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Enumerations Of Trees And Forests Related To Branching Processes And Random Walks
 Microsurveys in Discrete Probability, number 41 in DIMACS Ser. Discrete Math. Theoret. Comp. Sci
, 1997
"... In a GaltonWatson 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 ..."
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Cited by 38 (15 self)
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In a GaltonWatson 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...
Chipfiring games on directed graphs
 J. ALGEBRAIC COMBIN
, 1992
"... 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 und ..."
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Cited by 16 (1 self)
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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.
Outage and Error Events in Bursty Channels
 IEEE Transactions on Communications
, 1998
"... 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 app ..."
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Cited by 8 (2 self)
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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 closedform 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. I. INTRODUCTION One of the key features of the wireless channel is its correlation. In fact, effects such as mobility...
The distribution of the maximum of a Lévy process with positive jumps of phasetype
 Proceedings of the Conference Dedicated to the 90th Anniversary of Boris Vladimirovich Gnedenko (Kyiv
, 2002
"... Consider a Levy process with nite intensity positive jumps of the phasetype 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 phasetype, and nd the distribu ..."
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Cited by 8 (0 self)
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Consider a Levy process with nite intensity positive jumps of the phasetype 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 phasetype, and nd the distribution of this random variable. Previous results (hyperexponential positive jumps) are obtained as a particular case.
On perpetual American put valuation and firstpassage in a regimeswitching model with jumps,” Finance and Stochastics
, 2008
"... In this paper we consider the problem of pricing a perpetual American put option in an exponential regimeswitching Lévy model. For the case of the (dense) class of phasetype jumps and finitely many regimes we derive an explicit expression for the value function. The solution of the corresponding f ..."
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Cited by 7 (3 self)
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In this paper we consider the problem of pricing a perpetual American put option in an exponential regimeswitching Lévy model. For the case of the (dense) class of phasetype jumps and finitely many regimes we derive an explicit expression for the value function. The solution of the corresponding first passage problem under a statedependent level rests on a path transformation and a new matrix WienerHopf factorization result for this class of processes.
Individual displacements for linear probing hashing with different insertion policies
 ACM Transactions on Algorithms
, 2005
"... 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 occ ..."
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Cited by 4 (1 self)
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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.
Selforganizing data structures with dependent accesses
 ICALP'96, LNCS 1099
, 1995
"... We consider selforganizing data structures in the case where the sequence of accesses can be modeled by a first order Markov chain. For the simplek and batchedkmovetofront schemes, explicit formulae for the expected search costs are derived and compared. We use a new approach that employs th ..."
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Cited by 4 (1 self)
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We consider selforganizing data structures in the case where the sequence of accesses can be modeled by a first order Markov chain. For the simplek and batchedkmovetofront 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 memoryfree moveforward 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 selforganizing data structures with respect to the expected search time are reversed, compared with independent accesses. Finally we look at selforganizing binary trees with the movetoroot rule and compare the expected search cost with the entropy of the Markov chain of accesses.
SCALING LIMITS OF RANDOM PLANAR MAPS WITH A UNIQUE LARGE FACE
"... Abstract. We study random bipartite planar maps defined by assigning nonnegativeweights toeach face ofamap. Weproof thatfor certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown t ..."
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Cited by 1 (0 self)
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Abstract. We study random bipartite planar maps defined by assigning nonnegativeweights toeach face ofamap. Weproof thatfor certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges n of the planar maps goes to infinity, the profile of distances to a marked vertex rescaled by n −1/2 is described by a Brownian excursion. The planar maps, with the graph metric rescaled by n −1/2, are then shown to converge in distribution towards Aldous ’ Brownian tree in the Gromov–Hausdorff topology. In the proofs we rely on the Bouttier–di Francesco–Guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large. 1.
Examples: Special Graphs and Trees
, 1996
"... 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 ..."
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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
First passage time density for the Ehrenfest model
, 2007
"... 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. ..."
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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.