In a Galton-Watson branching process with offspring distribution (p 0 ; p 1 ; : : :) started with k individuals, the distribution of the total progeny is identical to the distribution of the first passage time to \Gammak for a random walk started at 0 which takes steps of size j with probability p j+1 for j \Gamma1. The formula for this distribution is a probabilistic expression of the Lagrange inversion formula for the coefficients in the power series expansion of f(z) k in terms of those of g(z) for f(z) defined implicitly by f(z) = zg(f(z)). The Lagrange inversion formula is the analytic counterpart of various enumerations of trees and forests which generalize Cayley's formula kn n\Gammak\Gamma1 for the number of rooted forests labeled by a set of size n whose set of roots is a particular subset of size k. These known results are derived by elementary combinatorial methods without appeal to the Lagrange formula, which is then obtained as a byproduct. This approach unifies an...

We consider the following (solitary) game: each node of a directed graph contains a pile of chips. A move consists of selecting a node with at least as many chips as its outdegree, and sending one chip along each outgoing edge to its neighbors. We extend to directed graphs several results on the undirected version obtained earlier by the authors, P. Shor, and G. Tardos, and we discuss some new topics such as periodicity, reachability and probabilistic aspects. Among the new results specifically concerning digraphs, we relate the length of the shortest period of an infinite game to the length of the longest terminating game, and also to the access time of random walks on the same graph. These questions involve a study of the Laplace operator for directed graphs. We show that for many graphs, in particular for undirected graphs, the problem whether a given position of the chips can be reached from the initial position is polynomial time solvable. Finally, we show how the basic properties of the “probabilistic abacus” can be derived from our results.

In this paper, the outage concept is discussed. Classic definitions, only based on the marginal statistics, are found not to be adequate for a proper characterization of the effect of channel impairments at the application level. A more flexible definition, which incorporates time parameters and appears suitable for the characterization of the performance of packet communications systems, is proposed. An analytical framework for the computation of various performance metrics is also presented. A Markov description is assumed for the channel, which allows closed-form solutions and can effectively model many situations of interest. Outage events are defined as being triggered by b consecutive channel errors, and being ended by g consecutive successes. A numerical example of application is given, and extensions of the analysis to more general definitions are discussed. Keywords---Outage, Minimum duration outage, Bursty errors, Correlated channel, Error statistics, Error events. I. INTR...

Consider a Levy process with nite intensity positive jumps of the phase-type and arbitrary negative jumps. Assume that the process either is killed at a constant rate or drifts to 1. We show that the distribution of the overall maximum of this process is also of phase-type, and nd the distribution of this random variable. Previous results (hyperexponential positive jumps) are obtained as a particular case.

Abstract. We study the distribution of the individual displacements in hashing with linear probing for three different versions: First Come, Last Come and Robin Hood. Asymptotic distributions and their moments are found when the the size of the hash table tends to infinity with the proportion of occupied cells converging to some α, 0 < α < 1. (In the case of Last Come, the results are more complicated and less complete than in the other cases.) We also show, using the diagonal Poisson transform studied by Poblete, Viola and Munro, that exact expressions for finite m and n can be obtained from the limits as m, n → ∞. We end with some results, conjectures and questions about the shape of the limit distributions. These have some relevance for computer applications. 1.

We consider self-organizing data structures in the case where the sequence of accesses can be modeled by a first order Markov chain. For the simple-k- and batched-k--move-to-front schemes, explicit formulae for the expected search costs are derived and compared. We use a new approach that employs the technique of expanding a Markov chain. This approach generalizes the results of Gonnet/Munro/Suwanda. In order to analyze arbitrary memory-free move-forward heuristics for linear lists, we restrict our attention to a special access sequence, thereby reducing the state space of the chain governing the behaviour of the data structure. In the case of accesses with locality (inert transition behaviour), we find that the hierarchies of self-organizing data structures with respect to the expected search time are reversed, compared with independent accesses. Finally we look at self-organizing binary trees with the move-to-root rule and compare the expected search cost with the entropy of the Markov chain of accesses.

8.928> x + E x T v = w w vx ; where w = X i X j w ij : (2) Specializing to the unweighted case, E v T x = 2jE(v; x)j + 1 (3) E v T x +E x T v = 2jEj: (4) 1 Proof. It is enough to prove (1), since (2) follows by adding the two expressions of the form (1). Because (v; x) is essential, we may delete all vertices of A(x; v) except x, and this does not affect the behavior of the chain up until time T x , because x must be the first visited vertex of A(x; v). After this deletio

Abstract We derive an explicit expression for the probability density of the first passage time to state 0 for the Ehrenfest diffusion model in continuous time.

Abstract. We study the simple random walk on the n-dimensional hypercube, in particular its hitting times of large (possibly random) sets. We give simple conditions on these sets ensuring that the properly-rescaled hitting time is asymptotically exponentially distributed, uniformly in the starting position of the walk. These conditions are then verified for percolation clouds with densities that are much smaller than (n log n) −1. A main motivation behind this paper is the study of the so-called aging phenomenon in the Random Energy Model (REM), the simplest model of a meanfield spin glass. Our results allow us to prove aging in the REM for all temperatures, thereby extending earlier results to their optimal temperature domain. 1.

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