Computational analysis of multi-server discrete-time queueing system with balking, reneging and synchronous vacations

Goswami, V.; Mund, G. B.

doi:10.1051/ro/2016025

Goswami, V.¹ ; Mund, G. B.¹

¹ School of Computer Application, KIIT University, 751024 Bhubaneswar, India

RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 2, pp. 343-358.

Résumé

This paper proposes a discrete-time multi-server queue with multiple synchronous vacations under balking and reneging. Arriving customers decide whether to join the system or balk on the basis of some state-dependent joining/balking probabilities, and renege according to a geometric distribution when servers are busy. The servers take a vacation together if there are no customers in the system at a service completion instant. When the servers are on vacation, an arriving customer activates an impatience timer which is geometrically distributed. The inter-arrival times, service times and vacation times are assumed to be independent and geometrically distributed. We obtain closed-form expressions and develop a computational algorithm for calculating the steady-state probabilities. Specifically, we establish the application of the proposed framework in analyzing a multi-server queueing system with synchronous vacation under balking and reneging. Applications of such models can be found in a wide variety of real-time systems including call centers, computer and communication systems, cloud computing, quality control and maintenance in industrial establishments. We develop a cost model to determine the optimal service rate. Various performance measures and numerical examples are sketched out to demonstrate the impact of the proposed method. Some special cases of the model have also been discussed. Finally, we show that in the limiting case the results converge to the corresponding continuous-time counterparts.

Reçu le : 2015-07-22
Accepté le : 2016-03-13

MR Zbl

DOI : 10.1051/ro/2016025

Classification : 60K25, 90B22
Mots-clés : Discrete-time, multi-server, balking, reneging, synchronous vacations, queueing, finite-buffer

Affiliations des auteurs :

Goswami, V. ¹ ; Mund, G. B. ¹

¹ School of Computer Application, KIIT University, 751024 Bhubaneswar, India

@article{RO_2017__51_2_343_0,
     author = {Goswami, V. and Mund, G. B.},
     title = {Computational analysis of multi-server discrete-time queueing system with balking, reneging and synchronous vacations},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {343--358},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {2},
     year = {2017},
     doi = {10.1051/ro/2016025},
     mrnumber = {3619708},
     zbl = {1367.60111},
     language = {en},
     url = {https://numdam.org/articles/10.1051/ro/2016025/}
}

TY  - JOUR
AU  - Goswami, V.
AU  - Mund, G. B.
TI  - Computational analysis of multi-server discrete-time queueing system with balking, reneging and synchronous vacations
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2017
SP  - 343
EP  - 358
VL  - 51
IS  - 2
PB  - EDP-Sciences
UR  - https://numdam.org/articles/10.1051/ro/2016025/
DO  - 10.1051/ro/2016025
LA  - en
ID  - RO_2017__51_2_343_0
ER  -

%0 Journal Article
%A Goswami, V.
%A Mund, G. B.
%T Computational analysis of multi-server discrete-time queueing system with balking, reneging and synchronous vacations
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2017
%P 343-358
%V 51
%N 2
%I EDP-Sciences
%U https://numdam.org/articles/10.1051/ro/2016025/
%R 10.1051/ro/2016025
%G en
%F RO_2017__51_2_343_0

Goswami, V.; Mund, G. B. Computational analysis of multi-server discrete-time queueing system with balking, reneging and synchronous vacations. RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 2, pp. 343-358. doi : 10.1051/ro/2016025. https://numdam.org/articles/10.1051/ro/2016025/

Bibliographie
Cité par

M. Abou-El-Ata and A. Hariri, The $M / M / c / N$ queue with balking and reneging. Comput. Oper. Res. 19 (1992) 713–716. | DOI | Zbl

R.O. Al-Seedy, A. El-Sherbiny, S. El-Shehawy and S. Ammar, Transient solution of the $M / M / c$ queue with balking and reneging. Comput. Math. Appl. 57 (2009) 1280–1285. | DOI | MR | Zbl

E. Altman and U. Yechiali, Analysis of customers impatience in queues with server vacations. Queueing Systems 52 (2006) 261–279. | DOI | MR | Zbl

E. Altman and U. Yechiali, Infinite-server queues with system’s additional tasks and impatient customers. Probab. Eng. Inform. Sci. 22 (2008) 477–493. | DOI | MR | Zbl

J.R Artalejo and O. Hernández-Lerma, Performance analysis and optimal control of the $G e o / G e o / c$ queue. Perform. Eval. 52 (2003) 15–39. | DOI

H. Bruneel and T. Maertens, A discrete-time queue with customers with geometric deadlines. Perform. Eval. 85 (2015) 52–70. | DOI

S.R. Chakravarthy, Analysis of a multi-server queue with markovian arrivals and synchronous phase type vacations. Asia-Pacific J. Oper. Res. 26 (2009) 85–113. | DOI | MR | Zbl

W. Chan and D. Maa, $G I / G e o m / N$ queue in discrete-time. INFOR 16 (1978) 232–252. | MR | Zbl

M. Chaudhry and U. Gupta, Numerical evaluation of state probabilities at different epochs in multiserver $G I / G e o m / m$ queue. Proc. of Advances on Methodological and Applied Aspects of Probability and Statistics (2001) 31–46. | MR | Zbl

M.L. Chaudhry, U.C. Gupta and V. Goswami, On discrete-time multiserver queues with finite buffer: $G I / G e o m / m / N$ . Comput. Oper. Res. 31 (2004) 2137–2150. | DOI | MR | Zbl

A. Choudhury and P. Medhi, Balking and reneging in multiserver markovian queuing system. Int. J. Math. Oper. Res. 3 (2011) 377–394. | DOI | MR | Zbl

A. Economou, A. Gómez-Corral and S. Kanta, Optimal balking strategies in single-server queues with general service and vacation times. Perform. Eval. 68 (2011) 967–982. | DOI

P. Gao, S. Wittevrongel and H. Bruneel, Discrete-time multiserver queues with geometric service times. Comput. Oper. Res. 31 (2004) 81–99. | DOI | Zbl

V. Goswami, Analysis of discrete-time multi-server queue with balking. Int. J. Manage. Sci. Eng. Manage. 9 (2014) 21–32.

V. Goswami, A discrete-time queue with balking, reneging, and working vacations. Int. J. Stochastic Anal. 2014 (2014). | MR | Zbl

V Goswami, Study of customers’ impatience in a $G I / M / 1 / N$ queue with working vacations. Int. J. Manage. Sci. Eng. Manage. 10 (2015) 144–154.

D. Guha, A.D. Banik, V. Goswami and S. Ghosh, Equilibrium balking strategy in an unobservable $G I / M / c$ queue with customers’ impatience. In Distributed Computing and Internet Technology. Springer (2014) 188–199.

J.-C. Ke and C.-H. Wu, Multi-server machine repair model with standbys and synchronous multiple vacation. Comput. Ind. Eng. 62 (2012) 296–305. | DOI

G. Koole and A. Mandelbaum, Queueing models of call centers: An introduction. Ann. Oper. Res. 113 (2002) 41–59. | DOI | MR | Zbl

R. Kumar, Economic analysis of an $M / M / c / N$ queuing model with balking, reneging and retention of reneged customers. OPSEARCH 50 (2013) 383–403. | DOI | MR | Zbl

M. Lozano and P. Moreno, A discrete time single-server queue with balking: economic applications. Appl. Econ. 40 (2008) 735–748. | DOI

R.L Rardin, Optimization in operations research. Vol. 166. Prentice Hall Upper Saddle River, NJ (1998).

N. Selvaraju and C. Goswami, Impatient customers in an $M / M / 1$ queue with single and multiple working vacations. Comput. Ind. Eng. 65 (2013) 207–215. | DOI

A. Shawky and M. El-Paoumy, The interarrival hyperexponential queues: $H_{k} / M / C / N$ with balking and reneging. Stochastics: Int. J. Probab. Stochastic Processs 69 (2000) 67–76. | MR | Zbl

P. Vijaya Laxmi, V. Goswami and K. Jyothsna, Discrete-time renewal input queue with balking and multiple working vacations. Int. J. Manage. Sci. Eng. Manage. 10 (2015) 224–236.

K.-H. Wang, W.-L. Chen and D.-Y. Yang, Optimal management of the machine repair problem with working vacation: Newtons method. J. Comput. Appl. Math. 233 (2009) 449–458. | DOI | MR | Zbl

C.-H. Wu and J.-C. Ke, Computational algorithm and parameter optimization for a multi-server system with unreliable servers and impatient customers. J. Comput. Appl. Math. 235 (2010) 547–562. | DOI | MR | Zbl

D.Q. Yue and Y.-P. Sun, Waiting time of $M / M / c / N$ queuing system with balking, reneging, and multiple synchronous vacations of partial servers. Systems Engineering-Theory Practice 28 (2008) 89–97. | DOI

D. Yue and W. Yue, Analysis of an $M / M / c / N$ queueing system with balking, reneging, and synchronous vacations. In Advances in Queueing Theory and Network Applications. Springer (2009) 165–180. | MR

D. Yue, Y. Zhang and W. Yue, Optimal performance analysis of an $M / M / 1 / N$ queue system with balking, reneging and server vacation. Int. J. Pure Appl. Math. 28 (2006) 101–115. | MR | Zbl

D. Yue, W. Yue and G. Xu, Analysis of customers’impatience in an $M / M / 1$ queue with working vacations. J. Ind. Manage. Optim. 8 (2012) 895–908. | DOI | MR | Zbl

Z. Zhang, Call centres with balking and abandonment: from queueing to queueing network models. Ph.D. thesis, University of Saskatchewan Saskatoon (2010). | MR

Cité par Sources :