A new upper bound of the completion time of the background task in a foreground-background system

A foreground-background scheduling system is a simple real-time pre-emptive scheduler, which is commonly used in uniprocessor embedded systems. In this system, there is a single background task of the lowest priority and multiple foreground tasks have higher priorities. Foreground tasks may have different levels of priorities. Foreground tasks are allowed to pre-empt the background task. The background task takes a longer time to complete its execution because of the frequent interruptions caused by the foreground tasks. The completion time of the background task is calculated using the utilisation of the processor by the foreground tasks. In this paper, a new upper bound formula of the completion time of the background task is derived. The proposed formula gives a closer upper bound to the exact completion time compared to the existing bounds in the case of few foreground tasks and even it gives the exact time in certain cases for the heavily utilised systems. In addition, the proposed upper bound is not a recursive formula like that of the existing response time analysis.