UELIABILITY OF THAFFK" MKASURKMKiXTS 



375 



cording to the laws govornino; the com})()iinding of variables the moment 

 generating function of the sum of /( variables //, when n is also \'ariable 

 with generating function C/(/), is (!{M{a)) where M(a) is the moment 

 generating function of //. 



Th(^ gen(M'ating function of a Poisson \ai-iable with average "a" is 



r""^' so that 



G(Mi(a)) = e 



— a+aJl//(a) 



G(Mu(a)) = e 



—arc+arcM ji(a) 



(12) 



These independcMit variables may be added })y multiplying their mo- 

 ment generating functions to give the m.g.f. of the total measurement 

 error of the carried load 



■jiTf \ —(a+arc)+aMi(,a)+arcMji(a) 



From (6), (11) and (13) the following parameters are found 

 ^ = 



(7„ = arc 



/ctnhl - 



c re 



\ c c re 



If, now, re is sufficiently large 



<^y — a/ arc 



rctnhl - 



(13) 

 (14) 



-)] 



(15) 



It is more convenient to deal with the standard deviation expressed 

 as per cent of the carried erlang load, the coefficient of variation. This 

 is done by multiplying both sides of equation (15) by ^ to convert the 

 time dimension from holding times to hours, dividing by T to convert 

 from erlang-hours to erlangs, dividing by a' to convert to proportion of 

 carried load, and multiplying by 100 to convert to per cent. Assuming 



a . 1 



—^ IS approximately —, : 



•ctnh 



2 



a'T 



V^ = 100 



When A'' observations are made this reduces further to 

 F. = 100 



v^hK0- 



1 -,. ,« 9 .3 



ISow ctnh(.r) = - + - - — + — • 



^ X 3 45 945 



ix' < tt') 



