In this paper, we propose the first exact Markov model for connection blocking analysis in elastic optical networks, based on the occupancy status of spectrum slices on all links due to arrivals and departures of various classes of connections in a network. Since the complexity of the exact Markov model grows exponentially with the link capacity, number of links, routes, and classes of demands, we further advance the state-of-the-art in computing approximate blocking probability in elastic optical networks and propose two novel approximations, i.e., load-independent and load-dependent. These approximations are used to compute state-dependent per-class connection setup rates in multi-class elastic optical networks with or without spectrum converters by taking into account the spectrum fragmentation factor in each state. We validate approximation analysis by exact and/or simulation results, and show that load-independent and load-dependent approximations can be more accurately used than previously proposed approximations, under a random-fit (RF) and a first-fit (FF) spectrum allocation policies. The approximate results match closely with the exact model, for smaller networks, and with the simulations under a variety of network scenarios.