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A Homogeneous PCS network with Markov Call Arrival Process and Phase Type Cell Residence Time

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Abstract

In this paper, the arrival of calls (i.e., new and handoff calls) in a personal communications services (PCS) network is modeled by a Markov arrival process (MAP) in which we allow correlation of the interarrival times among new calls, among handoff calls, as well as between these two kinds of calls. The PCS network consists of homogeneous cells and each cell consists of a finite number of channels. Under the conditions that both cell's residence time and the requested call holding time possess the general phase type (PH) distribution, we obtain the distribution of the channel holding times, the new call blocking probability and the handoff call failure probability. Furthermore, we prove that the cell residence time is PH distribution if and only if

the new call channel holding time is PH distribution; or

the handoff call channel holding time is PH distribution; or

the call channel holding time is PH distribution;

provided that the requested call holding time is a PH distribution and the total call arrival process is a MAP. Also, we prove that the actual call holding time of a non-blocked new call is a mixture of PH distributions. We then developed the Markov process for describing the system and found the complexity of this Markov process. Finally, two interesting measures for the network users, i.e., the duration of new call blocking period and the duration of handoff call blocking period, are introduced; their distributions and the expectations are then obtained explicitly.

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References

  1. A.S. Alfa and M.F. Neuts, Modelling vehicular traffic using the discrete time Markovian arrival process, Transportation Science 29(2) (1995) 109–117.

    Google Scholar 

  2. S. Asmussen, Phase-type distributions and related point processes: Fitting and recent advances, in: Matrix-Analytic Methods in Stochastic Models, eds. S.R. Chakravarthy and A.S. Alfa (Marcel Dekker, New York, 1996) pp. 137–149.

    Google Scholar 

  3. V.A. Bolotin, Modeling call holding time distribution for CCS network design and performance analysis, IEEE Journal on Selected Areas in Communications 12(3) (1994) 433–438.

    Google Scholar 

  4. ETSI, Digital European telecommunications services and facilities requirements specification, Technical report, ETSI, DI/RES 3002, European Telecommunications Standards Institute (1991).

  5. Y.G. Fang and I. Chlamtac, Teletraffic analysis and mobility modeling of PCS networks, IEEE Transactions on Communications 47(7) (1999) 1062–1072.

    Google Scholar 

  6. Y.G. Fang, I. Chlamtac and Y.B. Lin, Call performance for a PCS network, IEEE Journal on Selected Areas in Communications 15(8) (1997) 1568–1581.

    Google Scholar 

  7. Y.G. Fang, I. Chlamtac and Y.B. Lin, Modeling PCS networks under general call holding time and cell residence time distributions, IEEE/ACM Transactions on Networking 6(5) (1997) 893–906.

    Google Scholar 

  8. Y.G. Fang, I. Chlamtac and Y.B. Lin, Channel occupancy times and handoff rate for mobile computing and PCS networks, IEEE Transactions on Computers 47(6) (1998) 679–692.

    Google Scholar 

  9. D.P. Gaver, P.A. Jacobs and G. Latouche, Finite birth-and death models in randomly changing environments, Advances in Applied Probability 16 (1984) 715–731.

    Google Scholar 

  10. A. Graham, Kronecker Products and Matrix Calculus with Applications (Ellis Horwood, Chichester, 1981).

    Google Scholar 

  11. R. Guerin, Channel occupancy time distribution in a cellular radio system, IEEE Transactions on Vehicular Technology 36 (1987) 89–99.

    Google Scholar 

  12. Guerin, Queueing-blocking system with two arrival streams and quard channels, IEEE Transactions on Communications 36(2) (1988) 153–163.

    Google Scholar 

  13. Q.M. He and M.F. Neuts, Markov chains with marked transitions, Stochastic Processes and their Applications 74 (1998) 37–52.

    Google Scholar 

  14. H. Heffes and D.M. Lucantoni, A Markov modulated characterization of packetized voice and date traffic and related statistical multiplexer performance, IEEE Journal on Selected Areas in Communications 4(6) (1986) 856–868.

    Google Scholar 

  15. J.S.M. Ho and I.F. Akyildiz, Mobile user location update and paging under delay constraints, Wireless Networks 1 (1995) 413–425.

    Google Scholar 

  16. D. Hong and S.S. Rappaport, Traffic model and performance analysis for cellular mobile radio telephone systems, IEEE Transactions on Communications 36(2) (1988) 153–163.

    Google Scholar 

  17. F. Khan and D. Zeghlache, Effect of cell residence time distribution on the performance of cellular mobile networks, in: Proc. IEEE VTC (May 1997) pp. 949–953.

  18. M.D. Kulavaratharasah and A.H. Aghvami, Teletraffic performance evaluation of microcellular personal communication networks (PCN's) with prioritized handoff procedures, IEEE Transactions on Vehicular Technology 48(1) (1999) 137–152.

    Google Scholar 

  19. D. Lam, D.C. Cox and J. Widom, Teletraffic modeling for personal communications services, IEEE Communication Magazine 35 (February 1997) pp. 79–87.

  20. A. Lang and J.L. Arthur, Parameter approximation for phase-type distributions, in: Matrix-Analytic Methods in Stochastic Models, eds. S.R. Chakravarthy and A.S. Alfa (Marcel Dekker, New York, 1996) pp. 151–206.

    Google Scholar 

  21. W. Li and A.S. Alfa, A PCS network with correlated arrival process and splitted-rate channels, IEEE Journal on Selected Areas in Communications 17(7) (1999) 1318–1325.

    Google Scholar 

  22. W. Li and A.S. Alfa, Channel reservation for handoff calls in a PCS network, IEEE Transactions on Vehicular Technology 49 (2000) 95–104.

    Google Scholar 

  23. Y.B. Lin, R. Anthony and D. Harasty, The sub-rate channel assignment strategy for PCS handoffs, IEEE Transactions on Vehicular Technology 45(1) (1996) 122–130.

    Google Scholar 

  24. Y.B. Lin and I. Chlamtac, Effective call holding times for a PCS network, IEEE Journal on Selected Areas on Communications (submitted).

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

    Google Scholar 

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

    Google Scholar 

  27. D.M. Lucantoni, K.S. Meier-Hellstern and M.F. Neuts, A singer-server queue with server vacations and a class of non-renewal arrival processes, Advances in Applied Probability 22 (1990) 676–705.

    Google Scholar 

  28. M.F. Neuts, A versatile Markovian point process, Journal of Applied Probability 16 (1979) 764–779.

    Google Scholar 

  29. M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models (The John Hopkins University Press, 1981).

  30. A.R. Noerpel, Y.B. Lin and H. Sherry, PACS: personal access communications system – A tutorial, IEEE Personal Communications 3 (1996) 32–43.

    Google Scholar 

  31. P.V. Orlik and S.S. Rappaport, A model for teletraffic performance and channel holding time characterization in wireless cellular communication with general session and dwell time distributions, IEEE Journal on Selected Areas in Communications 16(5) (1998) 788–803.

    Google Scholar 

  32. P.V. Orlik and S.S. Rappaport, Teletraffic performance and mobility modeling of cellular communications with mixed platforms and highly variable mobilities, Proceedings of the IEEE 86(7) (1998) 1464–1479.

    Google Scholar 

  33. S.S. Rappaport, The multiple-call handoff problem in high-capacity cellular communications systems, IEEE Transactions on Vehicular Technology 40(3) (1991) 546–557.

    Google Scholar 

  34. S.S. Rappaport, Blocking, handoff and traffic performance for cellular communications with mixed platforms, IEE Proceedings 140(5) (1993) 389–401.

    Google Scholar 

  35. C. Rose and R. Yates, Location uncertainly in mobile networks: A theoretical framework, IEEE Communication Magazine 35 (February 1997) 94–101.

  36. G. Ruiz, T.L. Doumi and J.G. Gardiner, Teletraffic analysis and simulation for nongeostationary mobile satellite systems, IEEE Transactions on Vehicular Technology 47(1) (1998) 311–320.

    Google Scholar 

  37. D.H. Shi, J. Guo and L. Liu, SPH-distributions and the rectangle-iterative algorithm, in: Matrix-Analytic Methods in Stochastic Models, eds. S.R. Chakravarthy and A.S. Alfa (Marcel Dekker, New York, 1996) pp. 207–224.

    Google Scholar 

  38. K. Sriram and W. Whitt, Characterizing superposition arrival processes in an ATMtransport network, IEEE Journal on Selected Areas in Communications 4(6) (1986) 359–367.

    Google Scholar 

  39. R. Steedman, The common air interface MPT 1375, in: Corsdless Telecommunications in Europe, ed. W.H. Tuttlebee (Springer-Verlag, New York, 1990).

    Google Scholar 

  40. M.M. Zonoozi and P. Dassanayake, User mobility modeling and characterization of mobility patters, IEEE Journal on Selected Areas in Communications 15(7) (1997) 239–252.

    Google Scholar 

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Alfa, A.S., Li, W. A Homogeneous PCS network with Markov Call Arrival Process and Phase Type Cell Residence Time. Wireless Networks 8, 597–605 (2002). https://doi.org/10.1023/A:1020329719692

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  • DOI: https://doi.org/10.1023/A:1020329719692

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