Folding algorithm: a computational method for finite QBD processes with level-dependent transitions

This paper presents a new computational method for steady state analysis of finite quasi-birth-death (QBD) processes with level-dependent transitions. The QBD state space is defined in two-dimension with N phases and K levels. Instead of formulating solutions in matrix-geometric form, the Folding-algorithm provides a technique for direct computation of πP=0, where P is the QBD generator which is an (NK)×(NK) matrix. Taking a finite sequence of fixed-cost binary reduction steps, the K-level matrix P is eventually reduced to a single-level matrix, from which a boundary vector is obtained. Each step halves the matrix size but keeps the QBD form. The solution π is expressed as a product of the boundary vector and a finite sequence of expansion factors. The time and space complexity for solving πP=0 is therefore reduced from O(N³K) and O(N²K) to O(N³ log₂ K) and O(N² log₂ K), respectively. The Folding-algorithm has a number of highly desirable advantages when it is applied to queueing analysis. First, the algorithm handles the multilevel control problem in finite buffer systems. Second, its total independence of the phase structure allows the algorithm to apply to more elaborate, multiple-state Markovian sources. Its computational efficiency, numerical stability and superior error performance are also distinctive advantages