Analysis of the RFB process

The stationary probabilities in the RFB process can be analyzed by applying the analysis in Section 3.5.2 recursively. Recall that, in the RFB process, the infinitesimal generator of the $i$-th process depend on the level of the $(i-1)$-th process for $2\leq i\leq m$. In Section 3.5.2, we analyzed the case of $m=2$ (the FB process).

We argue, by induction, that all the $m$ processes that constitute the RFB process can be approximated by finite-phase QBD processes (1D Markov chains) via the approach in Section 3.5.2. Then, the stationary probabilities in the $m$ processes can be obtained by analyzing the stationary probabilities in the 1D Markov chains. By our assumption, the first process is a finite-phase QBD process that does not depend on other processes, which proves the base case. Suppose that the $i$-th process is approximated by a QBD process with a finite number of phases, $B_i$. The QBD process $B_i$ typically has an infinite number of levels. However, by the analysis in Section 3.5.2, $B_i$ can be approximated by a QBD process with a finite number of levels, $\widetilde B_i$, such that $B_i$ and $\widetilde B_i$ have stochastically similar effect on the $(i+1)$-th process, $B_{i+1}$. Now, using $\widetilde B_i$, process $B_{i+1}$ can be approximated by a QBD process with a finite number of phases. This completes our argument.