Abstract
This paper illustrates new methodologies. that are under study for the analysis of complex degradable systems. The basic idea is to include into a single dependability model both the random variation of the system configuration in time, and the effective performance level of the system in each configuration. In order to characterize the system behaviour, cumulative measures are defined. The evaluation of the distribution function of these measures is a difficult task for the inherent computational complexity. This paper surveys the possibility of solving the above models by approximating the resulting non-markovian stochastic process by means of a suitably generated Markov chain.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
R.E. Barlow and F. Proschan. Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York, 1975.
M.D. Beaudry. Performance-related reliability measures for computing systems. IEEE Transactions on Computers, C-27: 540–547, 1978.
A. Bobbio and K.S. Trivedi. An aggregation technique for the transient analysis of stiff Markov chains. IEEE Transactions on Computers, C-35: 803–814, 1986.
A. Bobbio and K.S. Trivedi. Computation of the distribution of the completion time when the work requirement is a PH random variable. Technical Report, Accepted with revision on: Stochastic Models, 1988.
A. Bobbio and K.S. Trivedi. Computing cumulative measures of stiff Markov chains using aggregation. Technical Report CS-42, Duke University Tech Report, 1987.
P.J. Courtois. Decomposability: Queueing and Computer System Applications. Academic Press, New York, 1977.
D.R. Cox. A use of complex probabilities in the theory of stochastic processes. Proceedings of the Cambridge Phylosophical Society, 51: 313–319, 1955.
D.R. Cox and H.D. Miller. The theory of stochastic processes. Chapman and Hall, London, 1965.
A. Cumani. Esp — A package for the evaluation of stochastic Petri nets with phase-type distributed transition times. In Proceedings International Workshop Timed Petri Nets, IEEE Comp Soc Press no. 674, Torino (Italy), 1985.
A. Cumani. On the canonical representation of homogeneous Markov processes modelling failure-time distributions. Microelectronics and Reliability, 22: 583–602, 1982.
L. Donatiello and B.R. Iyer. Analysis of a composite performance reliability measure for fault tolerant systems. JACM, 34: 179–199, 1987.
L. Donatiello and B R. Iyer Closed-form solution for system availability distribution. IEEE Transactions on Reliability, R-36: 45–47, 1987.
E. De Souza e Silva and R Gail. Calculating cumulative operational time distributions of repairable computer systems. IEEE Transactions on Computers, C-35: 322–332, 1986.
D.P. Gaver. A waiting line with interrupted service, including priorities. Journal of the Royal Statistical Society, B24: 73–90, 1962.
A. Goyal, S. Lavenberg, and K.S. Trivedi. Probabilistic modeling of computer system availability. Annals of Operations Reserch, 8: 285–306, 1987.
A. Goyal and A.N. Tantawi. A measure of guaranteed availability and its numerical evaluation. IEEE Transactions on Computers, C-37: 25–32, 1988.
W.K. Grassmann. Means and variances of time averages in markovian environment. European J of Operations Research, 31: 132–139, 1987.
D. Gross and D. Miller. The randomization technique as a modeling tool and solution procedure for transient Markov processes. Operations Research, 32: 343–361, 1984.
R.A. Howard. Dynamic Probabilistic Systems, Volume II: Semi-Markov and Decision Processes. John Wiley and Sons, New York, 1971.
B.R. Iyer, L. Donatiello, and P. Heidelberger. Analysis of performability for stochastic models of fault-tolerant systems. IEEE Transactions on Computers, C-35: 902–907, 1986.
V.G. Kulkarni, V.F. Nicola, and K. Trivedi. The completion time of a job on a multi-mode system. Advances in Applied Probability, 19: 932–954, 1987.
V.G. Kulkarni, V.F. Nicola, and K. Trivedi. On modeling the performance and reliability of multi-mode computer systems. The Journal of Systems and Software, 6: 175–183, 1986.
B. Melamed and M. Yadin. Randomization procedure in the computation of cumulative-time distributions over discrete state Markov processes. Operations Research, 32: 926–944, 1984.
J.F. Meyer. Closed form solution of performability. IEEE Transactions on Computers, C-31: 648–657, 1982.
J.F. Meyer. On evaluating the performability of degradable systems. IEEE Transactions on Computers, C-29: 720–731, 1980.
M.F. Neuts. Matriz Geometric Solutions in Stochastic Models. John Hopkins University Press, Baltimore, 1981.
B.D. Plateau, J.M. Fourneau, and K.H. Lee. A package for solving complex Markov models of parallel systems. In Proceedings Fourth International Conference on Modelling Techniques and Tools for Computer Performance Evaluation, pages 341–360, 1988.
B.D. Plateau and S.K. Tripathi. Performance analysis of synchronization for two communicating processes. Performance Evaluation, 8: 305–320, 1988.
A. Reibman and K.S. Trivedi. Numerical transient analysis of Markov models. Computers and Operations Research, 15: 19–36, 1988.
M.L. Shooman. Probabilistic reliability: an engineering approach. Mc Graw Hill, 1968.
R. Smith, K. Trivedi, and A.V. Ramesh. Performability analysis: measures, an algorithm and a case study. IEEE Transactions on Computers, C-37: 406–417, 1988.
U. Sumita, J.G. Shanthikumar, and Y. Masuda. Analysis of fault tolerant computer systems. Microelectronics and Reliability, 27: 65–78, 1987.
K. Trivedi, A. Reibman, and R. Smith. Transient analysis of Markov and Markov reward models. In P. Courtois G. Iazeolla and O. Boxma eds., editors, Computer Performance and Reliability, pages 535–545, North Holland, 1988.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1989 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bobbio, A., Roberti, L., Vaccarino, E. (1989). Computing Cumulative Measures in Reward Stochastic Processes by a Phase-Type Approximation. In: Colombari, V. (eds) Reliability Data Collection and Use in Risk and Availability Assessment. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83721-0_55
Download citation
DOI: https://doi.org/10.1007/978-3-642-83721-0_55
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-83723-4
Online ISBN: 978-3-642-83721-0
eBook Packages: Springer Book Archive