432 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



imagine y waiting positions, whore y is so large that few calls are re- 

 jected.* Assume that the offered load is a erlangs, and that the calls have 

 exponential conversation holding times of unit average duration. Finally \ 

 let the average return time for calls which have advanced to the waiting > 

 positions, be 1/s times that of the unit conversation time. The statistical j 

 equilibrium equation can then be written for the probability j\m, n) (j 

 that m calls are in progress on the x trunks and n calls are waiting on 

 the y storage positions: ■ 



/(w, n) = aj{m — 1, n) dt + s(w + l)/(m — 1, n + 1) dt ''■) 



+ (m + \)J{m + 1, n) dt + a/(.r, n - 1) dH^ (2) 



+ [1 - (a*** + sn**) dt - m dt]f(m, n) ^ 



where ^ m ^ .-r, ^ w ^ //, and the special limiting situations are 

 recognized by: 



■* Include term only when m — x 



**■ Omit sn when m = x 



*** Omit a when m = x and n = y 



Equation (2) reduces to 



(a*** + snifif + m)f{m; n) = af{m — 1, n) 1 



+ s(n + l)/(m - 1, w + 1) (3) 



+ (m + l)/(w + 1, n) + af(x, n - !)•, 



Solution of (3) is most easily effected for moderate values of x and y 

 by first setting f(x, ?/) = 1 .000000 and solving for all other /(/?? , ?? ) in 



X y 



terms of /(o:, ?/). Normahzing through zl 11f(m, n) = 1.0, then gives 



m=0 n=0 



the entire f(m, n) array. 



The proportion of time "NC exists," will, of course be 



Z Six, n) (4) 



n=0 



and the load carried is 



L = Xl X wi/(m, n) (5) 



The proportion of call attempts meeting NC, including all re-trials 



* The quant itjr y can also be chosen so that some calls are rejected, thus roughly 

 describing those calls abandoned after the first attempt. 



