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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

Veena Goswami 1, and  Pikkala Vijaya Laxmi 2,

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.
Keywords: Finite buffer, Working vacation., Bulk arrival, Waiting time, Queue.
Mathematics Subject Classification: Primary: 60K25; Secondary: 90B22, 68M2.
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), 201-209. 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), 1701-1710. 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), 830-833. 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), 356-367. 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), 706-712. doi: 10.1016/j.orl.2005.11.006.  Google Scholar

[6]

B. T. Doshi, Queueing systems with vacations - A survey, Queueing Syst., 1 (1986), 29-66. doi: 10.1007/BF01149327.  Google Scholar

[7]

B. T. Doshi, Single server queues with vacations, Stochastic Analysis of Computer and Communication Systems, H. Takagi (Editor), Elsevier Science Publishers, 1990, 217-265.  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), 817-828. Google Scholar

[9]

G. Latouche and V. Ramaswami, "Introduction to Matrix Analytic Methods in Stochastic Modelling," SIAM $&$ ASA, Philadelphia, 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), 2715-2730. 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), 595-600. 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), 219-234. 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), 41-52. 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, New York, 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), Narosa Publishers (1998), 206-214. Google Scholar

[17]

D. Wu and H. Takagi, $M$/$G$/$1$ queue with multiple working vacations, Performance Evaluaton, 63 (2006), 654-681. 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), 1243-1253. doi: 10.1016/j.cie.2008.07.013.  Google Scholar

show all references

References:
[1]

Y. Baba, Analysis of $GI$/$M$/$1$ queue with multiple working vacations, Oper. Res. Lett., 33 (2005), 201-209. 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), 1701-1710. 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), 830-833. 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), 356-367. 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), 706-712. doi: 10.1016/j.orl.2005.11.006.  Google Scholar

[6]

B. T. Doshi, Queueing systems with vacations - A survey, Queueing Syst., 1 (1986), 29-66. doi: 10.1007/BF01149327.  Google Scholar

[7]

B. T. Doshi, Single server queues with vacations, Stochastic Analysis of Computer and Communication Systems, H. Takagi (Editor), Elsevier Science Publishers, 1990, 217-265.  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), 817-828. Google Scholar

[9]

G. Latouche and V. Ramaswami, "Introduction to Matrix Analytic Methods in Stochastic Modelling," SIAM $&$ ASA, Philadelphia, 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), 2715-2730. 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), 595-600. 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), 219-234. 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), 41-52. 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, New York, 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), Narosa Publishers (1998), 206-214. Google Scholar

[17]

D. Wu and H. Takagi, $M$/$G$/$1$ queue with multiple working vacations, Performance Evaluaton, 63 (2006), 654-681. 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), 1243-1253. doi: 10.1016/j.cie.2008.07.013.  Google Scholar

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