Wavelength converter sharing in asynchronous optical packet/burst switching: An exact blocking analysis for markovian arrivals

In this paper, we study the blocking probabilities in a wavelength division multiplexing-based asynchronous bufferless optical packet/burst switch equipped with a bank of tuneable wavelength converters dedicated to each output fiber line. Wavelength converter sharing, also referred to as partial wavelength conversion, corresponds to the case of a number of converters shared amongst a larger number of wavelength channels. In this study, we present a probabilistic framework for exactly calculating the packet blocking probabilities for optical packet/burst switching systems utilizing wavelength converter sharing. In our model, packet arrivals at the optical switch are first assumed to be Poisson and later generalized to the more general Markovian arrival process to cope with very general traffic patterns whereas packet lengths are assumed to be exponentially distributed. As opposed to the existing literature based on approximations and/or simulations, we formulate the problem as one of finding the steady-state solution of a continuous-time Markov chain with a block tridiagonal infinitesimal generator. To find such solutions, we propose a numerically efficient and stable algorithm based on block tridiagonal LU factorizations. We show that exact blocking probabilities can be efficiently calculated even for very large systems and rare blocking probabilities, e.g., systems with 256 wavelengths per fiber and blocking probabilities in the order of 10^-40. Relying on the stability and speed of the proposed algorithm, we also provide a means of provisioning wavelength channels and converters in optical packet/burst switching systems.