Delay bounds for packetizing time-varying fluid policies with speedup and lookahead in single server systems

We consider an online, processor sharing system with a single server in a packet-switched network where the processor multiplexes packets from multiple input ports onto the same output link. Under a fluid model, in which incoming data is treated as infinitely divisible, the multiplexer allocates its output capacity to serve fractions of packets from all input ports simultaneously (which we call a fluid policy). However, due to the packet nature of the network traffic, a multiplexer scheduler must approximate this fluid policy by a schedule in which only whole packets are sent (which we call a packetized policy). We assume that at each time instant, the aggregate service rate demanded by a fluid policy is no greater than the output link capacity. Then, for a single server with N input ports, and a scheduler using speedup s of at least 2 and a lookahead window of L ≥ 0 time steps, we prove bounds on worst-case maximum additional delay D. In particular, we show that (⌈Ne^-s⌉ - L - 2)⁺ ≤ D ≤ min {(⌈Ne^1-s⌉ - L)⁺, ⌈Ne ^-s⌉}. In the case when no lookahead is used (L = 0), a tighter bound can be obtained: (⌈Ne^-s⌉ - 2)⁺ ≤ D ≤ ⌈Ne^-s⌉.