Continuous-Time QBD Processes with Continuous Phase Variable

In this paper, we consider a continuous-time level-dependent QBD process with a continuous phase variable. We extend the censoring technique to be able to deal with the QBD process. We define a new version for the R- and G-measures, and provide two types of RG-factorizations in terms of these measures. If the matrix of generalized density functions of the QBD process is in L2([0,+∞)2), which is a space of square integrable bivariate real functions, then we provide orthonormal representations for the R- and G-measures, which lead to the matrix structure of the RG-factorizations. Based on this, we can provide an algorithmic framework for computing main performance measures of the QBD process. Furthermore, we introduce a continuous phase type (CPH) distribution, and then analyze a CPH/CPH/1 queue as an interesting example.