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.

Asymptotic expansions and converging factors V. Lommel, Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions

1959 ◽  
Vol 249 (1257) ◽  
pp. 284-292 ◽  
Keyword(s):  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Lommel Functions ◽  
Special Cases

A theory of Lommel functions is developed, based upon the methods described in the first four papers (I to IV) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘basic converging factors’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions.

Analysis of the M x/G/1 queue by N-policy and multiple vacations

1994 ◽  
Vol 31 (02) ◽  
pp. 476-496
Author(s):  
Ho Woo Lee ◽  
Soon Seok Lee ◽  
Jeong Ok Park ◽  
K. C. Chae
Keyword(s):  
Performance Measures ◽  
Queue Length ◽  
Time Distribution ◽  
Queueing System ◽  
Random Variables ◽  
System Size ◽  
The Other ◽  
Waiting Time Distribution ◽  
Multiple Vacations ◽  
Linear Cost

We consider an Mx /G/1 queueing system with N-policy and multiple vacations. As soon as the system empties, the server leaves for a vacation of random length V. When he returns, if the queue length is greater than or equal to a predetermined value N(threshold), the server immediately begins to serve the customers. If he finds less than N customers, he leaves for another vacation and so on until he finally finds at least N customers. We obtain the system size distribution and show that the system size decomposes into three random variables one of which is the system size of ordinary Mx /G/1 queue. The interpretation of the other random variables will be provided. We also derive the queue waiting time distribution and other performance measures. Finally we derive a condition under which the optimal stationary operating policy is achieved under a linear cost structure.

scholarly journals Transient analysis of queueing system with two heterogeneous servers

2019 ◽  
Vol 1333 ◽  
pp. 032039
Author(s):  
L I Korolkova ◽  
N Mashrabov ◽  
A M Murzin ◽  
A V Panfilov ◽  
Ju L Siuskina
Keyword(s):  
Transient Analysis ◽  
Queueing System ◽  
Heterogeneous Servers

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.

scholarly journals Some Novel Fractional Integral Inequalities over a New Class of Generalized Convex Function

2022 ◽  
Vol 6 (1) ◽  
pp. 42
Author(s):  
Soubhagya Kumar Sahoo ◽  
Muhammad Tariq ◽  
Hijaz Ahmad ◽  
Bibhakar Kodamasingh ◽  
Asif Ali Shaikh ◽  
...  
Keyword(s):  
Applied Sciences ◽  
Fractional Integral ◽  
Bessel Functions ◽  
Fractional Operators ◽  
Modified Bessel Functions ◽  
New Class ◽  
Special Means ◽  
Special Cases ◽  
Related Integral ◽  
Increasing Function

The comprehension of inequalities in convexity is very important for fractional calculus and its effectiveness in many applied sciences. In this article, we handle a novel investigation that depends on the Hermite–Hadamard-type inequalities concerning a monotonic increasing function. The proposed methodology deals with a new class of convexity and related integral and fractional inequalities. There exists a solid connection between fractional operators and convexity because of its fascinating nature in the numerical sciences. Some special cases have also been discussed, and several already-known inequalities have been recaptured to behave well. Some applications related to special means, q-digamma, modified Bessel functions, and matrices are discussed as well. The aftereffects of the plan show that the methodology can be applied directly and is computationally easy to understand and exact. We believe our findings generalise some well-known results in the literature on s-convexity.

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.

Asymptotic expansions and converging factors IV. Confluent hypergeometric, parabolic cylinder, modified Bessel, and ordinary Bessel functions

1959 ◽  
Vol 249 (1257) ◽  
pp. 270-283 ◽  
Keyword(s):  
Exponential Type ◽  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Parabolic Cylinder ◽  
Modified Bessel Functions ◽  
Confluent Hypergeometric Functions ◽  
Special Cases ◽  
Parabolic Cylinder Functions

A theory of confluent hypergeometric functions is developed, based upon the methods described in the first three papers (I, II and III) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘ basic converging factors ’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of exponential-type integrals, parabolic cylinder functions, modified Bessel functions, and ordinary Bessel functions.

The analysis of discrete time Geom/Geom/1 queue with single working vacation and multiple vacations (Geom/Geom/1/SWV+MV)

2018 ◽  
Vol 52 (1) ◽  
pp. 95-117 ◽  
Author(s):  
Qingqing Ye ◽  
Liwei Liu
Keyword(s):  
Discrete Time ◽  
System Size ◽  
Stochastic Decomposition ◽  
Model Parameters ◽  
Working Vacation ◽  
Two Phase ◽  
Multiple Vacations ◽  
Special Cases ◽  
The Matrix ◽  
Matrix Geometric Solution

In this article, we consider a discrete-time Geom/Geom/1 queue with two phase vacation policy that comprises single working vacation and multiple vacations, denoted by Geom/Geom/1/SWV+MV. For this model, we first derive the explicit expression for the stationary system size by the matrix-geometric solution method. Next, we obtain the stochastic decomposition structures of system size and the sojourn time of an arbitrary customer in steady state. Moreover, the regular busy period and busy cycle are analyzed by limiting theorem of alternative renewal process. Besides, some special cases are presented and the relationship between the Geom/Geom/1/SWV+MV queue and its continuous time counterpart is investigated. Finally, we perform several experiments to illustrate the effect of model parameters on some performance measures.

Analysis of the Mx/G/1 queue by N-policy and multiple vacations

1994 ◽  
Vol 31 (2) ◽  
pp. 476-496 ◽  
Author(s):  
Ho Woo Lee ◽  
Soon Seok Lee ◽  
Jeong Ok Park ◽  
K. C. Chae
Keyword(s):  
Performance Measures ◽  
Queue Length ◽  
Time Distribution ◽  
Queueing System ◽  
Random Variables ◽  
System Size ◽  
The Other ◽  
Waiting Time Distribution ◽  
Multiple Vacations ◽  
Linear Cost

We consider an Mx/G/1 queueing system with N-policy and multiple vacations. As soon as the system empties, the server leaves for a vacation of random length V. When he returns, if the queue length is greater than or equal to a predetermined value N(threshold), the server immediately begins to serve the customers. If he finds less than N customers, he leaves for another vacation and so on until he finally finds at least N customers. We obtain the system size distribution and show that the system size decomposes into three random variables one of which is the system size of ordinary Mx/G/1 queue. The interpretation of the other random variables will be provided. We also derive the queue waiting time distribution and other performance measures. Finally we derive a condition under which the optimal stationary operating policy is achieved under a linear cost structure.

scholarly journals A Discrete-TimeGeo/G/1Retrial Queue withJVacations and Two Types of Breakdowns

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Feng Zhang ◽  
Zhifeng Zhu
Keyword(s):  
Performance Measures ◽  
Generating Functions ◽  
Queueing System ◽  
System Size ◽  
Stochastic Decomposition ◽  
Size Distributions ◽  
System State ◽  
State Distribution ◽  
Retrial Queueing ◽  
Orbit Size

This paper is concerned with a discrete-timeGeo/G/1retrial queueing model withJvacations and two types of breakdowns. If the orbit is empty, the server takes at mostJvacations repeatedly until at least one customer appears in the orbit upon returning from a vacation. It is assumed that the server is subject to two types of different breakdowns and is sent immediately for repair. We analyze the Markov chain underlying the considered queueing system and derive the system state distribution as well as the orbit size and the system size distributions in terms of their generating functions. Then, we obtain some performance measures through the generating functions. Moreover, the stochastic decomposition property and the corresponding continuous-time queueing system are investigated. Finally, some numerical examples are provided to illustrate the effect of vacations and breakdowns on several performance measures of the system.

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