Reliability Analysis of the Proportional Mean Departure Time Model

In this article, themean lifetime of an individual whose lives lost based on a function of the time before which the individual has passed away is considered. e measure is used to construct a semi-parametric model called proportional mean departure time model. Examples are given and evidences are gathered to show that the model is a proper alternative for the proportional mean past lifetime model. Closure properties of the model concerning several stochastic orders and a number of reliability properties are established. Finally, the model is extended to entertain random amounts of the parameter and establish a proportional mean departure time frailty model. Further stochastic properties using several stochastic orders are developed in the context of the frailty model.


Introduction
Let X be a non-negative random variable (r.v.) having absolutely continuous cumulative distribution function (c.d.f.) F and probability density function (p.d.f.) f. Consider a situation where X is the time-to-failure of a lifespan and assume that it has been realized that the item has failed before the time point t. In literature, there has been growing interest in the study of reliability measures in reversed time and their applications. e amount of X, after knowing at the time t that a past failure has happened, is then the time of departure among lives lost before the time t. e mean departure time (m.d.t.) function of X is de ned by which is valid for all t ≥ 0 for which F(t) > 0. e conditional random variable X (t) (t − X|X ≤ t), where (X|A) denotes the conditional random variable X given that the event A has happened, is well-known in the literature as the past lifetime or the inactivity time (see, e.g., Di Crescenzo and Longobardi [1] and Kayid and Ahmad [2]). e random variable X (t) is also called the revered residual life (see Nanda et al. [3]). e concept of mean past lifetime (m.p.l.) or mean inactivity time (m.i.t.) is closely related to the mean departure time function given in (1). e m.i.t (or the m.p.l.) function provides the expected time elapsed since the failure of a subject given that he/she has failed before the time of observation. e m.i.t. function is given by where F(t) > 0. e inactivity time X (t) and the measure (2) have been very useful in science and engineering contexts. ey have many applications in various disciplines such as reliability theory, survival analysis, risk theory, and actuarial studies, among others (cf. Ortega [4], Izadkhah and Kayid [5], Jayasinghe and Zeephongsekul [6], Kayid and Izadkhah [7], Kayid et al. [8], Kayid and Izadkhah [9], Bhattacharyya et al. [10], Balmert et al. [11], Khan et al. [12], and Kayid and Alrasheedi [13]). e inactivity time at random times is one of the important notions in reliability and queuing theory (see, e.g., Kayid et al. [14] and Kundu and Patra [15]). e reversed hazard rate (r.h.r.) function of X, as another measure related to the inactivity time, is given by e m.d.t. function is connected with the r.h.r. and the m.i.t. function as follows: where m X ′ (t) � (d/dt)m X (t) and the m.d.t. function is related to the m.i.t. function as follows: e m.i.t. function (2) characterizes the underlying c.d.f. uniquely. By (5), it follows that the m.d.t. function also determines the underlying distribution in a unique way. It is said that X w is a weighted version of X which has p.d.
where w is a non-negative function such that E[w(X)] < + ∞. As pointed out in Equations (2.3) in Sunoj and Maya [16] by taking the weight function w(x) � x, the c.d.f. of X can be recovered from the m.d.t. function m X as Let X and Y have m.i.t. functions M X and M Y , respectively. Asadi and Berred [17] constructed a model by holding the two m.i.t. functions in a proportional relation so that For all t ≥ 0 and θ > 0 is the constant of proportionality. e r.v. X plays the role of the independent variable and the r.v. Y is the dependent variable whose distribution depends on θ and also it depends on the distribution of X. is model is called the proportional mean past lifetime (PMPL) model. ere is one difficulty, as the authors believe and, further, mathematical strategies confirms it, with the PMPL model given in (7) which makes it somewhat controversial to use this model in applied situations. e question is that to what extent the model (7) is useful to model lifetime events. Repeatedly encountered in survival analysis applications, consider a situation where X and Y with c.d.f.s F and G are supported in [0, +∞) and also X and Y have finite means. Let us assume that the identity in (7) holds true.
which shows that X and Y are identical in distribution. e conclusion is that every set of non-negative r.v.s (X, Y) with finite means having unbounded supports which satisfies Equation (7) has to have identical components in distribution and θ is not a parameter in this case. erefore, the model (7) cannot be useful to model lifetime events on [0, +∞) which obviously surrounds the applicability of the model (7). Finding an alternative model of (7) is, therefore, necessary as there are many cases in which the lifetime data are left censored and modeling data in the framework of past failures is needed.
is article aims to introduce an appropriate alternative model for the proportional mean past lifetime model. We show by some examples that the new model is applicable for modeling random lifetimes with unbounded supports. e role of the parameter of the model which is added to the original distribution is to extend the existing model and provided flexibility to adjust the mean departure time function of life spans. e theory of stochastic orders is used to provide some comparison results to address the question of whether the reliability of a device is either improved or deteriorated under the setup of the model. roughout the article, we will not use the terms "increasing" and "decreasing" in the strict case and will take these properties equivalently as "non-decreasing" and "non-increasing" behaviors, respectively. e organization of the article will be in the following order. In Section 2, we introduce and illustrate the proportional mean departure time model with some examples. In Section 3, we consider the closure properties of the model with respect to some well-known stochastic orders and also a number of reliability classes of lifetime distributions. In Section 4, the model is extended to the case where the parameter of the model is a random variable and some stochastic ordering properties are investigated. Finally, in Section 5, we conclude the article with some illustrative statements to describe our contribution and also we will add some remarks on possible future studies.

Proportional Mean Departure Time Model
In this section, we introduce and describe the PMDT model and study the advantages of this model over the PMPL model. Further distributional properties of the model are investigated. We present some examples which fulfill the PMDT model. functions m X and m Y , respectively. en, it is said that X and Y satisfy the PMDT model whenever For all t ≥ 0 and θ is a positive parameter. e parameter θ is called the departure index. e departure index θ in Equation (9) represents the magnitude of departure time of individuals whose lives lost before t with lifetime Y, in average, relative to the departure time of individuals whose lives lost before t with lifetime X. e parameter θ is, therefore, a conditional deterioration rate of Y with respect to X. Unlike the m.i.t function which is not originally an increasing function of time, the m.d.t. function is monotonically increasing in time, i.e., it certainly holds that for all t 1 ≤ t 2 ∈ R + , m X (t 1 ) ≤ m X (t 2 ). If we consider the ratio of two m.d.t. functions as a parameter then a new model, which we call it proportional mean departure time (PMDT) model, is constructed. e assumption that the ratio of two increasing functions which are a reliability characteristic associated with two different distributions coincides with a horizontal line is a more appropriate assumption than the same for two arbitrary reliability measures. In a bivariate setup, when the sample X 1 , . . . , X n copies of X and the sample Y 1 , . . . , Y m copies of Y is available then the model (9) may be used and the parameter θ is estimated by the estimation of m.d.t. functions of Y divided by the estimation of m.d.t. function of X. e model (7) in situations where it is applicable (when X and Y have finite supports) may not be proper in the two sample setting.
is is because the shapes of m.i.t. functions in the model (7) have to be the same since θ does not depend on time and in spite of that the set on samples on X and Y may not exhibit similar shapes as the distributions of X and Y may not belong to the same class of lifetime distributions, e.g., the increasing mean inactivity time (IMIT) class. erefore, as the m.d.t. functions of X and Y are always increasing independent of the distributions of X and Y, thus the model (9) may be preferable.
Note that m X (t) � E(X|X ≤ t) is the expected time at which individuals whose deaths happened before the time t departed. e time t may be considered as the first time one realizes that a failure or death has occurred in the past. In the PMDT model, the ratio m Y (t)/m X (t) is independent of the observation time t which induces that the m.d.t. function of X relative to the m.d.t. function of Y is independent of the process of observation of past failures. From a mathematical perspective, Indicating that relative mean departure time functions remain unchanged during time. Specifically, when r.v.s X and Y have finite means, the choices of t 1 � t and t 2 � +∞ in (10) lead to By appealing to (6) for r.v.s X and Y satisfying the PMDT model one has For an unspecified m X associated with a general distribution, the formula (12) may not provide a closed form for the c.d.f. of the r.v. Y relative to the c.d.f. of X. It is not hard to verify that (12) is a valid c.d.f. if the following conditions are satisfied: where t > 0 and the expectations are assumed to exist and they are finite. e r.v. X (resp. Y) is said to be the lengthbiased version of X (resp. Y). We will denote by F and G the c.d.f.s of X and Y, respectively. e r.h.r. functions of X and Y are given by h e concept of length-biased distribution as a typical weighted distribution has been very useful in survival analysis, etiologic studies, and marketing research (see, e.g., Wang [18], Simon [19], and Nowell and Stanley [20]). Length-biased sampling arises when a component which is already in use is sampled at a fixed time and then allowed to fail (see Scheaffer [21]).

Proposition 1.
e random lifetimes X and Y satisfy the PMDT model if, and only if, (i) or (ii) below holds: Proof. First, we prove the assertion (i). By Equations (2.5) in Sunoj and Maya [16], for all t > 0, , and , for all t > 0 which is equivalent to (i). By applying the identities in Equations (2.4) in Sunoj and Maya [16], for all t > 0 we have and

Journal of Mathematics
From (10), it follows that m Y (t) � θm X (t), for all t ≥ 0, if, and only if, For all t ≥ 0. is is also equivalent to (ii) and hence the proof is completed.
In survival analysis, the length-biased samples frequently come up where random sampling from X and Y with respective target distributions F and G cannot be conducted. In such situations, the available data follow the associated length-biased distributions F and G. Proposition 1 illustrates that in the setup of the PMDT model, the ratio of c.d.f.s G and F (or the ratio of the r.h.r. functions h Y and h X ) of the original distributions could be estimated under length-biased sampling as well as that estimated under random sampling. If the objective is the estimation of G using lengthbiased samples on Y and F is either known or predetermined, then Proposition 1 is again useful to present an estimator of G.
We present an example to introduce two distributions fulfilling the PMDT model.
, and where } have length-biased distributions associated with the underlying distributions of X and Y, respectively. It is plain to show that T X and and let X and Y satisfy E(X) < 1/λ and E(Y) < 1/λ. en, the denominators in the r.h.r. functions given in (17) are positive and X and Y satisfy the PMDT model. e following result determines that a limiting property of the r.h.r. functions of two r.v.s with the PMDT model is a characteristic for equality in distribution of the two r.v.s. Theorem 1. Let X and Y having finite means and r.h.r. functions h X and h Y , respectively, and denote lim t⟶0 only if, they are equal in distribution. (iii) If l 1 (l 2 + 1)/l 2 (l 1 + 1) ≠ l 3 (l 4 + 1)/l 4 (l 3 + 1) then, X and Y do not satisfy the PMDT model.
Proof. We only prove the assertion (i) and the assertion (iii). e assertion (ii) can be proved similarly as done for assertion (i). From (9), θ � m Y (t)/m X (t), for all t > 0, which is independent of t. Denote by M X and M Y , the m.i.t. functions of X and Y, respectively. One has Using L'Hopital's rule, By a similar method, we also have We conclude that θ � 1 and this means that m Y (t) � m X (t), for all t > 0, and since the m.d.t. function uniquely determines the distribution it follows that F(t) � G(t), for all t > 0. It can also be shown under the condition of assertion (iii) that there is no value of θ to fulfill the PMDT model.

Remark 1.
The results of Theorem 1 (i) and eorem 1 (ii) show that if either the right limits of length-biased r.h.r. functions given by th X (t) and th Y (t) at the point 0 or the limits of th X (t) and th Y (t) at +∞ do not depend on F and G, respectively, then the PMDT model is not a meaningful model. For instance, if X has an exponential distribution with mean 1/λ and also Y has an exponential distribution with mean 1/η then by the L'Hopital's rule, and analogously, We can also observe that erefore, l 1 � l 2 and also l 3 � l 4 thus according to the result of eorem 1 (i) and eorem 1 (ii), if X and Y satisfy the PMDT model then X and Y are equal in distribution, i.e., λ � η.

Closure Properties With Respect to Some Reliability Classes and Stochastic Orders
In this section, sufficient conditions to get the closure property of the PMDT model with respect to the reversed hazard rate (mean inactivity time) order and also sufficient conditions to establish the closure property of the model with respect to four reliability classes related to the inactivity time will be presented. Closure properties of models in reliability and survival analysis have attracted the attention of many researchers in the recent past decades (see, e.g., Crescenzo [22], Abouammoh and Qamber [23], Nanda et al. [24], Nanda and Das [25], Kayid et al. [26], and Jarrahiferiz et al. [27] among others). Below the definition of three stochastic orders are given (see Shaked and Shanthikumar [28] and Ahmad and Kayid [29]). We will use the convention a/0 � +∞ for a > 0 when a statement on the monotonicity of a ratio is made.

Definition 2.
Suppose that X and Y are two non-negative r.v.s with absolutely continuous c.d.f.s F and G, p.d.f.s f and g, r.h.r. functions h X and h Y , and m.i.t. functions M X and M Y , respectively. en, it is said that X is smaller than Y in the (i) likelihood ratio order (denoted by X ≤ lr Y) when- It has been proved that X ≤ lr Y⇒X ≤ rhr Y⇒X ≤ mit Y. e following result establishes a necessary and sufficient condition on the parameter θ in the model (9) to get X is smaller than Y in terms of the r.h.r. (or the m.i.t.) order.
Theorem 2. Let X and Y satisfy the PMDT model. X ≤ rhr ( ≤ mit )Y if, and only if, θ ≥ 1 .
Proof. It is enough to show that θ ≥ 1 implies X ≤ rhr Y, and also that X ≤ mit Y yields θ ≥ 1. From (4) , for all t > 0, which is non-negative for θ ≥ 1 since m X ′ (t) > 0, for all t > 0 because m X (t) � E(X|X ≤ t) is increasing in t ≥ 0 and also it is readily proved that θm X ′ (t)/(t − θm X (t)) is increasing in θ, and thus θm X ′ (t)/(t − θm X (t)) ≥ m X ′ (t)/ (t − m X (t)), for any t ≥ 0.
is demonstrates that θ ≥ 1 gives h Y (t) ≥ h X (t), for all t > 0, i.e., X ≤ rhr Y. It remains to prove that X ≤ mit Y provides that θ ≥ 1. From (5), we can get e following example shows that the reversed hazard rate order in eorem 2 cannot be replaced by the likelihood ratio order which is stronger than the reversed hazard rate order and, therefore, it is stronger than the mean inactivity time order. It is said that a non-negative r.v. T has Lomax distribution with parameters α > 0 and λ > 0 whenever it has survival function (s.f.) H(t) � 1/(1 + (t/λ)) α and we write T ∼ L(α, λ). Example 2. Let X ∼ L(2, 1) and also let Y have c.d.f.
is means that the result of eorem 2 cannot be strengthened to the case when likelihood ratio order is used instead for stochastic comparison of X and Y.
In Section 4, in eorem 7, we present sufficient conditions under which it is concluded that X ≤ lr Y. Below, the definitions of some reliability classes of lifetime distributions are given.

Journal of Mathematics
Definition 3. Suppose that X is a non-negative r.v. with absolutely continuous c.d.f. F, the r.h.r. function h X and the m.i.t. function M X . en, it is said that X has (i) a decreasing reversed hazard rate (denoted as X ∈ DRHR) distribution when h X (t) is decreasing in t > 0 (see Ahmad and Kayid [29]). (ii) an increasing mean inactivity time (denoted as X ∈ IMIT) distribution when M X (t) is increasing in t > 0 (see Kayid and Ahmad [2]). (iii) a decreasing proportional reversed hazard rate (denoted as X ∈ DPRHR) distribution when Prh X (t) � th X (t) is decreasing in t > 0 (see Oliveira and Torrado [30]). (iv) a strong mean inactivity time (denoted as X ∈ SIMIT) distribution when M X (t)/t is increasing in t > 0 (see Kayid and Izadkhah [7]).
e foregoing reliability classes are connected as follows: According to the PMDT model, we assume that m Y (t) � θm X (t), for all t > 0 in which θ is a positive parameter (see Equation (8)). We investigate whether the reliability properties of the DPRHR, the SIMIT, the DRHR, and the IMIT of X are inherited by the same reliability properties of Y and vice versa. e next technical lemma is useful to establish closure properties with respect to the foregoing reliability classes. Lemma 1. Let X ∈ SIMIT such that 0 < θ < 1 (resp. θ > 1 ) and assume that m X is differentiable. en the function c defined by is an increasing (resp. a decreasing) non-negative function in t > 0.
Proof. We want to prove that z/ztc(t, θ) ≥ ( ≤ )0, for all t > 0 and for any θ ∈ (0, 1) (θ ∈ (1, ∞)), we can get where a� sgn b means that a and b have the same sign. On the other hand, when M X (t) is the m.i.t. function of X, one has , for all t > 0, (31) and thus X is SIMIT if, and only if, t/m X (t) is increasing in t > 0, which holds if, and only if, d/dt(t/m X (t)) ≥ 0, for all t > 0. Hence, if X is SIMITand θ < 1, then c(t, θ) is increasing in t > 0 and if X is SIMIT and θ > 1, then c(t, θ) is decreasing in t > 0. e proof now is completed. e class of SIMIT distributions includes many standard distributions, for example, if X ∼ U(0, 1), X ∼ Beta(2, 2), or X ∼ Beta(1, 3) then M X (t)/t � 1/2, M X (t)/t � (2 − t)/ (6 − 4t), and M X (t)/t � (3 − t)/(6 − 3t) which are increasing functions over t ∈ (0, 1] and thus X has a SIMIT distribution. Theorem 3. Let X and Y satisfy the PMDT model as given in (9). en, Proof. Let us prove the assertion (i). We can see that where θ � 1 − θ ∈ (− ∞, 1). It can be concluded now that M X (t)/t is increasing in t > 0 or equivalently X has a SIMIT distribution, if, and only if, M Y (t)/t is increasing in t > 0 or equivalently Y has a SIMIT distribution. To prove the assertion (ii), note that from the proof of assertion (i), we have us, it suffices to prove that d/dtM X (t) ≥ 0, for all t > 0 implies that d/dtM Y (t) ≥ 0, for all t > 0. For 0 < θ < 1 observe that ≥ 0, for all t > 0.
e proof of assertion (ii) is thus complete. To prove the assertion (iii), notice that from (4) the r.h.r. function of X is written as h X (t) � m X ′ (t)/t − m X (t) and further the r.h.r. function of Y so that Y and X satisfy the PMDT model given in (9), can be written as From Lemma 1 we know that if X ∈ SIMIT, then for θ > 1, c(t, θ) is a non-negative decreasing function in t > 0 and since X ∈ DRHR thus h X (t) is also a non-negative decreasing function in t > 0 and so is h e proof of this assertion is also complied. It remains to prove the last assertion. Let us write By assumption, X ∈ DPRHR which further implies that X ∈ SIMIT and also we have by assumption that θ > 1. Hence, Lemma 1 concludes that c(t, θ) is a non-negative decreasing function in t > 0. Since X ∈ DPRHR thus Prh X (t) is also a non-negative decreasing function in t > 0 and so is Prh e proof of assertion (iii) is obtained.
Recently, some authors have shown their interest in stochastic comparisons of random lifetimes according to reversed average intensity (r.a.i.) function (see, for instance, Rezaei and Khalaf [31], Kundu and Ghosh [32], and Buono et al. [33]). e r.a.i. function of X is given by Below is the definition of r.a.i. stochastic order.

Definition 4.
e r.v.s X and Y with respective r.a.i. functions L X and L Y satisfy the r.a.i. order (denoted by X ≤ rai Y) In the next result, we build the r.a.i. order between X and Y which satisfy the PMDT model given in (9) under some sufficient conditions.

Theorem 4. Let X have a SIMIT distribution and let
Proof. We can see that X ≤ rai Y, if, and only if, for all t > 0, which holds equivalently if, It is known from Lemma 1 that if X is SIMIT, then for every 0 < θ < 1, c(x, θ) ≥ c(t, θ), for all x ≥ t > 0. us the inequality in (38) holds true and hence the proof. (39) e result of eorem 3 can be used to conclude that.
e result we presented in eorem 4 can also be accompanied with the following implication: Y ≤ rai X when, Y is SIMIT and θ > 1.
(40) e following example illustrates an application of eorem 4.
In Example 8, it was shown that m Y (t) � θm X (t). It can be seen that X has an m.i.t. function M X (t) � t(t+ 1)/t + 2, thus M X (t)/t � (t + 1)/(t + 2) is increasing in t > 0, so X has a SIMIT distribution. Notice that from eorem 3 (i), since X is SIMIT, thus Y is also SIMIT. It can be observed that , for all t > 0 and for any 0 < θ < 2, θ ≠ 2.

The Model with Random Departure Index
In recent past decades, frailty models have been frequently used in survival analysis to handle the influence of the covariates on the lifetime variable (see, e.g., Hougaard [34] and Hanagal [35]). In this section, the PMDT model with From which the corresponding p.d.f. is derived as Given a predetermined distribution function to be a choice for the c.d.f. of X, the family of distributions generated by (41) provides a way to add a parameter to the family of distributions of X. In the context of statistical inference, the arisen model could be examined using real data sets in different scenarios to model lifetime events. For a given value of θ, since m(t|θ) � θm 0 (t) is a mean departure time function thus m(t|θ) ≤ t, for all t > 0, thus θ ≤ t/m 0 (t) for all t ≥ 0. erefore, g(t|θ) � 0, for all θ > t/m 0 (t) which means that when F is predetermined and, therefore, t/m 0 (t) is fixed then for a value θ satisfying θ > t/m 0 (t) the PMDT model does not hold. is consideration may be useful before proceeding to do a statistical inference on the model. For example, prior to fitting the PMDT model to a real data set, we might want to test whether θ ≤ t/m 0 (t), for all t ≥ 0.
To integrate the effect of a random variable Θ which is random, we have to consider the unconditional r.v. Y * with a mixture distribution according to (13) which has s.f.
Remark 3. We may notice that in the PMDT model, in the fixed level of the departure index parameter, the values of θ and the c.d.f. F cannot be independently determined. e identity m(t|θ) � θm 0 (t) is meaningful whenever θ ≤ t/m 0 (t), for all t > 0. It is straightforward that t/m 0 (t) is a functional of F and can be written as θ t (F) � t/m 0 (t). For instance, when θ � 3 and F satisfying the inequality θ t (F) < 3 for some choices of t in the support of F, it is concluded that the PMDT model does not hold. In spite of that, when θ ∈ (0, 1] the selection of F does not depend on the choice of θ. is is because the inequality θ ≤ 1 ≤ θ t (F), holds for all t ≥ 0 and for all lifetime distributions F with m.d.t. function m 0 without any further consideration. e random pair (Y * , Θ) is assumed to follow the joint p.d.f g(y, θ) and joint c.d. f. G(y, θ). In the case θ is a realization of the r.v. Θ, the m.d.t. function may be written as where g(y|θ) and G(y|θ) are the conditional p.d.f. and the conditional c.d.f. of Y given Θ � θ, respectively, as given in (42)   erefore, in view of (43), Y * follows a mixture of distributions with proportional mean departure time functions with respect to the mixing distribution K(θ) for the random departure index Θ. In fact, when we are uncertain about the amount of θ in the model, we consider it to be in a dynamical state. Frequencies of values of θ in different intervals construct an empirical probability distribution for θ which converges to K as the observations on θ increase. e c.d.f. (43) takes an average of distributions, having proportional mean departure time functions, with fixed level of the departure index parameter θ with respect to the c.d.f. K as a mixing distribution.In a dynamic population, the departure index parameter varies from one individual to another and that is the value of Θ. Let us assume that an individual enters our investigation randomly. e amount of θ for this randomly chosen subject is considered to be a realization of Θ.
e lifetime of this individual then follows the c.d.f. (43). e amount the function G(t|θ) � P(Y * ≤ t|Θ � θ) takes is the probability for failure before time t for an individual with fixed departure index θ. In statistical Bayesian analysis, an inference strategy is done using the conditional likelihood function of an unknown parameter given data which follow a mixture model. Here, we present the density function of Θ given a single observation on Y * . Specifically, given that Y * � t, the density of Θ is obtained as e density function of Θ among individuals whose lives lost prior to time t is In applied probability, there are always methods to infer on population without data. By means of concepts arisen by theory of stochastic orderings we can partially infer on θ conditional on the events Y * ≤ t and Y * � t. e shapes the density function of Θ|Y * � t and density function of Θ|Y * � t have are complicated. erefore, in the Bayesian setting, the likelihood equation to derive the maximum likelihood estimations of θ probably fails. For this reason, it may be more appropriate to investigate some stochastic ordering properties in terms of the posterior distribution of Θ among individuals with a certain departure time t and also the posterior distribution of Θ among the individuals whose lives lost before time t. e likelihood ratio order is utilized here to build a stochastic ordering property. e likelihood ratio order is stronger than the hazard rate order and that is stronger than the usual stochastic order (cf. Shaked and Shanthikumar [28]).