In this section, we analyze a simplest FB process shown in Figure 3.13. Our goal is to reduce the FB process (2D Markov chain) to a QBD process with a finite number of phases (1D Markov chain) which closely approximates the FB process.
Observe that the infinite number of phases in the FB process
stems from the infinite number of states in the background
process, (Figure 3.13(a)). This motivates us to
approximate by a Markov chain with a finite number of states,
(Figure 3.23(a)). In process , all the states in levels of process is
aggregated into two states labeled . That is, the sojourn
time in levels , , is approximated by a two-phase PH
distribution (as defined in Section 2.2) with parameters:
By replacing the background process by , the 2D Markov
chain (Figure 3.13(c)) is reduced to a 1D Markov chain shown
in Figure 3.23(c). More formally,
the 1D Markov chain is a QBD process, and its generator matrix, , is obtained via the generator
matrix of ,
, as follows.
First, observe that
In general, the stationary probabilities in the 1D Markov chain can immediately be translated into the stationary probabilities in the foreground process. On the other hand, the stationary probabilities in the background process needs to be analyzed independently, since its state space is aggregated in the 1D Markov chain. Observe, however, that the background process is easy to analyze, since it is simply a birth-and-death process without dependency on the foreground process.