# American Institute of Mathematical Sciences

• Previous Article
A novel active DRX mechanism in LTE technology and its performance evaluation
• JIMO Home
• This Issue
• Next Article
Cross-layer modeling and optimization of multi-channel cognitive radio networks under imperfect channel sensing
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
 [1] John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti. Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021004 [2] Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 [3] Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451 [4] Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Recent progresses in the theory of nonlinear nonlocal problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : i-i. doi: 10.3934/dcdss.2020446 [5] Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225 [6] W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349 [7] Sohana Jahan. Discriminant analysis of regularized multidimensional scaling. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 255-267. doi: 10.3934/naco.2020024 [8] Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327 [9] Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1 [10] Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 [11] Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034 [12] Israa Mohammed Khudher, Yahya Ismail Ibrahim, Suhaib Abduljabbar Altamir. Individual biometrics pattern based artificial image analysis techniques. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2020056 [13] Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185 [14] Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197 [15] Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205 [16] Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137 [17] Xiaoyi Zhou, Tong Ye, Tony T. Lee. Designing and analysis of a Wi-Fi data offloading strategy catering for the preference of mobile users. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021038

2019 Impact Factor: 1.366