Various kinds of one-dimensional transport processes occur in the cell. One such class involves the transport of molecular cargo to and from the cell wall to the interior on filaments. Prominent examples include dynein and kinesin on microtubules and myosin on actin. We will consider a toy model that might prove useful in studying cellular transport. After a quick review of finite Markov processes, we will describe a one-dimensional lattice model of particle transport with reservoirs on both ends called the TASEP (totally asymmetric simple exclusion process). In the mid 90s, this model was solved exactly by Derrida et. al. using a novel technique known as the matrix ansatz. Using this approach, the phase diagram can be computed exactly.Time permitting, we will define a generalised model with two species of particles, which we will also solve using the matrix ansatz and calculate its phase diagram. This part is joint work with J. Lebowitz and E. R. Speer.