AbstractAnN × Nswitching element with output queueing, as used in a large ATM switching network, is considered. All the inlets of the switching element are synchronized on ‘minislots’, where a minislot is the fixed‐length time unit for the transmission of one ‘minicell’. When entering the switching network, an ATM cell is converted into a ‘minicell‐train’, consisting of a fixed number of minicells. Using an active/silent model, it is assumed that on each inlet of the switching element, the number of minicelltrains in an active period and the length of a silent period are both geometrically distributed, and the arriving minicell‐trains are uniformly distributed among all the outlets. The performance of the switching element can be obtained by analysing one single output queue, which is modeled as a discrete‐time single‐server queuing system with train arrivals. In this paper, an upper bound and an approximate expression for the mean queue length are derived. More importantly, an analytical method is developed to obtain a tight upper bound and a good approximation for the tail distribution of the queue length. This analytical method is very useful in buffer dimens