Role of recovery profile dependency in time‐dependent resilience

Resilience assessment of civil structures and infrastructure systems plays a vital role in quantitatively measuring the object's ability to withstand and recover from disruptive events (e.g., natural hazards). The performance of the object (structure or system) may degrade with time, and the external loads often display nonstationary characteristics with time‐varying occurrence rate and/or magnitude. In this context, the resilience would be dependent on the duration of the reference period of interest, known as time‐dependent resilience. This article investigates the effect of recovery profile dependency on the time‐dependent resilience of structures and systems. The term “recovery profile dependency” refers to the association of the post‐hazard recovery processes in the aftermath of different load events within the object's service life, and is reflected through the following three aspects: (1) the temporal correlation between the remaining functionalities associated with different load events; (2) the load–recovery interaction (i.e., the scenario that a load event occurs before the full recovery of the degraded functionality due to the previous load event), and (3) the resource allocation strategy in the presence of limited resources, which affects the expeditiousness of each recovery process. In particular, for (2), the concept of average ratio of interaction is proposed to measure the frequency of load‐recovery interaction. The evaluation of time‐dependent resilience is demonstrated through two examples. It is shown that, the time‐dependent resilience is sensitive to the impact of load–recovery interaction and the selection of resource allocation strategy, but is insensitive to the temporal correlation of the remaining functionalities.


INTRODUCTION
Natural and human-caused hazardous events, such as earthquakes, storms and bushfires, may trigger significant threats to the serviceability of civil structures and infrastructure systems.For example, according to the National Oceanic and Atmospheric Administration (NOAA), there were 18 weather/climate disaster events in 2022 that affected US, with losses exceeding $1 billion each. 1 In Australia, since the 2019-20 Black Summer bushfires until 2022, the Insurance Council of Australia has declared 11 catastrophes and two Significant Events, with which insurers have paid out more than $13 billion in natural disaster claims. 2Reliability 3,4 and resilience [5][6][7][8] analyses have been powerful tools to quantitatively measure the ability of an object (structure or system) to withstand disruptive events.The former focuses on the safety level of the object, featured by the probability that a limit state function is not violated, while the latter additionally incorporates the aspects of preparedness, robustness and recovery of the object.Mathematically, let R be the resistance (see, e.g., Song et al. 9 for methods on obtaining the resistance of a system consisting of multiple structures), and S the load event.Taking into account the uncertainties associated with both R and S, the reliability is determined by P(R > S), in which P( ) denotes the probability of the event in the brackets.Next, in terms of resilience assessment, Figure 1a illustrates the resilience problem associated with a single hazardous event.The object's functionality, Q(t), reduces from 1 to Q r due to a hazard occurring at time t f , and subsequently is restored to the pre-hazard state until time t r .A widely-used tool for resilience quantification (R s ) was developed by Attoh-Okine et al., 10 taking a form of Note that the resilience R s in Equation ( 1) is a dimensionless measure, and is a random variable if Q(t) itself is a stochastic process.The functionality Q(t) takes a value between 0 and 1, and thus R s also varies within [0, 1].In the presence of the uncertainty associated with Q(t), the mean value of R s is referred to as mean resilience, denoted by R s .The model in Equation (1) was applied in later studies for resilience assessment.For example, in the work by Decò et al., 11 the pre-event assessment of seismic resilience of bridges was conducted based on Equation (1), taking into account the uncertainties of bridge damage, recovery process and resourcefulness (costs).Vishwanath and Banerjee 12 assessed the life-cycle resilience of deteriorating reinforced concrete bridges in the presence of earthquake excitations, revealing a declining tread of bridge resilience due to aging effects.
Comparing the definitions of reliability and resilience, it is observed that the mean resilience is a generalized form of reliability.To demonstrate this point, assume that there is no resource to restore the functionality in the aftermath of a hazardous event in Equation (1), and Q(t) takes a value of either 1 (survival) or 0 (failure).With this, R s in Equation ( 1) equals 1 if the object survives the load event, and 0 otherwise, with which the mean value of R s (i.e., mean resilience) equals reliability from a mathematical point of view.Considering the performance deterioration of structures or systems due to severe operating or environmental conditions in service, as well as the non-stationary characteristics of the hazardous events, 3,13 both reliability and resilience are dependent on the duration of the reference period of interest (say, [0, T]), known as time-dependent reliability and time-dependent resilience, respectively.The assessment of time-dependent reliability has been extensively discussed in the literature (one may refer to Wang et al. 14 for a review of methods on time-dependent reliability analysis).For the time-dependent resilience of a structure or system during its service life of [0, T], denoted by R s (0, T), one would need to consider the multiple occurrences of hazardous events, as illustrated in Figure 1c.Suppose that there are totally n loads within [0, T], each occurring at time t f ,i for i = 1, 2, … n.As a result of load i, the functionality of the object (structure or system) degrades to Q r,i , followed by a restoration process until time t r,i .For the purpose of resilience quantification, a straightforward approach is to generalize the resilience model in Equation ( 1) by replacing the interval The resilience model in Equation ( 2) can be treated as the weighted summation of the resiliences associated with the occurrence of each hazardous event, and thus is referred to as summation-based resilience model (this accounts for the subscript "sum" used in Equation ( 2)).Illustratively, for the case in Figure 1c, if the reduced functionality is always fully restored before the occurrence of the next load event (i.e., t r,i ≤ t f ,i+1 for ∀i), then it follows that, in which R s,i is the resilience measure associated with the ith load event (computed according to Equation ( 1)).Wang and Zhang 15 evaluated the life-cycle seismic resilience of a power grid system based on Equation (2).The mean value and variance of the resilience measure were derived in a closed form, assuming a Poisson process for the seismic loads.Yang and Frangopol 16 evaluated the resilience loss within the life-cycle reference period as the summation of the resilience losses associated with different load events (taking a similar form of Equation ( 2)), where the number of loads is modeled by a renewal process.The disadvantage of the resilience model in Equation ( 2) is that, it cannot reflect the sensitivity of resilience to low/zero functionality.To demonstrate this point, consider a simple example that the functionality of a structure equals 1 for t ∈ [0, t 1 ] and 0 for t ∈ (t 1 , T] due to the occurrence of a load event at time t 1 , where 0 < t 1 < T. The structure's post-hazard functionality is no longer restored from time t 1 due to the lack of resources, and thus the resilience is expected to be zero for the service period of [0, T].However, applying Equation (2) yields a resilience of t 1 ∕T, which is greater than zero, implying that the structure is still partially resilient.Another approach to formulating the time-dependent resilience is to use the multiplication-based resilience model, 17,18 with which the time-dependent resilience for [0, T] equals the product of the resiliences associated with each load event, that is, Wang and Ayyub 18 derived a closed form solution for the mean resilience (i.e., mean value of R s,mul (0, T)) of an aging structure using Equation (4), based on the following three assumptions: (1) The occurrence of load events is modeled by a Poisson process.(2) The functionality of the post-hazard structure will be fully restored before the occurrence of the next hazardous event.
(3) The recovery processes associated with different loads are statistically independent.However, the resilience model in Equation ( 4) is only applicable to the case where at least one recovery measure is conducted.With this regard, a generalized resilience index was proposed by Wang 19 for functionality-sensitive resilience quantification, which releases the requirement on the number of recovery interventions.
In the analysis of time-dependent resilience, the object may suffer from multiple load events (which lead to multiple recovery processes).It is an essential ingredient to take into account the recovery profile dependency (i.e., the association between different recovery processes) in resilience assessment during the object's service period.One example is F I G U R E 2 Illustration of load-recovery interaction in the assessment of time-dependent resilience.illustrated in Figure 2, where the first load event occurs at time t f ,1 , leading to a reduced functionality to Q r,1 .Subsequently, the functionality is restored via maintenance measures.The second load event occurs at time t f ,2 before the full recovery of the functionality due to the first load.The interaction between the load and recovery processes (called load-recovery interaction) at time t f ,2 -which further leads to the correlation between the recovery processes associated with the first and second load events-should be incorporated in the evaluation of time-dependent resilience.The scenario in Figure 2 is referred to as the occurrence of sequential events.From a view of practical engineering, this may apply to the main-shock-after-shock sequence, 20 or the sequential tropical cyclones (TCs). 21Another real-world example is that, in terms of the erosion and accretion processes of artificial dunes at coastal areas due to multiple occurrences of storms, the performance schematic of a particular dune segment over time is featured by the load(storm)-recovery interaction, 22 although the storm occurrence process itself may be reasonably modeled by a Poisson process.The dependency of different recovery processes can also be reflected through the temporal correlation arising from the sequence of remaining functionalities and that between the resources allocated to the recovery processes in the aftermath of different load events.However, the impact of the recovery profile dependency on time-dependent resilience has yet to be addressed.
The aim of this article is to investigate the impact of recovery profile dependency on time-dependent resilience.The remainder of this paper is organized as follows.In Section 2, a measure for time-dependent resilience is discussed, which has benefited from that in Wang, 19 and can overcome the disadvantages of both summation-and multiplication-based resilience models.In Section 3, a special case of the time-dependent resilience is presented, based on a Poisson model for load occurrence and some independence assumptions on the resiliences associated with each load event.The evaluation of time-dependent resilience considering the impact of recovery profile dependency is presented in Section 4, followed by two examples in Section 5. Concluding remarks are finally formulated in Section 6.

MEASURING TIME-DEPENDENT RESILIENCE
For a reference period of [0, T], this article adopts the model in Wang 19 to evaluate the time-dependent resilience of an object (structure or system), with which the resilience, R s (0, T), is determined according to The resilience model in Equation ( 5) takes a similar form of the summation-based one in Equation ( 2), and at the same time is sensitive to low-zero functionality.Note that R s (0, T) itself is a random variable in the presence of a stochastic functionality Q(t).The mean resilience will be used in the following, based on Equation ( 5), which takes a form of, where E( ) denotes the mean value of the variable in the brackets.With the definition in Equation ( 6), the time-dependent resilience reduces to time-dependent reliability if there is no resource to restore the post-hazard functionality, 19 as explained in the following.Let T f be the time to failure.Based on Equation (5), and thus, R s (0, , where R l (0, T) is the time-dependent reliability for a reference period of [0, T].In some occasions, it is more convenient to use the term "mean nonresilience", denoted by nR s (0, T), which equals 1 − R s (0, T).

A SPECIAL CASE WITH INDEPENDENCE ASSUMPTIONS
In this section, a special case of the resilience model in Equation ( 6) is discussed, assuming: (1) a Poisson process for the occurrence of loads; (2) full restoration of post-hazard functionality before the occurrence of the next load; and (3) statistically independent recovery profiles associated with each load event.The assumptions herein enable the resilience in Equation ( 6) to be evaluated in a closed-form solution.In the next section, the three assumptions will be released so as to formulate a more robust tool for resilience analysis.Within a reference period of [0, T], let N be the number of loads, and R s,T,i the resilience associated with the ith load event (i = 1, 2, … N) evaluated by Equation ( 6) (i.e., by assuming that the ith load event is the only event occurring within the service period of [0, T]).Suppose that the ith load occurs at time t f ,i , followed by a recovery process of the functionality until time t r,i .In terms of R s,T,i , Taking the mean value of R s,T,i , the mean resilience is With some specific configuration of the recovery profile, a closed-form solution for R s,T,i can be derived.For example, as illustrated in Figure 1(b), if the reduced functionality is Q r due to the occurrence of a load at time t, and the recovery process is linear with a rate of K, R s,T,i is also a function of t, written as R s,T,i (t).The duration of recovery, Δt, equals (1 − Q r )∕K.The recovery rate is dependent on the remaining functionality in the presence of given resources, expressed by a function h as follows: Note that nonlinear recovery processes have also been used in the previous studies, including exponential and trigonometric models. 23,24][32][33] Next, the relationship between R s,T,i and R s (0, T) is investigated.According to Equation ( 6), one has, It is observed from Equation ( 11) that the time-dependent resilience in Equation ( 6) takes a similar form of the multiplication-based resilience in Equation ( 4).Modeling the load occurrence as a Poisson process with a time-varying occurrence rate of (t), the probability mass function (PMF) of N is Using the law of total expectation, Equation (11) becomes in which R s (t) is the mean resilience (evaluated at the temporal scale of [0, T]) associated with a single load event occurring at time t.Substituting the PMF of N in Equation ( 12) into Equation ( 13) yields, which is further simplified as follows: Equation (15) shows that the time-dependent resilience in Equation ( 6) can be evaluated explicitly based on some assumptions on the load process and recovery profiles.In particular, if the recovery processes in the aftermath of each load event are linear (as illustrated in Figure 1(b)), one can use Equation ( 10) to evaluate R s (t).That is, substituting Equation (10)  into Equation (15), it follows that, Recall that in Equation ( 7), it was shown that the mean resilience is a generalized form of time-dependent reliability.This point can also be verified through examining Equation (15).If the object's post-hazard state is either survival or failure, then and there is no resource to support recovery of functionality, R s (t) in Equation ( 15) equals the probability of survival at time t conditional on the occurrence of one load event, and thus Equation (15) reduces to the formula for time-dependent reliability. 13,14t is noted that the three assumptions used in this section are the same as those in the multiplication-based resilience model in Wang and Ayyub 18 (as introduced in Section 1), and as a result, Equation (15) takes a similar form of that in Wang and Ayyub. 18This observation implies that Equation ( 6) is a "unified" version of both summation-and multiplication-based resilience models.

TIME-DEPENDENT RESILIENCE FOR GENERAL CASES
In this section, the assessment of time-dependent resilience for general cases is discussed, releasing the assumptions in Section 3.

Load process
The Poisson process has been widely used to model the occurrence of loads. 13,14Other models such as renewal process 34 are also appropriate candidates for use in resilience analysis in Equation (11).When the PMF of the number of loads cannot be expressed in a closed form, one may employ Monte Carlo simulation to find the distribution of the load number for use in resilience assessment.
Temporal correlation often exists between the load intensities due to common causes, which may affect the time-dependent resilience.6][37] The effect of temporal correlation in loads on the time-dependent reliability has been investigated in previous works. 35,38It is thus an interesting topic to examine the sensitivity of resilience to the load temporal correlation.To do so, the temporal correlation of loads is reflected through the correlation between the remaining functionalities associated with each load event.In the presence of N loads, let {Q r,1 , Q r,2 , … Q r,N } be the sequence of (correlated) remaining functionalities.The correlation structure can be described by a correlation decay model (e.g., the exponential law model). 36,39,40Mathematically, let  ij be the correlation between Q r,i and Q r,j at times t i and t j , respectively.It follows that, 38  ij = exp where L c is the correlation length, 1( ) is an indication function, 1(t) = t if there occurs a load event at time t and 0 otherwise.Given the sequence of occurrence times, one can use the well-established Nataf transformation method to generate correlated variables. 4,41

Load-recovery interaction
Recall the second assumption in Section 3, with which the post-hazard functionality is fully restored before the occurrence of the next event.This is reasonable for the case of a relatively expeditious recovery rate and a large time lag between different load events.For more general cases (e.g., a long duration of recovery process), however, such an assumption may result in a biased evaluation of resilience.Consider a simple example that within a reference period of [0, T], there are totally two loads occurring at times t f ,1 and t f ,2 , respectively.Following the first load, the post-hazard functionality is expected to be fully restored at time t r,1 in the absence of the second load event, but is interrupted at time t f ,2 as t f ,2 < t r,1 .
In such a case, if ignoring the load-recovery interaction, R s,T,1 would be underestimated since exp On the other hand, R s,T,2 is overestimated because the remaining functionality Q r,2 is negatively affected by the incomplete recovery process associated with the first load event.Combining both effects on R s,T,1 and R s,T,2 , one cannot directly determine whether ignoring the load-recovery interaction will underestimate or overestimate the time-dependent resilience; however, the role of load-recovery interaction plays a vital role and thus should be taken into account in resilience assessment.With this regard, a new concept of "average ratio of interaction" (ARI) is defined to feature the occurrence frequency of load-recovery interaction.By definition, this item equals E(N int ∕N), in which N int is the number of interactions.Clearly, as this ratio approaches zero, ignoring the load-recovery interaction will lead to negligible error to the resilience assessment.

Dependence of recovery profiles
In the derivation of Equation (15), it has been assumed that the recovery profiles associated with each load event are statistically independent.In practice, it is often the case that the total resource for an object within its service life is predetermined (e.g., at the design stage of the object), and as a result, a strategy for resource allocation for different load events is applicable.Conceptually, for the ith load event (i = 1, 2, … N), if the allocated resource is  i , the mean resilience R s,T,i is a function of both t i and  i , denoted by g(t i ,  i ).The resource allocation problem then becomes: where  tot is the total resource for the life cycle of the object.It is an open question to explicitly optimize this resource allocation problem; however, Equation ( 6) provides a tool for comparing the efficiency of candidate allocation strategies.On the other hand, in the optimal strategy, each  i may be mutually dependent, implying the dependency of the recovery profiles.

Modified versions of the resilience model
The resilience model in Equation ( 6) can be modified slightly in the presence of additional considerations of the resilience problem.For example, if one additionally treats the case of Q(t) <  (i.e., the functionality of the object is smaller than a predefined threshold , where 0 ≤  ≤ 1) as nonfunctional, the resilience in Equation ( 6) is rewritten as follows: in which With the additionally-introduced threshold , similar to Equation ( 19), Equation ( 16) is rewritten as follows: Equation ( 19) again provides a unified tool for reliability and resilience analyses, if the limit state function for reliability is that the resistance is beyond a threshold R 0 (R 0 is the initial resistance).As illustrated in Figure 3, in the context of reliability, if the time-varying resistance R(t) degrades below R 0 at time t  < T, the object is deemed to fail within the service period of [0, T] (this is known as Type-II failure in Wang and Zhang 38 ).Similarly, in terms of resilience analysis, redefine the functionality Q  (t) as follows: it takes a value of 1 if R(t)∕R 0 ≥ , and degrades to 0 upon R(t)∕R 0 is below .With this, based on Equation (19), it follows that, R s (0, T) = E [R s (0, T)] = P(t  > T) = R l,II (0, T), where R l,II (0, T) is the Type-II reliability for a service period of [0, T].
Next, another modified version of Equation ( 15) is discussed.In the aftermath of a hazardous event, there may be an idle time before the restoration measure is conducted.In such an idle time, the functionality could be temporarily zero (or at a very low level); however, the object (structure or system) is still deemed as resilient due to the subsequent scheduled repair measure.In such a case, one would need to exclude the set of idle time in the evaluation of resilience.That is, Equation ( 15) is revised as follows: in which  t is the set of idle time.

Numerical example
In this section, the applicability of the resilience model in Equation ( 6) is examined through a numerical example.The occurrence of loads is modeled by a Poisson process with a time-varying occurrence rate of (t) = c 0 + c  ⋅ t, where c 0 and c  are two constants, t is in years, and c  ∕c 0 = 0.02 (so that the occurrence rate doubles at the end of 50 years).Conditional on the occurrence of a load event at time t, the remaining functionality Q r follows a Beta distribution.To reflect the non-stationarity in load intensities and aging effects, the mean value of Q r decreases linearly with time from 0.8 at the initial time to 0.3 at the end of 50 years.The coefficient of variation (COV) of Q r is constantly 0.2 over the reference periods.In the aftermath of each load event, the recovery rate K is modeled as , where  h is a time-invariant parameter.For example, if  h = 1 and Q r = 0.5, then it needs 5 months to recover the functionality to pre-hazard state; if  h = 1 and Q r = 0, the recovery duration would be 5 years.First, the accuracy of Equation ( 16) is examined through a comparison with Equation ( 6).One can compute Equation ( 16) numerically and evaluate Equation ( 6) based on Monte Carlo simulation.For the latter, a flowchart of generating samples of resilience is presented in Figure 4 for a single simulation run.In the step of "adjust the Q r sequence" F I G U R E 4 Flowchart of simulating sample values of resilience and 'average ratio of interaction (ARI) for a single simulation run.
in Figure 4, Q r,i (i.e., the remaining functionality at the occurrence of the ith event) is adjusted to be 1 to account for the potential impact of the incomplete recovery process associated with the previous load event (Q * i−1 is the recovered functionality immediately before the occurrence of the ith event, and may take a value that is less than 1). Figure 5 presents the mean nonresiliences for reference periods up to 50 years.The initial occurrence rate, c 0 , equals 0.1 in Figure 5a (i.e., on average one load event occurs over 10 years) and 0.5 in Figure 5(b).The legend "Poisson-based" refers to the resilience model in Equation ( 16) (see the lines with triangle symbols), whilst Equation ( 6) has been evaluated through 10 5 replications of simulation.It is observed that, the time-dependent mean nonresilience increases with the duration of service period due to the accumulation of risks.A greater value of c 0 (i.e., greater occurrence frequency of loads) results in a larger nonresilience.For example, with  h = 1, the mean nonresilience for a reference period of 50 years is 0.0277 if c 0 = 0.1, which becomes 0.1788 (6.4 times) if c 0 = 0.5.Further, comparing Equations ( 16) and ( 6), the results are close to each other with a sufficiently large  h (with which the duration of recovery is so short that the occurrence frequency of load-recovery interaction is low).However, with a smaller value of  h , the Poisson-based model overestimates the nonresilience, as explained in Equation ( 18).This difference is enhanced by a greater value of c 0 , due to a relatively smaller time gap between two successive load events.The ARI, as introduced in Section 4.2, is examined in Figure 6 for the two cases in Figure 5.With a fixed reference period, a greater value of  h or a smaller value of c 0 leads to a smaller ARI (occurrence rate of load-recovery interaction).On the other hand, as the value of ARI approaches zero, the difference in resiliences associated with Equations ( 16) and ( 6) becomes negligible.
The impact of threshold  (see Equation ( 19)) on mean resilience is presented in Figure 7, assuming that  h = 1.The additional consideration of  means that Q(t) is replaced by a zero value at any time t once it is smaller than , as shown in Equation ( 20) (Recall that in Figure 5,  has been set as 0).For both cases in Figure 7, a greater value of  means a stricter requirement on the object's functionality and thus leads to a greater mean nonresilience.For example, if c 0 = 0.1,

(A) (B)
F I G U R E 5 Mean nonresiliences for reference periods up to 50 years obtained by Equations ( 6) and ( 16).

(A) (B)
F I G U R E 6 ARI for the two cases in Figure 5.
(A) (B) the mean nonresilience for 50 years increases from 0.0277 to 0.5534 (approximately 20 times) as  varies from 0 to 0.3.This effect is enhanced by a greater value of c 0 .The effects of non-stationarity in load occurrence frequency and robustness of the object (reflected through the mean value of remaining functionality) on time-dependent nonresilience are presented in Figure 8, assuming that  h = 1.The legends "0.2, 0.3, and 0.4" correspond to the cases that the mean value of post-hazard remaining functionality degrades linearly (from 0.8 at the initial time) to 0.2, 0.3, and 0.4, respectively at the end of 50 years.The lines without triangle symbols are obtained with c  = 0 (i.e., a time-invariant occurrence rate of loads).For both cases in Figure 8, with a more severe deterioration of the mean value of Q r , the mean nonresilience becomes larger.Further, an increasing trend in the occurrence rate of loads leads to a greater mean nonresilience, indicating the importance of properly predicting the future changing scenarios of load occurrence in resilience assessment.For example, with c 0 = 0.1 and E(Q r ) = 0.3 at the end of 50 years, the mean nonresilience nR s (0, 50) associated with c  ∕c 0 = 0.02 (i.e., the occurrence rate doubles within 50 years) is 1.76 times that associated with c  = 0.
The sensitivity of mean nonresilience to the temporal correlation in the remaining functionalities is investigated in Figure 9, where c 0 = 0.1 and  h = 1.The correlation length L c (see Equation 17) equals 0, 3, 10, and 200 (in years), respectively, with which the correlation of remaining functionalities associated with two successive load events with a time lag of 1 year equals 0 (independent), 0.717, 0.905, and 0.995 (approximately fully correlated).The results in Figure 9 (A) demonstrate that the selection of L c has a negligible impact on the time-dependent nonresilience.This is explained by the similarity between Equation ( 6) and the summation-based resilience model in Equation (2).Theoretically, the correlation arising from a variable sequence does not affect the mean value of the summation of the variables.The observation from Figure 9 suggests that in the resilience assessment employing Equation ( 6), one can simply assume an independent sequence of remaining functionalities.
Next, the impact of different resource allocation strategies on the time-dependent nonresilience is investigated.Recall that in the derivation of Equation ( 16), the same h function has been used for each recovery process.In this example, h(Q r ) =  h (0.2 + 2Q r ), where the coefficient  h is representative of the recovery rate, and is a function of the allocated resource , written as K = f () (f is a monotonically increasing function of  since a greater amount of resource is expected to result in greater recovery expeditiousness).For illustration purpose, assume that f () = , which indicates that, with unit resource ( = 1),  h = 1.The following two strategies will be examined in the presence of limited resource R tot for the whole life cycle of the object (where m is a positive integer).
1. Apply  tot ∕m for each of the first m load events, leaving no more resource for the other load events (starting from the (m + 1)th).2. Apply  r ∕m for each load event, where  r is the remaining resources immediately before the occurrence of the load event (e.g., for the first load event,  r =  tot , and for the second load event,  r =  tot −  tot ∕m).
Assume that the total resource equals R tot = E(N) ⋅  h , and that m = ⌊E(N)∕a⌋, in which a is a factor that determines m, and the symbol ⌊ ⌋ denotes the greatest integer that does not exceed the number inside.The dependence of mean nonresilience on the resource allocation strategy is shown in Figure 10, where the legend "sx∕a" means "Strategy x (x = 1, 2), and the value of a after the slash symbol".For each strategy, a greater value of a (i.e., a smaller value of m) does not necessarily mean a greater nonresilience.For example, with Strategy 1, c 0 = 0.5 and  h = 1 (see Figure 10b), for a reference period of 50 years, the case of a = 1 results in the smallest mean nonresilience, followed by those associated with a = 0.5 and a = 2, respectively.In fact, the non-monotonicity of nonresilience as a function of a indicates that the optimal solution of m depends on the configuration of the resilience problem.For each case in Figure 10, considering the six candidates together (x = 1, 2, and a = 0.5, 1, 2), the minimum nonresilience has been marked by an arrow.With different values of c 0 and  h , solution is not necessarily by the same strategy.For example, in Figure 10c, for reference periods of 30 and 40 years, the best strategies are s2/0.5 and s1/0.5, respectively.The results in Figure 10 demonstrate that, one can conveniently evaluate the efficiency of resource allocation strategies by applying Equation (6).It is worthy of future works to investigate the optimal solution to the resource allocation problem to achieve greatest resilience with limited resource.

Resilience of a building portfolio subjected to sequential TCs
In this section, the resilience of a building portfolio exposed to sequential TCs (i.e., two cyclones that make landfall close together) is examined.The occurrence of sequential cyclones has been witnessed in many US locations, and has an increasing frequency in the future due to the impacts of climate change. 21For example, Hurricane Ida (2021) and Hurricane Nicholas (2021) struck Louisiana, USA within 15 days of each other. 42,43The latter further worsened the (existing) damages caused by Hurricane Ida due to the short time difference between the two events. 42n this example, a residential building portfolio adopted from the Centerville virtual community 44 is considered to demonstrate the impact of load-recovery interaction on resilience.The portfolio consists of 1856 single-family, one-story wood buildings, with an average size of 130 m 2 .The buildings are exposed to the TC wind hazards.Given the occurrence of a TC event, a building may suffer from one of the following four damage states: none, moderate, severe and complete.These states are defined by three generalized capacities, denoted by  1 ,  2 , and  3 , respectively.The cumulative distribution function (CDF) of  i (i = 1, 2, and 3) takes exactly the same shape as the fragility curve that defines the post-cyclone damage state, 45 which is typically modeled by a lognormal distribution. 46,47Corresponding to the four damage states, the statistics of the generalized capacities are adopted from the hurricane model released by the Federal Emergency Management Agency (FEMA). 48The mean values of  1 ,  2 , and  3 are 49.24,56.13, and 63.02 m/s, respectively, while the COV of the three capacities is identically 0.11.Given the occurrence of moderate, severe, or complete damage state, it takes 120, 360, and 720 days on average to fully recover the building's functionality. 48Assume that the recovery duration Recovery trajectories of a building portfolio exposed to sequential TCs.
caused by the first TC (i.e., at time  1 = 0) equals 22.2%.However, the damage ratios due to the second TC (evaluated at time  2 ) are 39.5%, 24.3%, and 22.3%, respectively, for Δ TC = 15, 150, and 500 days.Similarly, in Figure 11(d-f), the damage ratio at time  1 = 0 equals 85.6%, which becomes 97.9%, 92.1% and 87.3% at time  2 due to the occurrence of the second TC with Δ TC being 15, 150, or 500 days.Next, the mean resilience of the building portfolio subjected to the two TC events is evaluated.For a reference period of 1000 days from  1 = 0, the mean resilience is computed according to Equation ( 6) by taking the average of the sampled resiliences, as presented in Figure 11.If ignoring the cyclone-recovery interaction, in particular when Δ TC takes a relatively small value, then the mean resilience would be overestimated.For example, for the case in Figure 11(a), the resilience for a reference period of 1000 days would become 0.929 (from 0.803) if assuming independence between the TC occurrence and the recovery process.Similarly, for Figure 11d, the resilience is increased from 0.172 to 0.439 by ignoring the cyclone-recovery interaction.This observation has been explained in Section 4.2, that is, the remaining functionality of the building portfolio associated with the second TC event (evaluated at time  2 ) is negatively affected by the incomplete recovery process associated with the first one, and this effect is dominant in the evaluation of resilience (note that this is different from the observation from Figure 5, where the impact of Equation ( 18) dominates).

CONCLUDING REMARKS
In this article, the role of recovery profile dependency in the time-dependent resilience of an object (an individual structure or a system consisting of multiple components) has been investigated.The feasibility of a resilience model is demonstrated, which has benefited from Wang. 19Some modified versions of the resilience model are also presented to fit more general cases (e.g., when a threshold (baseline) for the functionality is applied, or when the idle time is excluded).The following conclusions can be drawn from this article.
1.With some independence assumptions on the recovery profiles associated with different load events, a closed-form solution is derived for time-dependent resilience analysis of an object (structure or system).The explicit formula provides a reasonable estimate for mean nonresilience if the occurrence frequency of load-recovery interaction is low.2. Both the occurrence of load-recovery interaction and the non-stationarity in load occurrence affect the time-dependent mean resilience significantly, and thus should be reasonably predicted for use in resilience assessment.If not considering the non-stationarity in load occurrence (with an increasing trend), the mean nonresilience will be underestimated.However, ignoring the impact of load-recovery interaction does not necessarily lead to an overestimated or underestimated mean resilience.3. The temporal correlation arising from the sequence of remaining functionalities has negligible impact on the time-dependent mean resilience, due to the similarity of the resilience model considered in this article and the summation-based one.4. The resource allocation strategy plays an essential role in the assessment of mean resilience.For scheduled or known strategies, the efficiency can be evaluated through comparing the resulted resiliences, with which the best option can be chosen from the candidates.It is an open question to mathematically optimize the resource allocation problem (i.e., to allocate the limited resource to the recovery processes associated with all the load events in the life cycle) to achieve the largest resilience.

F I G U R E 1
Illustration of resilience and time-dependent resilience problems.(a) Resilience associated with a single hazardous event.(b)As of (a) but with a linear post-hazard recovery process.(c) Time-dependent resilience with multiple occurrences of hazardous events.

3
Dependence of reliability or resilience on the threshold .(a) Reliability and (b) resilience.

8
Effect of load occurrence frequency on time-dependent nonresilience.F I G U R E 9Impact of temporal correlation of Q r on time-dependent nonresilience.