The Queue Distribution in Multiprocessor Systems with the Waiting Time Restriction

Abstract

A queueing system model is considered, consisting of m (m ≥ 1) servicing devices and a maximum number of tasks with n (n ≥ 1) in the waiting queue. Each task is characterized by three random parameters (ν, β, ω), where ν is the number of servicing devices required to perform the task, β is the maximum time required to complete the task and ω is the possible time that the task can wait before assigning to run, after which it leaves the system without service. Tasks are accepted for service in the order of their entry into the system, i.e., FIFO (First-In-First-Out) discipline is used. In paper the equations are obtained for the state probabilities of the system in the stationary mode, which can serve as an assessment for real multiprocessor systems using MPI and OpenMP technologies.

A Computational Approach for Evaluating Steady-State Probabilities of a Multiprocessor Queueing System with a Waiting Time Restriction

“…In the previous publications [1], [3], three lemmas were derived and established that are instrumental to this current analysis. These lemmas provide key insights and results that will be utilized in this section to advance the derivation of supporting probabilities.…”

“…For a more detailed formulation, consider a computing system consisting of m (m ≥ 1) computing processors (or cores). It is assumed that the number of tasks that can be queued is limited by a certain number of waiting slots: n (n ≥ 1) [1]. The reason for denying the service can be the impossibility of placing a task in the queue and the impossibility of serving with user-defined constraints(time, number of processors, etc.).…”

About Virtual Waiting Time in a Multiprocessor System

“…Jobs will be serviced in the order they arrive in the system, i.e., FIFO discipline is used [1], [2]. The system consists of m processors (cores, cluster nodes, etc.,) dedicated to task servicing, with a queue capable of accommodating up to n tasks awaiting service [3].…”

Steady-State Probabilities and Task Rejection/Failure Probability Estimation in the Multiprocessor Queueing System

“…The investigation involved a computing system consisting of m (m ≥ 1) processors and a queue capable of accommodating a maximum of n (n ≥ 1) tasks [2], [3]. By considering various operational scenarios, where i (1 ≤ i ≤ m) tasks are being serviced and j (1 ≤ j ≤ n) tasks are waiting in the queue, a computational approach was proposed to evaluate the P i,j steady-state probabilities in a multiprocessor queueing system.…”