Closed-form representations of the laplace transforms of bessel funtions

In this paper we find the waiting time distribution in the transient domain and the busy period distribution of the GI G/1 queue. We formulate the problem as a two-dimensional Lindley process and then transform it to a Hilbert factorization problem. We achieve the solution of the factorization problem for the GI/R/1, R/G/1 queues, where R is the class of distributions with rational Laplace transforms. We obtain simple closed-form expressions for the Laplace transforms of the waiting time distribution and the busy period distribution. Furthermore, we find closed-form formulae for the first two moments of the distributions involved.

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A correlated queue with infinitely many servers

Journal of Applied Probability ◽

10.1017/s0021900200106357 ◽

1986 ◽

Vol 23 (01) ◽

pp. 155-165 ◽

Cited By ~ 2

Author(s):

C. Langaris

Keyword(s):

Steady State ◽

Closed Form ◽

Service Time ◽

Transient State ◽

Laplace Transforms ◽

Bivariate Distribution ◽

Numerical Calculations ◽

Output Process ◽

Negative Exponential ◽

Steady State Probabilities

This work analyses a queueing mechanism with infinitely many servers in which the interarrival interval T preceding the arrival of a customer and his service time S are assumed correlated. A bivariate distribution with negative exponential marginals is used and the Laplace transforms pn (z) of the system probabilities in transient state are obtained. Closed-form expressions for the steady-state probabilities pn and their moments are given too. Finally the output process is investigated and conclusions are drawn from numerical calculations.

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Transient and busy period analysis of the GI G/1 Queue as a Hilbert factorization problem

Journal of Applied Probability ◽

10.1017/s0021900200042789 ◽

1991 ◽

Vol 28 (04) ◽

pp. 873-885 ◽

Cited By ~ 1

Author(s):

Dimitris J. Bertsimas ◽

Julian Keilson ◽

Daisuke Nakazato ◽

Hongtao Zhang

Keyword(s):

Closed Form ◽

Waiting Time ◽

Time Distribution ◽

Busy Period ◽

Laplace Transforms ◽

Waiting Time Distribution ◽

Period Analysis ◽

Factorization Problem ◽

Period Distribution ◽

Lindley Process

In this paper we find the waiting time distribution in the transient domain and the busy period distribution of the GI G/1 queue. We formulate the problem as a two-dimensional Lindley process and then transform it to a Hilbert factorization problem. We achieve the solution of the factorization problem for the GI/R/1, R/G/1 queues, where R is the class of distributions with rational Laplace transforms. We obtain simple closed-form expressions for the Laplace transforms of the waiting time distribution and the busy period distribution. Furthermore, we find closed-form formulae for the first two moments of the distributions involved.

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The Integral Mittag-Leffler, Whittaker and Wright Functions

Mathematics ◽

10.3390/math9243255 ◽

2021 ◽

Vol 9 (24) ◽

pp. 3255

Author(s):

Alexander Apelblat ◽

Juan Luis González-Santander

Keyword(s):

Closed Form ◽

Hypergeometric Functions ◽

Special Functions ◽

Laplace Transforms ◽

Graphical Form ◽

Generalized Hypergeometric Functions ◽

Infinite Sums ◽

First Time ◽

Mathematical Literature

Integral Mittag-Leffler, Whittaker and Wright functions with integrands similar to those which already exist in mathematical literature are introduced for the first time. For particular values of parameters, they can be presented in closed-form. In most reported cases, these new integral functions are expressed as generalized hypergeometric functions but also in terms of elementary and special functions. The behavior of some of the new integral functions is presented in graphical form. By using the MATHEMATICA program to obtain infinite sums that define the Mittag-Leffler, Whittaker, and Wright functions and also their corresponding integral functions, these functions and many new Laplace transforms of them are also reported in the Appendices for integral and fractional values of parameters.

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A correlated queue with infinitely many servers

Journal of Applied Probability ◽

10.2307/3214124 ◽

1986 ◽

Vol 23 (1) ◽

pp. 155-165 ◽

Cited By ~ 10

Author(s):

C. Langaris

Keyword(s):

Steady State ◽

Closed Form ◽

Service Time ◽

Transient State ◽

Laplace Transforms ◽

Bivariate Distribution ◽

Numerical Calculations ◽

Output Process ◽

Negative Exponential ◽

Steady State Probabilities

This work analyses a queueing mechanism with infinitely many servers in which the interarrival interval T preceding the arrival of a customer and his service time S are assumed correlated. A bivariate distribution with negative exponential marginals is used and the Laplace transforms pn (z) of the system probabilities in transient state are obtained. Closed-form expressions for the steady-state probabilities pn and their moments are given too. Finally the output process is investigated and conclusions are drawn from numerical calculations.

Download Full-text