420 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1953 



that a minimum should occur for N = 240, when n = 6.81 and m = 

 3.56. In Table VII the minimum occurs when n = 6 and m = 4. It 

 fails to occur at n = 7 because 240 is not exactly divisible by 7. Except 

 for this situation, the minimum would have occurred as predicted. 



RECTANGULAR ARRAY 



Referring to Fig. 1, if there were A^ inputs and M outputs, a simple 

 rectangular array would result which would be capable of sustaining up 

 to A^ or Mj whichever is the lesser, simultaneous connections without 

 blocking. The number of crosspoints is: 



C(l) = NM (20) 



N INPUTS AND M OUTPUTS IN A THREE-STAGE ARRAY 



For the case of a three-stage switching array with N inputs and M 

 outputs, let there be n inputs per input switch and m outputs per out- 

 put switch. A particular input to be able to connect without blocking 

 under the worst set of conditions to a particular output will require 

 (n — l) + (m— 1) + 1 available paths. Thus by providing for that 

 many intermediary switches, a non-blocking switching array is obtained. 

 The number of crosspoints is : 



C(3) = {n + m- 1)\n + M + —1 (21) 



Differentiating this equation first with respect to n and then to m 

 yields two partial differential equations w^hose solution indicates that a 

 minimum is reached when n = m. Replacing m by n in equation (21), 

 the equation for the number of crosspoints becomes: 



C(3) = (2n - 1) [at + ilf + ^] 



(22) 



Solving for the minimum number of crosspoints gives the following 

 expression : 



, ATM „ , NM „ ..-„, 



When N — M this equation reduces to equation (8). 



The three-way relationships of n, N and M are shown in Fig. 8. 



