Fluid limit of an asynchronous optical packet switch with shared per link full range wavelength conversion

Abstract

We consider an asynchronous all optical packet switch (OPS) where each link consists of N wavelength channels and a pool of C ≤ N full range tunable wavelength converters. Under the assumption of Poisson arrivals with rate λ (per wavelength channel) and exponential packet lengths, we determine a simple closed-form expression for the limit of the loss probabilities Ploss(N) as N tends to infinity (while the load and conversion ratio σ=C/N remains fixed). More specifically, for σ ≤ λ2 the loss probability tends to (λ2-σ)/λ(1+λ), while for σ > λ2 the loss tends to zero. We also prove an insensitivity result when the exponential packet lengths are replaced by certain classes of phase-type distributions. A key feature of the dynamical system (i.e., set of ODEs) that describes the limit behavior of this OPS switch, is that its right-hand side is discontinuous. To prove the convergence, we therefore had to generalize some existing result to the setting of piece-wise smooth dynamical systems.