A multi-station system for reducing congestion in high-variability queues

We consider the problem of allocating processing times in a multi-station series system characterized by high variability. Servers are arranged into multiple stations in series with the objective of minimizing the waiting time through a truncation scheme. Every station has a threshold on the amount of time spent servicing a job. A job being served at Station i that has a processing time exceeding this station's threshold is forwarded to Station i+1. Otherwise, the job completes its service at Station i and leaves the system. We develop an analytical model to determine the optimal system configuration, in terms of the number of stations, the corresponding thresholds, and the number of servers at each station, for a given number of servers facing Poisson demand. In order to simplify the computation of the thresholds, we assume that the load is balanced among different stations. We justify this assumption with numerical and analytical evidence. Our analytical and numerical results indicate that, under high traffic intensity, our series system performs better than the standard M/G/c and other variants having two stations only, when the shape of the service time distribution allows reducing variability by multiple truncations. Our results also indicate that having a moderate number of stations is beneficial as this offers a good trade-off between variance reduction and idle time downstream. Two by-products of our work are (i) comparing the accuracy of several M/G/c waiting time approximations via simulation, with service times following a family of balanced hyperexponential distributions, and (ii) testing the appropriate M/G/c simulation length. We find that approximations by Whitt (1989, 1993) to be adequate.