Intuitively, an FB process consists of a background (QBD) process and a foreground (QBD) process, where the behavior (infinitesimal generator) of the foreground process can depend on the level of the background process. Additionally, we require that there exists a level, , in the background process such that the infinitesimal generator of the foreground process does not change while the background process is in levels .
Consider a simpler case where the foreground and background
processes are homogeneous birth-and-death processes. The background
process, , has a fixed generator matrix, (see Figure 3.13(a)):
Recall the 2D Markov chains shown in Figure 3.1. These Markov chains are FB processes. In Figure 3.1(a), the background process tracks the number of donor jobs, and the foreground process tracks the number of beneficiary jobs. In Figure 3.1(b), the background process tracks the number of high priority jobs, and the foreground process tracks the number of low priority jobs. Also, recall the 2D Markov chain shown in Figure 3.4(c). This 2D Markov chain is an FB process, where the background process is a QBD process (and not a birth-and-death process). Again, the background process tracks the number of high priority jobs, and the foreground process tracks the number of low priority jobs.
In general, an FB process is defined by a vector of generator matrices . Here, denotes the generator matrix of the foreground process when the background process is in level . We require that and satisfy the following characteristics: