The Non-Saturated Multiserver Job Queuing Model with Two Job Classes: a Matrix Geometric Analysis

Datacenters comprise large quantities of processors, memory, and input/output modules. These resources are shared among requests (jobs) submitted by datacenter users. Jobs differ in their frequency of arrivals, demand for resources, and execution times. Resource sharing generates contention, especially in heavily loaded systems, that must therefore implement effective scheduling policies for incoming jobs. The First-In First-Out (FIFO) policy is often used for batch jobs, but may produce under-utilization of resources, in terms of wasted servers. This is due to the fact that a job that requires many resources can block jobs arriving later that could be served because they require fewer resources. The mathematical construct often used to study this problem is the Multiserver Job Queuing Model (MJQM), where servers represent resources which are requested and used by jobs in different quantities. Unfortunately, very few explicit results are known for the MJQM, especially at realistic system loads (i.e., before saturation). In this paper, we propose the first exact analytical model of the non-saturated MJQM in case of two classes of customers with exponentially distributed service times and an arbitrary number of identical servers. Our analysis is based on the matrix geometric method. Our results provide insight into datacenter dynamics, thus supporting the design of more complex schedulers, capable of improving performance and energy consumption within large datacenters.