A Taylor series approach for coupled queueing systems with intermediate load

Abstract

We focus on the numerical analysis of a coupled queueing system with Poisson arrivals and exponentially distributed service times. Such a system consists of multiple queues served by a single server. Service is synchronised meaning that there is a departure from every queue upon service completion and there is no service whenever one of the queues is empty. It was shown before that the terms in the Maclaurin series expansion of the steady-state distribution of this queueing system when the service rate is sent to 0 (overload) can be calculated efficiently. In the present paper we extend this approach to lower loads. We focus on a sequence of Taylor series expansions of the stationary distribution around increasing service rates. For each series expansion, we use Jacobi iteration to calculate the terms in the series expansion where the initial solution is the approximation found by the preceding series expansion. As the generator matrix of the queueing system at hand is sparse, the numerical complexity of a single Jacobi iteration is O(N MK), where N is the order of the series expansion, K is the number of queues and M is the size of the state space. Having a good initial solution reduces the number of Jacobi iterations considerably, meaning that we can find a sequence of good approximations of the steady state probabilities fast.