The linear geodesic property is not generally preserved by a FIFO queue

If a FIFO queue is fed by several input streams that jointly satisfy a sample path large deviation principle (LDP) with "linear geodesics," then the cumulative departures (up to a large time) also satisfy the LDP with a rate function which depends in a relatively simple way on the rate function corresponding to the inputs: this was demonstrated in a recent paper by the second author. It suggests the possibility of an iterative scheme which would allow one to determine the large deviation behavior of more complicated networks. To do this, however, one would require that the linear geodesic property be preserved: in this paper we demonstrate that in general it is not preserved. This is true even in the case of a single input stream.