On the equivalence of flows in queuing networks

The following surprising result has recently been proved: the arrival and departure processes at a node in a Jackson network of M/M/1 nodes have the same interarrival time distribution, under equilibrium [1]. This result, which has been generalized to the case of M/M/1 node in a quasireversible network [2], led the authors of [1] to conjecture the equivalence in law of those processes. It will be shown that this equivalence does not hold in general. The necessary and sufficient conditions for the equivalence of the flows in a closed network of two M/M/s queues will be given (see [3], [4]). The technique used is the martingale approach to the analysis of networks of queues and it will be briefly discussed (see [5]-[6]).