New Stability Results for Multiserver-job Models via Product-form Saturated Systems

Multiserver-job (MSJ) models are increasingly common in today's datacenters. In these models, each job runs on multiple servers concurrently, for some duration. The most common service ordering for jobs is First-Come First-Served (FCFS). Unfortunately, MSJ FCFS models are hard to analyze, and even the stability region of MSJ FCFS models is not well understood. Stability has only been analyzed in the case where all jobs have independent exponentially-distributed durations with the same mean (the "single duration" setting). This does not allow, however, for modeling the common situation where jobs with higher server need (number of servers required) also have higher expected duration. This paper provides the first analysis of stability for MSJ FCFS models in the two-class setting, where each class of jobs has its own server need and its own exponentially-distributed duration. To analyze stability, we make use of the saturated system, whose throughput determines the stability region. While the saturated system has been useful in deriving stability regions in the past, it has never been applied in settings where jobs occupy multiple servers. By looking at the saturated system in a new light, we find that its solution has an attractive product form in the two-class setting, and a different product form in the single-duration setting, both novel results. Besides solving our problem, this product form may also serve as a gateway to analyzing other complex models.