Generalization of the Pollaczek-Khinchin formula for throughput analysis of input-buffered switches

Many switch architectures with buffers placed at input ports suffer from the head-of-line blocking (HOL) problem and thus can not achieve 100% throughput. For an input-buffered switch, the number of HOL packets is often characterized by the Lindley equation for a discrete-time queue, i.e. q(t+1)=(q(t)-F)⁺+a(t), where q(t) is the number of HOL packets at time t, a(t) is the number of new HOL packets at time t, and F is the maximum number of HOL packets that can depart per unit of time. As the total number of HOL packets is bounded in a switch, it places an upper limit on the expected number of HOL packets. Thus, the maximum throughput is the utilization that makes the expected HOL packets equal to the upper limit. For the case with F=1, the expected number of HOL packets can be found via the Pollaczek-Khinchin formula and the maximum throughput can be solved by a quadratic equation as reported in [M.J. Karol et al. (1987), C. Kolias et al. (1996), G. Thomas (1997)]. One of the main contributions of this paper is that we derive a generalized Pollaczek-Khinchin formula for the case F>1. Such a formula is then used for finding the maximum throughput of several input-buffered switches. For the case F≫1, numerical computation of the maximum throughput becomes difficult. For large F, we present several bounds and approximations for the throughput. Numerical studies and simulation results confirm that our approximation methods work well.