On occupation times for quasi-Markov processes

1981 ◽  
Vol 18 (1) ◽  
pp. 297-301 ◽  
Author(s):  
Lennart Bondesson
Keyword(s):  
Markov Process ◽  
Laplace Transform ◽  
Markov Processes ◽  
Joint Distribution ◽  
Stieltjes Transform ◽  
Occupation Times ◽  
The Laplace Transform ◽  
The Times

In this note the joint distribution for the times in an interval [0, t] spent in the states 1, 2, ···, N in a standard quasi-Markov process of order N is considered. An expression for the Laplace transform with respect to t of the Laplace–Stieltjes transform of this joint distribution is derived.

On occupation times for quasi-Markov processes

1981 ◽  
Vol 18 (01) ◽  
pp. 297-301 ◽  
Author(s):  
Lennart Bondesson
Keyword(s):  
Markov Process ◽  
Laplace Transform ◽  
Markov Processes ◽  
Joint Distribution ◽  
Stieltjes Transform ◽  
Occupation Times ◽  
The Laplace Transform ◽  
The Times

In this note the joint distribution for the times in an interval [0, t] spent in the states 1, 2, ···, N in a standard quasi-Markov process of order N is considered. An expression for the Laplace transform with respect to t of the Laplace–Stieltjes transform of this joint distribution is derived.

A temporal factorization at the maximum for certain positive self-similar Markov processes

2020 ◽  
Vol 57 (4) ◽  
pp. 1045-1069
Author(s):  
Matija Vidmar
Keyword(s):  
Markov Process ◽  
Laplace Transform ◽  
Markov Processes ◽  
Absorption Time ◽  
The Laplace Transform ◽  
Self Similar

AbstractFor a spectrally negative self-similar Markov process on $[0,\infty)$ with an a.s. finite overall supremum, we provide, in tractable detail, a kind of conditional Wiener–Hopf factorization at the maximum of the absorption time at zero, the conditioning being on the overall supremum and the jump at the overall supremum. In a companion result the Laplace transform of this absorption time (on the event that the process does not go above a given level) is identified under no other assumptions (such as the process admitting a recurrent extension and/or hitting zero continuously), generalizing some existing results in the literature.

Analysis of the fluid weighted fair queueing system

2003 ◽  
Vol 40 (1) ◽  
pp. 180-199 ◽  
Author(s):  
Fabrice Guillemin ◽  
Ravi Mazumdar ◽  
Alain Dupuis ◽  
Jacqueline Boyer
Keyword(s):  
Laplace Transform ◽  
Performance Measures ◽  
Joint Distribution ◽  
Queueing System ◽  
Laplace Transforms ◽  
Fair Queueing ◽  
Mean Waiting Time ◽  
The Laplace Transform ◽  
The Mean

We analyse in this paper the fluid weighted fair queueing system with two classes of customers, who arrive according to Poisson processes and require arbitrarily distributed service times. In a first step, we express the Laplace transform of the joint distribution of the workloads in the two virtual queues of the system by means of unknown Laplace transforms. Such an unknown Laplace transform is related to the distribution of the workload in one queue provided that the other queue is empty. We explicitly compute the unknown Laplace transforms by means of a Wiener—Hopf technique. The determination of the unknown Laplace transforms can be used to compute some performance measures characterizing the system (e.g. the mean waiting time for each class) which we compute in the exponential service case.

A note on functions of Markov processes with an application to a sequence of χ2 statistics

1973 ◽  
Vol 10 (4) ◽  
pp. 895-900
Author(s):  
Murray A. Cameron
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Joint Distribution ◽  
Sufficient Condition ◽  
Simple Derivation ◽  
Reverse Process

A sufficient condition for a function of a Markov process to be Markovian is obtained by considering a reverse process of the original Markov process. An application of this result provides a simple derivation of the joint distribution of a sequence of Pearson χ2 statistics previously obtained by Zaharov, Sarmanov and Sevast'ianov (1969).

The supremum distribution of a Lévy process with no negative jumps

1977 ◽  
Vol 9 (02) ◽  
pp. 417-422 ◽  
Author(s):  
J. Michael Harrison
Keyword(s):  
Markov Process ◽  
Laplace Transform ◽  
Stationary Distribution ◽  
Analytical Methods ◽  
Simple Formula ◽  
Elementary Proof ◽  
Levy Process ◽  
Storage Process ◽  
Independent Increments ◽  
The Laplace Transform

Let be a process with stationary, independent increments and no negative jumps. Let be this same process modified by a reflecting barrier at zero (a storage process). Assuming that – and denote by ψ(s) the exponent function of X. A simple formula is derived for the Laplace transform of as a function of W(0). Using the fact that the distribution of M is the unique stationary distribution of the Markov process W, this yields an elementary proof that the Laplace transform of M is µs/ψ(s). If it follows that These surprisingly simple formulas were originally obtained by Zolotarev using analytical methods.

A result on the Laplace transform associated with the sticky Brownian motion on an interval

2020 ◽  
pp. 2150031
Author(s):  
Shiyu Song
Keyword(s):  
Brownian Motion ◽  
Boundary Conditions ◽  
Laplace Transform ◽  
Laplace Transforms ◽  
Perturbation Approach ◽  
Integrated Process ◽  
Modified Bessel Function ◽  
Airy Functions ◽  
Occupation Times ◽  
The Laplace Transform

In this paper, we study the joint Laplace transform of the sticky Brownian motion on an interval, its occupation time at zero and its integrated process. The perturbation approach of Li and Zhou [The joint Laplace transforms for diffusion occupation times, Adv. Appl. Probab. 45 (2013) 1049–1067] is adopted to convert the problem into the computation of three Laplace transforms, which is essentially equivalent to solving the associated differential equations with boundary conditions. We obtain the explicit expression for the joint Laplace transform in terms of the modified Bessel function and Airy functions.

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.

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.

New identities for the Wright and the Mittag-Leffler functions using the Laplace transform

2014 ◽  
Vol 07 (03) ◽  
pp. 1450038 ◽  
Author(s):  
Alireza Ansari ◽  
Amirhossein Refahi Sheikhani
Keyword(s):  
Laplace Transform ◽  
Inverse Laplace Transform ◽  
Stieltjes Transform ◽  
The Laplace Transform ◽  
Integral Identities

In this paper, we state three theorems for the inverse Laplace transform and using these theorems we obtain new integral identities involving the products of the Wright and Mittag-Leffler functions. The relationships of these integral identities with the Stieltjes transform are also given.

Analysis of the fluid weighted fair queueing system

2003 ◽  
Vol 40 (01) ◽  
pp. 180-199
Author(s):  
Fabrice Guillemin ◽  
Ravi Mazumdar ◽  
Alain Dupuis ◽  
Jacqueline Boyer
Keyword(s):  
Laplace Transform ◽  
Performance Measures ◽  
Joint Distribution ◽  
Queueing System ◽  
Laplace Transforms ◽  
Fair Queueing ◽  
Mean Waiting Time ◽  
The Laplace Transform ◽  
The Mean

We analyse in this paper the fluid weighted fair queueing system with two classes of customers, who arrive according to Poisson processes and require arbitrarily distributed service times. In a first step, we express the Laplace transform of the joint distribution of the workloads in the two virtual queues of the system by means of unknown Laplace transforms. Such an unknown Laplace transform is related to the distribution of the workload in one queue provided that the other queue is empty. We explicitly compute the unknown Laplace transforms by means of a Wiener—Hopf technique. The determination of the unknown Laplace transforms can be used to compute some performance measures characterizing the system (e.g. the mean waiting time for each class) which we compute in the exponential service case.

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