A Computational Approach for Evaluating Steady-State Probabilities and Virtual Waiting Time of a Multiprocessor Queuing System

Abstract

This scientific paper explores the operation of a multiprocessor task servicing system. Tasks are received into the system at random intervals and are characterized by several stochastic parameters, including the number of processors required for their execution, the maximum allowable busy time for these processors, and the permissible waiting time in the task queue. The organization of task servicing in this system follows a first-in, first-out (FIFO) approach, ensuring uninterrupted processing. The key servicing process involves periodically selecting the first task in the queue and assessing its feasibility for immediate execution. If the task meets the necessary criteria, it is dispatched for processing. This process continues iteratively until a task is found, the parameters of which prevent immediate servicing. It is important to note that tasks in the queue have a limited window of time within which they can be serviced; otherwise, they may exit the system without service.

This paper focuses on systems characterized by exponential distributions for random variables related to task arrivals, servicing times, and waiting restrictions. A system of equations is derived that describes the system’s steady-state behavior. These equations enable the calculation of probabilities associated with the system’s various states. Additionally, the paper provides insights into the probability distributions of virtual waiting times for tasks that arrive in the system at any given moment.