A New General-Purpose Method for the Computation of the Interval Availability Distribution

Published Online:https://doi.org/10.1287/ijoc.1120.0539

References

  • Abramowitz M, Stegun IA (1970) Handbook of Mathematical Functions (Dover, New York).Google Scholar
  • Barlow RE, Proschan F (1981) Statistical Theory of Reliability and Life Testing. Probability Models (McArdle Press, Silver Spring, MD).Google Scholar
  • Buchholz P (1994) Exact and ordinary lumpability in finite Markov chains. J. Appl. Probab. 31(1):59–75.CrossrefGoogle Scholar
  • Carrasco JA (2003) Computation of bounds for transient measures of large rewarded Markov models using regenerative randomization. Comput. Oper. Res. 30(7):1005–1035.CrossrefGoogle Scholar
  • Carrasco JA (2004a) Solving large interval availability models using a model transformation approach. Comput. Oper. Res. 31(6):807–861.CrossrefGoogle Scholar
  • Carrasco JA (2004b) Transient analysis of some rewarded Markov models using randomization with quasistationarity detection. IEEE Trans. Comput. 53(9):1106–1120.CrossrefGoogle Scholar
  • Carrasco JA (2005) Transient analysis of large Markov models with absorbing states using regenerative randomization. Comm. Statist.—Simulation Comput. 34(4):1027–1052.CrossrefGoogle Scholar
  • Carrasco JA (2006) Two methods for computing bounds for the distribution of cumulative reward for large Markov models. Performance Eval. 63(12):1165–1195.CrossrefGoogle Scholar
  • Carrasco JA (2011) An efficient and numerically stable method for computing bounds for the interval availability distribution. INFORMS J. Comput. 23(2):268–283.LinkGoogle Scholar
  • Dahlquist G, Bjőrck Å (1974) Numerical Methods (Prentice-Hall, Englewood Cliffs, NJ).Google Scholar
  • de Souza e Silva E, Gail HR (1986) Calculating cumulative operational time distributions of repairable computer systems. IEEE Trans. Comput. 35(4):322–332.CrossrefGoogle Scholar
  • de Souza e Silva E, Gail HR (1989) Calculating availability and performability measures of repairable computer systems using randomization. J. ACM 36(1):171–193.CrossrefGoogle Scholar
  • de Souza e Silva E, Gail HR (1998) An algorithm to calculate transient distributions of cumulative rate and impulse based reward. Comm. Statist.—Stochastic Models 14(3):509–536.CrossrefGoogle Scholar
  • Donatiello L, Grassi V (1991) On evaluating the cumulative performance distribution of fault-tolerant computer systems. IEEE Trans. Comput. 40(11):1301–1307.CrossrefGoogle Scholar
  • Fox BL, Glynn PW (1988) Computing Poisson probabilities. Comm. ACM 31(4):440–445.CrossrefGoogle Scholar
  • Goyal A, Tantawi AN (1988) A measure of guaranteed availability and its numerical evaluation. IEEE Trans. Comput. 37(1):25–32.CrossrefGoogle Scholar
  • Grassmann W (1987) Means and variances of time averages in Markovian environments. Eur. J. Oper. Res. 31(1):132–139.CrossrefGoogle Scholar
  • Grassmann WK (1977a) Transient solutions in Markovian queuing systems. Comput. Oper. Res. 4(1):47–53.CrossrefGoogle Scholar
  • Grassmann WK (1977b) Transient solutions in Markovian queues. An algorithm for finding them and determining their waiting-time distributions. Eur. J. Oper. Res. 1(6):396–402.CrossrefGoogle Scholar
  • Grassmann WK (1993) Rounding errors in certain algorithms involving Markov chains. ACM Trans. Math. Software 19(4):496–508.CrossrefGoogle Scholar
  • Gross D, Miller DR (1984) The randomization technique as a modeling tool and solution procedure for transient Markov processes. Oper. Res. 32(2):343–361.LinkGoogle Scholar
  • Higham NJ (2002) Accuracy and Stability of Numerical Algorithms (SIAM, Philadelphia).CrossrefGoogle Scholar
  • IEEE (1985) IEEE Standard for Binary Floating-Point Arithmetic (IEEE Computer Society Press, New York).Google Scholar
  • Islam SMR, Ammar HH (1989) Performability of the hypercube. IEEE Trans. Reliability 38(5):518–526.CrossrefGoogle Scholar
  • Kijima M (1997) Markov Processes for Stochastic Modeling (Chapman & Hall, London).CrossrefGoogle Scholar
  • Knüsel L (1986) Computation of the Chi-square and Poisson distribution. SIAM J. Sci. Statist. Comput. 7(3):1022–1036.CrossrefGoogle Scholar
  • Melamed B, Yadin M (1984) Randomization procedures in the computation of cumulative-time distributions over discrete state Markov processes. Oper. Res. 32(4):926–944.LinkGoogle Scholar
  • Nabli H, Sericola B (1996) Performability analysis: A new algorithm. IEEE Trans. Comput. 45(4):491–494.CrossrefGoogle Scholar
  • Pattipati KR, Li Y, Blom HA (1993) A unified framework for the performability evaluation of fault-tolerant computer systems. IEEE Trans. Comput. 42(2):312–326.CrossrefGoogle Scholar
  • Qureshi MA, Sanders WH (1996) A new methodology for calculating distributions of reward accumulated during a finite interval. IEEE Annual Sympos. Fault-Tolerant Comput., Sendai, Japan, June (IEEE Computer Society Press, Los Alamitos, CA), 116–125.CrossrefGoogle Scholar
  • Rácz S, Tari A, Telek M (2002) MRMSolve: Distribution estimation of large Markov reward models. Field T, Harrison PG, Bradley JT, Harder U, eds. Computer Performance Evaluation/TOOLS, Lecture Notes in Computer Science (Springer, New York), 71–81.CrossrefGoogle Scholar
  • Reibman A, Trivedi K (1988) Numerical transient analysis of Markov models. Comput. Oper. Res. 15(1):19–36.CrossrefGoogle Scholar
  • Reibman A, Trivedi K (1989) Transient analysis of cumulative measures of Markov model behavior. Comm. Statist.—Stochastic Models 5(4):683–710.CrossrefGoogle Scholar
  • Ross SM (1983) Stochastic Processes (John Wiley & Sons, New York).Google Scholar
  • Rubino G, Sericola B (1992) Interval availability analysis using operational periods. Performance Eval. 14(3–4):257–272.CrossrefGoogle Scholar
  • Rubino G, Sericola B (1993) Interval availability distribution computation. IEEE International Sympos. Fault-Tolerant Comput., Toulouse, France, June (IEEE Computer Society Press, Los Alamitos, CA), 48–55.CrossrefGoogle Scholar
  • Rubino G, Sericola B (1995) Interval availability analysis using denumerable Markov processes: Application to multiprocessor subject to breakdowns and repair. IEEE Trans. Comput. 44(2):286–291.CrossrefGoogle Scholar
  • Sericola B (1990) Closed form solution for the distribution of the total time spent in a subset of a homogeneous Markov process during a finite observation period. J. Appl. Probab. 27(2):713–719.CrossrefGoogle Scholar
  • Sericola B (1999) Availability analysis of repairable computer systems and stationarity detection. IEEE Trans. Comput. 48(11):1166–1172.CrossrefGoogle Scholar
  • Smith R, Trivedi KS, Ramesh AV (1988) Performability analysis: Measures, an algorithm, and a case study. IEEE Trans. Comput. 37(4):406–417.CrossrefGoogle Scholar
  • Suñé V, Carrasco JA (2005) Efficient implementations of the randomization method with control of the relative error. Comput. Oper. Res. 32(5):1089–1114.CrossrefGoogle Scholar
  • Suñé V, Carrasco JA, Nabli H, Sericola B (2010) Comment on “Performability analysis: A new algorithm”. IEEE Trans. Comput. 59(1):137–138.CrossrefGoogle Scholar
  • Takács L (1957) On certain sojourn time problems in the theory of stochastic processes. Acta Mathematica Hungarica 8(1–2):169–191.CrossrefGoogle Scholar
  • van Moorsel APA, Sanders WH (1994) Adaptive uniformization. Comm. Statist.—Stochastic Models 10(3):619–647.CrossrefGoogle Scholar
  • van Moorsel APA, Sanders WH (1997) Transient solution of Markov models by combining adaptive and standard uniformization. IEEE Trans. Reliability 46(3):430–440.CrossrefGoogle Scholar
INFORMS site uses cookies to store information on your computer. Some are essential to make our site work; Others help us improve the user experience. By using this site, you consent to the placement of these cookies. Please read our Privacy Statement to learn more.