scholarly journals Lattice Bessel functions and their applications to a transient analysis of queueing networks

1987 ◽  
Vol 26 ◽  
pp. 222-223
Author(s):  
William A. Massey
Keyword(s):  
Queueing Networks ◽  
Transient Analysis ◽  
Bessel Functions

A sample path analysis of the M/M/1 queue

1989 ◽  
Vol 26 (02) ◽  
pp. 418-422 ◽  
Author(s):  
Francois Baccelli ◽  
William A. Massey
Keyword(s):  
Generating Functions ◽  
Queueing Networks ◽  
Transient Analysis ◽  
Bessel Functions ◽  
Sample Path ◽  
Laplace Transforms ◽  
Sample Path Analysis ◽  
Transient Distribution ◽  
The Past ◽  
Novel Approach

The exact solution for the transient distribution of the queue length and busy period of the M/M/1 queue in terms of modified Bessel functions has been proved in a variety of ways. Methods of the past range from spectral analysis (Lederman and Reuter (1954)), combinatorial arguments (Champernowne (1956)), to generating functions coupled with Laplace transforms (Clarke (1956)). In this paper, we present a novel approach that ties the computation of these transient distributions directly to the random sample path behavior of the M/M/1 queue. The use of Laplace transforms is minimized, and the use of generating functions is eliminated completely. This is a method that could prove to be useful in developing a similar transient analysis for queueing networks.

A sample path analysis of the M/M/1 queue

1989 ◽  
Vol 26 (2) ◽  
pp. 418-422 ◽  
Author(s):  
Francois Baccelli ◽  
William A. Massey
Keyword(s):  
Generating Functions ◽  
Queueing Networks ◽  
Transient Analysis ◽  
Bessel Functions ◽  
Sample Path ◽  
Laplace Transforms ◽  
Sample Path Analysis ◽  
Transient Distribution ◽  
The Past ◽  
Novel Approach

The exact solution for the transient distribution of the queue length and busy period of the M/M/1 queue in terms of modified Bessel functions has been proved in a variety of ways. Methods of the past range from spectral analysis (Lederman and Reuter (1954)), combinatorial arguments (Champernowne (1956)), to generating functions coupled with Laplace transforms (Clarke (1956)). In this paper, we present a novel approach that ties the computation of these transient distributions directly to the random sample path behavior of the M/M/1 queue. The use of Laplace transforms is minimized, and the use of generating functions is eliminated completely. This is a method that could prove to be useful in developing a similar transient analysis for queueing networks.

Transient Analysis of a Two-Heterogeneous Severs Queue with Impatient Behaviour and Multiple Vacations

2018 ◽  
Vol 6 (1) ◽  
pp. 69-84
Author(s):  
Jia Xu ◽  
Liwei Liu ◽  
Taozeng Zhu
Keyword(s):  
Generating Functions ◽  
Transient Analysis ◽  
Bessel Functions ◽  
Queueing System ◽  
System Size ◽  
Heterogeneous Servers ◽  
Modified Bessel Functions ◽  
Multiple Vacations ◽  
Special Cases ◽  
Mean And Variance

AbstractWe consider anM/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtain explicit expressions for the time dependent probabilities, mean and variance of the system size at timetby employing probability generating functions, continued fractions and properties of the modified Bessel functions. Finally, two special cases are provided.

Transient analysis of state-dependent queueing networks via cumulant functions

2001 ◽  
Vol 38 (04) ◽  
pp. 841-859 ◽  
Author(s):  
Timothy I. Matis ◽  
Richard M. Feldman
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Queueing Networks ◽  
Transient Analysis ◽  
Queueing Network ◽  
Partial Differential ◽  
Cumulant Generating Function ◽  
State Dependent ◽  
Intensity Functions ◽  
First Moment

A new procedure that generates the transient solution of the first moment of the state of a Markovian queueing network with state-dependent arrivals, services, and routeing is developed. The procedure involves defining a partial differential equation that relates an approximate multivariate cumulant generating function to the intensity functions of the network. The partial differential equation then yields a set of ordinary differential equations which are numerically solved to obtain the first moment.

Transient analysis of state-dependent queueing networks via cumulant functions

2001 ◽  
Vol 38 (4) ◽  
pp. 841-859 ◽  
Author(s):  
Timothy I. Matis ◽  
Richard M. Feldman
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Queueing Networks ◽  
Transient Analysis ◽  
Queueing Network ◽  
Partial Differential ◽  
Cumulant Generating Function ◽  
State Dependent ◽  
Intensity Functions ◽  
First Moment

A new procedure that generates the transient solution of the first moment of the state of a Markovian queueing network with state-dependent arrivals, services, and routeing is developed. The procedure involves defining a partial differential equation that relates an approximate multivariate cumulant generating function to the intensity functions of the network. The partial differential equation then yields a set of ordinary differential equations which are numerically solved to obtain the first moment.

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