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Combinatorics of geometrically distributed random variables: Inversions and a parameter of Knuth
 Annals of Combinatorics
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The number of distinct values of some multiplicity in sequences of geometrically distributed . . .
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Reliability benefit of network coding
, 2007
"... The capacity benefit of network coding has been extensively studied in wired and wireless networks. Moreover, it has been shown that network coding improves network reliability by reducing the number of packet retransmissions in lossy networks. However, the extent of the reliability benefit of netwo ..."
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The capacity benefit of network coding has been extensively studied in wired and wireless networks. Moreover, it has been shown that network coding improves network reliability by reducing the number of packet retransmissions in lossy networks. However, the extent of the reliability benefit of network coding is not known. In this work, we characterize the reliability benefit of network coding for reliable multicasting. In particular, we show that the expected number of transmissions using linkbylink ARQ compared to network coding to send a packet from the multicast source to K receivers scales as Θ( log K log log K).
Combinatorics of Geometrically Distributed Random Variables: Value and Position of the rth LefttoRight Maximum
 Discrete Math
"... For words of length n, generated by independent geometric random variables, we consider the average value and the average position of the rth lefttoright maximum, for fixed r and n !1. 1. ..."
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For words of length n, generated by independent geometric random variables, we consider the average value and the average position of the rth lefttoright maximum, for fixed r and n !1. 1.
The Number of Distinct Values in a Geometrically Distributed Sample
"... For words of length n, generated by independent geometric random variables, we consider the average and variance of the number of distinct values (= letters) that occur in the word. We then generalise this to the number of values which occur at least b times in the word. 1. ..."
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Cited by 11 (4 self)
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For words of length n, generated by independent geometric random variables, we consider the average and variance of the number of distinct values (= letters) that occur in the word. We then generalise this to the number of values which occur at least b times in the word. 1.
The first descent in samples of geometric random variables and permutations
"... For words of length n, generated by independent geometric random variables, we study the average initial and end heights of the first strict and weak descents in the word. Higher moments and limiting distributions are also derived. In addition we compute the average initial and end height of the fir ..."
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Cited by 5 (3 self)
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For words of length n, generated by independent geometric random variables, we study the average initial and end heights of the first strict and weak descents in the word. Higher moments and limiting distributions are also derived. In addition we compute the average initial and end height of the first descent for a random permutation of n letters.
Probability calculus for silent elimination : A method for medium access control
"... A probability problem arising in the context of medium access control in wireless networks is considered. It is described as a problem with n urns, each one having one ball at time 0. Each ball leaves its urn after a geometrically distributed time. Then there is a first time T such that no departure ..."
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A probability problem arising in the context of medium access control in wireless networks is considered. It is described as a problem with n urns, each one having one ball at time 0. Each ball leaves its urn after a geometrically distributed time. Then there is a first time T such that no departures take place at the times T +1, T +2,..., T + k, where k is fixed. The focus is on the probability distribution of (XT, ST, T), where XT is the number of balls that leave their urns at time T and ST is the number of balls remaining there at that time. Efficient recursion formulas are derived. Asymptotics and continuous time approximations are considered. For k = ∞, T is the maximum of n geometrically distributed variables. This case has earlier got a large literature.
drecords in geometrically distributed random variables
 Discrete Mathematics & Theoretical Computer Science
"... We study d–records in sequences generated by independent geometric random variables and derive explicit and asymptotic formulæ for expectation and variance. Informally speaking, a d–record occurs, when one computes the d–largest values, and the variable maintaining it changes its value while the seq ..."
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We study d–records in sequences generated by independent geometric random variables and derive explicit and asymptotic formulæ for expectation and variance. Informally speaking, a d–record occurs, when one computes the d–largest values, and the variable maintaining it changes its value while the sequence is scanned from left to right. This is done for the “strict model, ” but a “weak model ” is also briefly investigated. We also discuss the limit q → 1 (q the parameter of the geometric distribution), which leads to the model of random permutations.
Asymptotic results for silent elimination
 DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE
"... Following the model of Bondesson, Nilsson, and Wikstrand, we consider randomly filled urns, where the probability of falling into urn i is the geometric probability (1 − q)q i−1. Assuming n independent random entries, and a fixed parameter k, the interest is in the following parameters: Let T be the ..."
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Cited by 1 (1 self)
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Following the model of Bondesson, Nilsson, and Wikstrand, we consider randomly filled urns, where the probability of falling into urn i is the geometric probability (1 − q)q i−1. Assuming n independent random entries, and a fixed parameter k, the interest is in the following parameters: Let T be the smallest index, such that urn T is nonempty, but the following k are empty, then: XT = number of balls in urn T, ST = number of balls in urns with index larger than T, and finally T itself. We analyse the recursions (that appeared earlier) precisely, and derive results about the joint distribution of a related urn model.
The first descent in samples of geometric random variables and permutations
"... For words of length n, generated by independent geometric random variables, we study the average initial and end heights of the first strict and weak descents in the word. Higher moments and limiting distributions are also derived. In addition we compute the average initial and end height of the fir ..."
Abstract
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For words of length n, generated by independent geometric random variables, we study the average initial and end heights of the first strict and weak descents in the word. Higher moments and limiting distributions are also derived. In addition we compute the average initial and end height of the first descent for a random permutation of n letters.