Asymptotic Stability of Gaver’s Parallel System Attended by a Cold Standby Unit and a Repairman with Multiple Vacations

Abstract

We investigate Gaver’s parallel system attended by a cold standby unit and a repairman with multiple vacations. By analysing the spectral distribution of the system operator and taking into account the irreducibility of the semigroup generated by the system operator we prove that the dynamic solution converges strongly to the steady state solution. Thus we obtain asymptotic stability of the dynamic solution of the system.

Share and Cite:

Haji, A. (2015) Asymptotic Stability of Gaver’s Parallel System Attended by a Cold Standby Unit and a Repairman with Multiple Vacations. World Journal of Engineering and Technology, 3, 277-283. doi: 10.4236/wjet.2015.33C041.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Gaver, D.P. (1963) Time to Failure and Availability of Paralleled System with Repair. IEEE Transactions on Reliability, 12, 30-38. http://dx.doi.org/10.1109/TR.1963.5218202
[2] Dhillon, B.S. and Anude, O.C. (1993) Common-Cause Failure Analysis of a Parallel System with Warm Standby. Microelectronics Reliability, 33, 1321-1342. http://dx.doi.org/10.1016/0026-2714(93)90133-j
[3] Vanderperre, E.J. and Makhanov, S.S. (2002) On Gaver’s Parallel System Sustained by a Cold Standby Unit and Attended by Two Repairmen. Operations Research Letters, 30, 43-48. http://dx.doi.org/10.1016/S0167-6377(02)00106-2
[4] Yue, D.Q., Zhu, J.L., Qin, Y.L. and Li, C.Y. (2006) The Gaver Parallel System Attended by a Cold Standby Unit and a Repairman with Multiple Vacations. Systems Engineering-Theory & Practice, 6, 59-68.
[5] Haji, A. and Yunus, B. (2015) Well-Posedness of Gaver’s Parallel System Attended by a Cold Standby Unit and a Repairman with Multiple Vacations. Journal of Applied Mathematics and Physics, 3, 821-827. http://dx.doi.org/10.4236/jamp.2015.37101
[6] Engel, K.-J. and Nagel, R. (2000) One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, 194, Springer-Verlag.
[7] Nagel, R. (1986) One-Parameter Semigroups of Positive Operators. Springer-Verlag. http://dx.doi.org/10.1007/BFb0074922
[8] Greiner, G. (1987) Perturbing the Boundary Conditions of a Generator. Houston J. Math., 13, 213-229.
[9] Haji, A. and Radl, A. (2007) A Semigroup Approach to Queueing Systems. Semi-group Forum, 75, 609-623. http://dx.doi.org/10.1007/s00233-007-0726-6

Copyright © 2022 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.