The sojourn time probability distribution function is first derived for a Markovian queue, with both batch arrivals and batch departures, that admits a product-form in networks of such queues. The reversed process is then considered and it is confirmed that the unconditional reversed sojourn time has the same distribution as the forwards sojourn time. Using conditional forward and reversed node sojourn times, a result is obtained for the sojourn time distribution on paths in networks of queues with a certain forward/reversed overtake-free property, which is satisfied by batched queues. This is a considerable generalisation of previous results that apply strictly to overtake-free paths in simple Jackson networks, and indeed the product-form result for the stationary probabilities itself is novel to the authors' best knowledge.
Information from pubs.doc.ic.ac.uk/batches-poster.