Joint Distribution of Distances in Large Random Regular Networks

2013 ◽  
Vol 50 (03) ◽  
pp. 861-870 ◽  
Author(s):  
Justin Salez
Keyword(s):  
Joint Distribution ◽  
Marginal Distribution ◽  
Weak Sense ◽  
Regular Graphs ◽  
Random Array ◽  
Point To Point ◽  
Regular Networks ◽  
Different Sources ◽  
Individual Entry

We study the array of point-to-point distances in random regular graphs equipped with exponential edge lengths. We consider the regime where the degree is kept fixed while the number of vertices tends to ∞. The marginal distribution of an individual entry is now well understood, thanks to the work of Bhamidi, van der Hofstad and Hooghiemstra (2010). The purpose of this note is to show that the whole array, suitably recentered, converges in the weak sense to an explicit infinite random array. Our proof consists in analyzing the invasion of the network by several mutually exclusive flows emanating from different sources and propagating simultaneously along the edges.

scholarly journals Joint Distribution of Distances in Large Random Regular Networks

2013 ◽  
Vol 50 (3) ◽  
pp. 861-870 ◽  
Author(s):  
Justin Salez
Keyword(s):  
Joint Distribution ◽  
Marginal Distribution ◽  
Weak Sense ◽  
Regular Graphs ◽  
Random Array ◽  
Point To Point ◽  
Regular Networks ◽  
Different Sources ◽  
Individual Entry

We study the array of point-to-point distances in random regular graphs equipped with exponential edge lengths. We consider the regime where the degree is kept fixed while the number of vertices tends to ∞. The marginal distribution of an individual entry is now well understood, thanks to the work of Bhamidi, van der Hofstad and Hooghiemstra (2010). The purpose of this note is to show that the whole array, suitably recentered, converges in the weak sense to an explicit infinite random array. Our proof consists in analyzing the invasion of the network by several mutually exclusive flows emanating from different sources and propagating simultaneously along the edges.

Distributions of the Longest Excursions in a Tied Down Simple Random Walk and in a Brownian Bridge

2007 ◽  
Vol 44 (04) ◽  
pp. 1056-1067 ◽  
Author(s):  
Andreas Lindell ◽  
Lars Holst
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Weak Convergence ◽  
Joint Distribution ◽  
Marginal Distribution ◽  
Brownian Bridge ◽  
Simple Random Walk ◽  
Marginal Distributions

Expressions for the joint distribution of the longest and second longest excursions as well as the marginal distributions of the three longest excursions in the Brownian bridge are obtained. The method, which primarily makes use of the weak convergence of the random walk to the Brownian motion, principally gives the possibility to obtain any desired joint or marginal distribution. Numerical illustrations of the results are also given.

Distributions of the Longest Excursions in a Tied Down Simple Random Walk and in a Brownian Bridge

2007 ◽  
Vol 44 (4) ◽  
pp. 1056-1067
Author(s):  
Andreas Lindell ◽  
Lars Holst
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Weak Convergence ◽  
Joint Distribution ◽  
Marginal Distribution ◽  
Brownian Bridge ◽  
Simple Random Walk ◽  
Marginal Distributions

Expressions for the joint distribution of the longest and second longest excursions as well as the marginal distributions of the three longest excursions in the Brownian bridge are obtained. The method, which primarily makes use of the weak convergence of the random walk to the Brownian motion, principally gives the possibility to obtain any desired joint or marginal distribution. Numerical illustrations of the results are also given.

Stationary joint distributions arising in the analysis of the M/G/1 queue by the method of the imbedded markov chain

1966 ◽  
Vol 3 (2) ◽  
pp. 512-520 ◽  
Author(s):  
J. H. Jenkins
Keyword(s):  
Markov Chain ◽  
Generating Functions ◽  
Joint Distribution ◽  
Marginal Distribution ◽  
Probability Generating Function ◽  
Joint Distributions ◽  
Left Behind ◽  
Markov Chain Theory ◽  
Imbedded Markov Chain ◽  
Chain Theory

SummaryProbability generating functions are used to relate the joint distribution of the numbers of customers left behind by two successive departing customers to the marginal distribution of the number left behind by each departing customer. A probability generating function is then found for the joint distribution of the numbers of customers arriving in two successive departure intervals using the joint distribution of the numbers of customers left behind by three successive departing customers. The results could be obtained from general Markov chain theory but the method used in this paper is quicker.

The maximum of a random walk whose mean path has a maximum

1985 ◽  
Vol 17 (1) ◽  
pp. 85-99 ◽  
Author(s):  
H. E. Daniels ◽  
T. H. R. Skyrme
Keyword(s):  
Integral Equation ◽  
Random Walk ◽  
Diffusion Approximation ◽  
Conditional Expectation ◽  
Joint Distribution ◽  
Marginal Distribution ◽  
Brownian Bridge ◽  
Distribution Of The Maximum

This paper discusses the joint distribution of the maximum and the time at which it is attained, of a random walk whose mean path is a curvilinear trend which itself has a maximum. A typical example of such a problem is the distribution of the maximum number of infectives present during the course of an epidemic. Another example where the random walk is constrained to terminate at 0 after a given time is provided by the distribution of the strength and breaking extension of a bundle of fibres.A diffusion approximation to the joint distribution is obtained for the general case of a Brownian bridge. In the commonest class of cases which includes the two examples mentioned, a certain integral equation has to be solved. Its solution enables the marginal distribution of the time to reach the maximum to be tabulated, and the marginal distribution of the maximum confirms the results previously obtained by Daniels (1974) and Barbour (1975). Of particular interest is the conditional expectation of the maximum for a given time of attainment which behaves asymmetrically.

On a storage allocation model with finite capacity

2016 ◽  
Vol 27 (5) ◽  
pp. 738-755 ◽  
Author(s):  
EUNJU SOHN ◽  
CHARLES KNESSL
Keyword(s):  
Random Walk ◽  
Steady State ◽  
Finite Number ◽  
Poisson Process ◽  
Joint Distribution ◽  
Marginal Distribution ◽  
Finite Capacity ◽  
Storage Allocation ◽  
Allocation Model ◽  
Numerical Studies

We consider a storage allocation model with a finite number of storage spaces. There are m primary spaces that are ranked {1,2,. . .,m} and R secondary spaces ranked {m + 1, m + 2,. . .,m + R}. Items arrive according to a Poisson process, occupy a space for a random exponentially distributed time, and an arriving item takes the lowest ranked available space. Letting N1 and N2 denote the numbers of occupied primary and secondary spaces, we study the joint distribution Prob[N1 = k, N2 = r] in the steady state. The joint process (N1, N2) behaves as a random walk in a lattice rectangle. We shall obtain explicit expressions for the distribution of (N1, N2), as well as the marginal distribution of N2. We also give some numerical studies to illustrate the qualitative behaviors of the distribution(s). The main contribution is to study the effects of a finite secondary capacity R, whereas previous studies had R = ∞.

Stationary joint distributions arising in the analysis of the M/G/1 queue by the method of the imbedded markov chain

1966 ◽  
Vol 3 (02) ◽  
pp. 512-520 ◽  
Author(s):  
J. H. Jenkins
Keyword(s):  
Markov Chain ◽  
Generating Functions ◽  
Joint Distribution ◽  
Marginal Distribution ◽  
Probability Generating Function ◽  
Joint Distributions ◽  
Left Behind ◽  
Markov Chain Theory ◽  
Imbedded Markov Chain ◽  
Chain Theory

Summary Probability generating functions are used to relate the joint distribution of the numbers of customers left behind by two successive departing customers to the marginal distribution of the number left behind by each departing customer. A probability generating function is then found for the joint distribution of the numbers of customers arriving in two successive departure intervals using the joint distribution of the numbers of customers left behind by three successive departing customers. The results could be obtained from general Markov chain theory but the method used in this paper is quicker.

The maximum of a random walk whose mean path has a maximum

1985 ◽  
Vol 17 (01) ◽  
pp. 85-99 ◽  
Author(s):  
H. E. Daniels ◽  
T. H. R. Skyrme
Keyword(s):  
Integral Equation ◽  
Random Walk ◽  
Diffusion Approximation ◽  
Conditional Expectation ◽  
Joint Distribution ◽  
Marginal Distribution ◽  
Brownian Bridge ◽  
Distribution Of The Maximum

This paper discusses the joint distribution of the maximum and the time at which it is attained, of a random walk whose mean path is a curvilinear trend which itself has a maximum. A typical example of such a problem is the distribution of the maximum number of infectives present during the course of an epidemic. Another example where the random walk is constrained to terminate at 0 after a given time is provided by the distribution of the strength and breaking extension of a bundle of fibres. A diffusion approximation to the joint distribution is obtained for the general case of a Brownian bridge. In the commonest class of cases which includes the two examples mentioned, a certain integral equation has to be solved. Its solution enables the marginal distribution of the time to reach the maximum to be tabulated, and the marginal distribution of the maximum confirms the results previously obtained by Daniels (1974) and Barbour (1975). Of particular interest is the conditional expectation of the maximum for a given time of attainment which behaves asymmetrically.

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