Necessary and Sufficient Conditions for Representing General Distributions by Coxians

A common analytical technique involves using a Coxian distribution to model a general distribution G, where the Coxian distribution agrees with G on the first three moments. This technique is motivated by the analytical tractability of the Coxian distribution. Algorithms for mapping an input distribution G to a Coxian distribution largely hinge on knowing a priori the necessary and sufficient number of phases in the representative Coxian distribution. In this paper, we formally characterize the set of distributions G which are well-represented by an n-phase Coxian distribution, in the sense that the Coxian distribution matches the first three moments of G. We also discuss a few common, practical examples.

[1]  Ramon Puigjaner,et al.  Computer Performance Evaluation , 2000, Lecture Notes in Computer Science.

[2]  David L. Peterson,et al.  Fractal Patterns In DASD I/O Traffic , 1996, Int. CMG Conference.

[3]  T. Altiok.  On the Phase-Type Approximations of General Distributions , 1985 .

[4]  Alan Scheller-Wolf,et al.  Analysis of cycle stealing with switching cost , 2003, SIGMETRICS '03.

[5]  W. Whitt.  On approximations for queues, I: Extremal distributions , 1984, AT&T Bell Laboratories Technical Journal.

[6]  A. Cumani.  On the canonical representation of homogeneous markov processes modelling failure - time distributions , 1982 .

[7]  A. David,et al.  The least variable phase type distribution is Erlang , 1987 .

[8]  Sally Floyd,et al.  Wide area traffic: the failure of Poisson modeling , 1995, TNET.

[9]  MarieRaymond.  Calculating equilibrium probabilities for (n)/Ck/1/N queues , 1980 .

[10]  W. J. Studden,et al.  Tchebycheff Systems: With Applications in Analysis and Statistics. , 1967 .

[11]  Alma Riska,et al.  Efficient fitting of long-tailed data sets into hyperexponential distributions , 2002, Global Telecommunications Conference, 2002. GLOBECOM '02. IEEE.

[12]  Sally Floyd,et al.  Wide-Area Traffic: The Failure of Poisson Modeling , 1994, SIGCOMM.

[13]  M. A. Johnson,et al.  Matching moments to phase distributions: nonlinear programming approaches , 1990 .

[14]  Alma Riska,et al.  Efficient fitting of long-tailed data sets into PH distributions ? , 1975 .

[15]  Ramin Sadre,et al.  Fitting World Wide Web request traces with the EM-algorithim , 2001, SPIE ITCom.

[16]  Ramin Sadre,et al.  Fitting World Wide Web request traces with the EM-algorithim , 2001, SPIE ITCom.

[17]  Raymond A. Marie,et al.  Calculating equilibrium probabilities for &lgr;(n)/Ck/1/N queues , 1980 .

[18]  Azer Bestavros,et al.  Self-similarity in World Wide Web traffic: evidence and possible causes , 1997, TNET.

[19]  K. Mani Chandy,et al.  Approximate Analysis of Central Server Models , 1975, IBM J. Res. Dev..

[20]  Mark Crovella,et al.  Self - similarity in World Wide Web: Evidence and possible causes , 1997 .

[21]  Anand Sivasubramaniam,et al.  An Integrated Approach to Parallel Scheduling Using Gang-Scheduling, Backfilling, and Migration , 2001, JSSPP.

[22]  Luis G. Vargas.  Review: Marcel F. Neuts, Matrix-geometric solutions in stochastic models, an algorithmic approach , 1983 .

[23]  M. Telek,et al.  Moment Bounds for Acyclic Discrete and Continuous Phase Type Distributions of Second Order , 2002 .

[24]  Moshe Sidi,et al.  Modeling and analysis of power-tail distributions via classical teletraffic methods , 2000, Queueing Syst. Theory Appl..

[25]  Anja Feldmann,et al.  Fitting Mixtures of Exponentials to Long-Tail Distributions to Analyze Network , 1998, Perform. Evaluation.

[26]  Marcel F. Neuts,et al.  Matrix-geometric solutions in stochastic models - an algorithmic approach , 1982 .

[27]  Anand Sivasubramaniam,et al.  An Integrated Approach to Parallel Scheduling Using Gang-Scheduling, Backfilling, and Migration , 2001, IEEE Trans. Parallel Distributed Syst..

[28]  Mor Harchol-Balter.  Task assignment with unknown duration , 2002, JACM.

[29]  Mark S. Squillante,et al.  Analysis of task assignment with cycle stealing under central queue , 2003, 23rd International Conference on Distributed Computing Systems, 2003. Proceedings..

[30]  Mor Harchol-Balter,et al.  Exploiting process lifetime distributions for dynamic load balancing , 1996, SIGMETRICS '96.

[31]  Mor Harchol-Balter,et al.  Evaluation of Task Assignment Policies for Supercomputing Servers: The Case for Load Unbalancing and Fairness , 2000, Proceedings the Ninth International Symposium on High-Performance Distributed Computing.

[32]  Marcel F. Neuts,et al.  Matrix-Geometric Solutions in Stochastic Models , 1981 .

[33]  Mor Harchol-Balter,et al.  A Closed-Form Solution for Mapping General Distributions to Minimal PH Distributions , 2003, Computer Performance Evaluation / TOOLS.

[34]  Ward Whitt,et al.  Approximating a point process by a renewal process , 1981 .

[35]  M. A. Johnson,et al.  A graphical investigation of error bounds for moment-based queueing approximations , 1991, Queueing Syst. Theory Appl..

[36]  Miklós Telek,et al.  PhFit: a general phase-type fitting tool , 2002, Proceedings International Conference on Dependable Systems and Networks.

[37]  M. Crovella,et al.  Heavy-tailed probability distributions in the World Wide Web , 1998 .

[38]  Alma Riska,et al.  Efficient fitting of long-tailed data sets into phase-type distributions , 2002, PERV.

[39]  M. A. Johnson,et al.  Matching moments to phase distributions: density function shapes , 1990 .

[40]  Vaidyanathan Ramaswami,et al.  Introduction to Matrix Analytic Methods in Stochastic Modeling , 1999, ASA-SIAM Series on Statistics and Applied Mathematics.

[41]  Mark A. McComb.  A Practical Guide to Heavy Tails , 2000, Technometrics.

[42]  Bruno Sericola,et al.  Transient Analysis of Stochastic Fluid Models , 1998, Perform. Evaluation.

[43]  Miklós Telek,et al.  PhFit: A General Phase-Type Fitting Tool , 2002, Computer Performance Evaluation / TOOLS.

[44]  Teunis J. Ott,et al.  Load-balancing heuristics and process behavior , 1986, SIGMETRICS '86/PERFORMANCE '86.

[45]  Raymond A. Marie,et al.  Calculating equilibrium probabilities for λ(n)/Ck/1/N queues , 1980, Performance.

[46]  J. Moreira,et al.  An Evaluation of Parallel Job Scheduling for ASCI Blue-Pacific , 1999, ACM/IEEE SC 1999 Conference (SC'99).