Transient and stationary characteristics of a packet buffer modelled as an MAP/SM/1/b system

Open access


A packet buffer limited to a fixed number of packets (regardless of their lengths) is considered. The buffer is described as a finite FIFO queuing system fed by a Markovian Arrival Process (MAP) with service times forming a Semi-Markov (SM) process (MAP /SM /1/b in Kendall’s notation). Such assumptions allow us to obtain new analytical results for the queuing characteristics of the buffer. In the paper, the following are considered: the time to fill the buffer, the local loss intensity, the loss ratio, and the total number of losses in a given time interval. Predictions of the proposed model are much closer to the trace-driven simulation results compared with the prediction of the MAP /G/1/b model.


Abate, J., Choudhury, G. and Whitt, W. (2000). An introduction to numerical transform inversion and its application to probability models, in W. Grassmann (Ed.), Computational Probability, International Series in Operations Research & Management Science, Vol. 24, Springer, New York, NY, pp. 257–323.

Chydziński, A. (2006). Queue size in a BMAP queue with finite buffer, in Y. Koucheryavy, J. Harju and V. Iversen (Eds.), Next Generation Teletraffic and Wired/Wireless Advanced Networking, Lecture Notes in Computer Science, Vol. 4003, Springer, Berlin/Heidelberg, pp. 200–210.

Chydziński, A.(2007). TimetoreachbuffercapacityinaBMAP queue, Stochastic Models 23(2): 195–209.

Chydziński, A. (2008). Packet loss process in a queue with Markovian arrivals, 7th International Conference on Networking, ICN 2008, Cancum, Mexico, pp. 524–529.

Chydziński, A. and Chr´ost, Ł. (2011). Analysis of AQM queues with queue size based packet dropping, International Journal of Applied Mathematics and Computer Science 21(3): 567–577, DOI: 10.2478/v10006-011-0045-7.

Dainotti, A., Pescape, A., Salvo Rossi, P., Iannello, G., Palmieri, F. and Ventre, G. (2006). Qrp07-2: An HMM approach to internet traffic modeling, IEEE Global Telecommunications Conference, GLOBECOM’06, San Francisco, CA, USA, pp. 1–6.

Dudin, A.N., Klimenok, V.I. and Tsarenkov, G.V. (2002). A single-server queueing system with batch Markov arrivals, semi-Markov service, and finite buffer: Its characteristics, Automation and Remote Control 63(8): 1285–1297.

Dudin, A.N., Shaban, A.A. and Klimenok, V.I. (2005). Analysis of a queue in the BMAP/G/1/N system, International Journal of Simulation Systems, Science & Technology 6(1–2): 13–22.

Emmert, B., Binzenh¨ofer, A., Schlosser, D. and Weiß, M. (2007). Source traffic characterization for thin client based office applications, in A. Pras and M. van Sinderen (Eds.), Dependable and Adaptable Networks and Services, Lecture Notes in Computer Science, Vol. 4606, Springer, Berlin/Heidelberg, pp. 86–94.

Fischer, W. and Meierhellstern, K. (1993). The Markov-modulated Poisson process (MMPP) cookbook, Performance Evaluation 18(2): 149–171.

Heyman, D. and Lucantoni, D. (2003). Modeling multiple IP traffic streams with rate limits, IEEE/ACM Transactions on Networking 11(6): 948–958.

Janowski, L. and Owezarski, P. (2010). Assessing the accuracy of using aggregated traffic traces in network engineering, Telecommunication Systems 43(3–4): 223–236.

Janssen, J. and Manca, R. (2006). Applied Semi-Markov Processes, Springer, Science+Business Media, New York, NY.

Klemm, A., Klemm, E., Lindemann, C. and Lohmann, M. (2003). Modeling IP traffic using the batch Markovian arrival process, Performance Evaluation 54(2): 149–173.

Kobayashi, H. and Ren, Q. (1992). A mathematical-theory for transient analysis of communication networks, IEICE Transactions on Communications E75B(12): 1266–1276.

Lucantoni, D.M. (1991). New results on the single server queue with a batch Markovian arrival process, Communications in Statistics: Stochastic Models 7(1): 1–46.

Lucantoni, D.M., Choudhury, G.L. and Whitt, W. (1994). The transient BMAP/G/L queue, Communications in Statistics. Stochastic Models 10(1): 145–182.

Meyer, C. (2000). Matrix Analysis and Applied Linear Algebra, Miscellaneous Titles in Applied Mathematics Series, Society for Industrial and Applied Mathematics, Philadelphia, PA.

Muscariello, L., Mellia, M., Meo, M., Marsan, M.A. and Cigno, R.L. (2005). Markov models of internet traffic and a new hierarchical MMPP model, Computer Communications 28(16): 1835–1851.

Neuts, M.F. (1966). The single server queue with poisson input and semi-Markov service times, Journal of Applied Probability 3(1): 202–230.

Paxson, V. and Floyd, S. (1995). Wide area traffic: The failure of Poisson modeling, IEEE/ACM Transactions on Networking 3(3): 226–244.

Rabiner, L. (1989). A tutorial on hidden Markov models and selected applications in speech recognition, Proceedings of the IEEE 77(2): 257–286.

Robert, S. and LeBoudec, J. (1996). On a Markov modulated chain exhibiting self-similarities over finite timescale, Performance Evaluation 27(8): 159–173.

Rusek, K., Janowski, L. and Papir, Z. (2011). Correct router interface modeling, Proceedings of the 2nd Joint WOSP/SIPEW International Conference on Performance Engineering, ICPE’11, Karlsruhe, Germany, pp. 97–102.

Rusek, K., Papir, Z. and Janowski, L. (2012). MAP/SM/1/b model of a store and forward router interface, 2012 International Symposium on Performance Evaluation of Computer and Telecommunication Systems (SPECTS), Genova, Italy, pp. 1–8.

Salvador, P., Pacheco, A. and Valadas, R. (2004). Modeling IP traffic: Joint characterization of packet arrivals and packet sizes using BMAPs, Computer Networks 44(3): 335–352.

Schwefel, H.-P., Lipsky, L. and Jobman, M. (2001). On the necessity of transient performance analysis in telecommunication networks, Teletraffic Engineering in the Internet Era: Proceedings of the International Teletraffic Congress, ITC-17, Salvador da Bahia, Brazil, pp. 1087–1099.

Sequeira, L., Fernandez-Navajas, J., Saldana, J. and Casadesus, L. (2012). Empirically characterizing the buffer behaviour of real devices, 2012 International Symposium on Performance Evaluation of Computer and Telecommunication Systems (SPECTS), Genova, Italy, pp. 1–6.

Weisstein, E.W. (2013). Correlation coefficient, From MathWorld—A Wolfram web resource, CorrelationCoefficient.html.

International Journal of Applied Mathematics and Computer Science

Journal of the University of Zielona Góra

Journal Information

IMPACT FACTOR 2017: 1.694
5-year IMPACT FACTOR: 1.712

CiteScore 2018: 2.09

SCImago Journal Rank (SJR) 2018: 0.493
Source Normalized Impact per Paper (SNIP) 2018: 1.361

Mathematical Citation Quotient (MCQ) 2017: 0.13

Cited By


All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 139 112 9
PDF Downloads 47 40 3