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Limiting availability of system with non-identical lifetime distributions and non-identical repair time distributions
ARTICLE IN PRESS
Statistics & Probability Letters 76 (2006) 729–736 www.elsevier.com/locate/stapro
Limiting availability of system with non-identical lifetime distributions and non-identical repair time distributions Jie Mi Department of Statistics, Florida International University, University Park, Miami, Florida 33199 USA Received 12 December 2003; received in revised form 19 September 2005 Available online 4 November 2005
Abstract The limiting availability, or steady state availability of a repairable system is studied. It is assumed that the lifetimes of system are independent but not necessarily identically distributed, and the times for repairing the failed systems are also independent but not necessarily identically distributed. It can be used to better model the real situation than the usual i.i.d. assumption. In this paper, sufficient conditions are given under which the limiting availability of the system exists and its closed-form expression is also derived. r 2005 Elsevier B.V. All rights reserved. MSC: primary 60K10; secondary 62N05 Keywords: Limiting availability; Dominated sequence of distribution functions; Laplace-Stieltjes transform; Tauberian theorem
1. Introduction Consider a system that has two states: ‘up’ or ‘down’. By ‘up’ we mean the system is still working and by ‘down’ we mean the system is not working; in the latter case the system is being repaired or replaced, depending on the maintenance policy. Suppose a new system starts operation at time t ¼ 0 and works until it fails. At the time of failure, the first up period is over and the first down period begins which ends when the repair or replacement is finished. The first up and down periods constitute the first cycle of the system. At the end of each subsequent down period a new cycle of the system is completed and the system will resume operating, and so on and so forth. Let U j and Dj denote the length of the jth up and down period, respectively. Basically U j is the lifetime of the system after the ðj 1Þth down period while Dj is the length of time needed to do repairing or replacement work. Letting 1 denote the up state, 0 the down state, and X ðtÞ be the state of the system at time t, then the (instant) availability of the system is defined as AðtÞ ¼ PðX ðtÞ ¼ 1Þ. Certainly it is desired to find the expression of AðtÞ, but this is too hard except for a few simple cases. In practice, engineers are more interested in the
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E-mail address: mi@fiu.edu. 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.10.004
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steady system availability Að1Þ limt!1 AðtÞ provided the limit exists, the steady system availability measures the extent to which the system will be available after it has been run for a long time. In many discussions fðU n ; Dn Þ; nX1g are assumed to be i.i.d. and it was shown that Að1Þ ¼ EðU 1 Þ= ðEðU 1 Þ þ EðD1 ÞÞ. For this result see, for example, Barlow and Proschan, 1981. The expression of AðtÞ is obtained in Sarkar and Chandhuri (1999) when U 1 and D1 follow gamma distribution and exponential distribution, respectively. The problem of finding AðtÞ and Að1Þ becomes more difficult when fðU n ; Dn Þ; nX1g are independent but not necessarily identically distributed. Mi (1995) considered other two quantities that can also measure system ¯ n , is the ratio of availability but are not exactly the same as AðtÞ defined in the above. One, denoted as A ¯n ¼ accumulated up time in the first n cycles to the total length of time in the n cycles A Pn Pn ¯ U =ð ðU þ D ÞÞ; the other, denoted as AðtÞ, is the ratio of the total up time in interval ½0; t to the j j j j¼1 j¼1 ¯ ¼ the total up timein ½0; t=t. Under some mild conditions, Mi (1995) length t of this time interval, i.e., AðtÞ ¯ ¯ and showed the asymptotic normality of An and AðtÞ. derived both limn!1 An and limt!1 AðtÞ, Biswas and Sarkar (2000) considered a similar problem. Let k be a predetermined positive integer. The system will experience the first k cycles. At the end of the ðk þ 1Þth up period, the system will be replaced (Model A) or be repaired perfectly (Model B). That paper assumed that U j and Dj follow distribution F j and G j , respectively,1pjpk þ 1. They then obtained the Fourier transform of the instantaneous system unavailability BðtÞ ¼ 1 AðtÞ. By using the inversion formula, they thus obtained closed-form expressions of AðtÞ for exponentially distributed U j and Dj ð1pjpk þ 1Þ for both Model A and Model B with k ¼ 1; 2, and also derived the steady system availability Að1Þ. In this paper, we will continue studying the system considered in Mi (1995). The paper is organized as follows. Some useful lemmas are given in the next section. In Section 3, different sufficient conditions are given under which the steady state availability of the system exists and its expression is derived.
2. Some useful lemmas We keep the same notation as in the previous section and consider the model studied in Mi (1995). In the following, we denote the distribution functions of U j and Dj as F j and G j , respectively. Mi (1995) introduced the concept that a collection of distributions on ½0; 1Þ is dominated. That is, let fF i ; i 2 Ig be a collection of distribution functions on ½0; 1Þ where I is an index set. If there exist a constant 0oLo1 and a distribution function F such that F¯ i ðtÞpLF¯ ðtÞ, 8i 2 I, 8tX0, where F¯ ðtÞ 1 F ðtÞ, then we say that the collection fF i ; i 2 Ig is dominated (by F). In the rest of the paper, we will assume the distribution functions fF j ; jX1g and fG j ; jX1g associated with up times fU j ; jX1g and down times fDj ; jX1g are dominated. According to Definition 1 in Mi (1995) it can be easily verified that if the sequences fF j ; jX1g and fG j ; jX1g are dominated, then the sequence fH j ; jX1g is also dominated, where H j ¼ F j G j is the convolution of F j and G j . P P Lemma 1. Suppose fF j ; jX1g and fG j ; jX1g are dominated and nj¼1 mj =n ! mo1, nj¼1 nj =n ! no1 with R1 R1 P ¯ j ðtÞ dt. Then for any tX0 limn!1 Pð n ðU i þ m þ n40 where mj ¼ EðU j Þ ¼ 0 F¯ j ðtÞ dt and nj ¼ EðDj Þ ¼ 0 G i¼1 Di ÞptÞ ¼ 0 and this convergence is uniform on any finite interval ½0; a. Proof. Since fF j ; jX1g andPfG j ; jX1g are dominated sequences of distribution functions on ½0; 1Þ it follows from Mi (1995) that ð1=nÞ ni¼1 ðU i þ Di Þ ! m þ n; a.s. This particularly implies that n 1X P ðU i þ Di Þ ! m þ n. n i¼1
(1)
For any given tX0 let n0 be large enough such that t mþn o ; n 2
8nXn0 .
(2)
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Then for any nXn0 we have ! ! n n X 1X mþn P ðU i þ Di Þpt pP ðU i þ Di Þ ðm þ nÞp n i¼1 2 i¼1 ! mþn 1 X n pP ðU þ Di Þ ðm þ nÞX n i¼1 i 2 Pn and thus limn!1 P i¼1 ðU i þ Di Þpt ¼ 0 by (1). The uniform convergence on ½0; a can be obtained from replacing (2) by a=noðm þ nÞ=2. & d
d
For any jX1 let random variables U i and Di satisfy U i ¼ U iþj1 , Di ¼ Diþj1 , 8iX1, i.e. U i ; Di have distribution functions F iþj1 , and G iþj1 , respectively. Denote the unavailability of the system associated with fðU i ; Di Þ; iX1g as Bj ðtÞ. Lemma 2. Suppose that fF i ðtÞ, G i ðtÞ, jX1g is dominated. Then (i) Bj ðtÞ is right-continuous in tX0; (ii) Bj ðtÞ has at most countable discontinuity points; (iii) Bj ðtÞ ¼
1 X ½ðH j H jþ1 H jþk1 Þ F jþk ðtÞ ðH j H jþk1 H jþk ÞðtÞ,
(3)
k¼0
where H i ¼ F i G i is the convolution of F i and G i ; and (iv) Bj ðtÞ is a continuous function if F j ðtÞ, Gj ðtÞ, jX1 are. Proof. Let binary random variable X ðtÞ be the state of the system. We have fX ðtÞ ¼ 0g ¼ fU 1 otpU 1 þ D1 g [ fX ðtÞ ¼ 0; U 1 þ D1 otg ¼ fU 1 otpU 1 þ D1 g [ fU 1 þ D1 þ U 2 otpU 1 þ D1 þ U 2 þ D2 g [ fX ðtÞ ¼ 0; U 1 þ D1 þ U 2 þ D2 otg. By induction it can be obtained that ( !) k kþ1 X X n fX ðtÞ ¼ 0g ¼ [ ðU m þ Dm Þ þ U kþ1 otp ðU m þ Dm Þ k¼0
( [
m¼1
X ðtÞ ¼ 0;
nþ1 X
)
m¼1
ðU m þ Dm Þot .
m¼1
Bj ðtÞ ¼ PðX ðtÞ ¼ 0Þ ¼ P X ðtÞ ¼ 0;
nþ1 X
! ðU m þ Dm Þot
m¼1
þ
n X
½ðH j H jþ1 H jþk1 Þ F jþk ðtÞ ðH j H jþk1 H jþk ÞðtÞ.
ð4Þ
k¼0
We For any given t0 X0 and 40, there exists sufficiently large n0 such that Pfirst prove (i). n0 þ1 P m¼1 ðU m þ Dm Þot o, 8t0 ptpt0 þ 1 by the uniform convergence in Lemma 1. Hence ! ! nX nX 0 þ1 0 þ1 P X ðtÞ ¼ 0; ðU m þ Dm Þot pP ðU m þ Dm Þot o; 8t0 ptpt0 þ 1. (5) m¼1
m¼1
For the fixed n0 , there exists 0odo1 such that n0 X ½ðH j H jþ1 H jþk1 Þ F jþk ðtÞ ðH j H jþk1 H jþk ÞðtÞ k¼0
½ðH j H jþ1 H jþk1 Þ F jþk ðt0 Þ ðH j H jþk1 H jþk Þðt0 Þo k¼0 n0 X
ð6Þ
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for any t 2 ½t0 ; t0 þ d since all F i ; G i , iX1 are right-continuous. We thus obtain from (4), (5), and (6) that jBj ðtÞ Bj ðt0 Þjp3; 8t0 ptpt0 þ d. This shows that Bj ðtÞ is right-continuous in tX0. Letting t ! 1 in (4) and applying Lemma 1, we see that (3) holds. Let D be the set of all the discontinuity points of F i ; Gi , iX1. Obviously D is a countable set. It can be shown in the same way as proving (i) that Bj ðtÞ is continuous at every point of ½0; 1ÞnD. Thus, Bj ðtÞ has at most countable discontinuity points. If, in particular, all F i ; G i , iX1 are continuous, then it is straight forward to see that Bj ðtÞ is also a continuous function of tX0. & Under the conditions of Lemma 2 the function 0pBj ðtÞp1 has at most countable many discontinuity points. That R 1 is, Bj ðtÞ is almost everywhere (in Lebesque measure) continuous. Thus it is well known that the integral 0 est dBj ðtÞ exists. This is the so-called the Laplace-Stieltjes transform of Bj ðtÞ and is denoted as B^ j ðsÞ, 8s40. P Lemma 2 implies that the unavailability of the original system fðU i ; Di Þ; iX1g is BðtÞ ¼ B1 ðtÞ ¼ 1 k¼0 ½ðH 1 H 2 H k Þ F kþ1 ðtÞ ðH 1 H k H kþ1 ÞðtÞ and the Laplace–Stieltjes transform of BðtÞ is given as " ! ! ! !# 1 k k kþ1 kþ1 Y Y Y X Y ^ ¼ B^ 1 ðsÞ ¼ BðsÞ F^ j ðsÞ F^ j ðsÞ G^ j ðsÞ F^ kþ1 ðsÞ G^ j ðsÞ k¼0
¼
j¼1
1 X
k Y
k¼0
j¼1
F^ j ðsÞ
!
j¼1 k Y
!
j¼1
j¼1
G^ j ðsÞ F^ kþ1 ðsÞð1 G^ kþ1 ðsÞÞ.
j¼1
Lemma 3. Let F i , iX1 be a sequence of distribution functions in ½0; 1Þ. Suppose fF i ; iX1g converges to F in distribution. Define F ðtÞ ¼ supiX1 F i ðtÞ and F ðtÞ ¼ inf iX1 F i ðtÞ, then both F ðtÞ and F ðtÞ are distribution functions on ½0; 1Þ. Proof. First we prove the result for F ðtÞ. Obviously F ðtÞX0 is non-decreasing in tX0 and F ð0Þ ¼ 0. Let us show that F ð1Þ ¼ 1. For any given 0oo1, there exists t0 2 C F \ ð0; 1Þ such that F ðt0 Þ41 , where C F is the set of continuity points of F. For the fixed 40 and t0 there exists n0 such that jF i ðt0 Þ F ðt0 Þjo, 8iXn0 . It implies that F i ðt0 Þ4F ðt0 Þ 41 2, 8iXn0 . Thus F ðt0 Þ ¼ supiX1 F i ðt0 ÞXsupiXn0 F i ðt0 ÞX1 2 and consequently F ð1Þ ¼ 1 since 40 is arbitrary. To prove F ðtÞ is right continuous, we first show that for any fixed n, the function F n ðtÞ supfF 1 ðtÞ; . . . ; F n ðtÞg is right continuous. By induction, it suffices to prove this result for n ¼ 2. For any chosen t0 X0, suppose F 2 ðt0 Þ ¼ F 2 ðt0 Þ. For any given 40, there exists d40 such that 0pF i ðtÞ F i ðt0 Þo, 8t0 ptpt0 þ d, i ¼ 1; 2 due to the right continuity of F i ðtÞ. Thus if t0 ptpt0 þ d then F 2 ðtÞ ¼ maxfF 1 ðtÞ; F 2 ðtÞg p maxfF 1 ðt0 Þ þ ; F 2 ðt0 Þ þ g ¼ F 2 ðt0 Þ þ ¼ F 2 ðt0 Þ þ . This means that 0pF 2 ðtÞ F 2 ðt0 Þp, 8t0 ptpt0 þ d and consequently F 2 ðtÞ is right-continuous. Now we need to show that F ðtÞ is right-continuous. Let t0 X0 and 40 be given. Consider the following two cases. Case 1: F ðt0 Þ ¼ F ðt0 Þ. For any given 40, there exists d1 40 such that 0pF ðtÞ F ðt0 Þp for any t0 ptpt0 þ d1 since F is right-continuous. We can also assume that t þ d1 2 C F , the set of continuity points of F ðÞ. Since fF i g converges to F in distribution, there exists n0 such that jF i ðt0 þ d1 Þ F ðt0 þ d1 Þjo, 8iXn0 . That is, F i ðt0 þ d1 ÞoF ðt0 þ d1 Þ þ , 8iXn0 and consequently supiXn0 F i ðt0 þ d1 ÞpF ðt0 þ d1 Þ þ . This further implies that supiXn0 F i ðtÞpF ðt0 þ d1 Þ þ , 8t0 ptpt0 þ d1 since supiXn0 F i ðtÞ is nondecreasing. With the fixed n0 there exists d2 40 such that F i ðtÞoF i ðt0 Þ þ , 8t0 ptpt0 þ d2 , 81pipn0 . Let d ¼ minfd1 ; d2 g40. Then for any t0 ptpt0 þ d we have F ðtÞ ¼ maxfsupfF 1 ðtÞ; . . . ; F n0 ðtÞg; supfF n0 ðtÞ; . . .ggp maxfF ðt0 Þ þ ; F ðt0 þ d1 Þ þ g ¼ F ðt0 þ d1 Þ þ pF ðt0 Þ þ þ ¼ F ðt0 Þ þ 2 ¼ F ðt0 Þ þ 2. This implies that 0pF ðtÞ F ðt0 Þp2, 8t0 ptpt0 þ d and thus F is right-continuous at t0 . Case 2: F ðt0 Þ ¼ F j ðt0 Þ for 1pjo1. In this case we can assume F j ðt0 Þ4F ðt0 Þ. For any given 40 there exists d1 40 such that t0 þ d1 2 C F and F j ðtÞoF j ðt0 Þ þ , F ðtÞoF ðt0 Þ þ , 8t0 ptpt0 þ d1 . For this chosen d1 40 there exists n0 such that jF i ðt0 þ d1 Þ F ðt0 þ d1 Þjo, 8iXn0 . Hence supiXn0 F i ðt0 þ d1 ÞpF ðt0 þ d1 Þ þ and consequently supiXn0 F i ðtÞpF ðt0 þ d1 Þ þ , 8t0 ptp0 þ d1 . For the given 40, there exists d2 40 such that F i ðtÞoF i ðt0 Þ þ , 81pipn0 ,8t0 ptpt0 þ d2 due to the right-continuity of F i , iX1. Let d ¼ minfd1 ; d2 g40.
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Then for any t0 ptpt0 þ d we have F ðtÞ ¼ max supfF 1 ðtÞ; . . . ; F n0 ðtÞg; sup F i ðtÞ iXn0
p maxfF j ðt0 Þ þ ; F ðt0 þ d1 Þ þ gpF j ðt0 Þ þ 2 ¼ F ðt0 Þ þ 2. That is 0pF ðtÞ F ðt0 Þp2, 8t0 ptpt0 þ d and so F ðÞ is right-continuous at t0 . From the above we see that F ðtÞ is a right-continuous function and hence must be a distribution function on ½0; 1Þ. In the following we consider F ðtÞ inf iX1 F i ðtÞ. Let us show F ð1Þ ¼ 1. For any given 40, there exists t0 2 C F such that F ðt0 Þ41 =2. There also exists sufficiently large n0 such that F i ðt0 Þ4F ðt0 Þ =2, 8iXn0 . This gives inf iXn0 F i ðt0 ÞXF ðt0 Þ =2X1 . For the same 40, there exists t1 40 such that F i ðt1 Þ41 , 1pipn0 . Therefore, if tX maxðt0 ; t1 Þ, then F ðtÞ ¼ inf F i ðtÞ ¼ min min F i ðtÞ; inf F i ðtÞ X minf1 ; 1 g ¼ 1 . iX1
iXn0
1pipn0
This certainly implies F ð1Þ ¼ 1. To show that F ðtÞ is right-continuous let t0 X0 be arbitrary. Case 1: F ðt0 Þ ¼ F ðt0 Þ. This gives F i ðt0 ÞXF ðt0 Þ, 8iX1. For any given 40, there exits d1 40 such that 0pF ðtÞ F ðt0 Þp, 8t0 ptpt0 þ d1 . For the given 40, there exists 0odod1 such that t0 þ d 2 C F and there exists n0 satisfying F i ðt0 þ dÞpF ðt0 þ dÞ þ , 8iXn0 and so F i ðtÞpF ðt0 þ dÞ þ , 8iXn0 , 8t0 ptpt0 þ d. Then for all t0 ptpt0 þ d we have F ðtÞp inf F i ðtÞpF ðt0 þ dÞ þ pðF ðt0 Þ þ Þ þ ¼ F ðt0 Þ þ 2 ¼ F ðt0 Þ þ 2. iXn0
That is, F ðÞ is right-continuous at t0 . Case 2: F ðt0 Þ ¼ F j ðt0 Þ for 1pjo1. For any given 40 there exists d40 such that 0pF j ðtÞ F j ðt0 Þp, 8t0 ptpt0 þ d. Thus, F ðtÞpF j ðtÞpF j ðt0 Þ þ ¼ F ðt0 Þ þ , 8t0 ptpt0 þ d and so F ðÞ is right-continuous at t0 . & Summarizing the above, we see that F ðtÞ is also right-continuous and consequently it must be a CDF on ½0; 1Þ.
3. Main results st
st
Theorem 1. Suppose that there exist CDFs F, G, and K m1 ; K m2 ; Lm1 ; Lm2 , mX1 satisfying (i) K m1 p F j p K m2 , st st 8jXm; (ii) Lm1 p G j p Lm2 , 8jXm; (iii) K m1 ! F ; K m2 ! F ; Lm1 ! G; Lm2 ! G in distribution as m ! 1; (iv) km1 ; km2 ! m; l m1 ; l m2 ! n as m ! 1, where m; n are the means associated with CDFs F ðÞ and GðÞ with 0omx þ no1, and km1 ; km2 ; l m1 ; l m2 are the means associated with CDFs K m1 ; K m2 ; Lm1 ; Lm2 , respectively. Then the steady state availability of the system exists and is given as Að1Þ ¼ limt!1 AðtÞ ¼ m=m þ n. Proof. It can be seen that the conditions (i) and (ii) show that fF j ; jX1g andfG j ; jX1g are dominated. From conditions (i) and (ii) we can also obtain K^ m1 ðsÞXF^ j ðsÞXK^ m2 ðsÞ, and L^ m1 ðsÞXG^ j ðsÞXL^ m2 ðsÞ; 8sX0; 8jXm. ^ ¼ From Lemma 2 the Laplace–Stieltjes transform of the unavailability of the system is given by BðsÞ P1 Q k Qk ^ B^ 1 ðsÞ ¼ k¼0 ð j¼1 F^ j ðsÞÞð j¼1 G j ðsÞÞF^ kþ1 ðsÞð1 G^ kþ1 ðsÞÞ. For any mX1 we have ! ! m 1 Y k k Y X ^ ¼ BðsÞ F^ j ðsÞ G^ j ðsÞ F^ kþ1 ðsÞð1 G^ kþ1 ðsÞÞ
þ
k¼0
j¼1
1 X
m 1 Y
k¼m
j¼1
! F^ j ðsÞ
j¼1 m 1 Y j¼1
! G^ j ðsÞ
k Y j¼m
! F^ j ðsÞ
k Y j¼m
! G^ j ðsÞ F^ kþ1 ðsÞð1 G^ kþ1 ðsÞÞ
ð7Þ
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¼
m 1 X
k Y
k¼0
j¼1
þ
m 1 Y
! F^ j ðsÞG^ j ðsÞ F^ kþ1 ðsÞð1 G^ kþ1 ðsÞÞ !
F^ j ðsÞG^ j ðsÞ ðK^ m1 ðsÞÞ2 L^ m1 ðsÞð1 L^ m2 ðsÞÞ
j¼1
1 1 K^ m1 ðsÞL^ m1 ðsÞ
.
It yields 1 L^ m2 ðsÞ ^ m1 ðsÞL^ m1 ðsÞ s!0þ 1 K
^ lim sup BðsÞp lim s!0þ
l m2 s þ oðsÞ l m2 ¼ ; þ l m1 sÞ þ oðsÞ km1 þ l m1
¼ lim
8mX1
s!0þ ðk m1
and consequently ^ lim sup BðsÞp lim
l m2 n . ¼ þ l m1 m þ n
(8)
m!1 km1
s!0þ
On the other hand, from (7) we see that ! ! m 1 Y k k X Y ^ BðsÞX F^ j ðsÞ G^ j ðsÞ F^ kþ1 ðsÞð1 G^ kþ1 ðsÞÞ k¼0
þ
j¼1 m 1 Y
j¼1
! ^ ^ F j ðsÞG j ðsÞ ðK^ m2 ðsÞÞ2 L^ m2 ðsÞð1 L^ m1 ðsÞÞ
j¼1
1 . ^ 1 K m2 ðsÞL^ m2 ðsÞ
^ ^ ^ ^ This inequality implies lim inf s!0þ BðsÞXlim s!0þ ð1 Lm1 ðsÞÞ=ð1 K m2 ðsÞLm2 ðsÞÞ ¼ l m1 =ðkm2 þ l m2 Þ and further l m1 n . (9) ¼ þ l m2 m þ n ^ ¼ n=ðm þ nÞ. The application of Tauberian Theorem (see Feller, Combining (8) and (9), we obtain lims!0þ BðsÞ ^ ¼ n=ðm þ nÞ and therefore Að1Þ ¼ limt!1 AðtÞ exists and 1971) gives Bð1Þ limt!1 BðtÞ ¼ lims!0þ BðsÞ Að1Þ ¼ 1 n=ðm þ mÞ ¼ m=ðm þ nÞ. & ^ lim inf BðsÞX lim
m!1 km2
s!0þ
Theorem 2. Suppose F j , G j , jX1, are dominated by a distribution function with finite mean. Moreover, F j ! F , G j ! G in distribution as j ! 1. If mj ! m, nj ! n, and 0om þ no1, then Að1Þ limt!1 AðtÞ exists and is given by Að1Þ ¼ m=ðm þ nÞ. Proof. For any mX1 define K m1 ðtÞ supjXm F j ðtÞ, K m2 ðtÞ ¼ inf jXm F j ðtÞ, Lm1 ðtÞ supjXm G j ðtÞ, and Lm2 ðtÞ ¼ inf jXm F j ðtÞ. By Lemma 3 we know that K m1 , K m2 , Lm1 , Lm2 , mX1 are distribution functions on st
½0; 1Þ. It can be seen that K m1 ðtÞXF j ðtÞ, 8tX0, 8jXm and so K m1 p F j , 8jXm. In the meantime, st
st
st
K m2 ðtÞpF j ðtÞ, 8tX0, 8jXm and thus F j p K m2 , 8jXm. This shows that K m1 p F j p K m2 ; 8jXm. Similarly it st
st
holds that Lm1 p G j p Lm2 ; 8jXm. For any t 2 C F it is easy to see that F ðtÞ ¼ limn!1 F n ðtÞ ¼ limm!1 inf jXm F j ðtÞ ¼ limm!1 supjXm F j ðtÞ and consequently K m1 ; K m2 ! F in distribution as m ! 1. Similarly it holds Lm1 ; Lm2 ! G in distribution as m ! 1. Obviously K m ðtÞ decreases in mX1 for any fixed tX0, and thus the survival function K¯ m1 ðtÞ increases in mX1. By the monotone convergence theorem it follows that km1 ! m as m ! 1. On the other hand, K¯ m2 ðtÞ, mX1, are dominated by a distribution function with finite mean, so by the dominance convergence theorem km2 ! m as m ! 1. Similarly, l m1 ; l m2 ! n as m ! 1. Now, the application of Theorem 1 yields the desired result. & Corollary. Suppose both sequences fF j ; jX1g and fG j ; jX1g are monotone according to the usual stochastic R1 R1 st ¯ dto1, ordering p. Let F j ! F and G j ! G in distribution as j ! 1. If m ¼ 0 F¯ ðtÞ dto1, n ¼ 0 GðtÞ
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R1 R1 ¯ 1 ðtÞ dto1 in the case of monotone increasing fF j g; fG j g, then m þ n40, and if m1 ¼ 0 F¯ 1 ðtÞ dto1, n1 ¼ 0 G the limiting availability Að1Þ exists and is given as Að1Þ ¼ limt!1 AðtÞ ¼ m=ðm þ nÞ. st
st
Proof. If F 1 p F 2 p . . . , and F j ! F in distribution, then fF j ; jX1g is dominated by F. Moreover, by the st st monotone convergence theorem it holds that limj!1 mj ¼ mo1. On the other hand, if F 1 p F 2 p . . ., then fF j ; jX1g is dominated by F 1 . Further, by the dominance convergence theorem it follows that limj!1 mj ¼ mo1 since in this case we have m1 o1. Similarly, we can see that the sequence fG j ; jX1g is also dominated and limj!1 nj ¼ no1. Applying Theorem 2, we obtain the desired result. & Theorem 3. Suppose that there exists pX1 such that F lpþj ðtÞ ¼ F j ðtÞ, G lpþj ðtÞ ¼ G j ðtÞ, 81pjpp, lX0, tX0. Let P P mj ; nj be the means associated with F j ; Gj , 1pjpp. If 0o pj¼1 mj þ pj¼1 nj o1, then the steady state availability Pp P of the system exists and is given as Að1Þ ¼ limt!1 AðtÞ ¼ j¼1 mj =ð pj¼1 ðmj þ nj ÞÞ. Proof. By (7) we have ^ ¼ BðsÞ
p1 Y j X j¼0
! ^ ^ F i ðsÞGi ðsÞ F^ jþ1 ðsÞð1 G^ jþ1 ðsÞÞ
i¼1
1ð
Qp
1 . F^ i ðsÞG^ i ðsÞÞ
(10)
i¼1
From (8) we have Pp p1 X nj 1 G^ jþ1 ðsÞ njþ1 Pp Pp j¼1 ¼ lim Qp ^ ^ s!0þ s!0þ 1 j¼1 ðmj þ nj Þ j¼1 ðmj þ nj Þ j¼0 j¼0 j¼1 F i ðsÞG i ðsÞ Pp Pp Therefore, Að1Þ ¼ 1 Bð1Þ ¼ j¼1 mj =ð j¼1 ðmj þ nj ÞÞ. & ^ ¼ lim BðsÞ
p1 X
Remark 1. The results of both Model A and Model B discussed in Biswas and Sarkar (2000) can be obtained as special cases of Theorem 3. Remark 2. Let fF ðt; yÞ; y 2 Yg be a family of lifetime distributions with a single parameter y where st Y ð1; 1Þ. Suppose that for this family it holds that y1 ; y2 2 Y, y1 oy2 implies F ðt; y1 Þ p F ðt; y2 Þ, i.e. st F ð; yÞ increases in y 2 Y according to the usual stochastic ordering p. Now let fyn g Y be a converging sequence with limn!1 yn ¼ y0 . Since fyn g must be a bounded sequence, we know that y inf nX1 yn and y supnX1 yn are finite. Actually, one of y and y must equal to one of yn ; nX1, and the other must equal to y0 since fyn ; nX1g is a converging sequence.We have y pyj py , 8jX1. If further y ; y 2 Y, then it follows st st that F ðt; y Þ p F ðt; yj Þ p F ðt; y Þ; 8jX1 That is, fF ðt; yj Þ; jX1g is a dominated sequence of distribution R1 functions on ½0; 1Þ. Moreover, if it is known that 0 F¯ ðt; yÞ dt, the mean associated with F ðt; yÞ, is a R1 continuous function of y, then we will also have limn!1 mn ¼ m0 , where mn 0 F¯ ðt; yn Þ dt and R1 m0 0 F¯ ðt; y0 Þ dt. Example 1. Assume that the uptime U j has Gamma distribution functionF j ðtÞ ¼ Gammaðt; aj ; bj Þ with pdf f j ðtÞ ¼
1 taj 1 et=bj ; Gðaj Þbj aj
jX1; t40.
Suppose that limj!1 aj ¼ a 2 ð0; 1Þ and limj!1 bj ¼ b 2 ð0; 1Þ. It is then evident that a ¼ inf jX1 aj 2 ð0; 1Þ, a ¼ supjX1 aj 2 ð0; 1Þ, b ¼ inf jX1 bj 2 ð0; 1Þ and b ¼ supjX1 bj 2 ð0; 1Þ. Note that 0oa0 pa00 o1 and st 0ob0 pb00 o1 imply that Gammaðt; a0 ; b0 Þ p Gamma ðt; a00 ; b00 Þ. Hence F j ðÞ ! Gammað; a; bÞ in distribution, st st and Gammað; a ; b Þ p F j ðÞ ppGammað; a ; b Þ; 8jX1. This and the fact that mj ¼ aj bj ! ab as j ! 1 show that fF j ; jX1g satisfies condition of Theorem 2. Now we assume that the repair time Dj has log-normal distribution G j ðtÞ ¼ LNðt; tj ; sÞ with pdf ( ) ðln t tj Þ2 1 gj ðtÞ ¼ pffiffiffiffiffiffi exp ; 8t40; 8jX1. 2s2 ts 2p
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st
st
We further assume limj!1 tj ¼ t 2 ð0; 1Þ. As the above we can obtain LNð; t ; sÞ p G j ðÞ p LNð; t ; sÞ; 8jX1, where t ¼ inf jX1 tj 2 ð0; 1Þ and tR ¼ supjX1 tj 2 ð0; 1Þ. In addition, it is true that G j ðÞ ! 1 LNð; t; sÞ GðÞ in distribution as j ! 1 and nj ¼ 0 F¯ j ðtÞ dt ¼ expðtj þ s2 =2Þ ! expðt þ s2 =2Þ ¼ n. Thus, fG j ; jX1g also satisfied all the conditions required in Theorem 2. Therefore, we conclude that the limiting availability of the system exists and is given as Að1Þ ¼
ab . ab þ expðt þ s2 =2Þ
References Barlow, R.E., Proschan, F., 1981. Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring, MD. Biswas, A., Sarkar, J., 2000. Availability of a system maintained through several imperfect repair before a replacement or a perfect repair. Statist. Probab. Lett. 50, 105–114. Feller, W., 1971. An Introduction to Probability Theory and its Applications. vol. 2, second ed., Wiley, New York. Mi, J., 1995. Limiting behavior of some measure of system availability. J. Appl. Probab. 32, 482–493. Sarkar, J., Chandhuri, G., 1999. Availability of a system with gamma life and exponential repair time under a perfect repair policy. Statist. Probab. Lett. 43, 189–196.
Statistics & Probability Letters 76 (2006) 729–736 www.elsevier.com/locate/stapro
Limiting availability of system with non-identical lifetime distributions and non-identical repair time distributions Jie Mi Department of Statistics, Florida International University, University Park, Miami, Florida 33199 USA Received 12 December 2003; received in revised form 19 September 2005 Available online 4 November 2005
Abstract The limiting availability, or steady state availability of a repairable system is studied. It is assumed that the lifetimes of system are independent but not necessarily identically distributed, and the times for repairing the failed systems are also independent but not necessarily identically distributed. It can be used to better model the real situation than the usual i.i.d. assumption. In this paper, sufficient conditions are given under which the limiting availability of the system exists and its closed-form expression is also derived. r 2005 Elsevier B.V. All rights reserved. MSC: primary 60K10; secondary 62N05 Keywords: Limiting availability; Dominated sequence of distribution functions; Laplace-Stieltjes transform; Tauberian theorem
1. Introduction Consider a system that has two states: ‘up’ or ‘down’. By ‘up’ we mean the system is still working and by ‘down’ we mean the system is not working; in the latter case the system is being repaired or replaced, depending on the maintenance policy. Suppose a new system starts operation at time t ¼ 0 and works until it fails. At the time of failure, the first up period is over and the first down period begins which ends when the repair or replacement is finished. The first up and down periods constitute the first cycle of the system. At the end of each subsequent down period a new cycle of the system is completed and the system will resume operating, and so on and so forth. Let U j and Dj denote the length of the jth up and down period, respectively. Basically U j is the lifetime of the system after the ðj 1Þth down period while Dj is the length of time needed to do repairing or replacement work. Letting 1 denote the up state, 0 the down state, and X ðtÞ be the state of the system at time t, then the (instant) availability of the system is defined as AðtÞ ¼ PðX ðtÞ ¼ 1Þ. Certainly it is desired to find the expression of AðtÞ, but this is too hard except for a few simple cases. In practice, engineers are more interested in the
Tel.: +1 305 348 2602; fax: +1 305 348 6895.
E-mail address: mi@fiu.edu. 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.10.004
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steady system availability Að1Þ limt!1 AðtÞ provided the limit exists, the steady system availability measures the extent to which the system will be available after it has been run for a long time. In many discussions fðU n ; Dn Þ; nX1g are assumed to be i.i.d. and it was shown that Að1Þ ¼ EðU 1 Þ= ðEðU 1 Þ þ EðD1 ÞÞ. For this result see, for example, Barlow and Proschan, 1981. The expression of AðtÞ is obtained in Sarkar and Chandhuri (1999) when U 1 and D1 follow gamma distribution and exponential distribution, respectively. The problem of finding AðtÞ and Að1Þ becomes more difficult when fðU n ; Dn Þ; nX1g are independent but not necessarily identically distributed. Mi (1995) considered other two quantities that can also measure system ¯ n , is the ratio of availability but are not exactly the same as AðtÞ defined in the above. One, denoted as A ¯n ¼ accumulated up time in the first n cycles to the total length of time in the n cycles A Pn Pn ¯ U =ð ðU þ D ÞÞ; the other, denoted as AðtÞ, is the ratio of the total up time in interval ½0; t to the j j j j¼1 j¼1 ¯ ¼ the total up timein ½0; t=t. Under some mild conditions, Mi (1995) length t of this time interval, i.e., AðtÞ ¯ ¯ and showed the asymptotic normality of An and AðtÞ. derived both limn!1 An and limt!1 AðtÞ, Biswas and Sarkar (2000) considered a similar problem. Let k be a predetermined positive integer. The system will experience the first k cycles. At the end of the ðk þ 1Þth up period, the system will be replaced (Model A) or be repaired perfectly (Model B). That paper assumed that U j and Dj follow distribution F j and G j , respectively,1pjpk þ 1. They then obtained the Fourier transform of the instantaneous system unavailability BðtÞ ¼ 1 AðtÞ. By using the inversion formula, they thus obtained closed-form expressions of AðtÞ for exponentially distributed U j and Dj ð1pjpk þ 1Þ for both Model A and Model B with k ¼ 1; 2, and also derived the steady system availability Að1Þ. In this paper, we will continue studying the system considered in Mi (1995). The paper is organized as follows. Some useful lemmas are given in the next section. In Section 3, different sufficient conditions are given under which the steady state availability of the system exists and its expression is derived.
2. Some useful lemmas We keep the same notation as in the previous section and consider the model studied in Mi (1995). In the following, we denote the distribution functions of U j and Dj as F j and G j , respectively. Mi (1995) introduced the concept that a collection of distributions on ½0; 1Þ is dominated. That is, let fF i ; i 2 Ig be a collection of distribution functions on ½0; 1Þ where I is an index set. If there exist a constant 0oLo1 and a distribution function F such that F¯ i ðtÞpLF¯ ðtÞ, 8i 2 I, 8tX0, where F¯ ðtÞ 1 F ðtÞ, then we say that the collection fF i ; i 2 Ig is dominated (by F). In the rest of the paper, we will assume the distribution functions fF j ; jX1g and fG j ; jX1g associated with up times fU j ; jX1g and down times fDj ; jX1g are dominated. According to Definition 1 in Mi (1995) it can be easily verified that if the sequences fF j ; jX1g and fG j ; jX1g are dominated, then the sequence fH j ; jX1g is also dominated, where H j ¼ F j G j is the convolution of F j and G j . P P Lemma 1. Suppose fF j ; jX1g and fG j ; jX1g are dominated and nj¼1 mj =n ! mo1, nj¼1 nj =n ! no1 with R1 R1 P ¯ j ðtÞ dt. Then for any tX0 limn!1 Pð n ðU i þ m þ n40 where mj ¼ EðU j Þ ¼ 0 F¯ j ðtÞ dt and nj ¼ EðDj Þ ¼ 0 G i¼1 Di ÞptÞ ¼ 0 and this convergence is uniform on any finite interval ½0; a. Proof. Since fF j ; jX1g andPfG j ; jX1g are dominated sequences of distribution functions on ½0; 1Þ it follows from Mi (1995) that ð1=nÞ ni¼1 ðU i þ Di Þ ! m þ n; a.s. This particularly implies that n 1X P ðU i þ Di Þ ! m þ n. n i¼1
(1)
For any given tX0 let n0 be large enough such that t mþn o ; n 2
8nXn0 .
(2)
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Then for any nXn0 we have ! ! n n X 1X mþn P ðU i þ Di Þpt pP ðU i þ Di Þ ðm þ nÞp n i¼1 2 i¼1 ! mþn 1 X n pP ðU þ Di Þ ðm þ nÞX n i¼1 i 2 Pn and thus limn!1 P i¼1 ðU i þ Di Þpt ¼ 0 by (1). The uniform convergence on ½0; a can be obtained from replacing (2) by a=noðm þ nÞ=2. & d
d
For any jX1 let random variables U i and Di satisfy U i ¼ U iþj1 , Di ¼ Diþj1 , 8iX1, i.e. U i ; Di have distribution functions F iþj1 , and G iþj1 , respectively. Denote the unavailability of the system associated with fðU i ; Di Þ; iX1g as Bj ðtÞ. Lemma 2. Suppose that fF i ðtÞ, G i ðtÞ, jX1g is dominated. Then (i) Bj ðtÞ is right-continuous in tX0; (ii) Bj ðtÞ has at most countable discontinuity points; (iii) Bj ðtÞ ¼
1 X ½ðH j H jþ1 H jþk1 Þ F jþk ðtÞ ðH j H jþk1 H jþk ÞðtÞ,
(3)
k¼0
where H i ¼ F i G i is the convolution of F i and G i ; and (iv) Bj ðtÞ is a continuous function if F j ðtÞ, Gj ðtÞ, jX1 are. Proof. Let binary random variable X ðtÞ be the state of the system. We have fX ðtÞ ¼ 0g ¼ fU 1 otpU 1 þ D1 g [ fX ðtÞ ¼ 0; U 1 þ D1 otg ¼ fU 1 otpU 1 þ D1 g [ fU 1 þ D1 þ U 2 otpU 1 þ D1 þ U 2 þ D2 g [ fX ðtÞ ¼ 0; U 1 þ D1 þ U 2 þ D2 otg. By induction it can be obtained that ( !) k kþ1 X X n fX ðtÞ ¼ 0g ¼ [ ðU m þ Dm Þ þ U kþ1 otp ðU m þ Dm Þ k¼0
( [
m¼1
X ðtÞ ¼ 0;
nþ1 X
)
m¼1
ðU m þ Dm Þot .
m¼1
Bj ðtÞ ¼ PðX ðtÞ ¼ 0Þ ¼ P X ðtÞ ¼ 0;
nþ1 X
! ðU m þ Dm Þot
m¼1
þ
n X
½ðH j H jþ1 H jþk1 Þ F jþk ðtÞ ðH j H jþk1 H jþk ÞðtÞ.
ð4Þ
k¼0
We For any given t0 X0 and 40, there exists sufficiently large n0 such that Pfirst prove (i). n0 þ1 P m¼1 ðU m þ Dm Þot o, 8t0 ptpt0 þ 1 by the uniform convergence in Lemma 1. Hence ! ! nX nX 0 þ1 0 þ1 P X ðtÞ ¼ 0; ðU m þ Dm Þot pP ðU m þ Dm Þot o; 8t0 ptpt0 þ 1. (5) m¼1
m¼1
For the fixed n0 , there exists 0odo1 such that n0 X ½ðH j H jþ1 H jþk1 Þ F jþk ðtÞ ðH j H jþk1 H jþk ÞðtÞ k¼0
½ðH j H jþ1 H jþk1 Þ F jþk ðt0 Þ ðH j H jþk1 H jþk Þðt0 Þo k¼0 n0 X
ð6Þ
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for any t 2 ½t0 ; t0 þ d since all F i ; G i , iX1 are right-continuous. We thus obtain from (4), (5), and (6) that jBj ðtÞ Bj ðt0 Þjp3; 8t0 ptpt0 þ d. This shows that Bj ðtÞ is right-continuous in tX0. Letting t ! 1 in (4) and applying Lemma 1, we see that (3) holds. Let D be the set of all the discontinuity points of F i ; Gi , iX1. Obviously D is a countable set. It can be shown in the same way as proving (i) that Bj ðtÞ is continuous at every point of ½0; 1ÞnD. Thus, Bj ðtÞ has at most countable discontinuity points. If, in particular, all F i ; G i , iX1 are continuous, then it is straight forward to see that Bj ðtÞ is also a continuous function of tX0. & Under the conditions of Lemma 2 the function 0pBj ðtÞp1 has at most countable many discontinuity points. That R 1 is, Bj ðtÞ is almost everywhere (in Lebesque measure) continuous. Thus it is well known that the integral 0 est dBj ðtÞ exists. This is the so-called the Laplace-Stieltjes transform of Bj ðtÞ and is denoted as B^ j ðsÞ, 8s40. P Lemma 2 implies that the unavailability of the original system fðU i ; Di Þ; iX1g is BðtÞ ¼ B1 ðtÞ ¼ 1 k¼0 ½ðH 1 H 2 H k Þ F kþ1 ðtÞ ðH 1 H k H kþ1 ÞðtÞ and the Laplace–Stieltjes transform of BðtÞ is given as " ! ! ! !# 1 k k kþ1 kþ1 Y Y Y X Y ^ ¼ B^ 1 ðsÞ ¼ BðsÞ F^ j ðsÞ F^ j ðsÞ G^ j ðsÞ F^ kþ1 ðsÞ G^ j ðsÞ k¼0
¼
j¼1
1 X
k Y
k¼0
j¼1
F^ j ðsÞ
!
j¼1 k Y
!
j¼1
j¼1
G^ j ðsÞ F^ kþ1 ðsÞð1 G^ kþ1 ðsÞÞ.
j¼1
Lemma 3. Let F i , iX1 be a sequence of distribution functions in ½0; 1Þ. Suppose fF i ; iX1g converges to F in distribution. Define F ðtÞ ¼ supiX1 F i ðtÞ and F ðtÞ ¼ inf iX1 F i ðtÞ, then both F ðtÞ and F ðtÞ are distribution functions on ½0; 1Þ. Proof. First we prove the result for F ðtÞ. Obviously F ðtÞX0 is non-decreasing in tX0 and F ð0Þ ¼ 0. Let us show that F ð1Þ ¼ 1. For any given 0oo1, there exists t0 2 C F \ ð0; 1Þ such that F ðt0 Þ41 , where C F is the set of continuity points of F. For the fixed 40 and t0 there exists n0 such that jF i ðt0 Þ F ðt0 Þjo, 8iXn0 . It implies that F i ðt0 Þ4F ðt0 Þ 41 2, 8iXn0 . Thus F ðt0 Þ ¼ supiX1 F i ðt0 ÞXsupiXn0 F i ðt0 ÞX1 2 and consequently F ð1Þ ¼ 1 since 40 is arbitrary. To prove F ðtÞ is right continuous, we first show that for any fixed n, the function F n ðtÞ supfF 1 ðtÞ; . . . ; F n ðtÞg is right continuous. By induction, it suffices to prove this result for n ¼ 2. For any chosen t0 X0, suppose F 2 ðt0 Þ ¼ F 2 ðt0 Þ. For any given 40, there exists d40 such that 0pF i ðtÞ F i ðt0 Þo, 8t0 ptpt0 þ d, i ¼ 1; 2 due to the right continuity of F i ðtÞ. Thus if t0 ptpt0 þ d then F 2 ðtÞ ¼ maxfF 1 ðtÞ; F 2 ðtÞg p maxfF 1 ðt0 Þ þ ; F 2 ðt0 Þ þ g ¼ F 2 ðt0 Þ þ ¼ F 2 ðt0 Þ þ . This means that 0pF 2 ðtÞ F 2 ðt0 Þp, 8t0 ptpt0 þ d and consequently F 2 ðtÞ is right-continuous. Now we need to show that F ðtÞ is right-continuous. Let t0 X0 and 40 be given. Consider the following two cases. Case 1: F ðt0 Þ ¼ F ðt0 Þ. For any given 40, there exists d1 40 such that 0pF ðtÞ F ðt0 Þp for any t0 ptpt0 þ d1 since F is right-continuous. We can also assume that t þ d1 2 C F , the set of continuity points of F ðÞ. Since fF i g converges to F in distribution, there exists n0 such that jF i ðt0 þ d1 Þ F ðt0 þ d1 Þjo, 8iXn0 . That is, F i ðt0 þ d1 ÞoF ðt0 þ d1 Þ þ , 8iXn0 and consequently supiXn0 F i ðt0 þ d1 ÞpF ðt0 þ d1 Þ þ . This further implies that supiXn0 F i ðtÞpF ðt0 þ d1 Þ þ , 8t0 ptpt0 þ d1 since supiXn0 F i ðtÞ is nondecreasing. With the fixed n0 there exists d2 40 such that F i ðtÞoF i ðt0 Þ þ , 8t0 ptpt0 þ d2 , 81pipn0 . Let d ¼ minfd1 ; d2 g40. Then for any t0 ptpt0 þ d we have F ðtÞ ¼ maxfsupfF 1 ðtÞ; . . . ; F n0 ðtÞg; supfF n0 ðtÞ; . . .ggp maxfF ðt0 Þ þ ; F ðt0 þ d1 Þ þ g ¼ F ðt0 þ d1 Þ þ pF ðt0 Þ þ þ ¼ F ðt0 Þ þ 2 ¼ F ðt0 Þ þ 2. This implies that 0pF ðtÞ F ðt0 Þp2, 8t0 ptpt0 þ d and thus F is right-continuous at t0 . Case 2: F ðt0 Þ ¼ F j ðt0 Þ for 1pjo1. In this case we can assume F j ðt0 Þ4F ðt0 Þ. For any given 40 there exists d1 40 such that t0 þ d1 2 C F and F j ðtÞoF j ðt0 Þ þ , F ðtÞoF ðt0 Þ þ , 8t0 ptpt0 þ d1 . For this chosen d1 40 there exists n0 such that jF i ðt0 þ d1 Þ F ðt0 þ d1 Þjo, 8iXn0 . Hence supiXn0 F i ðt0 þ d1 ÞpF ðt0 þ d1 Þ þ and consequently supiXn0 F i ðtÞpF ðt0 þ d1 Þ þ , 8t0 ptp0 þ d1 . For the given 40, there exists d2 40 such that F i ðtÞoF i ðt0 Þ þ , 81pipn0 ,8t0 ptpt0 þ d2 due to the right-continuity of F i , iX1. Let d ¼ minfd1 ; d2 g40.
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Then for any t0 ptpt0 þ d we have F ðtÞ ¼ max supfF 1 ðtÞ; . . . ; F n0 ðtÞg; sup F i ðtÞ iXn0
p maxfF j ðt0 Þ þ ; F ðt0 þ d1 Þ þ gpF j ðt0 Þ þ 2 ¼ F ðt0 Þ þ 2. That is 0pF ðtÞ F ðt0 Þp2, 8t0 ptpt0 þ d and so F ðÞ is right-continuous at t0 . From the above we see that F ðtÞ is a right-continuous function and hence must be a distribution function on ½0; 1Þ. In the following we consider F ðtÞ inf iX1 F i ðtÞ. Let us show F ð1Þ ¼ 1. For any given 40, there exists t0 2 C F such that F ðt0 Þ41 =2. There also exists sufficiently large n0 such that F i ðt0 Þ4F ðt0 Þ =2, 8iXn0 . This gives inf iXn0 F i ðt0 ÞXF ðt0 Þ =2X1 . For the same 40, there exists t1 40 such that F i ðt1 Þ41 , 1pipn0 . Therefore, if tX maxðt0 ; t1 Þ, then F ðtÞ ¼ inf F i ðtÞ ¼ min min F i ðtÞ; inf F i ðtÞ X minf1 ; 1 g ¼ 1 . iX1
iXn0
1pipn0
This certainly implies F ð1Þ ¼ 1. To show that F ðtÞ is right-continuous let t0 X0 be arbitrary. Case 1: F ðt0 Þ ¼ F ðt0 Þ. This gives F i ðt0 ÞXF ðt0 Þ, 8iX1. For any given 40, there exits d1 40 such that 0pF ðtÞ F ðt0 Þp, 8t0 ptpt0 þ d1 . For the given 40, there exists 0odod1 such that t0 þ d 2 C F and there exists n0 satisfying F i ðt0 þ dÞpF ðt0 þ dÞ þ , 8iXn0 and so F i ðtÞpF ðt0 þ dÞ þ , 8iXn0 , 8t0 ptpt0 þ d. Then for all t0 ptpt0 þ d we have F ðtÞp inf F i ðtÞpF ðt0 þ dÞ þ pðF ðt0 Þ þ Þ þ ¼ F ðt0 Þ þ 2 ¼ F ðt0 Þ þ 2. iXn0
That is, F ðÞ is right-continuous at t0 . Case 2: F ðt0 Þ ¼ F j ðt0 Þ for 1pjo1. For any given 40 there exists d40 such that 0pF j ðtÞ F j ðt0 Þp, 8t0 ptpt0 þ d. Thus, F ðtÞpF j ðtÞpF j ðt0 Þ þ ¼ F ðt0 Þ þ , 8t0 ptpt0 þ d and so F ðÞ is right-continuous at t0 . & Summarizing the above, we see that F ðtÞ is also right-continuous and consequently it must be a CDF on ½0; 1Þ.
3. Main results st
st
Theorem 1. Suppose that there exist CDFs F, G, and K m1 ; K m2 ; Lm1 ; Lm2 , mX1 satisfying (i) K m1 p F j p K m2 , st st 8jXm; (ii) Lm1 p G j p Lm2 , 8jXm; (iii) K m1 ! F ; K m2 ! F ; Lm1 ! G; Lm2 ! G in distribution as m ! 1; (iv) km1 ; km2 ! m; l m1 ; l m2 ! n as m ! 1, where m; n are the means associated with CDFs F ðÞ and GðÞ with 0omx þ no1, and km1 ; km2 ; l m1 ; l m2 are the means associated with CDFs K m1 ; K m2 ; Lm1 ; Lm2 , respectively. Then the steady state availability of the system exists and is given as Að1Þ ¼ limt!1 AðtÞ ¼ m=m þ n. Proof. It can be seen that the conditions (i) and (ii) show that fF j ; jX1g andfG j ; jX1g are dominated. From conditions (i) and (ii) we can also obtain K^ m1 ðsÞXF^ j ðsÞXK^ m2 ðsÞ, and L^ m1 ðsÞXG^ j ðsÞXL^ m2 ðsÞ; 8sX0; 8jXm. ^ ¼ From Lemma 2 the Laplace–Stieltjes transform of the unavailability of the system is given by BðsÞ P1 Q k Qk ^ B^ 1 ðsÞ ¼ k¼0 ð j¼1 F^ j ðsÞÞð j¼1 G j ðsÞÞF^ kþ1 ðsÞð1 G^ kþ1 ðsÞÞ. For any mX1 we have ! ! m 1 Y k k Y X ^ ¼ BðsÞ F^ j ðsÞ G^ j ðsÞ F^ kþ1 ðsÞð1 G^ kþ1 ðsÞÞ
þ
k¼0
j¼1
1 X
m 1 Y
k¼m
j¼1
! F^ j ðsÞ
j¼1 m 1 Y j¼1
! G^ j ðsÞ
k Y j¼m
! F^ j ðsÞ
k Y j¼m
! G^ j ðsÞ F^ kþ1 ðsÞð1 G^ kþ1 ðsÞÞ
ð7Þ
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¼
m 1 X
k Y
k¼0
j¼1
þ
m 1 Y
! F^ j ðsÞG^ j ðsÞ F^ kþ1 ðsÞð1 G^ kþ1 ðsÞÞ !
F^ j ðsÞG^ j ðsÞ ðK^ m1 ðsÞÞ2 L^ m1 ðsÞð1 L^ m2 ðsÞÞ
j¼1
1 1 K^ m1 ðsÞL^ m1 ðsÞ
.
It yields 1 L^ m2 ðsÞ ^ m1 ðsÞL^ m1 ðsÞ s!0þ 1 K
^ lim sup BðsÞp lim s!0þ
l m2 s þ oðsÞ l m2 ¼ ; þ l m1 sÞ þ oðsÞ km1 þ l m1
¼ lim
8mX1
s!0þ ðk m1
and consequently ^ lim sup BðsÞp lim
l m2 n . ¼ þ l m1 m þ n
(8)
m!1 km1
s!0þ
On the other hand, from (7) we see that ! ! m 1 Y k k X Y ^ BðsÞX F^ j ðsÞ G^ j ðsÞ F^ kþ1 ðsÞð1 G^ kþ1 ðsÞÞ k¼0
þ
j¼1 m 1 Y
j¼1
! ^ ^ F j ðsÞG j ðsÞ ðK^ m2 ðsÞÞ2 L^ m2 ðsÞð1 L^ m1 ðsÞÞ
j¼1
1 . ^ 1 K m2 ðsÞL^ m2 ðsÞ
^ ^ ^ ^ This inequality implies lim inf s!0þ BðsÞXlim s!0þ ð1 Lm1 ðsÞÞ=ð1 K m2 ðsÞLm2 ðsÞÞ ¼ l m1 =ðkm2 þ l m2 Þ and further l m1 n . (9) ¼ þ l m2 m þ n ^ ¼ n=ðm þ nÞ. The application of Tauberian Theorem (see Feller, Combining (8) and (9), we obtain lims!0þ BðsÞ ^ ¼ n=ðm þ nÞ and therefore Að1Þ ¼ limt!1 AðtÞ exists and 1971) gives Bð1Þ limt!1 BðtÞ ¼ lims!0þ BðsÞ Að1Þ ¼ 1 n=ðm þ mÞ ¼ m=ðm þ nÞ. & ^ lim inf BðsÞX lim
m!1 km2
s!0þ
Theorem 2. Suppose F j , G j , jX1, are dominated by a distribution function with finite mean. Moreover, F j ! F , G j ! G in distribution as j ! 1. If mj ! m, nj ! n, and 0om þ no1, then Að1Þ limt!1 AðtÞ exists and is given by Að1Þ ¼ m=ðm þ nÞ. Proof. For any mX1 define K m1 ðtÞ supjXm F j ðtÞ, K m2 ðtÞ ¼ inf jXm F j ðtÞ, Lm1 ðtÞ supjXm G j ðtÞ, and Lm2 ðtÞ ¼ inf jXm F j ðtÞ. By Lemma 3 we know that K m1 , K m2 , Lm1 , Lm2 , mX1 are distribution functions on st
½0; 1Þ. It can be seen that K m1 ðtÞXF j ðtÞ, 8tX0, 8jXm and so K m1 p F j , 8jXm. In the meantime, st
st
st
K m2 ðtÞpF j ðtÞ, 8tX0, 8jXm and thus F j p K m2 , 8jXm. This shows that K m1 p F j p K m2 ; 8jXm. Similarly it st
st
holds that Lm1 p G j p Lm2 ; 8jXm. For any t 2 C F it is easy to see that F ðtÞ ¼ limn!1 F n ðtÞ ¼ limm!1 inf jXm F j ðtÞ ¼ limm!1 supjXm F j ðtÞ and consequently K m1 ; K m2 ! F in distribution as m ! 1. Similarly it holds Lm1 ; Lm2 ! G in distribution as m ! 1. Obviously K m ðtÞ decreases in mX1 for any fixed tX0, and thus the survival function K¯ m1 ðtÞ increases in mX1. By the monotone convergence theorem it follows that km1 ! m as m ! 1. On the other hand, K¯ m2 ðtÞ, mX1, are dominated by a distribution function with finite mean, so by the dominance convergence theorem km2 ! m as m ! 1. Similarly, l m1 ; l m2 ! n as m ! 1. Now, the application of Theorem 1 yields the desired result. & Corollary. Suppose both sequences fF j ; jX1g and fG j ; jX1g are monotone according to the usual stochastic R1 R1 st ¯ dto1, ordering p. Let F j ! F and G j ! G in distribution as j ! 1. If m ¼ 0 F¯ ðtÞ dto1, n ¼ 0 GðtÞ
ARTICLE IN PRESS J. Mi / Statistics & Probability Letters 76 (2006) 729–736
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R1 R1 ¯ 1 ðtÞ dto1 in the case of monotone increasing fF j g; fG j g, then m þ n40, and if m1 ¼ 0 F¯ 1 ðtÞ dto1, n1 ¼ 0 G the limiting availability Að1Þ exists and is given as Að1Þ ¼ limt!1 AðtÞ ¼ m=ðm þ nÞ. st
st
Proof. If F 1 p F 2 p . . . , and F j ! F in distribution, then fF j ; jX1g is dominated by F. Moreover, by the st st monotone convergence theorem it holds that limj!1 mj ¼ mo1. On the other hand, if F 1 p F 2 p . . ., then fF j ; jX1g is dominated by F 1 . Further, by the dominance convergence theorem it follows that limj!1 mj ¼ mo1 since in this case we have m1 o1. Similarly, we can see that the sequence fG j ; jX1g is also dominated and limj!1 nj ¼ no1. Applying Theorem 2, we obtain the desired result. & Theorem 3. Suppose that there exists pX1 such that F lpþj ðtÞ ¼ F j ðtÞ, G lpþj ðtÞ ¼ G j ðtÞ, 81pjpp, lX0, tX0. Let P P mj ; nj be the means associated with F j ; Gj , 1pjpp. If 0o pj¼1 mj þ pj¼1 nj o1, then the steady state availability Pp P of the system exists and is given as Að1Þ ¼ limt!1 AðtÞ ¼ j¼1 mj =ð pj¼1 ðmj þ nj ÞÞ. Proof. By (7) we have ^ ¼ BðsÞ
p1 Y j X j¼0
! ^ ^ F i ðsÞGi ðsÞ F^ jþ1 ðsÞð1 G^ jþ1 ðsÞÞ
i¼1
1ð
Qp
1 . F^ i ðsÞG^ i ðsÞÞ
(10)
i¼1
From (8) we have Pp p1 X nj 1 G^ jþ1 ðsÞ njþ1 Pp Pp j¼1 ¼ lim Qp ^ ^ s!0þ s!0þ 1 j¼1 ðmj þ nj Þ j¼1 ðmj þ nj Þ j¼0 j¼0 j¼1 F i ðsÞG i ðsÞ Pp Pp Therefore, Að1Þ ¼ 1 Bð1Þ ¼ j¼1 mj =ð j¼1 ðmj þ nj ÞÞ. & ^ ¼ lim BðsÞ
p1 X
Remark 1. The results of both Model A and Model B discussed in Biswas and Sarkar (2000) can be obtained as special cases of Theorem 3. Remark 2. Let fF ðt; yÞ; y 2 Yg be a family of lifetime distributions with a single parameter y where st Y ð1; 1Þ. Suppose that for this family it holds that y1 ; y2 2 Y, y1 oy2 implies F ðt; y1 Þ p F ðt; y2 Þ, i.e. st F ð; yÞ increases in y 2 Y according to the usual stochastic ordering p. Now let fyn g Y be a converging sequence with limn!1 yn ¼ y0 . Since fyn g must be a bounded sequence, we know that y inf nX1 yn and y supnX1 yn are finite. Actually, one of y and y must equal to one of yn ; nX1, and the other must equal to y0 since fyn ; nX1g is a converging sequence.We have y pyj py , 8jX1. If further y ; y 2 Y, then it follows st st that F ðt; y Þ p F ðt; yj Þ p F ðt; y Þ; 8jX1 That is, fF ðt; yj Þ; jX1g is a dominated sequence of distribution R1 functions on ½0; 1Þ. Moreover, if it is known that 0 F¯ ðt; yÞ dt, the mean associated with F ðt; yÞ, is a R1 continuous function of y, then we will also have limn!1 mn ¼ m0 , where mn 0 F¯ ðt; yn Þ dt and R1 m0 0 F¯ ðt; y0 Þ dt. Example 1. Assume that the uptime U j has Gamma distribution functionF j ðtÞ ¼ Gammaðt; aj ; bj Þ with pdf f j ðtÞ ¼
1 taj 1 et=bj ; Gðaj Þbj aj
jX1; t40.
Suppose that limj!1 aj ¼ a 2 ð0; 1Þ and limj!1 bj ¼ b 2 ð0; 1Þ. It is then evident that a ¼ inf jX1 aj 2 ð0; 1Þ, a ¼ supjX1 aj 2 ð0; 1Þ, b ¼ inf jX1 bj 2 ð0; 1Þ and b ¼ supjX1 bj 2 ð0; 1Þ. Note that 0oa0 pa00 o1 and st 0ob0 pb00 o1 imply that Gammaðt; a0 ; b0 Þ p Gamma ðt; a00 ; b00 Þ. Hence F j ðÞ ! Gammað; a; bÞ in distribution, st st and Gammað; a ; b Þ p F j ðÞ ppGammað; a ; b Þ; 8jX1. This and the fact that mj ¼ aj bj ! ab as j ! 1 show that fF j ; jX1g satisfies condition of Theorem 2. Now we assume that the repair time Dj has log-normal distribution G j ðtÞ ¼ LNðt; tj ; sÞ with pdf ( ) ðln t tj Þ2 1 gj ðtÞ ¼ pffiffiffiffiffiffi exp ; 8t40; 8jX1. 2s2 ts 2p
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st
st
We further assume limj!1 tj ¼ t 2 ð0; 1Þ. As the above we can obtain LNð; t ; sÞ p G j ðÞ p LNð; t ; sÞ; 8jX1, where t ¼ inf jX1 tj 2 ð0; 1Þ and tR ¼ supjX1 tj 2 ð0; 1Þ. In addition, it is true that G j ðÞ ! 1 LNð; t; sÞ GðÞ in distribution as j ! 1 and nj ¼ 0 F¯ j ðtÞ dt ¼ expðtj þ s2 =2Þ ! expðt þ s2 =2Þ ¼ n. Thus, fG j ; jX1g also satisfied all the conditions required in Theorem 2. Therefore, we conclude that the limiting availability of the system exists and is given as Að1Þ ¼
ab . ab þ expðt þ s2 =2Þ
References Barlow, R.E., Proschan, F., 1981. Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring, MD. Biswas, A., Sarkar, J., 2000. Availability of a system maintained through several imperfect repair before a replacement or a perfect repair. Statist. Probab. Lett. 50, 105–114. Feller, W., 1971. An Introduction to Probability Theory and its Applications. vol. 2, second ed., Wiley, New York. Mi, J., 1995. Limiting behavior of some measure of system availability. J. Appl. Probab. 32, 482–493. Sarkar, J., Chandhuri, G., 1999. Availability of a system with gamma life and exponential repair time under a perfect repair policy. Statist. Probab. Lett. 43, 189–196.