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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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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...
First passage and recurrence distributions
 Trans. Amer. Math. Soc
, 1952
"... designated by 0, 1, 2, • • • , and with transition probabilities independent of time. Letting xo, xu ■ ■ ■ be the states after 0, 1, • • • steps, we define (1.1) PM(i,j) = P(xn =j\x0=i), n 0, 1,. • •, where P(^4ß) stands for the conditional probability of A, given B. We as ..."
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Cited by 36 (0 self)
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designated by 0, 1, 2, • • • , and with transition probabilities independent of time. Letting xo, xu ■ ■ ■ be the states after 0, 1, • • • steps, we define (1.1) PM(i,j) = P(xn =j\x0=i), n 0, 1,. • •, where P(^4ß) stands for the conditional probability of A, given B. We as
Parking Functions, Empirical Processes, and the Width of Rooted Labeled Trees
"... This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking f ..."
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Cited by 19 (6 self)
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This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking functions and the empirical processes of mathematical statistics. Our result turns out to be a consequence of the strong convergence of empirical processes to the Brownian bridge (Komlos, Major and Tusnady, 1975).
Transient Behavior of the M/G/1 Workload Process
, 1992
"... In this paper we describe the timedependent moments of the workload process in the M/G/1 queue. The k th moment as a function of time can be characterized in terms of a differential equation involving lower moment functions and the timedependent serveroccupation probability. For general initial ..."
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Cited by 19 (9 self)
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In this paper we describe the timedependent moments of the workload process in the M/G/1 queue. The k th moment as a function of time can be characterized in terms of a differential equation involving lower moment functions and the timedependent serveroccupation probability. For general initial conditions, we show that the first two moment functions can be represented as the difference of two nondecreasing functions, one of which is the moment function starting at zero. The two nondecreasing components can be regarded as probability cumulative distribution functions (cdf's) after appropriate normalization. The normalized moment functions starting empty are called moment cdf's; the other normalized components are called momentdifference cdf's. We establish relations among these cdf's using stationaryexcess relations. We apply these relations to calculate moments and derivatives at the origin of these cdf's. We also obtain results for the covariance function of the stationary workload process. It is interesting that these various timedependent characteristics can be described directly in terms of the steadystate workload distribution. Subject classification: queues, transient results: M/G/1 workload process. queues, busyperiod analysis: M/G/1 queue. In this paper, we derive some simple descriptions of the transient behavior of the classical M/G/1 queue. In particular, we focus on the workload process {W(t) : t 0} (also known as the unfinished work process and the virtual waiting time process), which is convenient to analyze because it is a Markov process. Our main results describe the timedependent probability that the server is busy, P(W(t) > 0), the timedependent moments of the workload process, E[W(t) k ], and the covariance function of the stationary ...
The Transient BMAP/G/1 Queue
, 2000
"... We derive the twodimensional transforms of the transient workload and queuelength distributions in the singleserver queue with general service times and a batch Markovian arrival process (BMAP). This arrival process includes the familiar phasetype renewal process and the Markov modulated Poisson ..."
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Cited by 11 (2 self)
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We derive the twodimensional transforms of the transient workload and queuelength distributions in the singleserver queue with general service times and a batch Markovian arrival process (BMAP). This arrival process includes the familiar phasetype renewal process and the Markov modulated Poisson process as special cases, and allows correlated interarrival times and batch sizes. Numerical results are obtained via twodimensional transform inversion algorithms based on the Fourierseries method. From the numerical examples we see that predictions of system performance based on transient and stationary performance measures can be quite different. n
Towards Reliable Modelling with Stochastic Process Algebras
 Department of Computer Science, University of Bristol, Bristol
, 1999
"... Abstract In this thesis, we investigate reliable modelling within a stochastic process algebra framework. Primarily, we consider issues of variance in stochastic process algebras as a measure of model reliability. This is in contrast to previous research in the field which has tended to centre aroun ..."
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Abstract In this thesis, we investigate reliable modelling within a stochastic process algebra framework. Primarily, we consider issues of variance in stochastic process algebras as a measure of model reliability. This is in contrast to previous research in the field which has tended to centre around mean behaviour and steadystate solutions. We present a method of stochastic aggregation for analysing generallydistributed processes. This allows us more descriptive power in representing stochastic systems and thus gives us the ability to create more accurate models. We improve upon two welldeveloped Markovian process algebras and show how their simpler paradigm can be brought to bear on more realistic synchronisation models. Now, reliable performance figures can be obtained for systems, where previously only approximations of unknown accuracy were possible. Finally, we describe reliability definitions and variance metrics in stochastic models and demonstrate how systems can be made more reliable through careful combination under stochastic process algebra operators. ii Acknowledgements My three years in the department in Bristol have been a lot of fun and the person I have most to thank for this is my friend and mentor, Neil Davies. I should also acknowledge the funding from NATS for my project and especially the help of Suresh Tewari (NATS) and Gordon Hughes (SSRC).
Classification of Location Problems
 Location Science
, 1996
"... There are several good reasons to introduce classification schemes for optimization problems including, for instance, the ability for concise problem statement opposed to verbal, often ambiguous, descriptions or simple data encoding and information retrieval in bibliographical information systems or ..."
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There are several good reasons to introduce classification schemes for optimization problems including, for instance, the ability for concise problem statement opposed to verbal, often ambiguous, descriptions or simple data encoding and information retrieval in bibliographical information systems or software libraries. In some branches like scheduling and queuing theory classification is therefore a widely accepted and appreciated tool. The aim of this paper is to propose a 5position classification which can be used to cover all location problems. We will provide a list of currently available symbols and indicate its usefulness in a  necessarily noncomprehensive  list of "classical" location problems. The classification scheme is in use since 1992 and has since proved to be useful in research, software development, classroom, and for overview articles. 1 Introduction In several branches of optimization classification schemes have been successfully introduced and are used by every ...
Queueing analysis of relational operators for continuous data streams
 In Proc. Int. Conf. on Information and Knowledge Management (CIKM
, 2003
"... Currently, stream data processing is an active area of research, which includes everything from algorithms and architectures for stream processing to modelling and analysis of various components of a stream processing system. In this paper, we present an analysis of relational operators used for str ..."
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Cited by 7 (4 self)
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Currently, stream data processing is an active area of research, which includes everything from algorithms and architectures for stream processing to modelling and analysis of various components of a stream processing system. In this paper, we present an analysis of relational operators used for stream processing using queueing theory and study behaviors of streaming data in a query processing system. Our approach enables us to compute the fundamental performance metrics of relational operators — select, project, and join over data streams. Furthermore, this approach establishes a way to find the probability distribution functions of both the number of tuples and the waiting time of tuples in the system. Finally, we designed and implemented a number of experiments to validate the accuracy and effectiveness of our analysis. We believe that the results of this paper are very useful for performance evaluation of a continuous query processing system over streaming data, for design of a stream query processing system, and for the understanding of the behaviors of streaming data. 1.
Susceptibility in subcritical random graphs
 125207. OF RANDOM GRAPHS WITH GIVEN VERTEX DEGREES 25
"... Abstract. We study the evolution of the susceptibility in the subcritical random graph G(n, p) as n tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the scaled fluctuations of the susceptibility around its ..."
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Abstract. We study the evolution of the susceptibility in the subcritical random graph G(n, p) as n tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the scaled fluctuations of the susceptibility around its deterministic limit converge to a Gaussian law. We further extend our results to higher moments of the component size of a random vertex, and prove that they are jointly asymptotically normal. 1.
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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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.