# Fleet Resilience: Evaluating Maintenance Strategies in Critical Equipment
## 2 Resilience Metrics
#### Resilience and Maintenance  
To the best of our knowledge, only one work [12] has proposed in an explicit manner the use of system-level availability as the parameter that allows performance evaluation in engineering systems. System-level availability reflects the relationship between the time a physical asset is available and the total or nominal time for operation. Availability is an important parameter linked to maintenance and its efficiency. This can be observed in Equation (1), in which the average times between failures and the average times to repair are related. \[A_{s}=\ \frac{\text{MTBF}}{\text{MTBF}+\text{MTTR}} \tag{1}\]  
The first parameter, MTBF, is closely aligned with the concept of reliability, i.e., the probability of an asset to be operational at a certain point in time, while maintainability (MTTR) reflects the probability that the repair will take place in a certain period of time. Still in reference to Equation (1), it can be seen that if the reliability of an asset is reduced, availability will be affected. On the other hand, if the maintainability of the asset decreases,  
Figure 2: The concept of gradual performance degradation process (adapted from [14]). or it suffers a worsening, that is, the time to repair an asset increases, the availability of such asset will also be reduced. For this reason, to promote increases in availability at the system level, efforts must be deployed to improve the reliability of its constituent elements and install efficient and expedited repair procedures. Figure 3 explains the concept of availability in a graphic form. The availability concept allows the calculation of resilience as an intrinsic characteristic of the engineered system, stemming from its structure, its level of organization and its maintenance resources. Therefore, it is possible to assess the effects on the system-level availability considering alterations in the system's topology and the definition of technical resources allocated to maintenance activities. The availability at the system level is estimated taking into consideration the structure (topology) of the system. That is, in order to compute the value of systemic availability, the connections among different pieces of equipment (i.e., serial, parallel) or the existence of redundant or stand-by equipment, must be considered. For example, this can be observed in the time diagram in Figure 4, which reflects the up and downtimes in two pieces of equipment arranged in parallel. For each of the aforementioned configurations, there are analytical models that allow obtaining or estimating the availability at the system level. Table 1 shows the models for system availability computation, considering a system with n equipment, arranged according to different topologies (serial, parallel and with partial redundancy or \(r\) over \(n\)). Figure 4: System availability according to different system topologies. Figure 3: Availability computation. Here, \(A_{s}\) represents the system's availability and \(A_{i}\) represents the equipment's availability. As already mentioned, the model proposed by [PERSON] et al. [12] is the unique evidence of an analytical model that relates the system resilience to the availability of the system in the presence of a given set of disruptive events. Considering Figure 5, we have Equation (2):  
\[\rho=\frac{A_{1}}{n\left(\ln(t_{1})\right)}\sum\limits_{i=1}^{n}\frac{A_{2}^{i }A_{3}^{i}}{\ln(t_{3}^{i}-t_{2}^{i})} \tag{2}\]  
where \(A_{1}\) corresponds to the steady state system's availability. This occurs from the initial moment, until the instant \(t_{1}\). At the instant \(t_{2}\), a sudden decrease in the system availability occurs. From there on, availability presents a transitory state or value \(A_{2}\). Then, the system recovers its availability until a new equilibrium state \(A_{3}\). This happens in \(t_{3}\). The number of availability drops is denoted by \"\(n\)\". Consequently, \(A_{2}^{i}\) and \(A_{3}^{i}\) represents the corresponding values associated with each of the \(i\) shocks (\(i\leqq n\)).