Randomization does not reduce the average delay in parallel packet switches

Switching cells in parallel is a common approach to building switches with very high external line rates and a large number of ports. A prime example is the parallel packet switch (PPS) in which a demultiplexing algorithm sends cells, arriving at rate R on N input-ports, through one of K intermediate slower switches, operating at rate r < R. In order to utilize the parallelism of the PPS, a key issue is to balance the load among the planes; since randomization is known as a successful paradigm to solve load balancing problems, it is tempting to design randomized demultiplexing algorithms that balance the load on the average. This paper presents lower bounds on the average queuing delay introduced by the PPS relative to an optimal work-conserving first-come first-serve (FCFS) switch for randomized demultiplexing algorithms that do not have full and immediate information about the switch status. These lower bounds are shown to be asymptotically optimal through a methodology for analyzing the maximal relative queuing delay by measuring the imbalance between the middle stage switches; clearly, this also bounds (from above) the average relative queuing delay. The methodology is used to devise new algorithms that rely on slightly outdated global information on the switch status. It is also used to provide, for the first time, a complete proof of the maximum relative queuing delay provided by the fractional traffic dispatch algorithm [S. Iyer and N. McKeown, in Proceedings of IEEE INFOCOM, IEEE Communications Society, New York, NY, 2001, pp. 1680-1687; D. Khotimsky and S. Krishnan, in Proceedings of the IEEE International Conference on Communications, IEEE Communications Society, New York, NY, 2001, pp. 100-106]. These optimal algorithms arc deterministic, proving that randomization does not reduce the relative queuing delay of the PPS.