13-15] RELIABILITY 721 



Mathematically, the multiplication of component reliabilities to obtain 

 system reliability can be represented as 



Rsystem = ^1 • ^2 " ^3 Rn- (13-18) 



Now, if it is further assumed that each component in the system fails 

 exponentially, we get from Equations 13-16 and 13-18 



Rsystem == ^-''^' " ^"''^^ " e'"'"' e'''''- (13-19) 



where T„ is the mean life of the nth component in the system. The mean 

 life of the system is then given by 



—^— = 1- + i^ + 1 + ... -F ^. (13-20) 



J system -I 1 -i 2 J3 -*« 



Since 1 /T is equal to r, the hourly failure rate, we can rewrite Equation 

 13-20 as 



rsystem = Tj + Ts + Tg + •.• + r„. (13-21) 



The actual process of predicting the reliability of electronic equipment 

 with as many as 5000 individual electrical and mechanical component 

 parts would be impractical if it were necessary to multiply together 5000 

 individual failure rates. The procedure generally used is to group all 

 similar components into a series of ten or twelve classes — capacitors, 

 resistors, receiving tubes, etc. Each class of components has an associated 

 failure rate per hour, based on past history for that component class; 

 for example, vacuum tubes in airborne equipment have shown a failure 

 rate of 200 X 10~^ per hour. Multiplying the failure rate for any given 

 component class by the number of components in that class will give the 

 failure contribution of the class. Performing this for all the classes and then 

 summing the failure contributions will give the overall system predicted 

 failure rate. Mathematically, this can be represented as 



rsystem = X.Ty + X^r^ + X,r, + ..- + X,r„ (13-22) 



where X is the number of individual components in a class and r is the class 

 failure rate. Substituting the calculated system failure rate in Equation 

 13-17 will allow the generation of a reliability prediction. 



As an example, let us consider the AI radar and fire-control system 

 already discussed. Such systems typically have something of the order of 

 200 vacuum tubes. Thus the expected failure rate from vacuum tubes 

 alone would be 



(200) (200 X 10-«) = 0.04 failures per hour. 



This failure rate exceeds the allowable system failure rate (from all 

 causes) previously derived as 0.035. Thus, it may be expected that consid- 



