Departure processes and busy periods of a tandem network

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.