284 IEEE TRANSACTIONS ON RELIABILITY, VOL. R-25, NO. 4, OCTOBER 1976

Bibliography for Reliability and Availability of Stochastic Systems

Shunji Osaki They provide the practioner with a basis for applying models toToshio Nakagawa design and maintenance problems.

Abstract-A brief review of stochastic models on system-reliability ACKNOWLEDGMENTSlists selected references on system reliability models using stochasticprocesses such as Markov chains, Markov processes, renewal processes, The authors thank the referee and Editor for their helpfuland semi-Markov (Markov renewal) processes. suggestions and comrnents. Their thanks are also to Amrit L.

Key Words-Bibliography, Availability, System reliability model, Goel and Kazuhira Okumoto of Syracuse University and toStochastic process Masaharu Takeda of Hiroshima University.

Reader Aids: BOOKSPurpose: BibliographySpecial math needed: None [1] ARINC Research Corp., W.H. von Alven (ed.), Reliability Engi-Results useful to: Reliability engineers, Theoreticians, Teachers neering. Englewood Cliffs, New Jersey: Prentice-Hall, 1964.

[2] R.E. Barlow, F. Proschan, Mathematical Theory ofReliability.New York: Wiley, 1965.

[31 R.E. Barlow, F. Proschan, Statistical Theory ofReliability andINTRODUCTION Life Testing. New York: Holt, Rinehart and Winston, 1975.

[4] I. Bazovsky, Reliability: Theory and Practice. Englewood Cliffs,Two ways of improving system reliability are redundancy New Jersey: Prentice-Hall, 1961.

a51 R. Billinton, Power System Reliability Evaluation. New York:and/or maintenance. In complex systems, reliability and avail- Gordon and Breach, 1970.ability can be enhanced and cost can be reduced through judi- [6] R. Billinton, R.J. Ringlee, A.J. Woods, Power-System Reliabilitycious design and maintenance policies; e.g., design for low Calculations. Cambridge, Massachusetts: The M.I.T. Press, 1973.failure rates, use redundancy, and provide for preventive main- [7] B.S. Blanchard, Jr., E.E. Lawery, Maintainability. New York:teanuerandteparseas,epi a ecslyado m8 McGraw-Hill, 1969.tenance and repair. Because repair can be costly and/or im- 818 S.R. Calabro, Reliability Principles and Practice. New York:practical, a policy of preventive maintenance is often prefer- McGraw-Hill, 1962.able. [9] D.R. Cox, Renewal Theory. London: Methuen, 1962.

Many mathematical models and methods are available to 1101 B. Epstein, Mathematical Models for System Reliability. Ilaifa,analyze complex systems. This bibliography is directed towards Israel: Publishing House of the Student Association, Technion-methods using stochastic processes. The behaviorof a 1-unit ~ Israel Institute of Technology, 1969.methods using stochastic processes. The behavior Of a l-uit [11] B.V. Gnedenko, Yu. K. Belyaev, A.D. Solovyev, Mathematical

repairable system can be described by a stochastic process if Methods ofReliabiity Theory. New York: Academic Press, 1969.the probability laws of system uptime and downtime are speci- [12] A.S. Goldman, T.B. Slattery, Maintainability. New York: Wiley,fied. If uptime and downtime are both exponentially distri- 1964.buted, then the stochastic behavior can be described by a con- [131 D. Grouchko (ed.), Operations Research and Reliability, Newtinuous Markov process with 2 states (up and down). If the

York: Gordon and Breach, 1969.tinuos Marovprcess ith 2state (up nd don). I the [141 R.T. Haviland, Engineering Reliability and Long Life Design.

uptime and downtime are arbitrarily distributed, then the Princeton, New Jersey: D. Van Nostrand, 1964.stochastic behavior can be described by a semi-Markov (Markov [15] R.A. Howard, Dynamic Probabilistic Systems. vols. I and II. Newrenewal) process with 2 states. If there are more units and York: Wiley, 1971.more states per unit, the description and analysis become much [16] W.G. Ireson (ed.), Reliability Handbook. New York: McGraw-Hill,more complicated. Typically such systems can be described by 1966.Markor cmpics,Mated. Typicallyesuhsystremsprcanesesoribeib [117] A.K.S. Jardine (ed.), Operational Research in Maintenance. NewMarkov chains, Markov processes, renewal processes, or semi- York: Manchester University Press, Manchester and Barnes &Markov (Markov renewal) processes. Stochastic process models Nobles, Inc., 1970.provide decision criteria which aid in establishing optimum [18] D.W. Jorgenson, J.J. McCall, R. Radner, Optimal Replacement

maintenancepolicies. Policy. Chicago: Rand McNally, 1967.

31 boksand82 rtiles re istd i ths suvey Th peiod [19] B.A. Kozlov, I.A. Ushakov, Reliability Handtbook. New York:31 boks nd 8artclesare istd inthissurvy. Te peiod Holt, Rinehart anld Winston, 1970.1958-1975 is covered for the books and the period 1963-1975 [201 N.R. Mann, R.E. Shtafer, N.D. Singpurwalla, Methods for Statisticalis covered for the articles. The earlier articles eliminated in the Analysis of Reliability and Life Data. New York: Wiley, 1974.bibliography will be cited in the books [2, 1 1, 19, 30] and [21] P.M. Morse, Queues, Inventories and Maintenance. New York:others. Symposium and conference Proceedings are generally Wiley, 1958.

omittd; iformltehnicl reorts,dissrtatons,and emo- [22] R.H. Myer (ed.), Reliability Engineering for Electronic Systems.omlttd; mormaltechcal rport, dlsertalons,and emo-New York: Wiley, 1964.randa are also omitted. All references are written in English. [23 ] E. Pieruschaka, Principles ofReliability. Englewood Cliffs, New

Jersey: Prentice-Hall, 1963.

OSAKI/NAKAGAWA: BIBLIOGRAPHY FOR RELIABILITY AND AVAILABILITY OF STOCHASTIC SYSTEMS 285

[24] A.M. Polovko, Fundamentals ofReliability Theory. New York: of two dissimilar elements," IEEE Trans. Rel., vol. R-13, pp.Academic Press, 1968. 14-22, Mar. 1964.

[251 N.H. Roberts, Mathematical Methods in Reliability Engineering. [181 J.M. Goodwin, E.W. Giese, "Reliability of spare part support forNew York: McGraw-Hill, 1965. a complex system with repair," Oper. Res., vol. 13, pp. 413423,

(26] G.H. Sandler, System Reliability Engineering. Englewood Cliffs, May-June 1965.New Jersey: Prentice-Hall, 1963. [19] A.K. Govil, "Mean time to system failure for a 4-unit redundant

[27] M.L. Shooman,Probabiistic Reliability: An Engineering repairable system," IEEE Trans. Rel., vol. R-23, pp. 56-57, Apr.Approach. New York: McGraw-Hill, 1968. 1974.

[28] D.J. Smith, Reliability Engineering. New York: Pitman, 1973. [201 M. Halperin, "Some waiting time distributions for redundant sys-[291 D.J. Smith, A.H. Babb, Maintainability Engineering. New York: tems with repair," Technometrics, vol. 6, pp. 2740, Feb. 1964.

Pitman, 1973. [21] R. Harris, "Reliability applications of a bivariate exponential dis-[30] M. Zelen (ed.), Statistical Theory ofReliabiity. Madison, Wiscon- tribution," Oper. Res., vol. 16, pp. 18-27, Jan.-Feb. 1968.

sin: University of Wisconsin Press, 1963. [22] L. Tin Htun, "Reliability prediction techniques for complex sys-[31] Proceedings of the Annual Reliability & Maintainability Sympo- tems," IEEE Trans. Rel., vol. 15, pp. 58-69, Aug. 1966.

sia and its predecessors (available from Annual Reliability & [23] T. Ito, C. Kawaguchi, "Reliability of special redundant systemsMaintainability Symposium; 6411 Chillum Place, NW; Washington, considering exchange time and repair time," IEEE Trans. Rel.,DC 20012 USA; or from the IEEE). vol. R-20, pp. 11-16, June 1971.

[241 D.L. Jaquette, S. Osaki, "Initial provisioning of a standby systemwith deteriorating and repairable spares," IEEE Trans. Rel., vol.

ARTICLES R-21, pp. 245-247, Nov. 1972.[25] I.W. Kabak, "System availability and some design implications,"

[1] M. Alam, V.V.S. Sarma, "Optimum maintenance policy for an Oper. Res., vol. 17, pp. 827-837, Sep.-Oct. 1969.equipment subject to deterioration and random failure," IEEE [26] M. Kodama, "Reliability analysis of a 2-dissimilar units redundantTrans. Systems, Man, and Cybernetics. vol. SMC4, pp. 172-175, system with Erlang-failure and general repair distributions,"Mar. 1974. Microelectronics and Reliabdity, vol. 13, pp. 523-528, Dec. 1974.

[2] T.F. Arnold, "The concept of coverage and its effect on the relia- [27] M. Kodama, H. Deguchi, "Reliability considerations for a 2-unitbility model of a repairable system," IEEE Trans. Computers, redundant system with Erlang-failure and general repair distribu-vol. C-22, pp. 251-254, Mar. 1973. tions," IEEE Trans. Rel., vol. R-23, pp. 75-81, June 1974.

[3] U.N. Bhat, "Reliability of an independent component, s-spare [28] M. Kodama, J. Fukuta, S. Takamatsu, "Mission reliability forsystem with exponential life times and general repair times," a 1-unit system with allowed down-time," IEEE Trans. Rel., vol.Technometrics, vol. 15, pp. 529-539, Aug. 1973. R-22, pp. 268-270, Dec. 1973.

[4] A. Birolini, "Some applications of regenerative stochastic proces- [29] J.M. Kontoleon, N. Kontoleon, "Reliability analysis of a systemses to reliability theory-Part one: Tutorial introduction;-Part subject to partial and catastrophic failures," IEEE Trans. Rel., vol.two: Reliability and availability of 2-item redundant systems", R-23, pp. 277-278, Oct. 1974.IEEE Trans. Rel., vol. R-23, pp. 186-194, Aug. 1974; and vol. [30] D.K. Kulshrestha, "Reliability of a parallel redundant complexR-24, pp. 336-339, Dec. 1975. system," Oper. Res., vol. 16, pp. 28-35, Jan.-Feb. 1968.

[5] M.H. Branson, B. Shah, "Reliability analysis of systems com- [31] D.K. Kulshrestha, "Reliability of a repairable multi-componentprised of units with arbitrary repair-time distributions," IEEE system with redundancy in parallel," IEEE Trans. Rel., vol. R-19,Trans. Rel., vol. R-20, pp. 217-223, Nov. 1971. pp. 50-52, May 1970.

[61 S.M. Brodi, O.N. Vlasenko, "Reliability of systems with a variable [32] M. Kumagai, "Reliability analysis for systems with repairs," J.utilization mode," Engineering Cybernetics, vol. 5, pp. 30-35, Oper. Res. Soc. Japan, vol. 14, pp. 53-71, Sep. 1971.Jan.-Feb. 1967. [33] T.J. Li, "On the calculation of system downtime distribution,"

[7] J.A. Buzacott, "Availability of priority standby redundant sys- IEEE Trans. Rel., vol. R-20, pp. 38-39, Feb. 1971.tems," IEEE Trans Rel., vol. R-20, pp. 60-63, May 1971. [34] B.H. Liebowitz, "Reliability considerations for a two element

[81 J.A. Buzacott, "Reliability analysis of a nuclear reactor fuel redundant system with generalized repair times," Oper. Res., vol.charging system," IEEE Trans. Rel., vol. R-22, pp. 88-91, June 14, pp. 233-241, Mar.-Apr. 1966.1973. [35] D.G. Linton, R.N. Braswell, "Laplace transforms for the two-unit

[9] D.K. Chow, "Reliability of some redundant systems with repair," cold-standby redundant system," IEEE Trans. Rel., vol. R-22,IEEE Trans. Rel., vol. R-22, pp. 223-228, Oct. 1973. pp. 105-108, June 1973.

[10] D.K. Chow, "Availability of some repairable computer systems," [36] D.G. Linton, J.G. Saw, "Reliability analysis of the k-out-of-n:FIEEE Trans. Rel., vol. R-24, pp. 64-66, Apr. 1975. system," IEEE Trans. Rel., vol. R-23, pp. 97-103, June 1974.

[11] M.TS. Dimitrov, "A limit theorem for a duplicated system with [37] H.F. Martz, Jr., "On single-cycle availability," IEEE Trans. Rel.,unrenewable redundancy," Engineering Cybernetics, vol. 10, pp. vol. R-20, pp. 21-23, Feb. 1971.816-819, Sep.-Oct. 1972. [381 M. Mazumdar, "Reliability of two-unit redundant repairable sys-

[12] F. Downton, "The reliability of multiplex systems with repair," tems when failures are revealed by inspections," SIAM J. Appl.J. Roy Statist. Soc., ser. B, vol. 28, pp. 459476, 1966. Math., vol. 19, pp. 637.647, Dec. 1970.

[13] M.R. Dyer, DH. Young, "A note on the reliability of a system [39] J.J. McCall, "Maintenance policies for stochastically failing equip-with spares which operates at discrete times," Technometrics, ment: A servey," Management ScL, vol. 11, pp. 493-524, Mar.vol. 12, pp. 702-705, 1970. 1965.

[14] J.J. Eiss, "The reliability of dependent parallel or standby N-unit [40] H. Mine, T. Asakura, "The effect of an age replacement to aredundancies," Oper. Res., vol. 16, pp. 1068-1083, Sep.-Oct. standby redundant system," J. Appt. Prob., vol. 6, pp. 516-523,1968. Dec. 1969.

[15] J. Fukuta, M. Kodama, "Mission reliability for a redundant re- [41] H. Mine, H. Kawai, "Preventive replacement of a 1-unit systempairable system with two dissimilar units," IEEE Thans. Ret., with a wearout state,"' IEEE Thans. Ret., vol. R-23, pp. 24-29,vol. R-23, pp. 280-282, Oct. 1974. Apr. 1974.

[16] D.P. Gayer, Jr., "Time to failure and availability of paralleled [42] H. Mine, H. Kawai, "An optimal maintenance policy for a 2-unitsystems with repair," IEEE Trans Rel., vol. R-12, pp. 30-38, parallel system with degraded states," IEEE Trans. Ret., vol. R-23,June 1963. pp. 81-86, June 1974.

[17] D.P. Gayer, Jr., "Failure time for a redundant repairable system [431 H. Mine, S. Osaki, "On failure time distributions for systems of

286 IEEE TRANSACTIONS ON RELIABILITY, OCTOBER 1976

dissimilar units," IEEE Trans. Rel., vol. R-18, pp. 165-168, Nov. [68] S. Osaki, T. Nakagawa, "On a two-unit standby redundant sys-1969. tem with standby failure," Oper. Res., vol. 19, pp. 510-523, Mar.-

[44] H. Mine, S. Osaki, T. Asakura, "Some considerations for multiple- Apr. 1971.unit redundant systems with generalized repair time distribu- [69] D.V. Rozhdestvenskiy, G.N. Fanarzhi, "Reliability of a duplicatedtions," IEEE Trans. Rel., vol. R-17, pp. 170-174, Sep. 1968. system with renewal and preventive maintenance," Engineering

[45] E.J. Muth, "Reliability of a system having a standby spare plus Cybernetics, vol. 8, pp. 475479, May-June 1970.multiple repair capability," IEEE Trans. Rel., vol. R-15, pp. 76-81, [70] V.V.S. Sarma, M. Alam, "Optimal maintenance policies forAug. 1966. machines subject to deterioration and intermittent breakdowns,"

[46] E.J. Muth, "A method for predicting system downtime," IEEE IEEE Trans. Systems, Man, and Cybernetics, vol. SMC-5, pp.Trans. Rel., vol. R-17, pp. 97-102, June 1968. 396-398, May 1975.

[47] E.J. Muth, "Excess time, a measure of system repairability," [71] S.P. Sethi, "Simultaneous optimization of preventive mainten-IEEE Trans. Rel., vol. R-19, pp. 16-19, Feb. 1970. ance and replacement policy for machines: Modern control

[481 T. Nakagawa, "The expected number of visits to state k before theory approach," AJIE Trans., vol. 5, pp. 156-163, June 1973.a total system failure of a complex system with repair mainten- [72] C.M. Shama, "Time dependent solution of a complex standby re-ance," Oper. Res., vol. 22, pp. 108-116, Jan.-Feb. 1974. dundant system under preemptive repeat repair discipline,"

[49] T. Nakagawa, A.L. Goel, "A note on availability for a finite inter- IEEE Trans. Rel., vol. R-23, pp. 283-285, Oct. 1974.val," IEEE Trans. Rel., vol. R-22, pp. 271-272, Dec. 1973. [73] A.D. Solovyev, "Asymptotic distribution of lifetime of a dupli-

[50] T. Nakagawa, S. Osaki, "Stochastic behaviour of a two-unit cated elements," Engineering Cybernetics, vol. 2, pp. 109-111,standby redundant system," INFOR, vol. 12, pp. 66-70, Feb. Sep.-Oct. 1964.1974. [74] S.K. Srinivasan, M.N. Gopalan, "Probabilistic analysis of a two-

[51] T. Nakagawa, S. Osaki, "Optimum preventive maintenance unit system with a warm standby and a single repair facility,"policies for a 2-unit redundant system," IEEE Trans. Rel., vol. Oper. Res., vol. 21, pp. 748-754, May-June 1973.R-23, pp. 86-91, June 1974. [75] S.K. Srinivasan, M.N. Gopalan, "Probabilistic analysis of a 2-unit

[52] T. Nakagawa, S. Osaki, "Optimum preventive maintenance poli- cold-standby system with a single repair facility," IEEE Trans.cies maximizing the mean time to the first system failure for a Rel., vol. R-22, pp. 250-254, Dec. 1973.two-unit standby redundant system," J. Optimization Theory [761 V.S. Srinivasan, "The effect of standby redundancy in system'sand Appl., vol. 14, pp. 115-129, July 1974. failure with repair maintenance," Oper. Res., vol. 14, pp. 1024-

[53] T. Nakagawa, S. Osaki, "Stochastic behavior of a two-dissimilar- 1036, Nov.-Dec. 1966.unit standby redundant system with repair maintenance," Micro- [77] V.S. Srinivasan, "First emptiness in the spare parts problem forelectronics and Reliabiity, vol. 13, pp. 143-148, Apr. 1974. repairable components," Oper. Res., vol. 16, pp. 407415, Mar.-

[54] T. Nakagawa, S. Osaki, "Stochastic behavior of a two-unit stand- Apr. 1968.by redundant system with imperfect switchover," IEEE Thans. [78] V.S. Srinivasan, "A standby redundant model with noninstan-Rel., vol. R-24, pp. 143-146, June 1975. taneous switchover," IEEE Trans. Rel., vol. R-17, pp. 175-178,

[55] T. Nakagawa, S. Osaki, "Stochastic behavior of a two-unit prior- Sep. 1968.ity standby redundant system with repair," Microelectronics and [79] E.J. Vanderperre, "The busy period of a repairman attaining aReliability, vol. 14, pp. 309-313, June 1975. (n + 1) unit parallel system," Revue Francaise d'Automatique,

[56] T. Nakagawa, S. Osaki, "A summary of optimum preventive Informatique et Recherche Operationnelle, 7 annee n° V-2, pp.maintenance policies for a two-unit standby redundant system," 124-126, May 1973.Zeitschrift fur Operations Research, vol. 20, to appear in 1976. [80] O.V. Viskov, "Reliability of a system under nonhomogeneous

[57] R. Natarajan, "A reliability problem with spares and multiple operating conditions," Engineering Cybernetics, vol. 4, pp. 93-95,repair facilities," Oper. Res., vol. 16, pp. 1041-1057, Sep.-Oct. Nov.-Dec. 1966.1968. [81] G.H. Weiss, "On certain redundant systems which operates at

[58] D.S. Nielsen, B. Runge, "Unreliability of a standby system with discrete times," Technometrics, vol. 4, pp. 69-74, 1962.repair and imperfect switching," IEEE Trans. Rel., vol. R-23, pp. [82] A.F. Zubova, "Idle duplication with repair for any distribution17-24, Apr. 1974. of flow of breadkdowns and time of repair," Engineering Cyber-

[59] A. Omar, Y. Nasr, "Nonloaded duplexing taking switching time netics, vol. 2, pp. 99-111, 1964.into account," Engineering Cybernetics, vol. 4, pp. 310-313,May-June 1966.

[60] S. Osaki, "Reliability analysis of a two-unit standby redundantsystem with priority," Canadian Oper. Res. Soc. J., vol. 8, pp. Manuscript received May 28, 1975; revised October 14, 1975, and March60-62, Mar. 1970. 30, 1976.

[61] S. Osaki, "System reliability analysis by Markov renewal pro-cesses," J. Oper. Res. Soc. Japan, vol. 12, pp. 127-188, May 1970.

[62] S. Osaki, "Renewal theoretic aspects of two-unit redundant sys- Shunji Osaki//Department of Industrial Engineering//Hiroshima Univer-tems," IEEE Trans. Rel., vol. R-19, pp. 105-110, Aug. 1970. sity//Hiroshima 730 JAPAN.

[63] S. Osaki, "On a two-unit standby-redundant system with imper-fect switchover," IEEE Trans. Rel., vol. R-21, pp. 20-24, Feb.1972.

[64] S. Osaki, "Reliability analysis of a two-unit standby-redundant Shunji Osaki was born in Nishio City, Aichi Prefecture, JAPAN on Jan-system with preventive maintenance," IEEE Trans. Re!., vol. uary 3, 1942. He received the B.S.E. degree from Nagoya Institute ofR-21, pp. 24-29, Feb. 1972. Technology in 1964, and the M.S. and Ph.D. degrees in applied mathie-

[65]l S. Osaki, "An intermittently used system with preventive main- matics and physics from Kyoto University in 1967 and 1970, respec-tenance," J. Oper. Res. Soc. Japan, vol. 15, pp. 102-111, June tively.1972. From 1970 to 1972 he was a Visiting Research Associate in the

[66] 5. Osaki, "Signal-flow graphs in reliability theory," Microe!ec- Department of Electrical Engineering, University of Southern California,tronics and Re!iability, vol. 13, pp. 539-541, Dec. 1974. Los Angeles. Since 1970 he has been working as anAssociate Professor

[67] 5. Osaki, T. Asakura, "A two-unit standby redundanty system in the Department of Industrial Engineering, Faculty of Engineering,with repair and preventive maintenance," J. App!. Prob., vol. 7, Hiroshima University. He is the author of numerous technical papers onpp. 641-648, Dec. 1970. reliability theory, biomathematics, and Markov decision processes.

OSAKI/NAKAGAWA: BIBLIOGRAPHY FOR RELIABILITY AND AVAILABILITY OF STOCHASTIC SYSTEMS 287

Toshio Nakagawa//Department of Mathematics//Meijyo University// in Syracuse, New York. Since 1967 he has been working in the Depart-Nagoya 468 JAPAN. ment of Mathematics, Faculty of Science and Engineering, Meijyo Univer-

sity in Nagoya. His research areas include reliability theory, maintain-Toshio Nakagawa was born in Yokkaichi City, Mie Prefecture, JAPAN ability demonstration, and biomathematics.on November 30, 1942. He received the B.S.E. and M.S. degrees bothfrom Nagoya Institute of Technology in 1965 and 1967, respectively.

From 1972 to 1974 he was a Research Associate in the Departmentof Industrial Engineering and Operations Research, Syracuse University ri n

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