By V. E. Benes
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Extra resources for Mathematical Theory of Connecting Networks and Telephone Traffic
Example text
Only in the state numbered 4 are there any blocked calls. I t is to be noticed that state 4 realizes essentially the same assignment of inlets to outlets as state 2, which has no blocked calls. T h e difference 40 1. HEURISTIC REMARKS AND MATHEMATICAL PROBLEMS I 0 0 0 o n n LO 0 FIG. 16. (Reduced) state diagram for the network shown in Fig. 15. between the two is that in state 2 all the traffic passes through one middle switch, leaving the other entirely free for any call that may arise. Clearly, then, this difference illustrates the “packing rule” that one should always put through a call using the most heavily loaded part of the network that will still accept the call.
By eliminating all the equilibrium probabilities except p , and p , , we find 1 p6 = h+3 ( 1 p7=h+3( + 8X2(l - a)16hpo 8X'%16hp0 240, 7 h 2 +-2h+2 4X216hp0 4hp6 + 4h+2 + + + 8h2(1- a)16hpo 4 x + r + APT) + (16h)'Apo 4hp7 4h+2 We have purposely not simplified the terms so that their origin can be verified. From these two equations we find that where X=A+3----- 2x3 > 0. x h+l 2x 2h+1 + + 7~ + 3 + 3x + 1 5 ~ 2 2x2 - + 2 ~ 3 2x2 w3+4x+3 2x3 + u" + +3 2 ~ 3 4~ 1. 48 HEURISTIC REMARKS A N D MATHEMATICAL PROBLEMS T h e coefficient of oi in f6 is 1 +------- h h+3---- 2x +1 This is positive, because x I f ______3-3 -__ 2x 3h 1 ~ + i X - L l 2X”+5h+3 =l--L-(---2x I 2h2 4h 3 + + + 1 However, Hence, W e shall now consider the problem of optimal routing in our (trivial) network from a different point of view.
T h e myriad choking “details” of the connecting network have been converted into a vast geometrical structure with special properties. T h e operating system describes a trajectory through the state diagram, moving between levels as calls begin and end. (ii) Any stochastic process describing the operation of the connecting network is a point moving randomly on the state diagram. T h e motion is only between adjacent levels. New calls put into the network correspond to jumps to the next higher level; hangups correspond to jumps to the next lower level.