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July  2015, 11(3): 829-848. doi: 10.3934/jimo.2015.11.829

## Performance analysis of buffers with train arrivals and correlated output interruptions

 1 SMACS Research Group, TELIN Department, Ghent University, Sint-Pietersnieuwstraat 41, B-9000 Gent 2 Supply Networks and Logistics Research Center, Department of Industrial Management, Ghent University, Technologiepark 903, B-9052 Zwijnaarde

Received  September 2013 Revised  May 2014 Published  October 2014

In this paper, we study a discrete-time buffer system with a time-correlated packet arrival process and one unreliable output line. In particular, packets arrive to the buffer in the form of variable-length packet trains at a fixed rate of exactly one packet per slot. The packet trains are assumed to have a geometric length, such that each packet has a fixed probability of being the last of its corresponding train. The output line is governed by a Markovian process, such that the probability that the line is available during a slot depends on the state of the underlying $J$-state Markov process during that slot.
First, we provide a general analysis of the state of the buffer system based on a matrix generating functions approach. This also leads to an expression for the mean buffer content. Additionally, we take a closer look at the distributions of the packet delay and the train delay. In order to make matters more concrete, we next present a detailed and explicit analysis of the buffer system in case the output line is governed by a $2$-state Markov process. Some numerical examples help to visualise the influence of the various model parameters.
Citation: Bart Feyaerts, Stijn De Vuyst, Herwig Bruneel, Sabine Wittevrongel. Performance analysis of buffers with train arrivals and correlated output interruptions. Journal of Industrial & Management Optimization, 2015, 11 (3) : 829-848. doi: 10.3934/jimo.2015.11.829
##### References:
 [1] M. M. Ali, X. Zhang and J. F. Hayes, A performance analysis of a discrete-time queueing system with server interruption for modeling wireless ATM multiplexer,, Performance Evaluation, 51 (2003), 1.   Google Scholar [2] E. Altman and A. Jean-Marie, The distribution of delays of dispersed messages in an $M/M/1$ queue,, Proceedings of IEEE INFOCOM '95 (Boston, (1995), 2.  doi: 10.1109/INFCOM.1995.515893.  Google Scholar [3] C. Blondia and O. Casals, Statistical multiplexing of VBR sources: A matrix-analytic approach,, Performance Evaluation, 16 (1992), 5.  doi: 10.1016/0166-5316(92)90064-N.  Google Scholar [4] H. Bruneel, On the behavior of buffers with random server interruptions,, Performance Evaluation, 3 (1983), 165.  doi: 10.1016/0166-5316(83)90001-9.  Google Scholar [5] H. Bruneel, Buffers with stochastic output interruptions,, Electronics Letters, 19 (1983), 735.  doi: 10.1049/el:19830501.  Google Scholar [6] H. Bruneel, Packet delay and queue length for statistical multiplexers with low-speed access lines,, Computer Networks and ISDN Systems, 25 (1993), 1267.  doi: 10.1016/0169-7552(93)90018-Y.  Google Scholar [7] H. Bruneel, Calculation of message delays and message waiting times in switching elements with slow access lines,, IEEE Transactions on Communications, 42 (1994), 255.  doi: 10.1109/TCOMM.1994.577026.  Google Scholar [8] B. D. Choi, D. I. Choi, Y. Lee and D. K. Sung, Priority queueing system with fixed-length packet-train arrivals,, IEE Proceedings-Communications, 145 (1998), 331.  doi: 10.1049/ip-com:19982288.  Google Scholar [9] A. Chydzinski, Time to reach buffer capacity in a BMAP queue,, Stochastic Models, 23 (2007), 195.  doi: 10.1080/15326340701300746.  Google Scholar [10] I. Cidon, A. Khamisy and M. Sidi, On queueing delays of dispersed messages,, Queueing Systems, 15 (1994), 325.  doi: 10.1007/BF01189244.  Google Scholar [11] I. Cidon, A. Khamisy and M. Sidi, Delay, jitter and threshold crossing in ATM systems with dispersed messages,, Performance Evaluation, 29 (1997), 85.  doi: 10.1016/S0166-5316(96)00006-5.  Google Scholar [12] J. Daigle, Message delays at packet-switching nodes serving multiple classes,, IEEE Transactions on Communications, 38 (1990), 447.  doi: 10.1109/26.52655.  Google Scholar [13] M. Dowell and P. Jarrat, A modified regula falsi method for computing the root of an equation,, BIT Numerical Mathematics, 11 (1971), 168.   Google Scholar [14] K. Elsayed and H. Perros, The superposition of discrete-time Markov renewal processes with an application to statistical multiplexing of bursty traffic sources,, Applied Mathematics and Computation, 115 (2000), 43.  doi: 10.1016/S0096-3003(99)00134-4.  Google Scholar [15] B. Feyaerts, S. De Vuyst, H. Bruneel and S. Wittevrongel, Analysis of discrete-time buffers with heterogeneous session-based arrivals and general session lengths,, Computers and Operations Research, 39 (2012), 2905.  doi: 10.1016/j.cor.2011.11.023.  Google Scholar [16] D. Fiems and H. Bruneel, A note on the discretization of Little's result,, Operations Research Letters, 30 (2002), 17.  doi: 10.1016/S0167-6377(01)00112-2.  Google Scholar [17] D. Fiems, B. Steyaert and H. Bruneel, Discrete-time queues with generally distributed service times and renewal-type server interruptions,, Performance Evaluation, 55 (2004), 277.  doi: 10.1016/j.peva.2003.08.004.  Google Scholar [18] H. R. Gail, S. L. Hantler, A. G. Konheim and B. A. Taylor, An analysis of a class of telecommunication models,, Performance Evaluation, 21 (1994), 151.  doi: 10.1016/0166-5316(94)90032-9.  Google Scholar [19] H. R. Gail, S. L. Hantler and B. A. Taylor, Spectral analysis of $M/G/1$ and $G/M/1$ type Markov chains,, Advances in Applied Probability, 28 (1996), 114.  doi: 10.2307/1427915.  Google Scholar [20] L. Hoflack, S. De Vuyst, S. Wittevrongel and H. Bruneel, Analytics traffic model of a web server,, Electronics Letters, 44 (2008), 61.  doi: 10.1049/el:20083020.  Google Scholar [21] L. Hoflack, S. De Vuyst, S. Wittevrongel and H. Bruneel, Discrete-time buffer systems with session-based arrival streams,, Performance Evaluation, 67 (2010), 432.  doi: 10.1016/j.peva.2009.12.007.  Google Scholar [22] G. U. Hwang and B. D. Choi, Closed-form expressions on the geometric tail behavior of statistical multiplexers with heterogeneous traffic,, IEEE Transactions on Communications, 46 (1998), 1575.   Google Scholar [23] F. Ishizaki, Decomposition property in a discrete-time queue with multiple input streams and service interruptions,, Journal of Applied Probability, 41 (2004), 524.  doi: 10.1239/jap/1082999083.  Google Scholar [24] F. Kamoun, Performance analysis of a discrete-time queueing system with a correlated train arrival process,, Performance Evaluation, 63 (2006), 315.   Google Scholar [25] F. Kamoun, Performance analysis of a non-preemptive priority queuing system subjected to a correlated Markovian interruption process,, Computers & Operations Research, 35 (2008), 3969.  doi: 10.1016/j.cor.2007.06.001.  Google Scholar [26] F. Kamoun, Performance evaluation of a queueing system with correlated packet-trains and server interruption,, Telecommunication Systems, 41 (2009), 267.   Google Scholar [27] K. Laevens and H. Bruneel, Delay analysis for discrete-time queueing systems with multiple randomly interrupted servers,, European Journal of Operations Research, 85 (1995), 161.  doi: 10.1016/0377-2217(93)E0148-Q.  Google Scholar [28] D. S. Lee, Analysis of a single server queue with semi-Markovian service interruption,, Queueing Systems, 27 (1997), 153.  doi: 10.1023/A:1019162014745.  Google Scholar [29] D. M. Lucantoni, K. S. Meier-Hellstern and M. Neuts, A single-server queue with server vacations and a class of non-renewal arrival processes,, Advances in Applied Probability, 22 (1990), 676.  doi: 10.2307/1427464.  Google Scholar [30] D. M. Lucantoni, New results on the single server queue with a batch Markovian arrival process,, Stochastic Models, 7 (1991), 1.  doi: 10.1080/15326349108807174.  Google Scholar [31] H. Masuyama and T. Takine, Stationary queue length in a FIFO single server queue with service interruptions and multiple batch Markovian arrival streams,, Journal of the Operations Research Society of Japan, 46 (2003), 319.   Google Scholar [32] C. D. Meyer, Matrix Analysis and Applied Linear Algebra,, SIAM, (2000).  doi: 10.1137/1.9780898719512.  Google Scholar [33] I. Mitrani, Modelling of Computer and Communication Systems,, Cambridge University Press, (1987).   Google Scholar [34] A. Mokhtar and M. Azizoglu, Analysis of state-dependent probabilistic server interruptions in discrete-time queues,, IEEE Communications Letters, 8 (2004), 544.  doi: 10.1109/LCOMM.2004.833827.  Google Scholar [35] M. Neuts, Structured Stochastic Matrices of $M/G/1$ type and Their Applications,, New York: Marcel Dekker, (1989).   Google Scholar [36] K. Sohraby, On the theory of general ON-OFF sources with applications in high-speed networks,, Proceedings of IEEE INFOCOM '93 (San Francisco, (1993), 401.  doi: 10.1109/INFCOM.1993.253336.  Google Scholar [37] J. Walraevens, S. Wittevrongel and H. Bruneel, A discrete-time priority queue with train arrivals,, Stochastic Models, 23 (2007), 489.  doi: 10.1080/15326340701471158.  Google Scholar [38] S. Wittevrongel and H. Bruneel, Correlation effects in ATM queues due to data format conversions,, Performance Evaluation, 32 (1998), 35.  doi: 10.1016/S0166-5316(97)00015-1.  Google Scholar [39] S. Wittevrongel, Discrete-time buffers with variable-length train arrivals,, Electronics Letters, 34 (1998), 1719.  doi: 10.1049/el:19981248.  Google Scholar [40] S. Wittevrongel, S. De Vuyst and H. Bruneel, Analysis of discrete-time buffers with general session-based arrivals,, 16th International conference on analytical and stochastic modelling techniques and applications (ASMTA) Madrid, 5513 (2009), 189.  doi: 10.1007/978-3-642-02205-0_14.  Google Scholar [41] Y. Xiong, H. Bruneel, Buffer behavior of statistical multiplexers with correlated train arrivals,, International Journal of Electronics and Communications, 51 (1997), 178.   Google Scholar

show all references

##### References:
 [1] M. M. Ali, X. Zhang and J. F. Hayes, A performance analysis of a discrete-time queueing system with server interruption for modeling wireless ATM multiplexer,, Performance Evaluation, 51 (2003), 1.   Google Scholar [2] E. Altman and A. Jean-Marie, The distribution of delays of dispersed messages in an $M/M/1$ queue,, Proceedings of IEEE INFOCOM '95 (Boston, (1995), 2.  doi: 10.1109/INFCOM.1995.515893.  Google Scholar [3] C. Blondia and O. Casals, Statistical multiplexing of VBR sources: A matrix-analytic approach,, Performance Evaluation, 16 (1992), 5.  doi: 10.1016/0166-5316(92)90064-N.  Google Scholar [4] H. Bruneel, On the behavior of buffers with random server interruptions,, Performance Evaluation, 3 (1983), 165.  doi: 10.1016/0166-5316(83)90001-9.  Google Scholar [5] H. Bruneel, Buffers with stochastic output interruptions,, Electronics Letters, 19 (1983), 735.  doi: 10.1049/el:19830501.  Google Scholar [6] H. Bruneel, Packet delay and queue length for statistical multiplexers with low-speed access lines,, Computer Networks and ISDN Systems, 25 (1993), 1267.  doi: 10.1016/0169-7552(93)90018-Y.  Google Scholar [7] H. Bruneel, Calculation of message delays and message waiting times in switching elements with slow access lines,, IEEE Transactions on Communications, 42 (1994), 255.  doi: 10.1109/TCOMM.1994.577026.  Google Scholar [8] B. D. Choi, D. I. Choi, Y. Lee and D. K. Sung, Priority queueing system with fixed-length packet-train arrivals,, IEE Proceedings-Communications, 145 (1998), 331.  doi: 10.1049/ip-com:19982288.  Google Scholar [9] A. Chydzinski, Time to reach buffer capacity in a BMAP queue,, Stochastic Models, 23 (2007), 195.  doi: 10.1080/15326340701300746.  Google Scholar [10] I. Cidon, A. Khamisy and M. Sidi, On queueing delays of dispersed messages,, Queueing Systems, 15 (1994), 325.  doi: 10.1007/BF01189244.  Google Scholar [11] I. Cidon, A. Khamisy and M. Sidi, Delay, jitter and threshold crossing in ATM systems with dispersed messages,, Performance Evaluation, 29 (1997), 85.  doi: 10.1016/S0166-5316(96)00006-5.  Google Scholar [12] J. Daigle, Message delays at packet-switching nodes serving multiple classes,, IEEE Transactions on Communications, 38 (1990), 447.  doi: 10.1109/26.52655.  Google Scholar [13] M. Dowell and P. Jarrat, A modified regula falsi method for computing the root of an equation,, BIT Numerical Mathematics, 11 (1971), 168.   Google Scholar [14] K. Elsayed and H. Perros, The superposition of discrete-time Markov renewal processes with an application to statistical multiplexing of bursty traffic sources,, Applied Mathematics and Computation, 115 (2000), 43.  doi: 10.1016/S0096-3003(99)00134-4.  Google Scholar [15] B. Feyaerts, S. De Vuyst, H. Bruneel and S. Wittevrongel, Analysis of discrete-time buffers with heterogeneous session-based arrivals and general session lengths,, Computers and Operations Research, 39 (2012), 2905.  doi: 10.1016/j.cor.2011.11.023.  Google Scholar [16] D. Fiems and H. Bruneel, A note on the discretization of Little's result,, Operations Research Letters, 30 (2002), 17.  doi: 10.1016/S0167-6377(01)00112-2.  Google Scholar [17] D. Fiems, B. Steyaert and H. Bruneel, Discrete-time queues with generally distributed service times and renewal-type server interruptions,, Performance Evaluation, 55 (2004), 277.  doi: 10.1016/j.peva.2003.08.004.  Google Scholar [18] H. R. Gail, S. L. Hantler, A. G. Konheim and B. A. Taylor, An analysis of a class of telecommunication models,, Performance Evaluation, 21 (1994), 151.  doi: 10.1016/0166-5316(94)90032-9.  Google Scholar [19] H. R. Gail, S. L. Hantler and B. A. Taylor, Spectral analysis of $M/G/1$ and $G/M/1$ type Markov chains,, Advances in Applied Probability, 28 (1996), 114.  doi: 10.2307/1427915.  Google Scholar [20] L. Hoflack, S. De Vuyst, S. Wittevrongel and H. Bruneel, Analytics traffic model of a web server,, Electronics Letters, 44 (2008), 61.  doi: 10.1049/el:20083020.  Google Scholar [21] L. Hoflack, S. De Vuyst, S. Wittevrongel and H. Bruneel, Discrete-time buffer systems with session-based arrival streams,, Performance Evaluation, 67 (2010), 432.  doi: 10.1016/j.peva.2009.12.007.  Google Scholar [22] G. U. Hwang and B. D. Choi, Closed-form expressions on the geometric tail behavior of statistical multiplexers with heterogeneous traffic,, IEEE Transactions on Communications, 46 (1998), 1575.   Google Scholar [23] F. Ishizaki, Decomposition property in a discrete-time queue with multiple input streams and service interruptions,, Journal of Applied Probability, 41 (2004), 524.  doi: 10.1239/jap/1082999083.  Google Scholar [24] F. Kamoun, Performance analysis of a discrete-time queueing system with a correlated train arrival process,, Performance Evaluation, 63 (2006), 315.   Google Scholar [25] F. Kamoun, Performance analysis of a non-preemptive priority queuing system subjected to a correlated Markovian interruption process,, Computers & Operations Research, 35 (2008), 3969.  doi: 10.1016/j.cor.2007.06.001.  Google Scholar [26] F. Kamoun, Performance evaluation of a queueing system with correlated packet-trains and server interruption,, Telecommunication Systems, 41 (2009), 267.   Google Scholar [27] K. Laevens and H. Bruneel, Delay analysis for discrete-time queueing systems with multiple randomly interrupted servers,, European Journal of Operations Research, 85 (1995), 161.  doi: 10.1016/0377-2217(93)E0148-Q.  Google Scholar [28] D. S. Lee, Analysis of a single server queue with semi-Markovian service interruption,, Queueing Systems, 27 (1997), 153.  doi: 10.1023/A:1019162014745.  Google Scholar [29] D. M. Lucantoni, K. S. Meier-Hellstern and M. Neuts, A single-server queue with server vacations and a class of non-renewal arrival processes,, Advances in Applied Probability, 22 (1990), 676.  doi: 10.2307/1427464.  Google Scholar [30] D. M. Lucantoni, New results on the single server queue with a batch Markovian arrival process,, Stochastic Models, 7 (1991), 1.  doi: 10.1080/15326349108807174.  Google Scholar [31] H. Masuyama and T. Takine, Stationary queue length in a FIFO single server queue with service interruptions and multiple batch Markovian arrival streams,, Journal of the Operations Research Society of Japan, 46 (2003), 319.   Google Scholar [32] C. D. Meyer, Matrix Analysis and Applied Linear Algebra,, SIAM, (2000).  doi: 10.1137/1.9780898719512.  Google Scholar [33] I. Mitrani, Modelling of Computer and Communication Systems,, Cambridge University Press, (1987).   Google Scholar [34] A. Mokhtar and M. Azizoglu, Analysis of state-dependent probabilistic server interruptions in discrete-time queues,, IEEE Communications Letters, 8 (2004), 544.  doi: 10.1109/LCOMM.2004.833827.  Google Scholar [35] M. Neuts, Structured Stochastic Matrices of $M/G/1$ type and Their Applications,, New York: Marcel Dekker, (1989).   Google Scholar [36] K. Sohraby, On the theory of general ON-OFF sources with applications in high-speed networks,, Proceedings of IEEE INFOCOM '93 (San Francisco, (1993), 401.  doi: 10.1109/INFCOM.1993.253336.  Google Scholar [37] J. Walraevens, S. Wittevrongel and H. Bruneel, A discrete-time priority queue with train arrivals,, Stochastic Models, 23 (2007), 489.  doi: 10.1080/15326340701471158.  Google Scholar [38] S. Wittevrongel and H. Bruneel, Correlation effects in ATM queues due to data format conversions,, Performance Evaluation, 32 (1998), 35.  doi: 10.1016/S0166-5316(97)00015-1.  Google Scholar [39] S. Wittevrongel, Discrete-time buffers with variable-length train arrivals,, Electronics Letters, 34 (1998), 1719.  doi: 10.1049/el:19981248.  Google Scholar [40] S. Wittevrongel, S. De Vuyst and H. Bruneel, Analysis of discrete-time buffers with general session-based arrivals,, 16th International conference on analytical and stochastic modelling techniques and applications (ASMTA) Madrid, 5513 (2009), 189.  doi: 10.1007/978-3-642-02205-0_14.  Google Scholar [41] Y. Xiong, H. Bruneel, Buffer behavior of statistical multiplexers with correlated train arrivals,, International Journal of Electronics and Communications, 51 (1997), 178.   Google Scholar
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