# American Institute of Mathematical Sciences

October  2010, 6(4): 929-944. doi: 10.3934/jimo.2010.6.929

## Analysis of a finite buffer general input queue with Markovian service process and accessible and non-accessible batch service

 1 Department of Applied Mathematics, Andhra University, Visakhapatnam - 530 003, India 2 Department of Mathematics, Andhra University, Visakhapatnam - 530 003, India

Received  January 2010 Revised  August 2010 Published  September 2010

Queues with Markovian service process ($MSP$) are mainly useful in modeling and performance analysis of telecommunication networks based on asynchronous transfer mode (ATM) environment. This paper analyzes a finite buffer single server batch service ($a, b)$ queue with general input and Markovian service process ($MSP$). The server accesses new arrivals even after service has started on any batch of initial number $a$. This operation continues till the service time of the ongoing batch is completed or the maximum accessible capacity $d ~(a\le d < b)$ of the batch being served is attained whichever occurs first. Using the embedded Markov chain technique and the supplementary variable technique we obtain the steady state queue length distributions at pre-arrival and arbitrary epochs. The primary focus is on the various performance measures of the steady state distribution of the batch service, special cases and also on numerical illustrations.
Citation: Pikkala Vijaya Laxmi, Obsie Mussa Yesuf. Analysis of a finite buffer general input queue with Markovian service process and accessible and non-accessible batch service. Journal of Industrial & Management Optimization, 2010, 6 (4) : 929-944. doi: 10.3934/jimo.2010.6.929
##### References:
 [1] F. J. Albores-Velasco and F. S. Tajonar-Sanabria, Anlysis of the $GI$/$MSP$/$c$/$r$ queueing system,, Information Processes, 4 (2004), 46. Google Scholar [2] 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 [3] A. D. Banik, U. C. Gupta and M. L. Chaudhry, Finite-buffer bulk service queue under Markovian service process: $GI$/$MSP^(a,b)$/$1$/$N$,, Stoch. Anal. Appl., 27 (2009), 500. doi: 10.1080/07362990902844157. Google Scholar [4] P. P. Bocharov, Stationary distribution of a finite queue with recurrent flow and Markovian service,, Automat. Remote Control, 57 (1996), 1274. Google Scholar [5] S. Chakravarthy, A finite capacity $GI$/$PH$/$1$ queue with group services,, Naval Res. Logist., 39 (1992), 345. doi: 10.1002/1520-6750(199204)39:3<345::AID-NAV3220390305>3.0.CO;2-V. Google Scholar [6] S. Chakravarthy, Analysis of a finite $MAP$/$G$/$1$ queue with group services,, Queueing Systems, 13 (1993), 385. doi: 10.1007/BF01149262. Google Scholar [7] M. L. Chaudhry and J. G. C. Templeton, "A First Course in Bulk Queues,", John Wiley, (1983). Google Scholar [8] J. H. Dshalalow, "Frontiers in Queueing: Models and Applications in Sciences and Engineering,", CRC press, (1997). Google Scholar [9] H. Gold and P. Tran-Gia, Performance analysis of a batch service queue arising out of manufacturing and system modelling,, Queueing Systems, 14 (1993), 413. doi: 10.1007/BF01158876. Google Scholar [10] V. Goswami, J. R. Mohanty and S. K. Samanta, Discrete-time bulk-service queues with accessible and non-accessible batches,, Appl. Math. Comput., 182 (2006), 898. doi: 10.1016/j.amc.2006.04.047. Google Scholar [11] V. Goswami and K. Sikdar, Discrete-time batch service $GI$/$Geo^(a,b)$/$1$/$N$ queue with accessible and non-accessible batches,, Internaional Journal of Mathematics in Operational Research, 2 (2010), 233. doi: 10.1504/IJMOR.2010.030818. Google Scholar [12] W. K. Grassmann, M. I. Taksar and D. P. Heyman, Regenerative analysis and steady state distributions for Markov chains,, Oper. Res., 33 (1985), 1107. doi: 10.1287/opre.33.5.1107. Google Scholar [13] D. Gross, J. F. Shortle, J. M. Thompson and C. M. Harris, "Fundamentals of Queueing Theory," 4th Edition,, John Wiley & Sons, (2008). Google Scholar [14] U. C. Gupta and A. D. Banik, Complete analysis of finite and infinite buffer $GI$/$MSP$/$1$ queue - A computational approach,, Oper. Res. Lett., 35 (2006), 273. doi: 10.1016/j.orl.2006.02.003. Google Scholar [15] U. C. Gupta and P. V. Laxmi, Analysis of $MAP$/$G^(a,b)$/$1$/$N$ queue,, Queueing Systems, 38 (2001), 109. doi: 10.1023/A:1010909913320. Google Scholar [16] G. Hébuterne and C. Rosenberg, Arrival and departure state distributions in the general bulk-service queue,, Naval Res. Logist., 46 (1999), 107. doi: 10.1002/(SICI)1520-6750(199902)46:1<107::AID-NAV7>3.0.CO;2-Y. Google Scholar [17] L. Kleinrock, "Queueing Systems - Theory,", Vol. I, (1975). Google Scholar [18] D. M. Lucantoni, New results on the single server queue with a batch Markovian arrival process,, Comm. Statist. Stochastic Models, 7 (1991), 1. doi: 10.1080/15326349108807174. Google Scholar [19] D. M. Lucantoni and V. Ramaswami, Efficient algorithms for solving non-linear matrix equations arising in phase type queues,, Comm. Statist. Stochastic Models, 1 (1985), 29. doi: 10.1080/15326348508807003. Google Scholar [20] J. Medhi, "Recent Developments in Bulk Queueing Models,", Wiley Eastern, (1984). Google Scholar [21] M. F. Neuts, A versatile Markovian point process,, J. Appl. Probab., 16 (1979), 764. doi: 10.2307/3213143. Google Scholar [22] M. F. Neuts, "Matrix-Geometric Solutions in Stochastic Models,", The John Hopkins University Press, (1981). Google Scholar [23] M. F. Neuts, "Structured Stochastic Matrices of $M$/$G$/$1$ Type and Their Applications,", Marcel Dekker, (1989). Google Scholar [24] R. Sivasamy, A bulk service queue with accessible and non-accessible batches,, Opsearch, 27 (1990), 46. Google Scholar [25] R. Sivasamy and N. Pukazhenthi, A discrete time bulk service queue with accessible batch: $(Geo)$/$NB^{(L,K)}$/$1$,, Opsearch, 46 (2009), 321. doi: 10.1007/s12597-009-0021-2. Google Scholar

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##### References:
 [1] F. J. Albores-Velasco and F. S. Tajonar-Sanabria, Anlysis of the $GI$/$MSP$/$c$/$r$ queueing system,, Information Processes, 4 (2004), 46. Google Scholar [2] 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 [3] A. D. Banik, U. C. Gupta and M. L. Chaudhry, Finite-buffer bulk service queue under Markovian service process: $GI$/$MSP^(a,b)$/$1$/$N$,, Stoch. Anal. Appl., 27 (2009), 500. doi: 10.1080/07362990902844157. Google Scholar [4] P. P. Bocharov, Stationary distribution of a finite queue with recurrent flow and Markovian service,, Automat. Remote Control, 57 (1996), 1274. Google Scholar [5] S. Chakravarthy, A finite capacity $GI$/$PH$/$1$ queue with group services,, Naval Res. Logist., 39 (1992), 345. doi: 10.1002/1520-6750(199204)39:3<345::AID-NAV3220390305>3.0.CO;2-V. Google Scholar [6] S. Chakravarthy, Analysis of a finite $MAP$/$G$/$1$ queue with group services,, Queueing Systems, 13 (1993), 385. doi: 10.1007/BF01149262. Google Scholar [7] M. L. Chaudhry and J. G. C. Templeton, "A First Course in Bulk Queues,", John Wiley, (1983). Google Scholar [8] J. H. Dshalalow, "Frontiers in Queueing: Models and Applications in Sciences and Engineering,", CRC press, (1997). Google Scholar [9] H. Gold and P. Tran-Gia, Performance analysis of a batch service queue arising out of manufacturing and system modelling,, Queueing Systems, 14 (1993), 413. doi: 10.1007/BF01158876. Google Scholar [10] V. Goswami, J. R. Mohanty and S. K. Samanta, Discrete-time bulk-service queues with accessible and non-accessible batches,, Appl. Math. Comput., 182 (2006), 898. doi: 10.1016/j.amc.2006.04.047. Google Scholar [11] V. Goswami and K. Sikdar, Discrete-time batch service $GI$/$Geo^(a,b)$/$1$/$N$ queue with accessible and non-accessible batches,, Internaional Journal of Mathematics in Operational Research, 2 (2010), 233. doi: 10.1504/IJMOR.2010.030818. Google Scholar [12] W. K. Grassmann, M. I. Taksar and D. P. Heyman, Regenerative analysis and steady state distributions for Markov chains,, Oper. Res., 33 (1985), 1107. doi: 10.1287/opre.33.5.1107. Google Scholar [13] D. Gross, J. F. Shortle, J. M. Thompson and C. M. Harris, "Fundamentals of Queueing Theory," 4th Edition,, John Wiley & Sons, (2008). Google Scholar [14] U. C. Gupta and A. D. Banik, Complete analysis of finite and infinite buffer $GI$/$MSP$/$1$ queue - A computational approach,, Oper. Res. Lett., 35 (2006), 273. doi: 10.1016/j.orl.2006.02.003. Google Scholar [15] U. C. Gupta and P. V. Laxmi, Analysis of $MAP$/$G^(a,b)$/$1$/$N$ queue,, Queueing Systems, 38 (2001), 109. doi: 10.1023/A:1010909913320. Google Scholar [16] G. Hébuterne and C. Rosenberg, Arrival and departure state distributions in the general bulk-service queue,, Naval Res. Logist., 46 (1999), 107. doi: 10.1002/(SICI)1520-6750(199902)46:1<107::AID-NAV7>3.0.CO;2-Y. Google Scholar [17] L. Kleinrock, "Queueing Systems - Theory,", Vol. I, (1975). Google Scholar [18] D. M. Lucantoni, New results on the single server queue with a batch Markovian arrival process,, Comm. Statist. Stochastic Models, 7 (1991), 1. doi: 10.1080/15326349108807174. Google Scholar [19] D. M. Lucantoni and V. Ramaswami, Efficient algorithms for solving non-linear matrix equations arising in phase type queues,, Comm. Statist. Stochastic Models, 1 (1985), 29. doi: 10.1080/15326348508807003. Google Scholar [20] J. Medhi, "Recent Developments in Bulk Queueing Models,", Wiley Eastern, (1984). Google Scholar [21] M. F. Neuts, A versatile Markovian point process,, J. Appl. Probab., 16 (1979), 764. doi: 10.2307/3213143. Google Scholar [22] M. F. Neuts, "Matrix-Geometric Solutions in Stochastic Models,", The John Hopkins University Press, (1981). Google Scholar [23] M. F. Neuts, "Structured Stochastic Matrices of $M$/$G$/$1$ Type and Their Applications,", Marcel Dekker, (1989). Google Scholar [24] R. Sivasamy, A bulk service queue with accessible and non-accessible batches,, Opsearch, 27 (1990), 46. Google Scholar [25] R. Sivasamy and N. Pukazhenthi, A discrete time bulk service queue with accessible batch: $(Geo)$/$NB^{(L,K)}$/$1$,, Opsearch, 46 (2009), 321. doi: 10.1007/s12597-009-0021-2. Google Scholar
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