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.

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.

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.

On the busy period of the modified GI/G/1 queue

1973 ◽  
Vol 10 (01) ◽  
pp. 192-197 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Laplace Transform ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Service Time Distribution ◽  
Duality Results ◽  
The Laplace Transform ◽  
Period Duration

Proceeding from duality results for the GI/G/1 queue, this paper obtains the probability of the number served in a busy period of aGI/G/1 system where customers initiating a busy period have a different service time distribution from other customers. Using duality arguments for processes with interchangeable increments, the Laplace transform of the busy period duration is found for a modified GI/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.

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.

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.

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.

Abstract: Stochastic Dominance in the Laplace Transformation Domain

1977 ◽  
Vol 12 (4) ◽  
pp. 639-639
Author(s):  
Stylianos Perrakis
Keyword(s):  
Laplace Transform ◽  
Stochastic Dominance ◽  
Laplace Transformation ◽  
Joint Distribution ◽  
Portfolio Returns ◽  
The Laplace Transform ◽  
Past Data ◽  
Selection Of

This paper examines the problem of the selection of first-, second-, and thirddegree undominated portfolios by using the properties of the Laplace transform (L-T) of the distributions of portfolio returns. It is assumed that the joint distribution of n interdependent prospects, as well as its Laplace transform, is known or may be estimated from past data. Next, it is shown that the L-T of the portfolio returns may be expressed very simply in terms of the L-T of the joint distribution. A theorem is then proved, which uses results from L-T theory and shows that stochastic dominance between two portfolios of first-, second- or third-degree may be expressed by inequalities between the L-T's of the portfolios and their derivatives. It is also shown through an example how this theorem may be used in finding undominated portfolios.

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.

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