44 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 



line finders busy and one call waiting. Hence the weighted delay distri- 

 bution, Pi(>t), is: 



PiOt) = Kc + 1)[1 + c/j - (c/j) exp (-t/H)] exp {-d/jH) (11) 



In the general case pni>t) is given by the following formula: 



PnOt) = F„+i(0 exp (t/H) (12) 



where F„+i(0 is given by Conny Palm.® The over-all delay distribution 

 is then : 



POO = PoOt) + PiOO + P2OO + 



(13) 



By making appropriate substitutions and summing the result, ex- 

 pression (13) becomes: 



POO = V(o) 



c! 



■ aexp i-t/H) a' exp (-21/ H) 



c+i ^ (c+j)(c + 2j)^ " 



exp [-ct/jH + (a/j)[i - exp (-t/H)]] (14) 



Expression (14) is equivalent to that of Riordan involving two in- 

 complete gamma functions as follows : 



P(>t) = p(>o)^t^/i>(«/i)^^P(-^/-^)l 



y(c/j, a/j) 



where the incomplete gamma function, 



y(N,x) = f x'^-'i 

 Jo 



dx 



(15) 



(16) 



The theoretical dial tone tester delay curves shown on Figs. 1(a) to 

 1(d), 2(a) to 2(d), and 3 to 6 were computed from expression (14), 

 using the following values of j and H for the classes of service studied, 

 these values being determined in a manner explained later: 



On Figs. 1(a), 1(c), 2(a), and 2(c), which show the per cent of dial tone 

 tests encountering delays greater than three seconds for various amounts 



* Equation 53, loc. cit. 



