298 BELL SYSTEM TECHNICAL JOURNAL 



and ifA</<r — A, we have effectively 



71(0 = f/^ F(t)dt (1.3-3) 



If we now average I(t) over all of the M intervals instead of only over 

 those having K arrivals, we get, as M — > oo , 



7(0 = £ PiK)T^) 



= V / F{t) dt (1.3-4) 



and this proves the first part of the theorem. We have used this rather 

 elaborate proof to prove the relatively simple (1.3-4) in order to illustrate a 

 method which may be used to prove more complicated results. Of course, 

 (1.3-4) could be established by noting that the integral is the average value 

 of the effect produced by one arrival, the average being taken over one 

 second, and that v is the average number of arrivals per second. 



In order to prove the second part, (1.2-3) of Campbell's theorem we first 

 compute P{t) and use 



W) - nor = im - 2 /(/)/(o + /(o 



= 7^) - Wf (1-3-5) 



From the definition (1.3-1) of //c(/), 



Averaging this over all values of /i , ^2 , • - • iK with t held fixed as in (1.3-2), 



iiio=tt r^--- r ^-^ Fit - t,)F{t - u 



k=l m=l Jo 1 Jo 1 



The multiple integral has two different values, li k = m its value is 



Jq 



dk 



JO T 



and ii k 7^ m its value is 



/ ''<' - '') T 1 ^(' - '"'> T 



