Reducing subtask dispersion in parallel queueing systems

In various real-world parallel processing systems, incoming tasks divide into several subtasks that are processed independently by parallel servers. Queueing networks are a natural way to represent the ow and processing of tasks and subtasks in such systems. Two useful classes of queueing network representations are split-merge and fork-join systems. There are two main metrics of interest in these systems: task response time and subtask dispersion. These metrics are in tension with each other: when one is reduced, it tends to lead to an increase in the other. Generally, using the fork-join paradigm leads to low task response times but high subtask dispersion, while using the split-merge paradigm leads to low subtask dispersion but moderate to high task response times. This thesis introduces methods for controlling subtask dispersion as well as for the trading o of subtask dispersion and task response time in parallel queueing systems. In the context of split-merge systems with generally distributed service times, we show how to control mean subtask dispersion by the application of judiciously-chosen delays to subtask processing and extend it to control percentiles of the distribution of subtask dispersion. Our analysis is based on extensions to the theory of heterogeneous order statistics. While solely focusing on the reduction of subtask dispersion leads to a large increase in task response time, together with a corresponding decrease in maximum sustainable system throughput, aiming to reduce a product of mean subtask dispersion and mean task response time leads to a marginal increase in task response time while dramatically improving mean subtask dispersion. Fork-join systems are widely deployed in the real world, but are notoriously more di cult to analyse. In the context of fork-join systems with heterogeneous exponentially distributed service times, we present an on-line technique which improves on both the mean task response time and mean subtask dispersion achievable in an equivalent split-merge system. For split-merge systems we validate our results analytically, while for fork-join systems we validate the solutions against simulations. We present case studies of di erent parts of our methodology in split-merge and fork-join systems with and without applications of the delays. These show the ability to reduce subtask dispersion while providing increasingly-sophisticated means to simultaneously control task response time.