Stability for Two-class Multiserver-job Systems

Multiserver-job systems, where jobs require concurrent service at many servers, occur widely in practice. Much is known in the dropping setting, where jobs are immediately discarded if they require more servers than are currently available. However, very little is known in the more practical setting where jobs queue instead. In this paper, we derive a closed-form analytical expression for the stability region of a two-class (non-dropping) multiserver-job system where each class of jobs requires a distinct number of servers and requires a distinct exponential distribution of service time, and jobs are served in first-come-first-served (FCFS) order. This is the first result of any kind for an FCFS multiserver-job system where the classes have distinct service distributions. Our work is based on a technique that leverages the idea of a "saturated" system, in which an unlimited number of jobs are always available. Our analytical formula provides insight into the behavior of FCFS multiserver-job systems, highlighting the huge wastage (idle servers while jobs are in the queue) that can occur, as well as the nonmonotonic effects of the service rates on wastage.