Continued Fraction Analysis of the Duration of an Excursion in an M/M/∞ System

1998 ◽  
Vol 35 (1) ◽  
pp. 165-183 ◽  
Author(s):  
Fabrice Guillemin ◽  
Didier Pinchon
Keyword(s):  
Laplace Transform ◽  
Hypergeometric Functions ◽  
Continued Fraction ◽  
Complex Analysis ◽  
Markov Property ◽  
Random Variable ◽  
Charlier Polynomials ◽  
Kummer’S Function ◽  
The Laplace Transform ◽  
Strong Markov

We show in this paper how the Laplace transform θ* of the duration θ of an excursion by the occupation process {Λt} of an M/M/∞ system above a given threshold can be obtained by means of continued fraction analysis. The representation of θ* by a continued fraction is established and the [m−1/m] Padé approximants are computed by means of well known orthogonal polynomials, namely associated Charlier polynomials. It turns out that the continued fraction considered is an S fraction and as a consequence the Stieltjes transform of some spectral measure. Then, using classic asymptotic expansion properties of hypergeometric functions, the representation of the Laplace transform θ* by means of Kummer's function is obtained. This allows us to recover an earlier result obtained via complex analysis and the use of the strong Markov property satisfied by the occupation process {Λt}. The continued fraction representation enables us to further characterize the distribution of the random variable θ.

Continued Fraction Analysis of the Duration of an Excursion in an M/M/∞ System

1998 ◽  
Vol 35 (01) ◽  
pp. 165-183
Author(s):  
Fabrice Guillemin ◽  
Didier Pinchon
Keyword(s):  
Laplace Transform ◽  
Hypergeometric Functions ◽  
Continued Fraction ◽  
Complex Analysis ◽  
Markov Property ◽  
Random Variable ◽  
Charlier Polynomials ◽  
Kummer’S Function ◽  
The Laplace Transform ◽  
Strong Markov

We show in this paper how the Laplace transform θ* of the duration θ of an excursion by the occupation process {Λ t } of an M/M/∞ system above a given threshold can be obtained by means of continued fraction analysis. The representation of θ* by a continued fraction is established and the [m−1/m] Padé approximants are computed by means of well known orthogonal polynomials, namely associated Charlier polynomials. It turns out that the continued fraction considered is an S fraction and as a consequence the Stieltjes transform of some spectral measure. Then, using classic asymptotic expansion properties of hypergeometric functions, the representation of the Laplace transform θ* by means of Kummer's function is obtained. This allows us to recover an earlier result obtained via complex analysis and the use of the strong Markov property satisfied by the occupation process {Λ t }. The continued fraction representation enables us to further characterize the distribution of the random variable θ.

Dirichlet Random Walks

2014 ◽  
Vol 51 (04) ◽  
pp. 1081-1099 ◽  
Author(s):  
Gérard Letac ◽  
Mauro Piccioni
Keyword(s):  
Random Walk ◽  
Laplace Transform ◽  
Dirichlet Process ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Random Variable ◽  
Infinite Divisibility ◽  
Stieltjes Transform ◽  
The Laplace Transform ◽  
Infinite Divisible Distributions

This paper provides tools for the study of the Dirichlet random walk inRd. We compute explicitly, for a number of cases, the distribution of the random variableWusing a form of Stieltjes transform ofWinstead of the Laplace transform, replacing the Bessel functions with hypergeometric functions. This enables us to simplify some existing results, in particular, some of the proofs by Le Caër (2010), (2011). We extend our results to the study of the limits of the Dirichlet random walk when the number of added terms goes to ∞, interpreting the results in terms of an integral by a Dirichlet process. We introduce the ideas of Dirichlet semigroups and Dirichlet infinite divisibility and characterize these infinite divisible distributions in the sense of Dirichlet when they are concentrated on the unit sphere ofRd.

scholarly journals Dirichlet Random Walks

2014 ◽  
Vol 51 (4) ◽  
pp. 1081-1099 ◽  
Author(s):  
Gérard Letac ◽  
Mauro Piccioni
Keyword(s):  
Random Walk ◽  
Laplace Transform ◽  
Dirichlet Process ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Random Variable ◽  
Infinite Divisibility ◽  
Stieltjes Transform ◽  
The Laplace Transform ◽  
Infinite Divisible Distributions

This paper provides tools for the study of the Dirichlet random walk in Rd. We compute explicitly, for a number of cases, the distribution of the random variable W using a form of Stieltjes transform of W instead of the Laplace transform, replacing the Bessel functions with hypergeometric functions. This enables us to simplify some existing results, in particular, some of the proofs by Le Caër (2010), (2011). We extend our results to the study of the limits of the Dirichlet random walk when the number of added terms goes to ∞, interpreting the results in terms of an integral by a Dirichlet process. We introduce the ideas of Dirichlet semigroups and Dirichlet infinite divisibility and characterize these infinite divisible distributions in the sense of Dirichlet when they are concentrated on the unit sphere of Rd.

M/M/∞ transience revisited

1997 ◽  
Vol 34 (04) ◽  
pp. 1061-1067 ◽  
Author(s):  
J. Preater
Keyword(s):  
Laplace Transform ◽  
Joint Distribution ◽  
Busy Period ◽  
Complex Analysis ◽  
Transient Characteristics ◽  
The Laplace Transform ◽  
Swept Area ◽  
The Way

We take a fresh look at some transient characteristics of an M/M/∞ queue, studied previously by Guillemin and Simonian using delicate complex analysis. Along the way we obtain the Laplace transform of the joint distribution of the duration, number of arrivals and swept area associated with a busy period of an M/M/1 queue.

The generalised state-dependent queue: the busy period Erlangian

1974 ◽  
Vol 11 (03) ◽  
pp. 618-623
Author(s):  
B. W. Conolly
Keyword(s):  
Laplace Transform ◽  
Generating Function ◽  
Busy Period ◽  
Continued Fraction ◽  
Queueing System ◽  
Joint Probability ◽  
Single Server ◽  
System State ◽  
State Dependent ◽  
The Laplace Transform

A continued fraction representation is presented of the Laplace transform of the generating function of the fundamental joint probability and density of busy period length measured in customers served and duration in time. The setting is the single server Erlang queueing system where the parameters of negative exponentially distributed arrival and service times have a general dependence on instantaneous system state.

The Laplace Transform of Hitting Times of Integrated Geometric Brownian Motion

2013 ◽  
Vol 50 (1) ◽  
pp. 295-299 ◽  
Author(s):  
Adam Metzler
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Hypergeometric Functions ◽  
Diffusion Processes ◽  
Geometric Brownian Motion ◽  
Hitting Times ◽  
Confluent Hypergeometric Functions ◽  
Change Of Variables ◽  
The Laplace Transform ◽  
Kummer's Equation

In this note we compute the Laplace transform of hitting times, to fixed levels, of integrated geometric Brownian motion. The transform is expressed in terms of the gamma and confluent hypergeometric functions. Using a simple Itô transformation and standard results on hitting times of diffusion processes, the transform is characterized as the solution to a linear second-order ordinary differential equation which, modulo a change of variables, is equivalent to Kummer's equation.

The Laplace Transform of Hitting Times of Integrated Geometric Brownian Motion

2013 ◽  
Vol 50 (01) ◽  
pp. 295-299 ◽  
Author(s):  
Adam Metzler
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Hypergeometric Functions ◽  
Diffusion Processes ◽  
Geometric Brownian Motion ◽  
Hitting Times ◽  
Confluent Hypergeometric Functions ◽  
Change Of Variables ◽  
The Laplace Transform ◽  
Kummer's Equation

In this note we compute the Laplace transform of hitting times, to fixed levels, of integrated geometric Brownian motion. The transform is expressed in terms of the gamma and confluent hypergeometric functions. Using a simple Itô transformation and standard results on hitting times of diffusion processes, the transform is characterized as the solution to a linear second-order ordinary differential equation which, modulo a change of variables, is equivalent to Kummer's equation.

M/M/∞ transience revisited

1997 ◽  
Vol 34 (4) ◽  
pp. 1061-1067 ◽  
Author(s):  
J. Preater
Keyword(s):  
Laplace Transform ◽  
Joint Distribution ◽  
Busy Period ◽  
Complex Analysis ◽  
Transient Characteristics ◽  
The Laplace Transform ◽  
Swept Area ◽  
The Way

We take a fresh look at some transient characteristics of an M/M/∞ queue, studied previously by Guillemin and Simonian using delicate complex analysis. Along the way we obtain the Laplace transform of the joint distribution of the duration, number of arrivals and swept area associated with a busy period of an M/M/1 queue.

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