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Clédel et al. J Surveill Secur Saf 2020;1:119­39 I http://dx.doi.org/10.20517/jsss.2020.08 Page 127 of 139

100%
Availability 1

3
2
0 1 2 3
Time

Figure 2. Availability of a system before, during, and after a shock [38] .

with the failure profile
/
∫ ∫
= P ( ) P ( ) (4)

and the recovery profile
/
∫ ∫
= P ( ) P ( ) (5)

To this performance loss, called systemic impact [26] , the authors added a recovery cost. This recovery cost
corresponds to resources expended in recovery efforts, and, once combined with the performance loss, it gives
the total loss due to a determined disruption, called recovery-dependent resilience [26] .

Babiceanu and Seker [41] evaluated separately the loss of performance in three phases: degradation of perfor-
mance from to , balanced degradation from to , and recovery of performance from to . The
evaluation is the same as the previous one: the integral of the difference between the original level and the
actual level of performance over a period.

Theresilienceofasystemtoaneventisevaluatedbyaresiliencefactorthatistheproductofthreeelements [15] : a
degradation ratio / , a partial recovery ratio / , and a speed factor / . corresponds to the maximum
acceptable value for and > implies that the system cannot recover from the disruption.

Cai et al. [38] used system availability instead of performance level. They defined availability as the ability to
be in a state of performing a function if required external resources are provided. This approach is similar to
the previously described ones in [15,26,41] and is depicted in Figure 2. The system begins at 100% of availability
and then progressively reaches a stable level 1 at time 1. Then, shocks impact the system at time 2 and
availability falls from 1 to 2. Resilience mechanisms handle these shocks such that availability reaches a post-
shock steady state 3 at time 3. Thus, resilience is measured as the product of availability before and after
shocks:

∑ .
(resilience) [38] : R = 1 ( 3 2 ) (6)
ln ( 1 )
=1 ln −
3 2
The authors claimed that the natural logarithm function is used to balance the availability and the recovery
process of the system.

Sterbenz et al. proposed another approach to evaluate network resilience [42] . A system is composed of several
layers: physical, link, topology, network path, end-to-end transport, and application. Each layer is represented