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In this paper, we study the blocking probabilities in a wavelength division multiplexing-based asynchronous bufferless optical burst switch equipped with a bank of tuneable wavelength converters that is shared per output link. The site of this bank is generally chosen to be less than the number of wavelengths on the link because of the relatively high cost of wavelength converters using current technologies; this case is referred to as partial wavelength conversion in the literature. We present a probabilistic framework for exactly calculating the blocking probabilities. Burst durations are assumed to be exponentially distributed. Burst arrivals are first assumed to be Poisson and later generalized to the more general phase-type distribution. Unlike 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. We propose a numerically efficient and stable solution technique based on block tridiagonal LU factorizations. We show that blocking probabilities can exactly and efficiently be found even for very large systems and rare blocking probabilities. Based on the results of this solution technique, we also show how this analysis can be used for provisioning wavelength channels and converters.