Summary
Assuming that the time-to-failure and repair time distributions areof phase type, a variety of reliability models with a small number of components may be studied in terms of finite-state Markov processes. Although the state spaces of these processes are typically large, their infinitesimal generators are highly structured. By utilizing the formalism of PH-distributions, it is possible to construct efficient algorithms to evaluate a large number of quantities of interest. Some new properties of PH-distributions are also established.
Zusammenfassung
Für Verteilungen der Lebensdauer und der Reparaturdauer vom Phasentyp lassen sich bei kleiner Komponentenanzahl eine Reihe von Zuverlässigkeitsmodellen mit Hilfe von Markoff-Prozessen mit endlichem Zustandsraum untersuchen. Zwar sind die Zustandsräume dieser Prozesse groß, doch sind die Matrizen der Übergangsraten stark strukturiert. Durch Anwendung des Formalismus der Verteilungen vom Phasentyp lassen sich effiziente Algorithmen zur Berechnung einer großen Anzahl interessierender Größen entwickeln. Daneben werden einige neue Eigenschaften der Verteilungen vom Phasentyp dargestellt.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bellman R (1960) Introduction to matrix analysis. McGrawHill, New York
Epstein B, Hosford J (1960) Reliability of some two-unit redundant systems. Proc. 6th National Symp. on Reliability and Quality Control, Washington, DC, pp 469–476
Gaver DP Jr (1963) Time to failure and availability of paralleled systems with repair. IEEE Trans Reliability R-12:30–38
Marcus M, Minc H (1964) A survey of matrix theory and matrix inequalities. Allyn and Bacon, Boston, MA
Neuts MF (1975) Probability distributions of phase type. Liber Amicorum Prof. Emeritus H. Florin, Dept. Math. Univ. Louvain, Belgium, pp 173–206
Neuts MF (1981) Matrix-geometric solutions in stochastic models. An algorithmic approach. The Johns Hopkins University Press, Baltimore, MD
Ohi F, Kodama M, Nishida T (1978) Several results on the dependent two-unit system with single repair facility. Trans. IECE Jpn E 61:710–716
Runnenburg JT (1965) On the use of the method of collective marks in queueing theory. Proc. Symp. on Congestion Theory, 1964. University of North Carolina Press, Chapel Hill, NC, pp 399–438
Author information
Authors and Affiliations
Additional information
This research was supported by the National Science Foundation under Grant No. ENG-7908351 and by the Air Force Office of Scientific Research under Grant No. AFOSR-77-3236
Rights and permissions
About this article
Cite this article
Neuts, M.F., Meier, K.S. On the use of phase type distributions in reliability modelling of systems with two components. OR Spektrum 2, 227–234 (1981). https://doi.org/10.1007/BF01721011
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01721011