Summary
Assume that the arcs of the underlying graph can take m + 1 states of reliability from complete failure (state 0) to perfect functioning (state m) where m is a non-negative integer.
For m = 1 an extensive literature about reliability problems on graphs is available, for m > 1 this paper gives bounds for the distribution function (and for the expectation, variance, etc. if necessary) of an intuitively appealing reliability measure of the graph in terms of properly chosen subgraphs based on a suitable two vertex connectivity notation.
A successive determination of lower and upper bounding distribution functions (and of bounds of the moments, if necessary) of the reliability measure is possible allowing improvements due to the choice of proper subgraph systems.
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.
References
BARLOW, R.E./F. PROSCHAN (1978): Statistische Theorie der Zuverlässigkeit, Verlag H. Deutsch.
BARLOW, R.E./A.S. WU (1978): Coherent Systems with Multi-State Components, Math. Operat. Res. 3, 275–281.
EL-NEWEIHI, E./F. PROSCHAN (1980): Multistate Reliability Models: A Survey, in KRISHNAIAH, P.R. (Ed.): Multivariate Analysis-V, North-Holland Publishing Company, 523–541.
EL-NEWEIHI, E./F. PROSCHAN/J. SETHURAMAN (1978): Multistate Coherent Systems, J. Appl. Prob. 15, 675–688.
ESARY, J.D./F. PROSCHAN/D.W. WALKUP (1967): Association of Random Variables, with Applications, Ann. Math. Statist. 38, 1466–1474
FARDIS, M.N./C.A. CORNELL (1981): Analysis of Coherent Multistate Systems, IEEE Transactions on Reliability, R-30, 117–122.
FRANK, O./W. GAUL (1982): On Reliability in Stochastic Graphs, Networks 12, 119–126.
GAEDE, K.W. (1977): Zuverlässigkeit: Mathematische Modelle, Verlag C. Hanser.
GAUL, W. (1978): Some Structural Properties of Project Digraphs, J. Comb., Inf. & Syst. Sc. 3, 217–222.
GAUL, W. (1983): Reliability-Estimation in Stochastic Graphs with Time-Associated Arc-Set Reliability Performance Processes, Diskussionspapier Nr. 56, Institut für Entscheidungstheorie und Unternehmensforschung, Universität Karlsruhe (TH).
GILBERT, E.N. (1959): Random Graphs, Ann. Math. Statist. 30, 1141–1144.
GRIFFITH, W.S. (1980): Multistate Reliability Models, J. Appl. Prob. 17, 735–744.
HARARY, F./F.Z. NORMAN/D. CARTWRIGHT (1965): Structural Models: An Introduction to the Theory of Directed Graphs, Wiley.
HARTUNG, J./B.ELPELT/K.H. KLÖSENER (1982): Statistik: Lehr-und Handbuch der angewandten Statistik, Kap. XIII, Oldenbourg Verlag.
JOGDEO, K: (1977): Association and Probability Inequalities, Ann. Statist. 5, 495–504.
LEE, C.Y. (1955): Analysis of Switching Networks, Bell System Techn. J. 34, 1287–1315.
MOORE, E.F./C.E. SHANNON (1956): Reliable Circuits Using Less Reliable Relays, J. of the Franklin Institute 262, 191–208 and 281–297.
ROSS, S. (1979): Mulitvalued State Component Systems, Ann. Prob. 7, 379–383.
SATYANARAYANA, A./A. PRABHAKAR (1978): New Topological Formula and Rapid Algorithm for Reliability Analysis of Complex Networks, IEEE Transactions on Reliability, R-27, 82–100.
WILKOV, R.S. (1972): Analysis and Design of Reliable Computer Networks, IEEE Transactions on Communications 20, 660–678.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1985 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gaul, W., Hartung, J. (1985). Multistate Reliability Problems for GSP-Digraphs. In: Neumann, K., Pallaschke, D. (eds) Contributions to Operations Research. Lecture Notes in Economics and Mathematical Systems, vol 240. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46534-5_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-46534-5_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-15205-7
Online ISBN: 978-3-642-46534-5
eBook Packages: Springer Book Archive