NON-BLOCKING SWITCHING SYSTEMS 413 



bers of switching stages and sizes of A^. The data of Table III are plotted 

 on Figure 5. The series of curves appear to be bounded by an envelope, 

 representing a minumum of crosspoints. The next section dealing with 

 minima indicates that points exist below this envelope. 



MOST FAVORABLE SIZE OF INPUT AND OUTPUT SWITCHES IN THE THREE- 

 STAGE ARRAY 



The foregoing derivations were for implicit relationships between n 



and iV, namely, n being the ( — - — \th root of N. To obtain minimum 



number of crosspoints a more general relationship is required. For the 

 three stage switching array this is : 



C(3) = (2n - 1) (2N -f J) (7) 



When n = N^^^ equation (7) reduces to equation (2). 

 For a given value of N, the minimum number of crosspoints occurs 

 when dC/dn = which gives: 



2n' - nN + N = (8) 



This equation has the following two pairs of integral values: 



n = 2, iNT = 16 and n = 3, N = 27 



As N approaches large values equation (8) can be approximated by: 



N = 2n2 (9) 



Graphs of equations (8) and (9) are shown in Fig. 6. In Table IV the 

 numbers of crosspoints are based on the nearest integral values of n for 

 given values of N. 



Where comparisons can be made, Table IV indicates fewer crosspoints 

 than does Table I. This fact can be realized in another manner. By 

 eliminating n in equations (7) and (9), the result for large values of N 

 is: 



C(3) = 4 (2f"N'^' - 4N (10) 



Equation (10) indicates fewer crosspoints than does equation (2). 



MOST FAVORABLE SWITCH SIZES IN THE FIVE-STAGE ARRAY 



If n be the number of inputs per input switch and outputs per output 

 smtch, and m be the number of inputs per smtch in the second stage 



