The effect of arrival model uncertainties on some optimal routing problems

The authors study the effect of uncertainties in modeling the arrival process on optimal routing in a system of parallel queues. For exponential service time distributions and Bernoulli routing, an asymptotic analysis of the error in optimal mean system delay due to an error in modeling the arrival process is presented for a heavy traffic condition. An asymptotic analysis of the error in optimal mean delay due to uncertainties in service time distribution for Poisson arrivals was reported where it was shown that when only the mean service time is known, the error in performance asymptotically vanishes as the traffic load approaches the system capacity. In contrast, under the assumption of a general renewal arrival process, the result is established that when only the mean arrival rate is known, the error in optimal mean delay due to uncertainties in the arrival model is unbounded as the traffic approaches the system capacity. However, when both first and second moments are known, the error asymptotically vanishes. These results provide additional insight into the optimization process in such a system of parallel queues which may be useful in designing optimal routing algorithms. Numerical examples corroborating the theoretical results are also presented