Applications of vector Riemann boundary value problems to analysis of queueing systems

Nonhomogeneous vector Riemann boundary value problems are introduced as a method of studying the generating functions of the steady-state queue-length probabilities. This method is applied to Markov-controlled random walks with finite perturbations that arise in analyses of semi-Markov queues with limited state-dependence. The solutions obtained often involve complex-analytic factorization components of certain matrix functions, but, in some cases, the solution is shown to depend only on selected elements of these components. Methods of finding such components are discussed, depending on the structure of the matrix functions and complex-analytic properties of their elements. In particular, a procedure is introduced to reduce the dimension of the problem and, in the case of special tridiagonal structures, to reduce the problem to scalar factorization. The technique introduced is applied to several special cases of semi-Markov queues, such as ‘¸±’ controlled and ‘queue-increase’ controlled systems, and to examples of limited state-dependence in semi-Markov queues such as ‘warm-up’ and ‘Bailey-type’ dependence with additional processing. Potential applications and implementation problems are discussed.