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

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 (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 innity (while the load and conversion ratio sigma= C/N remains fxed). More specifically, for sigma<lambda^ 2 the loss probability tends to (lambda^2 -sigma)/lambda(1+lambda), while for sigma > lambda^2 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.