Survival Analysis and Reliability. A Note on the Mean Residual Life Function of a Parallel System

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1 Communcaons n Sascs Theory and Mehods, 34: , 2005 Copyrgh Taylor & Francs, Inc. ISSN: prn/ x onlne DOI: /STA Survval Analyss and Relably A Noe on he Mean Resdual Lfe Funcon of a Parallel Sysem MAJID ASADI 1 AND ISMIHAN BAYRAMOGLU (BAIRAMOV 2 1 Deparmen of Sascs, Unversy of Isfahan, Isfahan, Iran 2 Deparmen of Mahemacs, Izmr Unversy of Economcs, Izmr, Turkey One of he mos mporan ypes of sysem srucures s he parallel srucure. In he presen arcle, we propose a defnon for he mean resdual lfe funcon of a parallel sysem and oban some of s properes. The proposed defnon measures he mean resdual lfe funcon of a parallel sysem conssng of n dencal and ndependen componens under he condon ha n, = 0 2 n 1, componens of he sysem are workng and oher componens of he sysem have already faled. I s shown ha, for he case where he componens of he sysem have ncreasng hazard rae, he mean resdual lfe funcon of he sysem s a nonncreasng funcon of me. Fnally, we wll oban an upper bound for he proposed mean resdual lfe funcon. Keywords Exponenal dsrbuon; Mean resdual lfe funcon; Order sascs; Parallel sysems. Mahemacs Subjec Classfcaon Prmary 62E10; Secondary 60E Inroducon An mporan mehod of mprovng he relably of a sysem s o buld redundancy o. A common srucure of redundancy s he k-ou-of-n sysems and an mporan specal case of k-ou-of-n sysems s parallel sysems. In he las few years, numerous papers have appeared n sascal and relably journals nvesgang he redundancy mporance of componens n complex sysems by usng he parallel sysem, seres sysem, or, n general a k-ou-of-n sysem (see e.g., Boland e al., 1991; Xe and La, 1996; Kuo and Prasad, A parallel sysem n, conssng of n componens, s a sysem whch funcons f and only f a leas one of s n componens funcons. Le T 1 T n denoe he lfemes of n componens Receved July 18, 2003, Acceped July 7, 2004 Address correspondence o Majd Asad, Deparmen of Sascs, Unversy of Isfahan, Isfahan 81744, Iran; E-mal: M.Asad@sc.u.ac.r 475

2 476 Asad and Bayramoglu conneced n a sysem wh parallel srucure. We assume ha T s are connuous, ndependen, and dencally dsrbued random varables wh common dsrbuon funcon F and survval funcon (or relably funcon F = 1 F. Le also T 1 n T 2 n T n n be he ordered lfemes of he componens. Then T n n represens he lfeme of parallel sysem n wh n componens. If we denoe he survval funcon of he sysem a me by S, we have S = P T n n = 1 F n > 0 (1.1 Assumng ha each componen of he sysem has survved up o me, he survval funcon of T gven ha T >, = 1 n,s F + x F x = (1.2 F Ths s he correspondng condonal survval funcon of he componens a age. From 1.2 we ge ha he mean resdual lfe (MRL funcon M of each componen s equal o M = E T T = 0 F x dx = F x dx F = 1 2 n The MRL funcon plays an mporan role n relably and survval analyss. I s well known ha he MRL funcon m characerzes he dsrbuon funcon F unquely (see, for example, Koz and Shanbhag, The MRL funcon of a sysem s closely relaed o he MRL of s componens. In he leraure he MRL funcon of a parallel sysem s defned as E T n n T n n > and some properes of ha have been obaned by, among ohers, Abouammoh and El-Neweh (1986. Recenly, Baramov e al. (2002, under he condon ha none of he componens of he sysem fals a me, defned he MRL of he parallel sysem n as M 1 n = E T n n T 1 n > (1.3 and obaned several properes of M 1 n. They have also shown ha, under some regulary condons, he survval funcon F can be represened as { F = exp d M 1 dx n x } n 0 M 1 n x M 1 n 1 x dx (1.4 where M 1 n 1 s he MRL of he parallel sysem n 1 havng n 1 componens. The am of he presen arcle s o exend he defnon of he MRL funcon proposed by Baramov e al. (2002 and explore s properes. We wll defne he MRL funcon of a sysem, under he condon ha T r n >,.e., n r + 1,

3 MRL Funcon of a Parallel Sysem 477 r = 1 2 n, componens of he sysem are sll workng. In oher words, we defne he MRL of he parallel sysem n as M r n = E T n n T r n > r = 1 2 n (1.5 We wll show ha M r n, for fxed n, s a decreasng funcon of r, r = 1 n. We wll also show ha, for he case where r = 1, M r n s an ncreasng funcon of n. If he componens of he sysem have an ncreasng hazard rae, s also shown ha M r n s a nonncreasng funcon of. Fnally, we wll oban an upper bound for Mn r. 2. Man Resuls Le M r n be he MRL funcon of he parallel sysem n. We assume ha he lfeme of he componens of he sysem are ndependen and dencally dsrbued wh common dsrbuon F. In he followng heorem, we frs oban a represenaon formula for M r n. Theorem 2.1. F > 0 If M r n s he MRL of he parallel sysem n defned as (1.5, hen for M r n = n j=1 1 j+1( n Mj j r = 1 2 n >0 where M j = F j x dx F j, = F F. Proof. To prove he heorem frs noe ha we have P T n n >x+ T r n > r 1 ( n n ( n = F 1 j+1 F n j F j x + (2.1 j j=1 Hence, f S x denoes he condonal survval of T n n a x + gven ha T r n s greaer han, hen Therefore, S x = P T n n >x+ T r n > = P T n n >x+ T r n > P T r n > F n j=1 = 1 j+1( n n j j j F F x + (2.2 F F n M r n = S x dx = 0 F n j=1 1 j+1( n n j j F F j + x dx 0 (2.3 F F n

4 478 Asad and Bayramoglu Hence, on akng M j = F j x dx F j and = F we ge F M r n = n j=1 1 j+1( n Mj j (2.4 Remark 2.1. To defne he MRL M r n and oban (2.4, one does no acually need o resrc he suppor of F on 0. In general, M r n can be defned for he dsrbuon funcons wh lef exremy a and rgh exremy b, respecvely. Remark 2.2. In Theorem 2.1, f we ake r = 1 hen n M 1 n = E T n n T 1 n > = =1 1 +1( n M (2.5 Hence, M r n can be wren as M r n = M 1 n (2.6 Tha s, M r n can be represened as a convex combnaon of M 1 n, = 0 1 r 1. Remark 2.3. Le A 1 A n be n evens. Then he represenaon formula for M 1 n n (2.5 s an analog of he well-known Boole formula, based on he prncple of ncluson and excluson, for he probably of occurrence of a leas one of he A, = 1 n. See Davd and Nagaraja (2003, p Remark 2.4. Le g n = E T n n T 1 n = be he regresson of T n n on T 1 n whch can be nerpreed as he bes predcor of a parallel sysem conssng of n dencal and ndependen componens knowng he me of he frs falure,.e., he me when he weakes componen fals. I s no dffcul o observe ha g n = n 1 F n 1 Therefore, M r n can be also represened as y F y F n 2 f y dy = M 1 n 1 + (2.7 M r n = [ g n+1 ] (2.8 Regresson of order sascs aroused neres of many sascan n recen years. A characerzaon of dsrbuons va lneary of regresson of order sascs was consdered frs by Ferguson (1967. Dembnska and Wesolowsk (1998 gve a complee soluon for non adjacen order sascs.

5 MRL Funcon of a Parallel Sysem 479 In he followng heorem, we wll show ha M 1 n s an ncreasng funcon of n for any >0. Theorem 2.2. M 1 n M 1 n 1, for any >0. Proof. We have n M 1 n M 1 n 1 = =1 1 +1( n n 1 ( n 1 M 1 +1 M =1 n 1 ( n 1 = 1 M +1 (2.9 Hence, o prove he resul, we need o show ha he rgh-hand sde of Eq. (2.9 s greaer han or equal o zero. Ths s equvalen o show ha n 1 ( n 1 1 F n 1 F +1 x dx 0 (2.10 Noe ha, for any >0, he lef-hand sde of (2.10 can be wren as ( F n 1 ( n 1 F n 2 F x n 1 F n 1 x ( n 1 F n 3 F 2 x 2 F x dx (2.11 whch s equal o ( F F x n 1 F x dx > 0 (2.12 Snce F s a non ncreasng funcon, he negran s non negave. Ths mples ha (2.12 s non negave. Hence, we ge M 1 n M 1 n 1 The heorem smply says ha he MRL of a parallel sysem havng n componens, under he condon ha all he componens of he sysem are workng, s greaer han or equal o he MRL of a parallel sysem havng n 1 componens, under he same condon. Inuvely, one expecs ha a parallel sysem conssng of n componens, n whch a me, n r + 1 of s componens are sll workng have, n average, more lfeme han a parallel sysem conssng of n componens of whch n r componens are sll workng. The followng heorem proves hs. In fac, he heorem shows ha, for any >0, M r n s a decreasng funcon of r. Theorem 2.3. M r 1 n M r n, for r = 1 n, and >0.

6 480 Asad and Bayramoglu Proof. We have r 2 n M r n = M r 1 = r 2 ( r 2 + r 1 ( r 1 M 1 n M 1 n [ ( ] n r 1 r 1 M 1 n r+1 r 1 ( r 2 n / M 1 n ( r 2 n = r 1 r 1 ( r 2 M 1 n M 1 n r+1 r 2 ( ( r 1 (2.13 Noe ha for = 1 r 1, we have n n r + 1. I s shown, n Theorem 2.2, ha M 1 n M 1 n 1. Ths mples ha M 1 n M 1 n r+1, whch n urn mples ha he rgh-hand sde of (2.13 s non negave. Hence we have he resul. In relably heory, modelng, and he sudy of he properes of a lfeme random varable, several classes of dsrbuons have been defned. One mporan class of lfe dsrbuons s he class of ncreasng falure rae (IFR dsrbuons. An absoluely connuous dsrbuon funcon F s sad o belong o class of IFR dsrbuons f he correspondng hazard rae r = f s non decreasng n. We refer he reader F o Barlow and Proschan (1975 for more deals on hs. In he followng heorem, we prove a resul showng ha when he componens of he sysem have a common IFR dsrbuon hen M r n s decreasng n. Theorem 2.4. Le T 1 T n denoe he lfemes of he componens of parallel sysem n. Assume ha T s are ndependen and have dencal absoluely connuous dsrbuon funcon F. IfF s IFR, hen M r n s a decreasng funcon of. Proof. The assumpon ha F s IFR mples ha for non negave values x and, he raon F x+ s a decreasng funcon n. Ths n urn mples ha for 0 < F 1 < 2 and n = 1 2 M 1 n 1 = 0 0 ( ( 1 1 F x+ 1 F 1 ( ( 1 1 F x+ 2 F 2 n dx n dx = M 1 n 2 (2.14 Tha s, M 1 n s a decreasng funcon of. Now, usng hs we show ha M r n = M 1 n (2.15

7 MRL Funcon of a Parallel Sysem 481 s a decreasng funcon of. Snce s an ncreasng funcon of, whou loss of generaly we assume ha =. Hence we show ha K = M 1 n s a decreasng funcon of. We have, on dfferenang boh sdes of hs las equaon, K = (( + ( ( 1 M 1 n ( M 1 n ( M 1 n ( ( /( 1 r (2.16 As M 1 n s decreasng n, he second erm n he rgh-hand sde s non posve. Hence, we need o show ha he frs erm n he rgh-hand sde s non posve. I can be shown afer some manpulaons ha ( r 1 ( ( n r 1 n 1 M 1 n ( r 1( n r 1 ( n = j j =j j=0 ( r 1 ( n ( r 1 ( n M 1 n 1 +j 1( M 1 n M1 n j (2.17 In Theorem 2.2, s shown ha M 1 n s an ncreasng funcon of n for n = 1 2. Ths mples ha M 1 n M 1 n j for j whch n urn shows ha he sum s non posve. Hence we ge ha K 0,.e., M r n s a decreasng funcon of. Ths complees he proof of he heorem. Example 2.1. Le T 1 T n denoe he lfemes of he componens of parallel sysem n.ift s have exponenal dsrbuon wh mean respecvely, hen usng he above heorem, M r n s a non ncreasng funcon of. In fac, can be shown, n hs case, ha M r n has he followng form: M r n = e 1 n j=1 1 j+1 j e 1 = 1 j e 1 n 1 j=1 j (2.18 e 1 In Fg. 1, we have presened he graph of he MRL funcon of a sysem conanng fve componens n whch he lfeme of he componens are assumed o be exponenal dsrbuon wh mean 1. I s seen ha when r ncreases hen he MRL of he sysem decreases and also M r n s a decreasng funcon of. Remark 2.5. Some oher examples whch are sasfed wh he resul of Theorem 2.4 are he Webull and Gamma dsrbuons wh shape parameer greaer han 1,

8 482 Asad and Bayramoglu Fgure 1. The MRL of a parallel sysem wh n = 5 exponenal componens. respecvely. Tha s, f he componens of he sysem are dsrbued as such dsrbuons hen he correspondng MRL funcons M r n of he sysem are decreasng funcons of me. In he followng heorem we fnd an upper bound for M r n. Theorem 2.5. Le M r n denoe he MRL of he parallel sysem n. Then, M r n nm 1 n 1 M 2 for any >0, and r = 1 2 n where M 1 = F x dx and M F 2 = F 2 x dx F 2 Proof. To prove he heorem, we frs show ha he resul s vald for r = 1. To do hs, we have o show ha M 1 n = n = Ths s equvalen o show ha n = M M nm n 1 M 2 (2.19 (( n n 1 M 2 2.

9 MRL Funcon of a Parallel Sysem 483 or n =3 1 M + n 1 n 2 M Noe ha he las nequaly s equvalen o ( (n 1 2 n 2 ( F n n 3 3 F x F n F x n 2 F 2 x dx F n 0 (2.20 Bu, can be easly shown ha he lef-hand sde of (2.20 s equal o n =1 F 1 ( F F x n 1 2 F x dx F n (2.21 whch s always non negave, leadng us o nequaly n (2.19. On he oher hand, usng Theorem 2.4 we have M r n M 1 n, for r = 1 2 n, mplyng ha M r n nm 1 n 1 M 2. Ths complees he proof of he heorem. Remark 2.6. In Theorem 2.5, n he case where r = 1, we can also oban a lower bound for he MRL funcon of he sysem. In oher words, we can show ha M 1 n nm 1 To prove hs, we have o show ha n = ( n M 2 2 M 0 (2.22 I can be shown ha he lef-hand sde of (2.22 s equal o ( n =3 F ( F F x n 3 +1 j=0 F j 3 x dx F n whch s always non negave. Hence, we have he lower bound as clamed. Remark 2.7. I s worh nong ha he bounds gven n Theorem 2.5 and Remark 2.6 are analogs of he bounds gven for he probably of occurrence of a leas one of A j n n evens A 1 A n of he sample space. I s known ha (see for example, Davd and Nagaraja, 2003, p. 127 ha for n evens A 1 A n for whch P A = P A 1 and P A A j = P A 1 A 2 for all j, j, np A 1 ( ( n n P A 2 1 A 2 P A np A 1 n 1 P A 1 A 2 =1

10 484 Asad and Bayramoglu Acknowledgmens The auhors would lke o hank he referee and he edor for valuable commens whch resuled n he mprovemen of he presenaon of hs arcle. The research was done when he frs auhor was vsng he Deparmen of Sascs, Dokuz Eylul Unversy, Izmr, Turkey. References Abouammoh, A., El-Neweh, E. (1986. Closure of he NBUE and DMRL classes under formaon of parallel sysems. Sas. Probab. Le. 4: Baramov, I., Ahsanullah, M., Akhundov, I. (2002. A resdual lfe funcon of a sysem havng parallel or seres srucures. J. Sas. Theor. Appl. 1(2: Barlow, R. E., Proschan, F. (1975. Sascal Theory of Relably and Lfe Tesng. New York: Hol, Rnehar, and Wnson. Boland, P. J., El-Neweh, E., Proschan, F. (1991. Redundancy mporance and allocaon of spares n coheren sysems. J. Sas. Plann. Inference 29: Davd, H. A., Nagaraja, H. N. (2003. Order Sascs. 3rd ed. New York: John Wley & Sons. Dembnska, A., Wesolowsk, J. (1998. Lneary of regresson for nonadjacen order sascs. Merka 48: Ferguson, T. S. (1967. On characerzng dsrbuons by properes of order sascs. Sakhya. Ser. A 29: Koz, S., Shanbhag, D. N. (1980. Some new approaches o probably dsrbuons. Adv. Appl. Prob. 12: Kuo, W., Prasad, V. R. (2000. An annoaed overvew of sysem relably opmzaon. IEEE Trans. Relabl. 49: Xe, M., La, C. D. (1996. On he ncrease of he expeced lfeme by parallel redundancy. Asa-Pasfc J. Opera. Res. 13: