Large deviation properties of constant rate data streams sharing a buffer with variable rate cross traffic

We consider a constant rate data stream which shares a buffer with a variable rate data stream. A first come first serve service discipline is applied at the buffer. After service at the first buffer the variable rate traffic leaves the system, whereas the constant rate traffic is sent to a second buffer. Both buffers provide non-idling service at constant rates and infinite waiting rooms. We model the behavior of the queue lengths as a function of the cumulative variable rate cross traffic arrivals. Under the assumption that the random variable rate cross traffic satisfies an appropriate sample path large deviation principle, we deduce a sample path large deviation principle for the induced queue length processes.

This allows us to investigate logarithmic large deviation asymptotics for the tail probabilities of the steady-state queue length distribution at the second buffer. We show that these asymptotics can be obtained as the solution of a two-dimensional minimization problem. We explicitly calculate rates and associated minimizing paths when the variable rate cross traffic consists of an increasing number of superimposed exponential on-off sources and compare them to related large buffer asymptotics for a single on-off source as cross traffic.

These results partially extend those of Ramanan and Dupuis [19] to more general rate functions. Also they complement our work [13] in which we investigated moderate deviations of this queueing network in critical loading.