Traffic overflow in closed queueing models

We consider a generalized birth death process that represents a multiserver (closed or open) overflow queueing system with n primary servers and c-n secondary servers. An arrival to the system joins a server of the primary group, if available, otherwise it overflows to the secondary group. If all servers are busy, arrivals are queued, provided the queue buffer is not full, and served as servers become free. Arrivals that find the queue buffer full are lost. Our overflow model differs from models in the open literature in that it combines state dependent arrival rates with group dependent service rates. We present a general formulation that allows a simple derivation of the joint stationary probabilities of i busy primary servers and j busy secondary severs. This main result easily lends itself to an efficient iterative algorithm to evaluate the joint probabilities. We apply our basic theorem to produce new results for several overflow models. A distinctive feature of our approach is that it uses transform-free analysis.