scholarly journals Functional analysis method for the M/G/1 queueing model with single working vacation

2018 ◽  
Vol 16 (1) ◽  
pp. 767-791 ◽  
Author(s):  
Ehmet Kasim ◽  
Geni Gupur
Keyword(s):  
Adjoint Operator ◽  
Steady State Solution ◽  
Time Dependent ◽  
Queueing Model ◽  
Working Vacation ◽  
Geometric Multiplicity ◽  
Resolvent Set ◽  
Multiplicity One ◽  
Vacation Period ◽  
Time Dependent Solution

AbstractIn this paper, we study the asymptotic property of underlying operator corresponding to the M/G/1 queueing model with single working vacation, where both service times in a regular busy period and in a working vacation period are function. We obtain that all points on the imaginary axis except zero belong to the resolvent set of the operator and zero is an eigenvalue of both the operator and its adjoint operator with geometric multiplicity one. Therefore, we deduce that the time-dependent solution of the queueing model strongly converges to its steady-state solution. We also study the asymptotic behavior of the time-dependent queueing system’s indices for the model.

scholarly journals Semigroup Methods for the M/G/1 Queueing Model with Working Vacation and Vacation Interruption

2016 ◽  
Vol 8 (5) ◽  
pp. 56 ◽  
Author(s):  
Ehmet Kasim
Keyword(s):  
Semigroup Theory ◽  
Steady State Solution ◽  
Linear Operators ◽  
Time Dependent ◽  
Queueing Model ◽  
Working Vacation ◽  
Positive Time ◽  
Vacation Interruption ◽  
Vacation Period ◽  
Time Dependent Solution

By using the strong continuous semigroup theory of linear operators we prove that the M/G/1 queueing model with working vacation and vacation interruption has a unique positive time dependent solution which satisfies probability conditions. When the both service completion rate in a working vacation period and in a regular busy period are constant, by investigating the spectral properties of an operator corresponding to the model we obtain that the time-dependent solution of the model strongly converges to its steady-state solution.

scholarly journals Further Research on the M/G/1 Retrial Queueing Model with Server Breakdowns

2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
Ehmet Kasim ◽  
Geni Gupur
Keyword(s):  
Spectral Properties ◽  
Imaginary Axis ◽  
Time Dependent ◽  
Queueing Model ◽  
Asymptotically Stable ◽  
Resolvent Set ◽  
Retrial Queueing ◽  
Time Dependent Solution

We study spectral properties of the operator which corresponds to the M/G/1 retrial queueing model with server breakdowns and obtain that all points on the imaginary axis except zero belong to the resolvent set of the operator and 0 is not an eigenvalue of the operator. Our results show that the time-dependent solution of the model is probably strongly asymptotically stable.

scholarly journals Performance Analysis of Retrial Queueing Model with Working Vacation, Interruption, Waiting Server, Breakdown and Repair

2021 ◽  
Vol 13 (3) ◽  
pp. 833-844
Author(s):  
P. Gupta ◽  
N. Kumar
Keyword(s):  
Cost Optimization ◽  
Probability Generating Function ◽  
Function Technique ◽  
Service Rate ◽  
Queueing Model ◽  
Working Vacation ◽  
Retrial Queueing ◽  
Server Breakdown ◽  
Vacation Interruption ◽  
Vacation Period

In this present paper, an M/M/1 retrial queueing model with a waiting server subject to breakdown and repair under working vacation, vacation interruption is considered. Customers are served at a slow rate during the working vacation period, and the server may undergo breakdowns from a normal busy state. The customer has to wait in orbit for the service until the server gets repaired. Steady-state solutions are obtained using the probability generating function technique. Probabilities of different server states and some other performance measures of the system are developed.  The variation in mean orbit size, availability of the server, and server state probabilities are plotted for different values of breakdown parameter and repair rate with the help of MATLAB software. Finally, cost optimization of the system is also discussed, and the optimal value of the slow service rate for the model is obtained.

scholarly journals Semigroup Method on aMX/G/1 Queueing Model

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Alim Mijit
Keyword(s):  
Functional Analysis ◽  
Time Dependent ◽  
Queueing Model ◽  
Semigroup Method ◽  
Time Dependent Solution

By using the Hille-Yosida theorem, Phillips theorem, and Fattorini theorem in functional analysis we prove that theMX/G/1 queueing model with vacation times has a unique nonnegative time-dependent solution.

scholarly journals On Eigenvalues of the Generator of aC0-Semigroup Appearing in Queueing Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Geni Gupur
Keyword(s):  
Queueing Theory ◽  
Infinite System ◽  
Integral Boundary Conditions ◽  
Steady State Solution ◽  
Queueing Model ◽  
Integral Boundary ◽  
Growth Bound ◽  
Essential Growth Bound ◽  
Essential Growth ◽  
Time Dependent Solution

We describe the point spectrum of the generator of aC0-semigroup associated with the M/M/1 queueing model that is governed by an infinite system of partial differential equations with integral boundary conditions. Our results imply that the essential growth bound of theC0-semigroup is 0 and, therefore, that the semigroup is not quasi-compact. Moreover, our result also shows that it is impossible that the time-dependent solution of the M/M/1 queueing model exponentially converges to its steady-state solution.

scholarly journals A Semigroup Approach to the System with Primary and Secondary Failures

2010 ◽  
Vol 2010 ◽  
pp. 1-33 ◽  
Author(s):  
Abdukerim Haji
Keyword(s):  
Functional Analysis ◽  
Steady State ◽  
Asymptotic Stability ◽  
Positive Solution ◽  
Steady State Solution ◽  
Linear Operators ◽  
Time Dependent ◽  
Well Posedness ◽  
Semigroup Approach ◽  
Time Dependent Solution

We investigate the solution of a repairable parallel system with primary as well as secondary failures. By using the method of functional analysis, especially, the spectral theory of linear operators and the theory ofC0-semigroups, we prove well-posedness of the system and the existence of positive solution of the system. And then we show that the time-dependent solution strongly converges to steady-state solution, thus we obtain the asymptotic stability of the time-dependent solution.

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