The Limit of the Blocking as Offered Load Decreases With Fixed Peakedness

A blocking system is a trunk group with no queueing; when an arriving call does not find an idle trunk immediately, it overflows from the system. Every trunk group mentioned here is assumed to be a blocking system and to have independent exponential service times with unit mean. Also, statistical equilibrium is always assumed. If the arrival stream to a trunk group is Poisson, the stream of overflow calls is termed a simple overflow process (SOP). An SOP is characterized by two parameters; generally, these are taken to be the intensity and peakedness, denoted here as a and z, respectively. (We follow Fredericks1 for terminology and notation as well as for proofs or references not given here.) 2911 A trunk group with SOP input is the model underlying the Equivalent Random method2 of sizing trunk groups that carry traffic whose peakedness is greater than unity. Algorithms based on the Equivalent Random method, or on approximations to it, are used to relate trunkgroup size to the parameters of the input traffic to achieve a given probability of blocking (blocking). When these algorithms iterate on a, with z (and possibly other parameters) fixed, to achieve a low blocking on a given trunk group, they have been found sometimes not to converge, most prominently for large values of z.3 This will certainly happen if there exists a minimum blocking, over all values of a, for the given values of the other parameters, and the target blocking is less than this minimum. An algorithm may fail also if it requires that the blocking be a monotone function of a when relevant values of a are in a region where this function is nonmonotone.