On Elevator polling with globally gated regime

We consider a polling system consisting of N queues and a single server where polling is performed according to an Elevator (scan) scheme. The server first serves queues in the "up" direction, i.e. in the order 1, 2,..., N - 1, N, and then serves these queues in the opposite ("down") direction, i.e. visiting them in the order N,N-1,...,2,1. The server then changes direction again, and so on. A globally gating regime is used each time the server changes direction. We show that, for this Elevator scheme, the expected waiting times in all channels are equal. This is the only known non-symmetric polling system that exhibits such a fairness phenomenon. We then discuss the problem of optimally ordering the queues so as to minimize some measure of variability of the waiting times.