In this paper, we study a passenger-taxi queueing systems in which nonzero matching time is considered. We assume that passengers and taxi drivers who arrive at a taxi station independently form two queues, and the waiting space for taxis is limited. We use a two-dimensional Markov process to model the system. By using matrix-analytic method, we give a sufficient condition that ensures the existence of steady-state probabilities and present an algorithm to calculate the joint steady-state probabilities. Further, we consider the tail distributions of sojourn times of passengers and taxi drivers, which can be computed by algorithms we present in this paper. And by running Matlab programs, we give the numerical results about the performance measures of the system.