Queueing System with Two Phases of Service and Service Rate Degradation

Abstract

In the paper, a queueing system with an unlimited number of servers and two phases of service with degradation in the service rate is studied. The problem of service rate degradation emerges in cloud nodes, where there is contention for hardware resources including computational resources such as CPU cores. In a node, we have a limited number of CPU cores that should execute potentially an unlimited number of processes (requests) in parallel. In our model, the term “server” means a process allocated in the node for execution. So, the number of “servers” is unlimited but their individual performances decrease because CPUs should switch between them during the execution. We consider processes executed in the node with two phases of life cycle that reflects periods with different activity of a process; e.g., in the first phase, the process may require intensive usage of CPU cores but low usage in the second one. Our model distinguishes the phases using different service parameters for them as well as different influence on the service rate degradation in the node. In the paper, two analytical methods are proposed: exact solving of the system of the local balance equation and the asymptotic analysis of the global balance equations. Formulas for the stationary probability distribution of the number of customers in the phases are obtained for both cases. Several numerical examples are provided that illustrate some properties and applicability of the obtained results.

Keywords

• queuing theory
• service rate degradation
• global and local balance equations
• asymptotic analysis