1020 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



3.2 Bounds and Asyynptotic Formula 



The Laplace transform of P{L) is . I 



p(s) = {s + (n - 1)X(1 - e-("^+«>^)pi (3-1) 



i 

 which has one real pole at a negative point s = —a, a < (n — 1)X. 



Again it is this pole which contributes the dominant term to both P(L) 



and F{L) for large L. We find 



^ —aL 



F{L) - ''^' 



(1 + [{n - 1)X - a]b){ii\ - a)' 



To bound P{L) by expressions of the form A exp( — aL) one finds 

 that .4 > 1 will gi\'e an upper bound and 



will give a lower bound. The corresponding bounds on F{L) are of the 

 form 



/^ nXA \ -n\L I n\A -aL 



(1-— le +— e . 



\ n\ — aj nh — a 



3.3 Exact Solution 



As in Part II an exact formula for F(L) may be given as a finite sum. 

 We now derive it from the Laplace transform, 



j\s) = (s + nXf (1 + nXpis)), 



of F{L). We may use (3-1) to expand /(s) into the series 



/r ^ 1 /i , ,,x V ((^^ - i)xe-^"^+-^y \ ,. _. 



lis) = — ■ — r U + nX 2^- — ,...,., >. (3-2) 



s + n\ [ t=o (s + (w — l)\y+^ J 



The identity 



(s + n\y\s -f (n - 1)X)"''~' 



= 7 i: (-A)-'^^(s + in - 1)X)-^'-^ + (-X)-^-^(s + n\r' 



X ;=0 



provides a partial fraction expansion for the k term of the series (3-2). 

 Transforming (3-2) term by term with the help of (3-3) we find 



F(L) = e~"'^[-(n - 1)]^"^"^+' 



[lU] k 



Z [-in-De-^'rZ 



This is the desired formula for F{L). 



+ ne'"'-'>''- "Z [-in -Dc^'Y' E ^~^^^.~ ^^^^' 



