Optical Burst Switching (OBS) has recently been proposed as a candidate architecture for the next generation optical Internet. The central idea behind OBS is the promise of optical technologies to enable switch reconfiguration in microseconds therefore providing a near-term optical networking solution with finer switching granularity in the optical domain. We study the blocking probabilities in a WDM-based optical burst switch  with a bank of Tuneable Wavelength Converters (TWC) that is shared per output line. 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 (PH) 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 (CTMC) 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 large systems and rare blocking probabilities.