Consider a single-server tandem queueing network with a Markovian arrival process (MAP) to the first station and exponential service times. Using the dimension-reduction and expanding blocks (DREB) scheme, we formulate the joint queue length process into a single-dimensional level-dependent quasi-birth-death (LDQBD) process with expanding blocks. This allows us to show that the departure process from each station is a MAP with infinite phases or an IMAP, which implies that the departure process of any IMAP/M/1 is an IMAP. It also allows us to study the inter-departure time and the busy period in each station easily.