Clos Networks: A correction of the Jacobaeus result

In 1950, C. Jacobaeus of LM Ericsson established a method for computing internal blocking probabilities for a point-to-point selection within an interconnection network. Jacobaeus result showed that a three stage network would be "nearly" free of internal blocking if the number of matrices in the second stage was equal to the number of inlets of a first stage matrix plus the number of outlets of a third stage matrix minus one. In 1953, C. Clos of Bell Labs showed that for such a number of second stage matrices referred as the "Clos number" the internal blocking probability should indeed be exactly zero under any traffic hypothesis. This meant that the Jacobaeus result, although quite precise, was not perfectly exact. In this paper we bring a correction to the Jacobaeus calculation, leading to a different result. This new result gives an exact zero internal blocking probability for a number of second stage matrices equal to the "Clos result" showing thus its better accuracy.