Analysis of GI/M(n)/1/N Queue with Single Working Vacation and Vacation Interruption
This paper presents a finite buffer renewal input single working vacation and vacation interruption queue with state dependent services and state dependent vacations, which has a wide range of applications in several areas including manufacturing, wireless communication systems. Service times during busy period, vacation period and vacation times are exponentially distributed and are state dependent. As a result of the finite waiting space, state dependent services and state dependent vacation policies, the analysis of these queueing models needs special attention. We provide a recursive method using the supplementary variable technique to compute the stationary queue length distributions at pre-arrival and arbitrary epochs. An efficient computational algorithm of the model is presented which is fast and accurate and easy to implement. Various performance measures have been discussed. Finally, some special cases and numerical results have been depicted in the form of tables and graphs.
<p>[1] B. T. Doshi, Queueing Systems with Vacations - A Survey, Queueing
Systems, Vol. 1 , pp. 29–66, 1986.
[2] N. Tian, N and Z. G. Zhang, Vacation Queueing Models: Theory and
Applications, Springer-Verlag, New York, 2006.
[3] L. D. Servi and S. G. Finn, The M/M/1 Queues with Working Vacations
(M/M/1/WV ), Perf. Eval., Vol. 50, pp. 41–52, 2002.
[4] W. Y. Liu, X. L. Xu, N. Tian, Stochastic decompositions in the M/M/1
queue with working vacations, Oper. Res. Lett., Vol. 35, pp. 596-600,
2007.
[5] N.Tian, X. Xhao and K. H. Wang, The M/M/1 Queue with Single
Working Vacation, Int. J. Inform. Manage. Sci., Vol. 19, pp. 621–634,
2008.
[6] Y. Baba, Analysis of a GI/M/1 Queue with Multiple Working Vacations,
Oper. Res. Lett., Vol. 33, 201–209, 2005.
[7] D. Wu and H. Takagi, M/G/1 queue with multiple working vacations,
Perform. Eval., Vol. 63, pp. 654–681, 2006.
[8] M. Zhang and Z. Hou, Performance analysis of M/G/1 queue with
working vacations and vacation interruption, Journal of Computational
and Applied Mathematics, Vol. 234, pp. 2977–2985, 2010.
[9] A. D. Banik, Analysis of Single Working Vacation in GI/M/1/N and
GI/M/1 Queueing Systems, Int. J. Oper. Res., Vol.7, 314–333, 2010.
[10] A. D. Banik, U. C. Gupta and S. S. Pathak, On the GI/M/1/N Queue
with Multiple Working Vacations - Analytic Analysis and Computation,
Appl. Math. Modell., Vol. 31, 1701–1710, 2007.
[11] J. Li and N. Tian, Performance Analysis of a GI/M/1 Queue with
Single Working Vacation, Appl. Math. Comput. , Vol. 217, pp. 4960–
4971, 2011.
[12] K. C. Chae, D. E. Lim and W. S. Yang, The GI/M/1 queue and the
GI/Geo/1 queue both with single working vacation, Perform. Eval.,
Vol. 66, pp. 356–367, 2009.
[13] Z. Zhang and X. Xu, Analysis for the M/M/1 Queue with Multiple
Working Vacations and N-Policy, Int. J. Inform. Manage. Sci., Vol. 19,
pp. 495–506, 2008.
[14] J. Li and N. Tian, The M/M/1 Queue with Working Vacations and
Vacation Interuptions, J. Syst. Sci. Syst. Eng., Vol. 16, pp. 121–127,
2007.
[15] J. Li, N. Tian and Z. Ma, Performence Analysis of GI/M/1 Queue with
Working Vacation and Vacation Interruption, Appl. Math. Modell., Vol.
32, pp. 2715–2730, 2008.
[16] G. Zhao, X. Du and N. Tian, GI/M/1 Queue with Setup Period and
Working Vacation and Vacation Interruption, Int. J. Inform. Manage.
Sci., Vol. 20, pp. 351–363, 2009.
[17] S. Guo and Z. Liu, AnM/G/1 Queue with Single Working Vacation and
Vacation Interruption under Bernoulli Schedule, Appl. Math. Modell.,
in press, 2012.
[18] Q. Wang and J. M. Peha, State Dependent Pricing and its Economic
Implications, Telecommu. Syst., Vol. 18, pp. 315–329, 2001.
[19] M. Kijima and N. Makimoto, A Unified Approach to GI/M(n)/1/K
and M(n)/G/1/K Queues via Finite Quasi-Birth-Death Processes,
Commun. Stat. Stoch. Models, Vol. 8, pp. 269–288, 1992.
[20] P. Yang, Unified Algorithm for Computing the Stationary Queue
Length Distribution in M(k)/G(n)/1/N and GI/M(k)/1/N Queues,
Queueing Systems, Vol. 17, pp. 383–401, 1994.
[21] X. Chao and R. Rahman, Anlysis and Computational Algorithem for
Queues with State Dependent Vacations I: G/M(n)/1/K, J. Syst.
Sci. Complex., Vol. 19, pp. 36–53, 2006.
[22] X. Chao and R. Rahman, Anlysis and Computational Algorithem for
Queues with State Dependent Vacations II: M(n)/G/1/K, J. Syst.
Sci. Complex., Vol. 19, pp. 191–210, 2006.</p>
<p>[1] B. T. Doshi, Queueing Systems with Vacations - A Survey, Queueing
Systems, Vol. 1 , pp. 29–66, 1986.
[2] N. Tian, N and Z. G. Zhang, Vacation Queueing Models: Theory and
Applications, Springer-Verlag, New York, 2006.
[3] L. D. Servi and S. G. Finn, The M/M/1 Queues with Working Vacations
(M/M/1/WV ), Perf. Eval., Vol. 50, pp. 41–52, 2002.
[4] W. Y. Liu, X. L. Xu, N. Tian, Stochastic decompositions in the M/M/1
queue with working vacations, Oper. Res. Lett., Vol. 35, pp. 596-600,
2007.
[5] N.Tian, X. Xhao and K. H. Wang, The M/M/1 Queue with Single
Working Vacation, Int. J. Inform. Manage. Sci., Vol. 19, pp. 621–634,
2008.
[6] Y. Baba, Analysis of a GI/M/1 Queue with Multiple Working Vacations,
Oper. Res. Lett., Vol. 33, 201–209, 2005.
[7] D. Wu and H. Takagi, M/G/1 queue with multiple working vacations,
Perform. Eval., Vol. 63, pp. 654–681, 2006.
[8] M. Zhang and Z. Hou, Performance analysis of M/G/1 queue with
working vacations and vacation interruption, Journal of Computational
and Applied Mathematics, Vol. 234, pp. 2977–2985, 2010.
[9] A. D. Banik, Analysis of Single Working Vacation in GI/M/1/N and
GI/M/1 Queueing Systems, Int. J. Oper. Res., Vol.7, 314–333, 2010.
[10] A. D. Banik, U. C. Gupta and S. S. Pathak, On the GI/M/1/N Queue
with Multiple Working Vacations - Analytic Analysis and Computation,
Appl. Math. Modell., Vol. 31, 1701–1710, 2007.
[11] J. Li and N. Tian, Performance Analysis of a GI/M/1 Queue with
Single Working Vacation, Appl. Math. Comput. , Vol. 217, pp. 4960–
4971, 2011.
[12] K. C. Chae, D. E. Lim and W. S. Yang, The GI/M/1 queue and the
GI/Geo/1 queue both with single working vacation, Perform. Eval.,
Vol. 66, pp. 356–367, 2009.
[13] Z. Zhang and X. Xu, Analysis for the M/M/1 Queue with Multiple
Working Vacations and N-Policy, Int. J. Inform. Manage. Sci., Vol. 19,
pp. 495–506, 2008.
[14] J. Li and N. Tian, The M/M/1 Queue with Working Vacations and
Vacation Interuptions, J. Syst. Sci. Syst. Eng., Vol. 16, pp. 121–127,
2007.
[15] J. Li, N. Tian and Z. Ma, Performence Analysis of GI/M/1 Queue with
Working Vacation and Vacation Interruption, Appl. Math. Modell., Vol.
32, pp. 2715–2730, 2008.
[16] G. Zhao, X. Du and N. Tian, GI/M/1 Queue with Setup Period and
Working Vacation and Vacation Interruption, Int. J. Inform. Manage.
Sci., Vol. 20, pp. 351–363, 2009.
[17] S. Guo and Z. Liu, AnM/G/1 Queue with Single Working Vacation and
Vacation Interruption under Bernoulli Schedule, Appl. Math. Modell.,
in press, 2012.
[18] Q. Wang and J. M. Peha, State Dependent Pricing and its Economic
Implications, Telecommu. Syst., Vol. 18, pp. 315–329, 2001.
[19] M. Kijima and N. Makimoto, A Unified Approach to GI/M(n)/1/K
and M(n)/G/1/K Queues via Finite Quasi-Birth-Death Processes,
Commun. Stat. Stoch. Models, Vol. 8, pp. 269–288, 1992.
[20] P. Yang, Unified Algorithm for Computing the Stationary Queue
Length Distribution in M(k)/G(n)/1/N and GI/M(k)/1/N Queues,
Queueing Systems, Vol. 17, pp. 383–401, 1994.
[21] X. Chao and R. Rahman, Anlysis and Computational Algorithem for
Queues with State Dependent Vacations I: G/M(n)/1/K, J. Syst.
Sci. Complex., Vol. 19, pp. 36–53, 2006.
[22] X. Chao and R. Rahman, Anlysis and Computational Algorithem for
Queues with State Dependent Vacations II: M(n)/G/1/K, J. Syst.
Sci. Complex., Vol. 19, pp. 191–210, 2006.</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:65373", author = "P. Vijaya Laxmi and V. Goswami and V. Suchitra", title = "Analysis of GI/M(n)/1/N Queue with Single Working Vacation and Vacation Interruption", abstract = "This paper presents a finite buffer renewal input single working vacation and vacation interruption queue with state dependent services and state dependent vacations, which has a wide range of applications in several areas including manufacturing, wireless communication systems. Service times during busy period, vacation period and vacation times are exponentially distributed and are state dependent. As a result of the finite waiting space, state dependent services and state dependent vacation policies, the analysis of these queueing models needs special attention. We provide a recursive method using the supplementary variable technique to compute the stationary queue length distributions at pre-arrival and arbitrary epochs. An efficient computational algorithm of the model is presented which is fast and accurate and easy to implement. Various performance measures have been discussed. Finally, some special cases and numerical results have been depicted in the form of tables and graphs.
", keywords = "State Dependent Service, Vacation Interruption, Supplementary
Variable, Single Working Vacation, Blocking Probability.", volume = "7", number = "4", pages = "720-7", }
