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 nite QBD-process with level-dependent transitions. The QBD state space is deened 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 P = 0, where P is the QBD generator which is an (NK)(NK) matrix. Taking a nite sequence of xed-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 is expressed as a product of the boundary vector and a nite sequence of expansion factors. The time and space complexity for solving P = 0 is therefore reduced from O(N 3 K) and O(N 2 K) to O(N 3 log 2 K)andO(N 2 log 2 K), respectively. The Folding-algorithm has a number of highly desirable advantages when it is applied to queueing analysis. First, the algorithm handles the multi-level control problem in nite buuer systems. Second, its total independence of the phase structure allows the algorithm to apply to more elaborate, multiple-state Markovian sources. Its computational eeciency, numerical stability and superior error performance are also distinctive advantages.