Comparison of two repairable systems

Statistics and Probability Letters 81 (2011) 446–450

Contents lists available at ScienceDirect

Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro

Comparison of two repairable systems Asok K. Nanda a,∗ , Amarjit Kundu b a

Department of Mathematical Sciences, IISER Kolkata, Mohanpur Campus, Mohanpur 741252, West Bengal, India

b

Department of Mathematics, Santipur College, Santipur, Dist. Nadia, West Bengal, India

article

info

Article history: Received 23 August 2010 Received in revised form 29 October 2010 Accepted 15 December 2010 Available online 22 December 2010 MSC: 60E15 60K10 Keywords: Ageing properties Minimal repair Repairable system

abstract Any repairable system improves (deteriorates) with time if the interarrival times of failure tend to get larger (smaller) in some sense. In this paper we consider two such repairable systems, and their performance in terms of several partial orderings of their respective interarrival times of failure are compared. The comparison of two systems’ improvement/deterioration under minimal repair policy has been characterized in terms of s-FR orders and also in terms of their shifted and dispersive versions. These results generalize some of the existing results in the literature and also provide some new results in this direction. © 2010 Elsevier B.V. All rights reserved.

1. Introduction A repairable system can improve or deteriorate with the passage of time. Any repairable system is said to improve (deteriorate) if the interarrival times of failure tend to get larger (smaller) in terms of several known partial orderings (cf. Ebrahimi (1989)). The comparison of interarrival times in terms of different partial orders viz. LR (Likelihood Ratio), FR (Failure Rate), ST (Stochastic), MRL (Mean Residual Life) orders has been discussed in Bagai and Jain (1994) and Deshpande and Singh (1995). Nanda (1997) has compared interarrival times with respect to s-FR and s-ST orders so that the results of Bagai and Jain (1994) and Deshpande and Singh (1995) become particular cases. For definitions of s-FR and s-ST orderings one may refer to Fagiuoli and Pellerey (1993) (see also Mukherjee and Chatterjee (1992)) where they have taken s to be any nonnegative integer. Nanda and Kundu (2010) compare interarrival times with respect to s-FR, s-ST and s-CX orders where they take s to be any nonnegative real number. They have also discussed improvement (deterioration) of a repairable system in terms of up s-FR and down s-FR orders. It is to be mentioned here that for s = 0, 1, 2, 3, s-FR order reduces respectively to LR, FR, MRL and VRL orderings. For definitions and different properties of different shifted LR and FR orderings one may refer to Lillo et al. (2000). Write N + = {1, 2, . . .}. Let {S (k) (n), n ∈ N + }, k = 1, 2, be two sequences of nonnegative random variables such that 0 ≡ S (k) (0) < S (k) (1) < · · · < S (k) (n) < · · ·. S (k) (n) is defined as the time of nth failure for the kth system. Let X (n) = S (1) (n) − S (1) (n − 1) and Y (n) = S (2) (n) − S (2) (n − 1), n ∈ N + . Then X (n) and Y (n) denote respectively the time between (n − 1)th failure and nth failure of the two systems. Again, let N (k) (t ) = sup{n : S (k) (n) ≤ t }, k = 1, 2. Thus, N (k) (t ) denotes the number of failures of the kth system by time t. Let Xsn−1 (n) denote the random variable [X (n)|S (1) (n − 1) = sn−1 , S (1) (n − 2) = sn−2 , . . . , S (1) (1) = s1 ], where 0 = s0 < s1 < · · · < sn−1 are real numbers and n = 1, 2, 3, . . . . Similarly, define Yln−1 (n) ≡ [Y (n)|S (2) (n − 1) =

∗

Corresponding author. E-mail address: [email protected] (A.K. Nanda).

0167-7152/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2010.12.008

A.K. Nanda, A. Kundu / Statistics and Probability Letters 81 (2011) 446–450

447

ln−1 , S (2) (n − 2) = ln−2 , . . . , S (2) (1) = l1 ], where 0 = l0 < l1 < · · · < ln−1 are also real numbers and n = 1, 2, 3, . . . . Further, write X0 (1) = X (1) and Y0 (1) = Y (1). In this paper, we compare interarrival times of failure of two repairable systems with respect to generalized orderings defined in Hu et al. (2004). The organization of this paper is as follows. In Section 2, we give different notations and definitions to be used in this paper. Some results related to the generalized stochastic orders for comparison of the interarrival times of failure of two repairable systems are given in Section 3. Here we have characterized different generalized stochastic orders in terms of improvement/deterioration comparison of two systems. Finally, Section 4 concludes. Throughout the paper the words increasing (resp. decreasing) and nondecreasing (resp. nonincreasing) are used interchangeably. Here we consider only the lifetime random variables. 2. Notations, definitions and preliminaries The following definition may be obtained in Hu et al. (2004). For s > 0, let (s)

γ ( x) =



(−x)(s−1) /Γ (s), 0,

x≤0 x > 0.

Let X be a random variable with distribution function F . Define

¯ sX (x) = E [γ (s) (x − X )] Φ ∫ ∞ (t − x)s−1 = dF (t ) (2.1) Γ ( s) x ¯ 0X (x) as the probability density function of X , if exists. Here s may be any nonnegative real number (not necessarily with Φ ¯ sX (x) takes the form restricted to be an integer). If s ∈ N + , then Φ ∫ ∞∫ ∞ ∫ ∞ ¯ sX (x) = Φ ... dF (t )dx1 . . . dxs−1 x

xs−1

x1

¯ sX (x) may be considered as the generalized iterated integral of the survival function of X . Clearly, for all x. For general s ≥ 0, Φ ¯ sX (x) = 0 lim Φ

x→∞

for all s ≥ 0,

(2.2)

and ∞

∫

¯ sX (u)du = Φ ¯ sX+1 (x). Φ

(2.3)

x

Definition 2.1. Let X and Y be two nonnegative random variables with distribution functions F and G respectively, and s be any nonnegative real number. X is said to be smaller than Y in (a) s-FR order (written as X ≤s−FR Y ) if

¯ sX (x) Φ ¯ sY (x) Φ

is decreasing in x ≥ 0;

(2.4)

(b) up shifted s-FR order (written as X ≤s−FR↑ Y ) if X − x ≤s−FR Y

for every x ≥ 0;

(2.5)

(c) down shifted s-FR order (written as X ≤s−FR↓ Y ) if X ≤s−FR [Y − x|Y > x] for every x ≥ 0;

(2.6)

(d) s-FR order of the dispersion-type (written as X ≤disp−(s−FR) Y ), if

[X − F −1 (p)|X > F −1 (p)] ≤s−FR [Y − G−1 (p)|Y > G−1 (p)],

(2.7)

where p ∈ (0, 1), and F −1 and G−1 are the respective right continuous inverses of F and G defined as F −1 (t ) = sup{x : F (x) ≤ t } and G−1 (t ) = sup{x : G(x) ≤ t }, for t ∈ [0, 1]. 3. Characterization results We start this section with the following definition. Definition 3.1. Let {N (k) (t )}, k = 1, 2 be two stochastic processes generated by a minimal repair policy. {N (1) (t )} is said to be better than {N (2) (t )} in P ordering if and only if Yln (n + 1) ≤P Xsn (n + 1)

for all 0 < ln < sn and n ∈ N + ,

where P could be any of the orderings discussed in Section 2.

448


The following lemma and proposition due to Nanda and Kundu (2010) will be used in proving the upcoming theorems. Lemma 3.1. Let {S (n), n ∈ N + } denote the failure times of a system undergoing minimal repair, and s be any nonnegative real number. Then, by writing Xsn (n + 1) ≡ Xsn , we have

¯ sX (1) (x + sn ) Φ , F¯X (1) (sn )

¯ sXsn (x) = Φ

for all x ≥ 0 and n ∈ N + , where F¯Z is the survival function of the random variable Z . Proposition 3.1. Let s be any nonnegative real number. For two nonnegative random variables X and Y , Y ≤s−FR↓ X if and only if

¯ sY (x) Φ ¯ sX (x + t ) Φ for all t ≥ 0.

is decreasing in x ≥ 0,

The following theorem characterizes the stochastic comparison in terms of their times to first failure. Theorem 3.1. Let {N (k) (t )}, k = 1, 2, be two stochastic processes generated by a minimal repair policy. Then {N (1) (t )} will be better than {N (2) (t )} in s-FR sense if and only if Y (1) ≤s−FR↓ X (1). Proof. Write Xsn (n + 1) ≡ Xsn and Yln (n + 1) ≡ Yln . {N (1) (t )} is said to be better than {N (2) (t )} in s-FR sense if and only if Yln ≤s−FR Xsn

for all 0 < ln < sn and n ∈ N + ,

which, by (2.4), is equivalent to the fact that

¯ sYln (x) Φ ¯ sXsn (x) Φ

is decreasing in x ≥ 0 for all 0 < ln < sn .

By Lemma 3.1, the above statement can equivalently be written as

¯ sY (1) (x + ln ) Φ ¯ sX (1) (x + sn ) Φ

is decreasing in x ≥ 0 for all 0 < ln < sn ,

which holds if and only if

¯ sY (1) (x) Φ ¯ sX (1) (x + t ) Φ

is decreasing in x ≥ 0 for all t ≥ 0.

This, by Proposition 3.1, gives the required result.

Corollary 3.1. Let {N (k) (t )}, k = 1, 2, be two stochastic processes generated by a minimal repair policy. Then {N (1) (t )} will be better than {N (2) (t )} in (a) (b) (c) (d)

LR sense if and only if Y (1) ≤LR↓ X (1); FR sense if and only if Y (1) ≤FR↓ X (1); MRL sense if and only if Y (1) ≤MRL↓ X (1); VRL sense if and only if Y (1) ≤VRL↓ X (1).

Remark 3.1. One-way implication of the results given in Corollary 3.1 may be obtained in Bagai and Jain (2006).

In the previous theorem we made comparison of two repairable systems in s-FR sense. The next theorem compares two repairable systems when interarrival times of failure of the two systems are ordered in disp-(s-FR) sense. The following proposition which will be used to prove Theorem 3.2 is due to Nanda and Kundu (2009). Proposition 3.2. Let s be any nonnegative real number. If X F F −1 (p)], then

¯ sX Φ

F −1 (p)

for all x ≥ 0.

(x) =

¯ sX (x + F −1 (p)) Φ , 1−p

−1 (p)

, p ∈ (0, 1), denotes the random variable [X − F −1 (p)|X >


449

Theorem 3.2. Let {sn , n ∈ N + } and {ln , n ∈ N + } be two increasing sequences of real numbers such that Yln ≤st Xsn , for ln < sn . Then Yln ≤disp-(s-FR) Xsn if and only if Y (1) ≤s−FR↓ X (1). −1 Fs (p)

Proof. Let the distribution functions of Xsn and Yln be Fsn (x) and Gln (x) respectively. Thus, if Xsn n random variable [Xsn − Fs−n 1 (p)|Xsn > Fs−n 1 (p)], then by Proposition 3.2 we have −1 Fs (p) n

¯ sXsn Φ

( x) = =

, p ∈ (0, 1) denotes the

¯ sXsn (x + Fs−1 (p)) Φ n

(3.8)

q

¯ sX (1) (x + sn + Fs−1 (p)) Φ n , qF¯ (sn )

(3.9)

¯ (x) is where F¯ (x) is the survival function of X (1) and q = 1 − p. The second equality follows from Lemma 3.1. Similarly, if G the survival function of Y (1), then −1 (p)

G l

Y n Φs ln

(x) =

¯

1 ¯ sY (1) (x + ln + G− Φ ln (p)) . ¯ (ln ) qG

(3.10)

Note, by (2.4) and (2.7), that, for all 0 < ln < sn , Yln ≤disp-(s-FR) Xsn if and only if −1 (p) G l Yl n n Φs −1 Fs (p) X n Φs sn

¯

(x)

¯

(x)


for p ∈ [0, 1], which, by (3.9) and (3.10), can equivalently be written as 1 ¯ sY (1) (x + ln + G− Φ ln (p))

¯ sX (1) (x + sn + Fs−n 1 (p)) Φ


1 −1 for p ∈ [0, 1]. As Yln ≤st Xsn , G− ln (p) ≤ Fsn (p) for all p ∈ [0, 1]. Thus, the above statement holds if and only if

¯ sY (1) (x) Φ ¯ sX (1) (x + t ) Φ


for t ≥ 0, which, by Proposition 3.1, gives Y (1) ≤s−FR↓ X (1).

Corollary 3.2. Let {N (k) (t )}, k = 1, 2, be two stochastic processes generated by a minimal repair policy, and Yln ≤st Xsn , where ln and sn are as defined earlier. Then {N (1) (t )} is better than {N (2) (t )} in (a) (b) (c) (d)

disp-LR sense if and only if Y (1) ≤LR↓ X (1). disp-FR sense if and only if Y (1) ≤FR↓ X (1). disp-MRL sense if and only if Y (1) ≤MRL↓ X (1). disp-VRL sense if and only if Y (1) ≤VRL↓ X (1).

Remark 3.2. That the condition of stochastic order between Xsn and Yln in Theorem 3.2 is only a sufficient condition, is shown in the following counterexample. But before the counterexample is given, we give a simple lemma without proof, which will be used in sequel. Lemma 3.2. For all t ≥ 0, √

e 1+

x +t −x

√

x+t

is decreasing in x.

Counterexample 3.1. Let F¯ (x) = e− by Lemma 3.2, Y (1) Φ2 X (1) Φ2 x

¯

¯

( x)

( + t)

√

x

√ ¯ (x) = e−x . Then, by (2.3), Φ ¯ 2X (1) (x) = 2(1 + x)e− and G


√

x

Y (1)

¯2 and Φ

(x) = e−x . So,

450


√

Fig. 1. Graph of exp( 0.15 −

√

x + 0.15) − exp(−x) versus x ∈ (0, 0.3).

√

√

¯ Yl (x) = e−x , for all 0 < ln < sn . If for all t ≥ 0, so that Y (1) ≤MRL↓ X (1). Again, by Lemma 3.1, F¯Xsn (x) = e sn − x+sn and G n we take sn = 0.15 and ln = 0.14, then, from Fig. 1, we can see that Yln st Xsn . Again, for all values of p ∈ (0, 1) and for the above values of sn and ln , we have −1 (p) G l Y n

¯ 2 ln Φ

−1 (p) F sn

¯ 2Xsn Φ

√ ( x) (x)

1 = e−(0.14−ln(1−p)) 2

e

 1+

√ e

= c (1 − p)

 1+

√

x+( 0.15−ln(1−p))2 −x

√

x + ( 0.15 − ln(1 − p))2

.

F¯ (0.15)

¯ (0.14) G

√

x+( 0.15−ln(1−p))2 −x

√

x + ( 0.15 − ln(1 − p))2

,

where c > 0 is a constant. By Lemma 3.2, the above equation is decreasing in x for all p ∈ (0, 1), giving Yln ≤disp-MRL Xsn . Thus, the condition Yln ≤st Xsn of Theorem 3.2 is only a sufficient condition. Theorems 3.1 and 3.2 together give the following. Remark 3.3. Let Yln ≤st Xsn , for 0 < ln < sn . Then {N (1) (t )} will be better than {N (2) (t )} in s-FR sense if and only if Yln ≤disp-(s-FR) Xsn . 4. Conclusions In this paper we compare the improvement/deterioration of two minimally repaired systems. We characterize the comparison in s-FR sense (so that likelihood ratio (LR), failure rate (FR), mean residual life (MRL), variance residual life (VRL) comparisons are special cases) in term of down s-FR order between the times of the first failures of the two systems under consideration. We also characterize the comparison in the sense of disp-(s-FR) order so that we get the characterization in the sense of disp-LR, disp-FR, disp-MRL and disp-VRL orders. The results are supported by examples. Acknowledgement The authors thank the referee for his/her suggestions that improve the presentation of the paper, and also for letting us know that the reference Bagai and Jain (2006) is presented in a conference. References Bagai, I., Jain, K., 1994. Improvement, deterioration and optimal replacement under age-replacement policy with minimal repair. IEEE Transactions on Reliability 43 (1), 156–162. Bagai, I., Jain, K., 2006. On comparison of two repairable systems. In: SCRA 2006-FIM XIII-Thirteenth International Conference of the Forum for Interdisciplinary Mathematics on Interdisciplinary Mathematical and Statistical Techniques, New University of Lisbon-Tomar Polytechnic Institute. Lisbon-Tomar, Portugal. Deshpande, J.V., Singh, H., 1995. Optimal replacement of improving and deteriorating repairable system. IEEE Transactions on Reliability 44 (3), 500–504. Ebrahimi, N., 1989. How to define system improvement and deterioration for a repairable system. IEEE Transactions on Reliability 38, 214–217. Fagiuoli, E., Pellerey, F., 1993. New partial orderings and applications. Naval Research Logistics 40, 829–842. Hu, T., Nanda, A.K., Xie, H., Zhu, Z., 2004. Properties of some stochastic orders: a unified study. Naval Research Logistics 51 (2), 193–216. Lillo, R.E., Nanda, A.K., Shaked, M., 2000. Some shifted stochastic orders. In: Limnios, N., Nikulin, M. (Eds.), Recent Advances in Reliability Theory: Methodology, Practice and Inference. Birkhäuser, Boston, USA, pp. 55–103. Mukherjee, S.P., Chatterjee, A., 1992. Stochastic dominance of higher orders and its implications. Communications in Statistics—Theory and Methods 21, 1977–1986. Nanda, A.K., 1997. On improvement and deterioration of a repairable system. IAPQR Transactions 22 (2), 107–113. Nanda, A.K., Kundu, A., 2009. On generalized stochastic orders of dispersion-type. Special Triennial Volume of Calcutta Statistical Association Bulletin 61, 155–182. Nanda, A.K., Kundu, A., 2010. On improvement and deterioration of a repairable system under generalized stochastic orders, Revised.