Stability Condition for the Multi-server Job Queuing Model: Sensitivity Analysis

A Multiserver Job Queuing Model (MJQM) is a queuing system that can be instrumental in the study of the dynamics of resource allocation in datacenters. The queue comprises a waiting line with infinite capacity and a large number of servers. In this paper, we look at the case of an infinite number of servers. Jobs are termed “multiserver” because each one is characterized by a resource demand in terms of number of simultaneously used servers and by a service duration. In a MJQM, jobs are clustered into classes, and a number of used servers is deterministically associated with each class. Instead, holding times are independent and identically distributed random variables whose distributions depend on the class of the job. We consider the case of just two job classes: “small” jobs use just one server, while “big” jobs use all servers in the system. The service discipline is First-Come-First-Served (FCFS). This means that if the job at the head-of-line (HOL) cannot enter service because the number of free servers is not sufficient to meet the job requirement, it blocks all subsequent jobs, even if there are sufficient free servers for them. Despite its importance, only few results exist for the MJQM, whose analysis is challenging, especially because the MJQM is not work-conserving. This implies that even the stability region of the MJQM is known only in special cases. In a previous work, we obtained a closed-form stability condition for MJQM with big and small jobs under the assumption of exponentially distributed service times for small jobs. In this paper, we compute the stability condition of MJQM with big and small jobs, with an infinite number of servers, considering different distributions of the service times of small jobs. Simulations are used to support the analytical results and to investigate the impact of service time distributions on the expected job waiting time before saturation.