Bimodal packet distribution in loss systems using maximum Tsallis entropy principle

A theoretical model of loss system is proposed and analysed within the framework of maximum Tsallis entropy principle. The study provides an explicit expression for state probability distribution of packets in presence of long-range dependent traffic. The unimodal state probability distribution corresponding to well-known Erlang's loss formula is recovered for Tsallis entropy parameter q = 1. As the parameter q is lowered from unity, it is shown that the state probability distribution makes a transition from unimodal to bimodal. The emergence of bimodality can be regarded as a consequence of long-range dependence. The implication of the model in the design of loss systems is discussed.

[1]  Karmeshu,et al.  Queue length distribution of network packet traffic: Tsallis entropy maximization with fractional moments , 2006, IEEE Communications Letters.

[2]  Don Towsley,et al.  Asymptotic behavior of a multiplexer fed by a long-range dependent process , 1999, Journal of Applied Probability.

[3]  Brian L. Mark,et al.  A maximum entropy approach to the analysis of loss systems , 1995, Proceedings of IEEE Singapore International Conference on Networks and International Conference on Information Engineering '95.

[4]  Edwin T. Jaynes,et al.  Prior Probabilities , 1968, Encyclopedia of Machine Learning.

[5]  Nicolas D. Georganas,et al.  Overflow and losses in a network queue with a self-similar input , 2000, Queueing Syst. Theory Appl..

[6]  C. Tsallis,et al.  Nonextensive Entropy: Interdisciplinary Applications , 2004 .

[7]  Walter Willinger,et al.  On the self-similar nature of Ethernet traffic , 1993, SIGCOMM '93.

[8]  Ravi Mazumdar,et al.  Cell loss asymptotics in buffers fed by heterogeneous long-tailed sources , 2000, Proceedings IEEE INFOCOM 2000. Conference on Computer Communications. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies (Cat. No.00CH37064).

[9]  Ness B. Shroff,et al.  Loss probability calculations and asymptotic analysis for finite buffer multiplexers , 2001, TNET.

[10]  Fumio Ishizaki,et al.  Loss Probability for a Finite Buffer Multiplexer with the M/G/∞ Input Process , 2005, Telecommunications Systems.

[11]  J. N. Kapur,et al.  Entropy optimization principles with applications , 1992 .

[12]  S. Abe,et al.  Necessity of q-expectation value in nonextensive statistical mechanics. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Hiroki Suyari.  Generalization of Shannon-Khinchin axioms to nonextensive systems and the uniqueness theorem for the nonextensive entropy , 2004, IEEE Transactions on Information Theory.

[14]  Karmeshu,et al.  q-Exponential product-form solution of packet distribution in queueing networks: maximization of Tsallis entropy , 2006, IEEE Communications Letters.

[15]  J. Naudts.  CONTINUITY OF A CLASS OF ENTROPIES AND RELATIVE ENTROPIES , 2002, math-ph/0208038.

[16]  Rodney W. Johnson.  Determining probability distributions by maximum entropy and minimum cross-entropy , 1979, APL '79.

[17]  Jawaharlal Karmeshu,et al.  Entropy Measures, Maximum Entropy Principle and Emerging Applications , 2003 .

[18]  Azer Bestavros,et al.  Self-similarity in World Wide Web traffic: evidence and possible causes , 1996, SIGMETRICS '96.

[19]  K. Sohraby,et al.  Control and Performance in Packet, Circuit, and ATM Networks [Book Review] , 1995, IEEE Communications Magazine.

[20]  Lisa Borland,et al.  Microscopic dynamics of the nonlinear Fokker-Planck equation: A phenomenological model , 1998 .

[21]  C. Tsallis.  Possible generalization of Boltzmann-Gibbs statistics , 1988 .