A multiple warm standby system with operational and repair times following phase-type distributions |
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| Institution: | 1. The Reliability, Risk, and Maintenance Research Laboratory (RRMR Lab), Mechanical and Industrial Engineering Department, Ryerson University, Toronto, Ontario, Canada;2. Distributed Systems & Multimedia Processing Laboratory (DSMP lab), Department of Computer Science, Ryerson University, Toronto, Ontario, Canada;1. Department of Production Engineering, Federal University of Pernambuco, Recife, Brazil;2. Cardiff Business School, Cardiff University, UK;3. RANDOM - Research Group on Risk and Decision Analysis in Operations and Maintenance, UFPE, Recife, Brazil;1. Department of Statistics, University of Isfahan, Isfahan 81744, Iran;2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O Box 19395-5746, Tehran, Iran;3. Department of Statistics, Shiraz University, Shiraz 71454, Iran |
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| Abstract: | We study a warm standby n-unit system. The system functions as long as there is one operative unit. When the unit online fails, a unit in standby becomes the new unit online, if any. When a unit fails it goes to repair. There is a repairman. The units are repaired following the arrival order. The unit online has a lifetime governed by a phase-time distribution. The repair times follow a phase-type distribution. The warm standby units have lifetimes exponentially distributed. All the other times are negligible. This system extends many others of frequent use in the literature. We show that this system is governed by a level-dependent quasi-birth-and-death process (LDQBD process). The availability, rate of occurrence of failures and other magnitudes of interest are calculated. The mathematical expressions are algorithmically and computationally implemented, using the Matlab programme. |
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