) an L 1-ergodic theorem for the unrestricted first-passage time from (0, 0) to the line X = n; and (iii) weakening of the boundedness restrictions on the underlying distribution in Hammersley and Welsh's weak renewal theorems for the cylinder reach processes.
12 citations
TL;DR: Besides probabilistic aspect of the model, the maximum likelihood estimation of model parameters is discussed, and its performance is illustrated through several numerical examples using both simulated and real–life data.
12 citations
TL;DR: This work introduces a novel class of Lyapunov functions (piecewise linear and nonincreasing in the length of heavy-tailed queues), whose drift analysis provides exponentially decaying upper bounds to queue-length tail asymptotics despite the presence of heavy tails.
Abstract: We consider switched queueing networks with a mix of heavy-tailed (i.e., arrival processes with infinite variance) and exponential-type traffic and study the delay performance of the max-weight policy, known for its throughput optimality and asymptotic delay optimality properties. Our focus is on the impact of heavy-tailed traffic on exponential-type queues/flows, which may manifest itself in the form of subtle rate-dependent phenomena. We introduce a novel class of Lyapunov functions (piecewise linear and nonincreasing in the length of heavy-tailed queues), whose drift analysis provides exponentially decaying upper bounds to queue-length tail asymptotics despite the presence of heavy tails. To facilitate a drift analysis, we employ fluid approximations, proving that if a continuous and piecewise linear function is also a “Lyapunov function” for the fluid model, then the same function is a “Lyapunov function” for the original stochastic system. Furthermore, we use fluid approximations and renewal theory i...
12 citations
TL;DR: In this article, the selective interaction of a stationary point process and a renewal process is studied, and equilibrium conditions for the resulting point process are given, the equilibrium counting distribution is derived, and an explicit expression for the rate of the process is determined.
Abstract: The selective interaction of a stationary point process and a renewal process is studied. Equilibrium conditions for the resulting point process are given, the equilibrium counting distribution is derived, and an explicit expression for the rate of the process is determined.
12 citations
01 Jan 1985
TL;DR: In this article, the renewal theory of Chapter VIII is used to justify and generalize the approximations suggested in IV.3.1 and IV.4.1 to the first passages of random walks to nonlinear boundaries.
Abstract: This chapter is concerned with first passages of random walks to nonlinear boundaries. Suitable generalizations of the renewal theory of Chapter VIII are developed in order to justify and generalize the approximations suggested in IV.3.
12 citations