446 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



case consisting of two subgroups of one trunk each and one common 

 overflow trunk. 



Instead of solving the enormously elaborate system of equations de- 

 scribing all the calls which could simultaneously be present in a large 

 multiple, several ingenious methods of convoluting the 



X 



6(n) = Z/(w, n) 



overflow distributions from the individual legs of a graded multiple have 

 been devised. For example, for the multiple of Fig. 10(a), the probability 

 of loss Pi as seen by a call entering subgroup number i, is approximately, 



Pi = 2 £ e.Ar)-rl^{z -r) +J: d.Ar) (9) 



r=0 z=y T—y 



in which \l/{z — r) is the probability of exactly z — r overflow calls being 

 present, or wanting to be present, on the alternate route from all the 

 subgroups except the zth, and with no regard for the numbers of calls 

 present in these subgroups. The ^x,i(^) = jiixi , r) term, of course, con- 

 templates all paths in the particular originating call's subgroup being 

 occupied, forcing the new call arriving in subgroup i to advance to the 

 alternate route. This corresponds to the method of solving graded mul- 

 tiples developed by E. C. Molina^ but has the advantage of overcoming 

 the artificial "no holes in the multiple" assumption which he made. 

 Similar calculating procedures have been suggested by Kosten.* These 

 computational methods doubtless yield useful estimates of the resulting 

 service, and for the limited numbers of multiple arrangements which 

 might occur in within-office switching trains (particularly ones of a sym- 

 metrical variety) such procedures might be practicable. But it would be 

 far too laborious to obtain the individual overflow distributions Q{n), 

 and then convolute them for the large variety of loads and multiple 

 arrangements expected to be met in toll alternate routing. 



7.2. Approximate Description of the Character of Overflow Traffic 



It was natural that various approximate procedures should be tried in 

 the attempt to obtain solutions to the general loss formula sufficiently 

 accurate for engineering and study purposes. The most ol^vious of these 

 is to calculate the lower moments or semi-invariants of the loads over- 

 flowing th(; sul)groups, and from them construct approximate fitting 



* Kosten gives the above approximation (9), which he calls Wb^, Jis an upper 

 limit to the blocking. He also gives a lower limit , Wr, in which z = // throughout 

 (References 4, 5). 



