Reliability Analysis of an n-unit Standby Repairable System with K Repair Facilities

Xianyun Meng College of Science, Yanshan University, Qinhuangdao 066004, China E-mail: xiaoxia820@gmail.com Hongxia Li College of Science, Yanshan University, Qinhuangdao 066004, China E-mail: xiaoxia209198282@163.com Ning Li College of Science, Yanshan University, Qinhuangdao 066004, China Abstract Redundancy or standby is a technique that has been widely applied to improving system reliability and availability in system design. In general, standby repairable system contains cold standby repairable system, hot standby repairable system and warm standby repairable system. In this paper, we shall deal with cold standby repairable system and warm standby repairable system with repair facilities. Under the assumptions that the failure time of unit and the repair time of failed unit are assumed to follow exponential distribution, we present a reliability analysis of an n-unit standby repairable system with k ( ) { } n k , , 2 , 1 L ∈ repair facilities.


Introduction
Standby repairable system contains cold standby repairable system, hot Standby repairable system and warm standby repairable system.(Jinhua Cao,1986).hasdiscussed the cold standby repairable system and the warm standby repairable system with only one repair facility, moreover, (Jianxiong Gu,2006) has discussed the cold standby repairable system with two repair facilities.In this paper, we shall deal with an n-unit standby repairable system with k repair facilities.Under the assumptions that the failure time of unit and the repair time of failed unit are assumed to follow exponential distribution, we obtain the steady-state quantities and average quantities by using the Markov process theory.

Cold standby repairable system
2.1 The assumptions of the model 2.1.1The system consists of n identical units and k repair facilities.At the beginning ( ) , all units are good, and one unit is working and others are under cold standby.The repair facility will repair the working one as soon as it fails.At the same time, the standby one begins to work .When the failed one has been repaired, it either begins to work again or becomes under cold standby.If the rest units fail while k units are still under repair, they must wait for repair.The switch is perfect, instantaneous.
2.1.2Assume that each unit after repair is 'as good as new'.

2.1.3
The failure time of unit and the repair time of failed unit are all exponentially distributed with parameter λ and μ respectively.The failure time distribution and the repair time distribution are denoted which, respectively, by The random variables are mutually independent.

2.2
The steady-state quantities and average quantities of the system.
Theorem 1: The system consists of n+ 1 no-identical state.Let ( ) denote the state that there are j failed units (which contains the units that they are under repair) in the system at time t .So, the steady-state availability of the system is ; the mean cycle time of the system is Where, ( The mean time to the first failure of the system is Where, ( ) Proof: By the definition of the cold standby repairable system, we can know that ( ) , The state n denotes that the system is down, and the rest states denote that the system is up.
, so, the stochastic process The transition probability form different states in the time t Δ is equation ( 5).
( ) ( ) Where, l ( ) And we get the matrix of transition probability that is matrix (6).

3.1The assumptions of the model
3.1.1The system consists of n identical units and k repair facilities.At the beginning ( ) , all units are good, and one unit is working and others are under warm standby.The repair facility will repair the working one as soon as it fails.At the same time, the warm standby one begins to work .When the failed one has been repaired, it either begins to work again or becomes under warm standby.If the rest units fail while k units are still under repair, they must wait for repair.The switch is perfect, instantaneous.
3.1.2Assume that each unit after repair is 'as good as new'.

3.1.3
The lifetime distribution of all units is ( )

3.2
The steady-state quantities and average quantities of the system.
Theorem 2: The system consists of n+ 1 no-identical state.Let ( ) denote the state that there are j failed units (which contains the units that they are under repair) in the system at time t .So, the steady-state availability of the system is ; the mean cycle time of the system is Where, The mean time to first failure of the system is Where, ( ) Proof: By the definition of the warm standby repairable system, we can know that ( ) The state n denotes the system is down, and the rest states denote the system is up.

Therefore
, so, the stochastic process ( )

{ }
, 0 x t t ≥ is Markov process with state space E .
The transition probability form different states during the time t Δ is equation ( 12).

(
The all random variables are mutually independent.