Related papers: Tail asymptotics for busy periods
Foss and Zachary (2003) and Foss, Palmowski and Zachary (2005) studied the probability of achieving a receding boundary on a time interval of random length by a random walk with a heavy-tailed jump distribution. They have proposed and…
In the context of communication networks, the framework of stochastic event graphs allows a modeling of control mechanisms induced by the communication protocol and an analysis of its performances. We concentrate on the logarithmic tail…
The asymptotics of the probability that the self-intersection local time of a random walk on $\Z^d$ exceeds its expectation by a large amount is a fascinating subject because of its relation to some models from Statistical Mechanics, to…
A common assumption when modeling queuing systems is that arrivals behave like a Poisson process with constant parameter. In practice, however, call arrivals are often observed to be significantly overdispersed. This motivates that in this…
We investigate what happens when an entire sample path of a smooth Gaussian process on a compact interval lies above a high level. Specifically, we determine the precise asymptotic probability of such an event, the extent to which the high…
We consider directed weighted graphs and define various families of path counting functions. Our main results are explicit formulas for the main term of the asymptotic growth rate of these counting functions, under some irrationality…
It has been recently suggested that a totally asymmetric exclusion process with two species on an open chain could exhibit spontaneous symmetry breaking in some range of the parameters defining its dynamics. The symmetry breaking is…
We consider random interlacements on Z^d, with d bigger or equal to 3, when their vacant set is in a strongly percolative regime. We derive an asymptotic upper bound on the probability that the random interlacements disconnect a box of…
We consider a queue to which only a finite pool of $n$ customers can arrive, at times depending on their service requirement. A customer with stochastic service requirement $S$ arrives to the queue after an exponentially distributed time…
A simple random walk on a graph is a sequence of movements from one vertex to another where at each step an edge is chosen uniformly at random from the set of edges incident on the current vertex, and then transitioned to next vertex.…
Let $Y=\sum_{k\ge 1} 1_{A_k}$ be an infinite sum of the indicators of independent events. We investigate a precise (as opposed to logarithmic) first-order asymptotic behavior of the tail probabilities $\mathbb{P}\{Y\ge n\}$ and the point…
The present paper is concerned with the stationary workload of queues with heavy-tailed (regularly varying) characteristics. We adopt a transform perspective to illuminate a close connection between the tail asymptotics and heavy-traffic…
The reflected process of a random walk or L\'evy process arises in many areas of applied probability, and a question of particular interest is how the tail of the distribution of the heights of the excursions away from zero behaves…
For a random walk on the integer lattice $\mathbb{Z}$ that is attracted to a strictly stable process with index $\alpha\in (1, 2)$ we obtain the asymptotic form of the transition probability for the walk killed when it hits a finite set.…
This note continues paper of Denisov and Wachtel (2010), where we have constructed a $k$-dimensional random walk conditioned to stay in the Weyl chamber of type $A$. The construction was done under the assumption that the original random…
We consider the problem of scheduling a queueing system in which many statistically identical servers cater to several classes of impatient customers. Service times and impatience clocks are exponential while arrival processes are renewal.…
We consider lattice walks in $\R^k$ confined to the region $0<x_1<x_2...<x_k$ with fixed (but arbitrary) starting and end points. The walks are required to be "reflectable", that is, we assume that the number of paths can be counted using…
This paper studies the tail probability of weighted sums of the form $\sum_{i=1}^n c_i X_i$, where random variables $X_i$'s are either independent or pairwise quasi-asymptotical independent with heavy tails. Using $h$-insensitive function,…
Consider a random walk $S_n=\sum_{i=1}^n X_i$ with independent and identically distributed real-valued increments $X_i$ of zero mean and finite variance. Assume that $X_i$ is non-lattice and has a moment of order $2+\delta$. For any $x\geq…
We prove a sample path large deviation principle (LDP) with sub-linear speed for unbounded functionals of certain Markov chains induced by the Lindley recursion. The LDP holds in the Skorokhod space $\mathbb{D}[0,T]$ equipped with the…