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October  2010, 6(4): 911-927. doi: 10.3934/jimo.2010.6.911

## Analysis of renewal input bulk arrival queue with single working vacation and partial batch rejection

 1 School of Computer Application, KIIT University, Bhubaneswar - 751 024, India 2 Department of Applied Mathematics, Andhra University, Visakhapatnam - 530 003, India

Received  December 2009 Revised  July 2010 Published  September 2010

This paper analyzes a finite buffer bulk arrival queueing system with a single working vacation and partial batch rejection in which the inter-arrival and service times are, respectively, arbitrarily and exponentially distributed. Using the supplementary variable and the imbedded Markov chain techniques, we obtain the system length distributions at pre-arrival and arbitrary epochs. We also present Laplace-Stiltjes transform of the actual waiting time distribution in the system. Finally, several performance measures and a variety of numerical results in the form of tables and graphs are discussed.
Citation: Veena Goswami, Pikkala Vijaya Laxmi. Analysis of renewal input bulk arrival queue with single working vacation and partial batch rejection. Journal of Industrial & Management Optimization, 2010, 6 (4) : 911-927. doi: 10.3934/jimo.2010.6.911
##### References:
 [1] Y. Baba, Analysis of $GI$/$M$/$1$ queue with multiple working vacations,, Oper. Res. Lett., 33 (2005), 201.  doi: 10.1016/j.orl.2004.05.006.  Google Scholar [2] A. D. Banik, U. C. Gupta and S. S. Pathak, On the $GI$/$M$/$1$/$N$ queue with multiple working vacations - Anaytic analysis and computation,, Appl. Math. Model., 31 (2007), 1701.  doi: 10.1016/j.apm.2006.05.010.  Google Scholar [3] P. J. Burke, Delays in single-server queues with batch input,, Oper. Res., 23 (1975), 830.  doi: 10.1287/opre.23.4.830.  Google Scholar [4] 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,, Performance Evaluaton, 68 (2009), 356.  doi: 10.1016/j.peva.2009.01.005.  Google Scholar [5] K. C. Chae, S. M. Lee and H. W. Lee, On stochastic decomposition in the $GI$/$M$/$1$ queue with single exponential vacation,, Oper. Res. Lett., 34 (2006), 706.  doi: 10.1016/j.orl.2005.11.006.  Google Scholar [6] B. T. Doshi, Queueing systems with vacations - A survey,, Queueing Syst., 1 (1986), 29.  doi: 10.1007/BF01149327.  Google Scholar [7] B. T. Doshi, Single server queues with vacations,, Stochastic Analysis of Computer and Communication Systems, (1990), 217.   Google Scholar [8] F. Karaesmen and S. M. Gupta, The finite capacity $GI$/$M$/$1$ with server vacations,, Journal of the Operational Research Society, 47 (1996), 817.   Google Scholar [9] G. Latouche and V. Ramaswami, "Introduction to Matrix Analytic Methods in Stochastic Modelling,", SIAM $&$ ASA, (1999).   Google Scholar [10] J. H. Li, N. S. Tian and Z. Y. Ma, Performance analysis of $GI$/$M$/$1$ queue with working vacations and vacation interruption,, Appl. Math. Model., 32 (2008), 2715.  doi: 10.1016/j.apm.2007.09.017.  Google Scholar [11] W. Liu, X. Xu and N. Tian, Some results on the M/M/1 queue with working vacations,, Oper. Res. Lett., 35 (2007), 595.  doi: 10.1016/j.orl.2006.12.007.  Google Scholar [12] K. Sikdar, U. C. Gupta and R. K. Sharma, The analysis of a finite-buffer general input queue with batch arrival and exponential multiple vacations,, Int. J. Oper. Res., 3 (2008), 219.  doi: 10.1504/IJOR.2008.016162.  Google Scholar [13] L. D. Servi and S. G. Finn, $M$/$M$/$1$ queue with working vacations ($M$/$M$/$1$/$WV$),, Performance Evaluaton, 50 (2002), 41.  doi: 10.1016/S0166-5316(02)00057-3.  Google Scholar [14] H. Takagi, "Queueing Analysis - A Foundation of Performance Evaluation : Volume 2, Finite Systems,", North Holland, (1993).   Google Scholar [15] N. Tian and Z. G. Zhang, "Vacation Queueing Models: Theory and Applications,", Springer-Verlag, (2006).   Google Scholar [16] P. Vijaya Laxmi and U. C. Gupta, A unified approach to analyze the $GI^X$/$M$/$1$/$N$ and $GI$/$E_k$/$1$/$N$ queues,, Proceedings of the International Conference on Stochastic Processes and Their Applications (eds. A. Vijayakumar and M. Sreenivasan), (1998), 206.   Google Scholar [17] D. Wu and H. Takagi, $M$/$G$/$1$ queue with multiple working vacations,, Performance Evaluaton, 63 (2006), 654.  doi: 10.1016/j.peva.2005.05.005.  Google Scholar [18] M. M. Yu, Y. H. Tang and Y. H. Fu, Steady state analysis and computation of the $GI^{[x]}$/$M^b$/$1$/$L$ queue with multiple working vacations and partial batch rejection,, Computers & Industrial Engineering, 56 (2009), 1243.  doi: 10.1016/j.cie.2008.07.013.  Google Scholar

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##### References:
 [1] Y. Baba, Analysis of $GI$/$M$/$1$ queue with multiple working vacations,, Oper. Res. Lett., 33 (2005), 201.  doi: 10.1016/j.orl.2004.05.006.  Google Scholar [2] A. D. Banik, U. C. Gupta and S. S. Pathak, On the $GI$/$M$/$1$/$N$ queue with multiple working vacations - Anaytic analysis and computation,, Appl. Math. Model., 31 (2007), 1701.  doi: 10.1016/j.apm.2006.05.010.  Google Scholar [3] P. J. Burke, Delays in single-server queues with batch input,, Oper. Res., 23 (1975), 830.  doi: 10.1287/opre.23.4.830.  Google Scholar [4] 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,, Performance Evaluaton, 68 (2009), 356.  doi: 10.1016/j.peva.2009.01.005.  Google Scholar [5] K. C. Chae, S. M. Lee and H. W. Lee, On stochastic decomposition in the $GI$/$M$/$1$ queue with single exponential vacation,, Oper. Res. Lett., 34 (2006), 706.  doi: 10.1016/j.orl.2005.11.006.  Google Scholar [6] B. T. Doshi, Queueing systems with vacations - A survey,, Queueing Syst., 1 (1986), 29.  doi: 10.1007/BF01149327.  Google Scholar [7] B. T. Doshi, Single server queues with vacations,, Stochastic Analysis of Computer and Communication Systems, (1990), 217.   Google Scholar [8] F. Karaesmen and S. M. Gupta, The finite capacity $GI$/$M$/$1$ with server vacations,, Journal of the Operational Research Society, 47 (1996), 817.   Google Scholar [9] G. Latouche and V. Ramaswami, "Introduction to Matrix Analytic Methods in Stochastic Modelling,", SIAM $&$ ASA, (1999).   Google Scholar [10] J. H. Li, N. S. Tian and Z. Y. Ma, Performance analysis of $GI$/$M$/$1$ queue with working vacations and vacation interruption,, Appl. Math. Model., 32 (2008), 2715.  doi: 10.1016/j.apm.2007.09.017.  Google Scholar [11] W. Liu, X. Xu and N. Tian, Some results on the M/M/1 queue with working vacations,, Oper. Res. Lett., 35 (2007), 595.  doi: 10.1016/j.orl.2006.12.007.  Google Scholar [12] K. Sikdar, U. C. Gupta and R. K. Sharma, The analysis of a finite-buffer general input queue with batch arrival and exponential multiple vacations,, Int. J. Oper. Res., 3 (2008), 219.  doi: 10.1504/IJOR.2008.016162.  Google Scholar [13] L. D. Servi and S. G. Finn, $M$/$M$/$1$ queue with working vacations ($M$/$M$/$1$/$WV$),, Performance Evaluaton, 50 (2002), 41.  doi: 10.1016/S0166-5316(02)00057-3.  Google Scholar [14] H. Takagi, "Queueing Analysis - A Foundation of Performance Evaluation : Volume 2, Finite Systems,", North Holland, (1993).   Google Scholar [15] N. Tian and Z. G. Zhang, "Vacation Queueing Models: Theory and Applications,", Springer-Verlag, (2006).   Google Scholar [16] P. Vijaya Laxmi and U. C. Gupta, A unified approach to analyze the $GI^X$/$M$/$1$/$N$ and $GI$/$E_k$/$1$/$N$ queues,, Proceedings of the International Conference on Stochastic Processes and Their Applications (eds. A. Vijayakumar and M. Sreenivasan), (1998), 206.   Google Scholar [17] D. Wu and H. Takagi, $M$/$G$/$1$ queue with multiple working vacations,, Performance Evaluaton, 63 (2006), 654.  doi: 10.1016/j.peva.2005.05.005.  Google Scholar [18] M. M. Yu, Y. H. Tang and Y. H. Fu, Steady state analysis and computation of the $GI^{[x]}$/$M^b$/$1$/$L$ queue with multiple working vacations and partial batch rejection,, Computers & Industrial Engineering, 56 (2009), 1243.  doi: 10.1016/j.cie.2008.07.013.  Google Scholar
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