Full Text in PDF - Gnedenko e-Forum

L.Knopik Some remarks on mean time between failures of repairable systems - RTA # 3-4, 2007, December - Special IssueIFRA ⊂ MTFR,where IFRA is increasing failure rate on average classof distributions. Preservation of life distribution classesunder reliability operations has been studied in [7], [8].The class MTFR is closed under the operations:(a) formation of a parallel system for absolutelycontinuous random variables,(b) formation of series system with identicallydistributed and absolutely continuous randomvariables,(c) weak convergence of distributions,(d) convolution.In this paper we propose new approximation andbounds, applicable for a wide range of systems. It isMTFR class of distributions, such that MTBF is nonincreasing.The class MTFR contains distributions withunimodal failure rate function. We analyze special caseof distribution with MTBF non-increasing as mixtureof an exponential distribution and Rayleigh’sdistributions. This distribution has unimodal failurerate function. However, it is not easy to obtaindistribution from mixtures with unimodal failure ratefunction [12]. The mixtures of two increasing linearfailures rate functions were studied in [4]. In [4]showed that a mixture of two distributions withincreasing linear failure rate functions does not givedistributions with unimodal failure rate function.In section 2, we introduce the proposed model ofmixture and we estimate their parameters for anexample of lifetime data [11]. In section 3 we proposea simple approximation of to MTBF for lifetimedistributions with MTBF non-increasing.2. Mixture of distributionsWe consider a mixture of two lifetimes X 1 and X 2 withdensities f 1 (t), f 2 (t), reliability functions R 1 (t), R 2 (t),failure rate functions r 1 (t), r 2 (t) and weights p andq=1– p, where 0 < p < 1. The mixed density is thenwritten asf (t ) = pf1( t ) + ( 1 − p ) f 2(t)and the mixed reliability function isR (t) =pR 1 (t) + (1-p)R 2 (t).The failure rate function of the mixture can be writtenas the mixturer(t)=(t ) r 1 (t)+[1 – (t)] r 2 (t),where (t)=pR 1 (t)/R(t).Moreover, from [3], we have under some mildconditions, thatlim r( t ) = lim min{ r ( t ),r ( t )} .t→∞t→∞1In the following propositions, we give some propertiesfor the mixture failure rate function.Proposition 1. For the first derivative of (t), we have’(t) = (t)[1 – (t)][r 2 (t) – r 1 (t)].Proposition 2. For the first derivative of r(t), we haver’(t)=[1– (t)]( – (t)(r 2 (t) – r 1 (t)) 2 +r’ 2 (t)+ (t)r’ 1 (t).Proposition 3. If X 1 is exponentially distributed withparameter , thenr’(t)=[1 – (t)](– (t)(r 2 (t) – ) 2 +r’ 2 (t)).We suppose that r 2 (t) =t, where >0. Consequently,the reliability function of X 2 is the reliability functionof Rayleigh’s distribution of the formαR 2 (t) =exp { − t 2 }22for t0.Proposition 4. If 2 , then r (t) is unimodal.Proof. By Proposition 1, we conclude that (t) isdecreasing for t ∈ (0, t 1 ), where t 1 = and isincreasing for t ∈ (t 1 , ). Hence, if t < t 1 , then(t)(r 2 (t) – ) 2 is decreasing from p 2 < to 0, and if t> t 1 then (t)(r 2 (t) – ) 2 is increasing from 0 to .Thus the equation (t) (r 2 (t) – ) 2 = has only onesolution t 2 and r (t) is increasing for t < t 2 anddecreasing for t > t 2 .Proposition 5. If 2 > , then there exist t 3 , and t 4 , t 3< t 1 < t 4 , such that r (t) decreases in (0, t 3 ), increases in(t 3, t 4 ) and decreases in (t 4 , ).Proof. Let h(t)= – (t)(r 2 (t) – ) 2 . It is easy to findthat h(0)= – p 2 0, and is decreasing from h(t 1 )=>0 toh()=–.Thus, there exist t 3 and t 4 , 0