Research Article Volume 9 Issue 2
1Department of Statistics, Maharaja’s College, India
2Department of Biostatistics, St. Thomas College, India
Correspondence: K.V. Jayamol, Department of Statistics, Maharaja’s College, Ernakulam, India, Tel 9447036746
Received: April 10, 2020 | Published: April 29, 2020
Citation: Jayamol KV, Jose KK. Shock models leading to G* class of lifetime distributions. Biom Biostat Int J. 2020;9(2):61-66. DOI: 10.15406/bbij.2020.09.00301
In this paper we study a stochastic ordering namely alternate probability generating function (a.p.g.f.a.p.g.f. ) ordering and its properties. The life distribution H(t)H(t) of a device subject to shocks governed by a Poisson process is considered as a function of the probabilities ˉPk¯¯¯Pk of surviving the first k shocks. Various properties of the discrete failure distribution PkPk are shown to be reflected in corresponding properties of the continuous life distribution H(t)H(t) . A certain cumulative damage model and various applications of these models in reliability modeling are also considered.
Keywords: lifetime distributions, probability and statistics
Stochastic orders and inequalities are being used at an accelerated rate in many diverse areas of probability and Statistics. For example, in statistical reliability theory, several concepts of partial orderings have been successfully used to develop various notions of ageing of non negative random variables. Ageing concept for discrete distributions were studied by various authors. See for example, Barlow and Proschan,1 Cai and Kalashnikov,2 Cai and Willmot,3 Lai and Xie,4 Shaked and Shanthikumar,5 Shaked et al.,6 Willmot and Cai,7 Willmot and Lin,8 Willmot et al.,9 and references therein. Using Laplace transform, various reliability classes have been characterized by different researches. For details, see Bryson and Siddiqui,10 Klefsjö,11,12 Shaked and Wong13 and the references there in. As a discrete analogue of Laplace transform ordering, Jayamol and Jose14 introduced ordering and class of lifetime distributions based on this ordering as follows.
Definition 1.1 Let denotes the probability mass function (p.m.f.p.m.f. ) of a non-negative integer-valued random variable , then the a.p.g.f.a.p.g.f. , G(⋅)G(⋅) of ff is defined as G(s)=E(1−s)X, 0<s≤1.G(s)=E(1−s)X, 0<s≤1. (1.1)
Definition 1.2 A non-negative integer-valued life distribution with mean μ belongs to the G*(ˉG*) class of lifetime distributions if and only if G(s)≤(≥)11+sμ, μ≥0,0<s≤1. (1.2)
It may be noted that the R.H.S. of the inequality (1.2) is the a.p.g.f. of a geometric distribution with p.m.f. f(x)=pqx,x=0,1,2,⋯ and with mean µ as that of f. For properties of G*(ˉG*) classes one may refer to Jayamol and Jose,15 Jayamol and Jose.14
For many equipments, useful life is often measured in discrete integer units, for example the number of copies a plain paper copier makes before a breakdown, the number of completed production runs in an automated assembly line before a malfunction occurs, etc. Even in situations where the time to failure is conceptually a continuous variable, one is often interested in measuring the life in suitably discretized work units successfully completed. For example, the number of days one needs to replace the batteries in an appliance under specified normal pattern of use is discrete. So as a discrete analogue of Laplace transform ordering, Jayamol and Jose14 introduced a new stochastic ordering namely alternate probability generating function (a.p.g.f. ) ordering. Some properties of this ordering are considered here.
As a discrete analogue of Laplace transform ordering introduced by Klefsjö,12 Jayamol and Jose14 defined a.p.g.f. ordering as follows.
Definition 2.1 Suppose X that Y and are two non-negative integer-valued random variables with p.m.f. s f1 and f2 and a.p.g.f.s G1(s) and G2(s) respectively. Then X is said to be smaller than Y (or equivalently, f1 is smaller than f2 ) in a.p.g.f. ordering if G1(s)≤G2(s), for 0≤s≤1 . It is denoted by X≤GY (or equivalently, we write f1≤Gf2 ).
In this context, we have the following theorems.
Theorem 2.1 Suppose that X and Y be two non-negative integer-valued random variables with respective p.m.f. s f1 and f2 . Then X≤GY implies E(X)≥E(Y) , provided the expectations exist.
Proof
If X≤GY then ∑∞x=0(1−s)xf1(x) ≤∑∞y=0(1−s)y f2(y)
Differentiating once with respect to s and letting s→0 , we get E(X)≥E(Y) .
Theorem 2.2 Let X and Y be two non-negative integer-valued random variables. If X≤GY then X+K≤GY+K for every K∈N+ .
Proof
If X≤GY,E(1−s)X≤E(1−s)y
Then we have,
E(1−s)X+K=E(1−s)k(1−s)X=(1−s)kE(1−s)X
≤(1−s)kE(1−s)Y =E(1−s)Y+K
Theorem 2.3 Let X1,X2,…,Xm be a set of independently distributed non-negative integer-valued random variables. Let Y1,Y2,…,Ym be another set of independently distributed non-negative integer-valued random variables. If Xi≤GYi for i=1,2...., m. Then ∑mi=1Xi≤G∑mi=1Yi
Proof
If Xi≤GYi then E(1−s)Xi≤E(1−s)Yi for i=1,2... , m.
Let X=∑mi=1Xi
and Y=∑mi=1Yi
E(1−s)X=E(1−s)X1…(1−s)Xm =E(1−s)X1…E(1−s)Xm
≤E(1−s)Yi...E(1−s)Ym=E(1−s)Y
Theorem 2.4 Let X1,X2,… be independently and identically distributed non-negative integer-valued random variables and let N1 and N2 be positive integer-valued random variables which are independent of Xi . Then N1≤GN2⇔∑N1i=1Xi≤G∑N2i=1Xi
Proof
We have the a.p.g.f.
GX1+X2+...+XN1(S)=∑∞i=1P[N1=i] GX1+X2+...+Xi(S)
∑∞i=1P[N1=i](GX1(S))i
∑∞i=1P[N1=i](1−(1−GX1(S)))i
=GN1(s),
where 0<s=(1−GX1(s))≤1.
N1≤GN2⇔GN1(s)≤GN2(s)
⇔GX1+X2+⋯+XN1(s)≤GX1+X2+⋯+XN2(s)
⇔∑N1i=1Xi≤G∑N2i=1Xi
Theorem 2.5 Let X and Y be two non-negative integer-valued random variables such that X≤GY. Let ˉPk and ˉQk be the survival functions of X and Y respectively. Then ∑∞k=0¯Pk(1−s)k≥∑∞k=0¯Qk(1−s)k
Proof
The stated result follows from the definition of a.p.g.f. ordering and from the equation
E(1−s)X=1−s∑∞k=0¯Pk(1−s)k (2.1)
In reliability analysis one may calculate the reliability of a complex system starting with the reliability of the components. If all components have life distributions belonging to a certain class, then one would like to conclude that the life distribution of the entire system belongs to the same, or a similar class. Shock models of this kind have been considered by a number of authors under all kinds of assumptions. The results center around proving that, subject to suitable assumptions on the point process {N(t)} of shocks, various discrete reliability characteristics of the {ˉP k} sequence, which arise naturally out of physical considerations are inherited by the continuous survival probability ˉF(t) . That is if the shock survival probabilities {ˉP k} belong to a discrete version of one of the life distribution classes, then under appropriate assumptions the continuous time survival probability belongs to the continuous version of that class. That is the life distribution H(t) of a device subject to shocks governed by a Poisson process is considered as a function of the probabilities ˉPk of surviving the first k shocks. Various properties of the discrete failure distribution Pk are shown to be reflected in the corresponding properties of the continuous life distribution H(t) . In the present paper we study some shock models leading to G* class. A certain cumulative damage model is also investigated. For that we consider the following definitions and Theorem.
Klefsjö12 introduced a class denoted by L , which consists of all distribution functions F, for which LF(s)≤11+sμ1(F), where LF(s) is the Laplace transform of F defined by LF(s)=∫∞0 e−sxdF(x), s≥0 and μk(F)=∫∞0xkdF(x) k=1,2....., Its dual class ˉL is obtained by reversing the inequality.
Definition 3.1 Suppose that X and Y are two non-negative integer-valued random variables with survival functions ˉPk and ˉQk respectively. Then X is said to be smaller than Y in a.p.g.f. ordering if ∑∞k=0¯Pk(1−s)k≥∑∞k=0¯Qk(1−s)k for 0<s≤1 .
Theorem 3.1 Let X be a non-negative integer-valued random variable with a.p.g.f. G(s) and survival function P[X>k]=ˉPk. Then for s∈(0, 1), X∈G*⇔∑∞k=0(1−s)kˉPk≥μ(1+sμ)
3.1 A Poisson shock modelAssume a device is subject to shocks occurring randomly in time according to a Poisson process with intensity λ Suppose if the device has the probability ˉPk of surviving k shocks, where 1= ˉP0≥ˉP2≥… , then the survival function of the device is given by,
ˉH(t)=∑∞k=0e−λt(λt)kk!ˉPk (3.1)
Esary et al.16 have shown that if {ˉPk}∞k=0 has the discrete Increasing Failure Rate (IFR), Increasing Failure Rate in Average (IFRA), New Better than Used in Expectation (NBUE) or Decreasing Mean Residual Life (DMRL) property, then this property will be reflected to ˉH(t) given by (3.1). Klefsjö11 has shown that a similar result holds for the Harmonically New Better than Used in Expectation (HNBUE) class. Shock models leading to GHNBUE (GHNWUE) classes are studied by A H N Ahmed.17 We now show that the same is true for G* class also.
Theorem 3.2 The survival function ˉH(t) in (3.1) is in L class if and only if {ˉPk}∞k=0 is in G* class.
Proof
Let μ be the mean of and H(t) be the mean of {ˉPk}∞k=0 . We have,μ=∫∞0ˉH(t)dt=∑∞k=0{∫∞0e−λt(λt)kk!dt}ˉPk=∑∞k=01λˉPk=mλ
Laplace transform of ˉH(t)
LˉH(S)=∫∞0e−stˉH(t)dt (3.2)∑∞k=0¯Pkλkk!∫∞0e−(s+λ)ttkdt
=1s+λ∑∞k=0(λs+λ)kˉPk (3.3)
H ∈ L class if and only if LˉH(s)≥μ1+sμ (3.4) that is if and only if,=1s+λ∑∞k=0(λs+λ)kˉPk≥μ1+sμ
∑∞k=0(λs+λ)kˉPk≥(s+λ)μ1+sμ ∑∞k=0(1−ss+λ)kˉPk≥m(s+λλ)1+smλ=mλs+λ+ss+λm≥m1+ss+λm (3.5)
(3.4) holds if and only if (3.5) holds. Hence from Theorem 3.1, we have the result.
Consider another device which is also subjected to shocks occurring randomly as events in a Poisson process with same constant intensity λ , and the device has probability ˉQk of surviving the first k shocks, where 1=ˉQ0≥ˉQ1≥… . The survival function of this device is given by
ˉF(t)=∑∞k=0e−λt(λt)kk!ˉQk (3.6)
Singh and Jain18 have shown that some partial orderings, namely likelihood ratio (LR) ordering, failure rate (FR) ordering, stochastic (ST) ordering, variable (V) ordering and mean residual life (MRL) ordering between the two shock survival probabilities ˉPk and ˉQk are preserved by the corresponding survival functions ˉH(t) and ˉF(t) of the devices. Here we extent this preservation property to a.p.g.f ordering.
Theorem 3.3 If ˉPk ≤G ˉQk then ˉH(t)≤LˉF(t) .
Proof
Let ˉPk ≤G ˉQk
. From (3.6), we have
∫∞0e−stˉF(t)dt=∫∞0e−st∑∞k=0e−λt(λt)kk!ˉQk
∑∞k=0λkk!ˉQk∫∞0e−(s+λ)ttkdt=1(s+λ)∑∞k=0(λs+λ)kˉQk
=1(s+λ)∑∞k=0(1−λs+λ)kˉQk
From Definition 3.1≤1s+λ∑∞k=0(1−ss+λ)kˉPk=∫∞0e−stˉH(t)dt
⇔ˉH(t)≤LˉF(t).
Remark 1 When λ is a random variable, denoted by Λ , whose distribution is Y, in this case ˉH(t) can be written as
ˉH(t)=∑∞k=0E(e−Λt(Λt)k))k! (3.7)
3.2 A Nonhomogeneous poisson shock modelSuppose that shocks occur according to a nonhomogeneous Poisson process with mean value function Ω(t) . If a device has the probability ˉPk of surviving the first k shocks, its survival function ˉH(t) is given by
ˉH(t)=∑∞k=0e−Ω(t)[Ω(t)]kk!ˉPk (3.8)
This shock model was studied by Hameed and Proschan.19 They proved that under suitable conditions on Ω(t) , the survival function ˉH(t) is IFR, IFRA, New Better than Used (NBU) NBUE or DMRL if {ˉPk}∞k=0 has the corresponding discrete property. We will now give a theorem for G* class.
Lemma 3.1 (Klefsjö)12: ˉH(t)=ˉH1[Ω(t)] belongs to L(ˉL) if ˉH1∈L(ˉL) class and Ω(t) is starshaped (antistarshaped).
Theorem 3.4 If {ˉPk}∞k=0 is in G* class and Ω(t) is starshaped then ˉH(t) in (3.8) belongs to L class.
Proof Let ˉH(t)=∑∞k=0e−ttkk!ˉPk Since {ˉPk}∞k=0 is in G* class, by Theorem 3.2 ˉH1(t) belongs to L class. Hence the result follows from Lemma 3.1.
3.3 A cumulative damage modelIn this section we study special model for the survival probability ˉPk . Suppose that a device is subjected to shocks. Every shock causes a random amount of damage. Suppose damage accumulates additively. The device fails when the accumulated damage exceeds a critical threshold Y which has the distribution F , where F(0−)=0 . If the damages X1,X2,…, , from successive shocks are independent and exponentially distributed with mean 1λ , and are independent of the threshold. Let be the number of shocks survived by the device. Then the survival probabilities are given by ˉPk(λ)=λk∫∞0xk−1(k−1)! e−λxˉF(x)dx for k=1,2,...,
ˉP0(λ)=1. (3.9)
Thus the probability function of N , P[N=k] is pk=∫∞0e−λx(λx)k−1(k−1)!dF(x),k≥1
The above cumulative damage model has been studied by Esary et al.16 for the NBU, IFR and IFRA cases. They proved that if F is NBUE, then {ˉPk}∞k=0 has the discrete NBUE property. Klefsjö11 proved that the same is true in the case of discrete HNBUE. We now claim that the result is true when {ˉPk}∞k=0 belongs to G* class.
Theorem 3.5 The survival probabilities ˉPk in (3.9) belongs to G* class for every λ>0 if F belongs to L class.
Proof
First observe that m be mean of ˉPk
is
m=∑∞k=0ˉPk=1+∑∞k=1λk∫∞0xk−1(k−1)!e−λxˉF(x)dx
=1+∫∞0λ∑∞k=0(λx)kk!e−λxˉF(x)dx
=1+∫∞0λe−λx+λxˉF(x)dx (3.10)
=1+λ∫∞0ˉFdx==1+λμ, where μ is the mean of FSo μ=m−1λ. (3.11)
Consider
∑∞k=0ˉPk(λ)(1−s)k=1+∑∞k=0(1−s)kλk∫∞0xk−1(k−1)!e−λxˉF(x)dx
=1+∫∞0e−λxˉF(x)(1−s)λ∑∞k=0((1−s)λx)kk!dx
=1+∫∞0e−λxˉF(x)(1−s)λeλ(1−s)xdx
=1+λ(1−s)∫∞0e−λsxˉF(x)dx
(3.12)
Let F∈L class, hence from (3.12) ∑∞k=0ˉPk(λ)(1−s)k≥1+λ(1−s)μ1+sλμ
=1+λ(1−s)m−1λ1+sλm−1λ (3.13)
=1+(1−s)(m−1)1+s(m−1)=m(1−s)+sm (3.14) ≥m1+sm.
Thus from the definition of G* class, we have the theorem.
Theorem 3.6 The survival probability ˉPk in (3.9) belongs to ˉG* class for every λ>0 if F belongs to ˉL class.
Proof
We have, from (3.1) and (3.12), for ˉPk∈ˉG*
1+λ(1−s)∫∞0e−λsxˉF(x)dx≤m1+sm≤m1−s+smHence from the definition of ˉL class the result follows.
Random minima and maxima
Let X1,X2,… be a sequence of non-negative integer-valued random variables which are independent and identically distributed. Let N1 be a positive integer-valued random variable which is independent of Xi' s . Denote X(1:N1)=min(X1,X2,…,XN1) and X(N1:N1)=max(X1,X2,…,XN1) (for details refer Gupta and Gupta,20 Rohatgi,21 Shaked and Wong13 and references there in). Since the Xi s are non-negative, the random variable X(N1:N1) arises naturally in reliability theory as the lifetime of a parallel system with a random number N1 of identical components with lifetimes X1,X2,…,XN1 . The random variable X(1:N1) arises naturally in transportation theory as the accident free distance of a shipment of explosives, where N1 of them are defectives which may explode and cause an accident after X1,X2,…,XN1 miles respectively. Let N2 be another positive integer-valued random variable which is also independent of the Xi and let X(1:N2)=min(X1,X2…XN2) and X(N2:N2)=max(X1,X2,…XN2).
Theorem 4.1 Let X1,X2,… be a sequence of non-negative integer-valued random variable which are independent and identically distributed. Let N1 and N2 be two positive integer-valued random variables which are independent of the X'is . Then the following results are true.
Proof
Let Pk
be the common distribution function of Xi
s, that is, Pk=P[Xi≤k] for i=1,2,…
and Pk(N1:N1)
denotes the distribution function of X(N1:N1)
. Then we have
Pk(N1:N1)=∑∞n=1(Pk)nP[N1=n]
∑∞n=1(1−ˉPk)nP[N1=n]
=GN1(ˉPk),0<ˉPk≤1, ˉPk=1−Pk
Similarly Pk(N2:N2)=GN2(ˉPk).
Also the survival function of X(1:N1), ˉPk(1:N1) is
ˉPk(1:N1)=∑∞n=1(ˉPk)nP[N1=n]
∑∞n=1(1−PK)nP[N1=n]
=GN1(P),0<Pk≤1
Similarly,
ˉPk(1:N2)=GN2(Pk)
N1≤GN2⇔ˉPk(1:N1)≤ˉPk(1:N2)
⇔X(1:N1)≤stX(1:N2)
N1≤GN2⇔ˉPk(N1:N1)≤ˉPk(N2:N2)
⇔X(N1:N1)≥stX(N2:N2)
Similar to continuous ageing classes, discrete classes can be classified according to various stochastic oderings. These discrete classes have been extensively used in different fields such as insurance, finance, reliability, survival analysis and others. In this paper, a. p. g. f. ordering, a discrete analogue of Laplace transform ordering and its properties and certain shock models leading to G* class are studied. It has been shown that a.p.g.f ordering between two shock survival functions ˉPk and ˉQk are preserved by survival function of the system. It has also been shown that it is necessary and sufficient for the survival function of the system to belong to L class is that the survival probability of surviving k shocks belongs to G* class, under the assumption that the shock occuring randomly in time according to a Poisson process. If the failure of the system is triggered by a sufficient number of shocks, we proved that the survival probability function is in G*(ˉG*) class only if the critical threshold is in L(ˉL) under the assumption that the damage is accumulated additively and the shocks do not damage the system unless the accumulated shocks exceeds a critical thershold. Finally stochastic ordering of random maxima and minima has studied in relation to a. p. g. f. ordering.
The authors thank the referee for pointing out some inadequacies that the earlier version of the manuscript had, and for valuable comments.
There is no conflicts of interest.
None.
©2020 Jayamol, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.
2 7