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**Properties of recurrent equations for the full-availability group with BPP traffic.**
*(English)*
Zbl 1264.68014

Summary: The paper proposes a formal derivation of recurrent equations describing the occupancy distribution in the full-availability group with multirate Binomial-Poisson-Pascal (BPP) traffic. The paper presents an effective algorithm for determining the occupancy distribution on the basis of derived recurrent equations and for the determination of the blocking probability as well as the loss probability of calls of particular classes of traffic offered to the system. A proof of the convergence of the iterative process of estimating the average number of busy traffic sources of particular classes is also given in the paper.

### MSC:

68M10 | Network design and communication in computer systems |

60K30 | Applications of queueing theory (congestion, allocation, storage, traffic, etc.) |

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\textit{M. Głąbowski} et al., Math. Probl. Eng. 2012, Article ID 547909, 17 p. (2012; Zbl 1264.68014)

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### References:

[1] | D. Staehle and A. Mäder, “An analytic approximation of the uplink capacity in a UMTS network with heterogeneous traffic,” in Proceedings of the 18th International Teletraffic Congress (ITC ’03), pp. 81-91, Berlin, Germany, 2003. |

[2] | M. Gł\cabowski, M. Stasiak, A. Wiśniewski, and P. Zwierzykowski, “Blocking probability calculation for cellular systems with WCDMA radio interface servicing PCT1 and PCT2 multirate traffic,” IEICE Transactions on Communications, vol. E92-B, no. 4, pp. 1156-1165, 2009. · doi:10.1587/transcom.E92.B.1156 |

[3] | G. A. Kallos, V. G. Vassilakis, and M. D. Logothetis, “Call-level performance analysis of a W-CDMA cell with finite population and interference cancellation,” The European Transactions on Telecommunications, vol. 22, no. 1, pp. 25-30, 2011. · doi:10.1002/ett.1446 |

[4] | M. Stasiak, M. Gł\cabowski, A. Wiśniewski, and P. Zwierzykowski, Modeling and Dimensioning of Mobile Networks, Wiley, 2011. |

[5] | M. Gł\cabowski, A. Kaliszan, and M. Stasiak, “Modeling product-form state-dependent systems with BPP traffic,” Performance Evaluation, vol. 67, no. 3, pp. 174-197, 2010. · doi:10.1016/j.peva.2009.10.002 |

[6] | F. P. Kelly, “Loss networks,” The Annals of Applied Probability, vol. 1, no. 3, pp. 319-378, 1991. · Zbl 0743.60099 · doi:10.1214/aoap/1177005872 |

[7] | J. W. Roberts, V. Mocci, and I. Virtamo, Broadband Network Teletraffic, Final Report of Action COST 242, Commission of the European Communities, Springer, Berlin, Germany, 1996. |

[8] | M. Pióro, J. Lubacz, and U. Körner, “Traffic engineering problems in multiservice circuit switched networks,” Computer Networks and ISDN Systems, vol. 20, pp. 127-136, 1990. · doi:10.1016/0169-7552(90)90018-N |

[9] | L. A. Gimpelson, “Analysis of mixtures of wide and narrow-band traffic,” IEEE Transactions on Communication Technology, vol. 13, no. 3, pp. 258-266, 1953. |

[10] | J. M. Aein, “A multi-user-class, blocked-calls-cleared, demand access model,” IEEE Transactions on Communications, vol. 26, no. 3, pp. 378-384, 1978. · Zbl 0368.94021 · doi:10.1109/TCOM.1978.1094081 |

[11] | V. B. Iversen, Teletraffic Engineering Handbook, ITU-D, Study Group 2, Question 16/2, Geneva, Switzerland, 2005. |

[12] | K. W. Ross, Multiservice Loss Models for Broadband Telecommunication Network, Springer, London, UK, 1995. · Zbl 1094.90514 |

[13] | G. M. Stamatelos and J. F. Hayes, “Admission-control techniques with application to broadband networks,” Computer Communications, vol. 17, no. 9, pp. 663-673, 1994. · doi:10.1016/0140-3664(94)90093-0 |

[14] | J. Conradt and A. Buchheister, “Considerations on loss probability of multi-slot connections,” in Proceedings of the 11th International Teletraffic Congress (ITC ’85), pp. 4.4B-2.1, Kyoto, Japan, 1985. |

[15] | V. B. Iversen, “The exact evaluation of multi-service loss systems with access control,” in Proceedings of the 7th Nordic Teletraffic Seminar (NTS ’87), pp. 56-61, Lund, Sweden, August 1987. |

[16] | M. E. Beshai and D. R. Manfield, “Multichannel services performance of switching networks,” in Proceedings of the 12th International Teletraffic Congress (ITC ’88), pp. 857-864, Elsevier, Torino, Italy, 1988. |

[17] | M. Stasiak, “An approximate model of a switching network carrying mixture of different multichannel traffic streams,” IEEE Transactions on Communications, vol. 41, no. 6, pp. 836-840, 1993. · Zbl 0800.94165 · doi:10.1109/26.231905 |

[18] | L. E. N. Delbrouck, “On the steady-state distribution in a service facility carrying mixtures of traffic with different peakedness factors and capacity requirements,” IEEE Transactions on Communications, vol. 31, no. 11, pp. 1209-1211, 1983. · doi:10.1109/TCOM.1983.1095768 |

[19] | J. S. Kaufman, “Blocking with retrials in a completely shared resource environment,” Performance Evaluation, vol. 15, no. 2, pp. 99-113, 1992. · Zbl 0762.90026 · doi:10.1016/0166-5316(92)90058-O |

[20] | H. L. Hartmann and M. Knoke, “The one-level functional equation of multi-rate loss systems,” The European Transactions on Telecommunications, vol. 14, no. 2, pp. 107-118, 2003. · doi:10.1002/ett.902 |

[21] | V. G. Vassilakis, I. D. Moscholios, and M. D. Logothetis, “Call-level performance modelling of elastic and adaptive service-classes with finite population,” IEICE Transactions on Communications, vol. E91-B, no. 1, pp. 151-163, 2008. · doi:10.1093/ietcom/e91-b.1.151 |

[22] | M. Gł\cabowski, A. Kaliszan, and M. Stasiak, “Asymmetric convolution algorithm for blocking probability calculation in full-availability group with bandwidth reservation,” IET Circuits, Devices and Systems, vol. 2, no. 1, pp. 87-94, 2007. · doi:10.1049/iet-cds:20070037 |

[23] | M. Gł\cabowski, A. Kaliszan, and M. Stasiak, “On the application of the asymmetric convolution algorithm in modeling of full-availability group with bandwidth reservation,” in Proceedings of the 20th International Teletraffic Congress (ITC ’07), Managing Traffic Performance in Converged Networks, L. Mason, T. Drwiega, and J. Yan, Eds., vol. 4516 of Lecture Notes in Computer Science, pp. 878-889, Springer, Ottawa, Canada, June 2007. |

[24] | J. S. Kaufman, “Blocking in a shared resource environment,” IEEE Transactions on Communications, vol. 29, no. 10, pp. 1474-1481, 1981. · doi:10.1109/TCOM.1981.1094894 |

[25] | J. W. Roberts, “A service system with heterogeneous user requirements-application to multi-service telecommunications systems,” in Proceedings of the International Conference on Performance of Data Communication Systems and Their Applications, G. Pujolle, Ed., pp. 423-431, North Holland, Amsterdam, The Netherlands, 1981. · Zbl 0541.90041 |

[26] | J. P. Labourdette and G. W. Hart, “Blocking probabilities in multitraffic loss systems: insensitivity, asymptotic behavior and approximations,” IEEE Transactions on Communications, vol. 40, no. 8, pp. 1355-1367, 1992. · Zbl 0825.90388 · doi:10.1109/26.156640 |

[27] | D. Tsang and K. W. Ross, “Algorithms to determine exact blocking probabilities for multirate tree networks,” IEEE Transactions on Communications, vol. 38, no. 8, pp. 1266-1271, 1990. · Zbl 0706.94028 · doi:10.1109/26.58760 |

[28] | Z. Dziong and J. W. Roberts, “Congestion probabilities in a circuit-switched integrated services network,” Performance Evaluation, vol. 7, no. 4, pp. 267-284, 1987. · doi:10.1016/0166-5316(87)90013-7 |

[29] | E. Pinsky and A. E. Conway, “Computational algorithms for blocking probabilities in circuit-switched networks,” Annals of Operations Research, vol. 35, no. 1, pp. 31-41, 1992. · Zbl 0760.90042 · doi:10.1007/BF02023089 |

[30] | G. A. Awater and H. A. B. van de Vlag, “Exact computation of time and call blocking probabilities in large, multi-traffic, multi-resource loss systems,” Performance Evaluation, vol. 25, no. 1, pp. 41-58, 1996. · Zbl 0875.68192 · doi:10.1016/0166-5316(93)00043-6 |

[31] | M. Gł\cabowski and M. Stasiak, “An approximate model of the full-availability group with multi-rate traffic and a finite source population,” in Proceedings of the 3rd Polish-German Teletraffic Symposium (PGTS ’04), P. Buchholtz, R. Lehnert, and M. Pióro, Eds., pp. 195-204, VDE Verlag, Dresden, Germany, September 2004. |

[32] | M. Gł\cabowski, “Modelling of state-dependent multirate systems carrying BPP traffic,” Annals of Telecommunications, vol. 63, no. 7-8, pp. 393-407, 2008. · doi:10.1007/s12243-008-0034-5 |

[33] | W. Rudin, Principles of Mathematical Analysis (International Series in Pure & Applied Mathematics), McGraw-Hill Science/Engineering/Math, New York, NY, USA, 1976. |

[34] | F. P. Kelly, “Notes on effective bandwidth,” Tech. Rep., University of Cambridge, 1996. · Zbl 0855.60084 |

[35] | L. Aspirot, P. Belzarena, P. Bermolen, A. Ferragut, G. Perera, and M. Simon, “Quality of service parameters and link operating point estimation based on effective bandwidths,” Performance Evaluation, vol. 59, no. 2-3, pp. 103-120, 2005. · doi:10.1016/j.peva.2004.07.006 |

[36] | S. Bodamer and J. Charzinski, “Evaluation of effective bandwidth schemes for self-similar traffic,” in Proceedings of the 13th ITC Specialist Seminar on IP Traffic Measurement, Modeling and Management, pp. 21.1-21.10, Monterey, Calif, USA, 2000. |

[37] | H. Holma and A. Toskala, WCDMA for UMTS: HSPA Evolution and LTE, John Wiley & Sons, New York, NY, USA, 2007. |

[38] | M. Gł\cabowski, “Continuous threshold model for multi-service wireless systems with PCT1 and PCT2 traffic,” in Proceedings of the 7th International Symposium on Communications and Information Technologies (ISCIT ’07), pp. 427-432, Sydney, Australia, October 2007. |

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