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This paper presents the distribution of the number of customers served during a busy period for special cases of the Geo/G/1 queue when initiated with m customers. We analyze the system under the assumptions of a late arrival system with delayed access and early arrival system policies. It is not easy to invert the functional equation for the number of customers served during a busy period except for the simple case Geo/Geo/1 queue, as stated by several researchers. Using the Lagrange inversion theorem, we give an elegant solution to this equation. We find the distribution of the number of customers served during a busy period for various service-time distributions such as geometric, deterministic, binomial, negative binomial, uniform, Delaporte, discrete phase-type and interrupted Bernoulli process. We compute the mean and variance of these distributions and also give numerical results. Due to the clarity of the expressions, the computations are very fast and robust. We also show that in the limiting case, the results tend to the analogous continuous-time counterparts.
In the previous chapter, we introduced distributions and density functions as two alternative methods to characterize the randomness of variates. In this chapter, we introduce the final method considered in this book, and relate it to the previous two: the moments of a variate (including mean, variance, and higher-order moments). The condition for the existence of moments is explained and justified mathematically. These moments can be summarized by means of "generating functions". We define moment-generating functions (m.g.f.s). Cumulants and their generating functions are introduced. Characteristic functions (c.f.s) always exist, even if some moments do not, and they identify uniquely the distribution of the variate, so we define c.f.s and the inversion theorem required to transform them into the c.d.f. of a variate. We also study the main inequalities satisfied by moments, such as those resulting from transformations (Jensen) or from comparing moments to probabilities (Markov, Chebyshev). We also show that the mean, median, and mode need not be linked by inequalities, as previously thought.
This chapter concerns the measurement of the dependence between variates, by exploiting the additional information contained in joint (rather than just marginal) distribution and density functions. For this multivariate context, we also generalize the third description of randomness seen earlier, i.e., moments and their generating functions. Joint moments and their generating functions are introduced, along with covariances, variance matrices, the Cauchy–Schwarz inequality, and joint c.f.s and their inversion into joint densities. We show how the law of iterated expectations makes use of conditioning when taking expectations with respect to more than one variate. We measure dependence via conditional densities, distributions, moments, and cumulants.
Within the online media universe, there are many underlying communities. These may be defined, for example, through politics, location, health, occupation, extracurricular interests or retail habits. Government departments, charities and commercial organisations can benefit greatly from insights about the structure of these communities; the move to customer-centred practices requires knowledge of the customer base. Motivated by this issue, we address the fundamental question of whether a sub-network looks like a collection of individuals who have effectively been picked at random from the whole, or instead forms a distinctive community with a new, discernible structure. In the former case, to spread a message to the intended user base it may be best to use traditional broadcast media (TV, billboard), whereas in the latter case a more targeted approach could be more effective. In this work, we therefore formalise a concept of testing for sub-structure and apply it to social interaction data. First, we develop a statistical test to determine whether a given sub-network (induced sub-graph) is likely to have been generated by sampling nodes from the full network uniformly at random. This tackles an interesting inverse alternative to the more widely studied “forward” problem. We then apply the test to a Twitter reciprocated mentions network where a range of brand name based sub-networks are created via tweet content. We correlate the computed results against the independent views of 16 digital marketing professionals. We conclude that there is great potential for social media based analytics to quantify, compare and interpret online brand allegiances systematically, in real time and at large scale.
In this paper we give an extension of the results of the generalized waiting time problem given by El-Desouky and Hussen (1990). An urn contains m types of balls of unequal numbers, and balls are drawn with replacement until first duplication. In the case of finite memory of order k, let ni be the number of type i, i = 1, 2, …, m. The probability of success pi = ni/N, i = 1, 2, …, m, where ni is a positive integer and Let Ym,k be the number of drawings required until first duplication. We obtain some new expressions of the probability function, in terms of Stirling numbers, symmetric polynomials, and generalized harmonic numbers. Moreover, some special cases are investigated. Finally, some important new combinatorial identities are obtained.
The probability h(n, m) that the block counting process of the Bolthausen-Sznitman n-coalescent ever visits the state m is analyzed. It is shown that the asymptotic hitting probabilities h(m) = limn→∞h(n, m), m ∈ N, exist and an integral formula for h(m) is provided. The proof is based on generating functions and exploits a certain convolution property of the Bolthausen-Sznitman coalescent. It follows that h(m) ∼ 1/log m as m → ∞. An application to linear recursions is indicated.
Combinatorial structures that compose and decompose give rise to Hopf monoids in Joyal's category of species. The Hadamard product of two Hopf monoids is another Hopf monoid. We prove two main results regarding freeness of Hadamard products. The first one states that if one factor is connected and the other is free as a monoid, their Hadamard product is free (and connected). The second provides an explicit basis for the Hadamard product when both factors are free.
The first main result is obtained by showing the existence of a one-parameter deformation of the comonoid structure and appealing to a rigidity result of Loday and Ronco that applies when the parameter is set to zero. To obtain the second result, we introduce an operation on species that is intertwined by the free monoid functor with the Hadamard product. As an application of the first result, we deduce that the Boolean transform of the dimension sequence of a connected Hopf monoid is nonnegative.
In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating function G and a maximum principle which, we prove, is satisfied by every fixed point of G. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.
Blecher [‘Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal’, Util. Math.88 (2012), 223–235] defined the combinatorial objects known as 1-shell totally symmetric plane partitions of weight
$n$
. He also proved that the generating function for
$f(n), $
the number of 1-shell totally symmetric plane partitions of weight
$n$
, is given by
In this brief note, we prove a number of arithmetic properties satisfied by
$f(n)$
using elementary generating function manipulations and well-known results of Ramanujan and Watson.
Let (Sk)k≥1 be the classical Bernoulli random walk on the integer line with jump parameters
p ∈ (0,1) and
q = 1 − p. The probability distribution of the sojourn
time of the walk in the set of non-negative integers up to a fixed time is well-known, but
its expression is not simple. By modifying slightly this sojourn time through a particular
counting process of the zeros of the walk as done by Chung & Feller [Proc.
Nat. Acad. Sci. USA 35 (1949) 605–608], simpler representations
may be obtained for its probability distribution. In the aforementioned article, only the
symmetric case (p = q = 1/2) is
considered. This is the discrete counterpart to the famous Paul Lévy’s arcsine law for
Brownian motion.
In the present paper, we write out a representation for this probability
distribution in the general case together with others related to the random walk subject
to a possible conditioning. The main tool is the use of generating functions.
We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times ln between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of ln / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.
A computer aided method using symbolic computations that enables the calculation of the
source terms (Boltzmann) in Grad’s method of moments is presented. The method is extremely
powerful, easy to program and allows the derivation of balance equations to very high
moments (limited only by computer resources). For sake of demonstration the method is
applied to a simple case: the one-dimensional stationary granular gas under gravity. The
method should find applications in the field of rarefied gases, as well. Questions of
convergence, closure are beyond the scope of this article.
The probability that two randomly selected phylogenetic trees of the same size are isomorphic is found to be asymptotic to a decreasing exponential modulated by a polynomial factor. The number of symmetrical nodes in a random phylogenetic tree of large size obeys a limiting Gaussian distribution, in the sense of both central and local limits. The probability that two random phylogenetic trees have the same number of symmetries asymptotically obeys an inverse square-root law. Precise estimates for these problems are obtained by methods of analytic combinatorics, involving bivariate generating functions, singularity analysis, and quasi-powers approximations.
Using elementary means, we derive an explicit formula for a3(n), the number of 3-core partitions of n, in terms of the prime factorization of 3n+1. Based on this result, we are able to prove several infinite families of arithmetic results involving a3(n), one of which specializes to the recent result of Baruah and Berndt which states that, for all n≥0, a3(4n+1)=a3(n).
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.
An occupancy model that has arisen in the investigation of randomized distributed schedules in all-optical networks is considered. The model consists of B initially empty urns, and at stage j of the process dj ≤ B balls are placed in distinct urns with uniform probability. Let Mi(j) denote the number of urns containing i balls at the end of stage j. An explicit expression for the joint factorial moments of M0(j) and M1(j) is obtained. A multivariate generating function for the joint factorial moments of Mi(j), 0 ≤ i ≤ I, is derived (where I is a positive integer). Finally, the case in which the dj, j ≥ 1, are independent, identically distributed random variables is investigated.
Methods using gambling teams and martingales are developed and applied to find formulae for the expected value and the generating function of the waiting time to observation of an element of a finite collection of patterns in a sequence generated by a two-state Markov chain of first, or higher, order.
We provide a simple set of sufficient conditions for the weak convergence of discrete-time, discrete-state Galton-Watson branching processes with immigration to continuous-time, continuous-state branching processes with immigration.
We consider a random walker on a d-regular graph. Starting from a fixed vertex, the first step is a unit step in any one of the d directions, with common probability 1/d for each one. At any later step, the random walker moves in any one of the directions, with probability q for a reversal of direction and probability p for any other direction. This model was introduced and first studied by Gillis (1955), in the case when the graph is a d-dimensional square lattice. We prove that the Gillis random walk on a d-regular graph is recurrent if and only if the simple random walk on the graph is recurrent. The Green function of the Gillis random walk will be also given, in terms of that of the simple random walk.
This paper investigates the rate of convergence to the probability distribution of the embedded M/G/1 and GI/M/n queues. We introduce several types of ergodicity including l-ergodicity, geometric ergodicity, uniformly polynomial ergodicity and strong ergodicity. The usual method to prove ergodicity of a Markov chain is to check the existence of a Foster–Lyapunov function or a drift condition, while here we analyse the generating function of the first return probability directly and obtain practical criteria. Moreover, the method can be extended to M/G/1- and GI/M/1-type Markov chains.