Moments of inter-level passage times in QBD processes

Specifically, we consider a QBD process on the state space
, which has generator matrix :

as defined in Section 3.2. We assume that the QBD process repeats after level ; i.e. , , and for all .

Our goal can be roughly stated as deriving the passage time required to get from state to level conditioned on the particular state first reached in level . To state our goal more precisely, let be the event that state is the first state reached in level when starting in , and let be the time to go from state to state . Then, our goal is to derive the matrix, , where is the -th moment of given event , for each and

Observe that

Hence, it suffices to derive two quantities:

The rest of this section is organized as follows. In Section 3.7.1, we introduce some notations that we will need along the way. In Section 3.7.2, we derive and for each and . (Matrix can also be obtained by the algorithm in Figure 3.12.) In Section 3.7.3, we mention some generalizations that Neuts' algorithm allows. In particular, we show an extension to the passage time from level to level for , which we will need in Section 3.8.