Dynamic Scaling and Growth Behavior of Queuing Network Normalization Constants

A sample dynamic scaling technique is shown that avoids both the overflow and underflow problems that are often encountered in the evaluation of normalization constants of closed product-form queuing networks W~th dynamic scaling, normalization constants for very large routing chain population sizes can be evaluated within the bounds of a relauvely small range of numbers. It is shown that the product-form solution possesses a local balance property and the M ~, M property with respect to routing chains. The relationships between normahzaUon constants of closed networks and certain equilibrium aggregate state probabdities in networks that permit external arrivals and departures are examined. The growth behavior of normalization constants is shown to be modeled by a birth-death process traversing over the set of chain population vectors