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In this correspondence, we develop fundamental convexity properties of unfinished work and packet waiting time in a queue serving general stochastic traffic. The queue input consists of an uncontrollable background process and a rate-controllable input stream. We show that any moment of unfinished work is a convex function of the controllable input rate. The convexity properties are then extended to address the problem of optimally routing arbitrary input streams over a collection of K queues in parallel with different (possibly time-varying) server rates (μ1(t),...,μK(t)). Our convexity results hold for stream-based routing (where individual packet streams must be routed to the same queue) as well as for packet-based routing where each packet is routed to a queue by probabilistic splitting. Our analysis uses a novel technique that combines sample path observations with stochastic equivalence relationships.