Overflow traffic from a trunk group with balking

A stream of telephone calls is submitted to a group of trunks, the first-choice group, according to a recurrent process. We allow balking on this trunk group; i.e., if a call finds k of the first-choice trunks busy it may be served, with probability pk, or may fail to be served, with probability qk. A call which fails to receive immediate service on the first-choice trunk group is submitted to a second-choice trunk group, the overflow group. We also allow balking on the overflow group. Calls which fail to receive immediate service on the overflow group are lost to the system. Holding times have negative-exponential distribution. We give methods for finding the joint distributions of numbers of busy trunks on the first-choice and overflow groups, at overflow instants (i.e., instants at which calls are submitted to the overflow group), at arrival instants, and at arbitrary instants. We consider the transient as well as the limiting distributions (and demonstrate the existence of the limiting distributions). The methods developed are illustrated by several examples. Numerical results are given for the blocking in the particular case that the first-choice group constitutes a random slip, while the overflow group is full-access (common).