S URVIVABILITY OF SYSTEMS UNDER MULTIPLE FACTOR IMPACT E DWARD K ORCZAK, G REGORY L EVITIN Adviser: Frank,Yeong-Sung Lin Present by Sean Chou 1

A GENDA Introduction Assumptions and model MSS survivability evaluation Computational example Conclusions 2

A GENDA Introduction Assumptions and model MSS survivability evaluation Computational example Conclusions 3

I NTRODUCTION Many technical systems operate under influence of external factors that may cause simultaneous damage of several system elements and lead to degradation or even termination of the mission/ function performed by the system. Survivability, the ability of a system to tolerate intentional attacks or accidental failures or errors 4

I NTRODUCTION In order to mitigate the impact of external factors, a multilevel protection is often used. The multilevel protection means that a subsystem and its inner level protection are in their turn protected by the protection of the outer level. Numerous studies were devoted to estimating the impact of external factors on the system’s survivability 5

I NTRODUCTION In the recent paper [1], a new algorithm for evaluation of the survivability of series-parallel systems with arbitrary (complex) structure of multilevel protection was presented, extending applicability of previous works. [1] Korczak E, Levitin G, Ben Haim H. Survivability of series–parallel systems with multilevel protection. Reliab Eng Sys Safe 2005;90:45–54. 6

I NTRODUCTION In many real situations the impacts can be characterized by several destructive factors (DF) affecting the system or its parts simultaneously. The groups of elements protected by different protections can overlap. Each protection can be effective against single or several DFs. However this fact was not considered in [1]. 7

A GENDA Introduction Assumptions and model MSS survivability evaluation Computational example Conclusions 8

A SSUMPTIONS AND MODEL 9

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A SSUMPTIONS AND MODEL Gj : : random performance rate of MSS element j The element can have Kj different states (from total failure up to perfect functioning) with performance rates gjk (1pkpKj). The performance distribution of each element when it is not affected by any DF is given as 11

A SSUMPTIONS AND MODEL Single elements or groups of elements can be protected. All the elements having the same protection compose a protection group (PG). Any PG or its part can belong to another PG. Oj : set of numbers of DFs that can destroy element j Yj, d : set of numbers of protections that protect element j against DF d 12

A SSUMPTIONS AND MODEL The system survives if its performance rate is not less than the minimal allowable level w. The MSS survivability is the probability that the system survives: 13

A SSUMPTIONS AND MODEL The presented model in which the system is always exposed to all the DFs can be directly used for two cases: 1) The system survivability is evaluated under assumption that the system is under single multifactor impact. 2) The system survivability is evaluated when external threats are continuously present. 14

A GENDA Introduction Assumptions and model MSS survivability evaluation Computational example Conclusions 15

MSS SURVIVABILITY EVALUATION The conditional performance distribution (1) of any system element j when it is not affected by any DF is represented by the u-function : When the element j is destroyed its performance is zeroed with probability 1. Therefore, the conditional performance distribution of destroyed element is represented by the u-function z0. 16

MSS SURVIVABILITY EVALUATION Let xm be the state of protection m( xm =1 if protection m is destroyed and xm = 0 if it survives). This condition can be represented by Boolean function bj,d(x): By convention the product over the empty set is equal to 1. Therefore bj;d ex = 0 if Yj;d ? +. 17

MSS SURVIVABILITY EVALUATION Element j survives if it is not destroyed by any DF from Oj. This condition can be represented by Boolean function Bj(x): 18

MSS SURVIVABILITY EVALUATION If all of the protections belonging to Yj,d for any DF d 2 Oj are destroyed element j is also destroyed and its performance distribution in this case can be represented by the u-function z0. The element performance distribution as a function of the states of the protections as 19

MSS SURVIVABILITY EVALUATION Applying these rules recursively, one obtains the final u-function of entire system: 20

A GENDA Introduction Assumptions and model MSS survivability evaluation Computational example Conclusions 21

C OMPUTATIONAL EXAMPLE A chemical reagent supply system consists of seven multi-state elements. Three DFs can incapacitate the system in the case of explosion: fire (DF 1) corrosion active gases (DF 2) voltage surge (DF 3). 22

C OMPUTATIONAL EXAMPLE Nine different protections are used to protect different groups of the elements. 23

C OMPUTATIONAL EXAMPLE The element protection sets Yj,d corresponding to different configurations 24

C OMPUTATIONAL EXAMPLE 25

C OMPUTATIONAL EXAMPLE 26

C OMPUTATIONAL EXAMPLE 27

C OMPUTATIONAL EXAMPLE 28 Fig. 2 presents the MSS survivability as a function of the demand for each configuration.

C OMPUTATIONAL EXAMPLE Observe that configuration A provides greater system survivability in the range of small demands while configuration B outperforms configuration A in the range of greater demands. This shows that when different protection configurations are compared the expected system demand should be taken into account. 29

A GENDA Introduction Assumptions and model MSS survivability evaluation Computational example Conclusions 30

C ONCLUSIONS This paper presents an adaptation of the numeric algorithm for evaluating the survivability of series-parallel systems with multilevel protection [1] to the case of multiple factor impacts. 31

Thanks for your listening. 32