An intelligent node of a distributed control system has four tasks, 1 4 (rate

An intelligent node of a distributed control system has four tasks, τ₁ – τ₄ (rate - monotonic priorities), with the corresponding execution periods p₁ = 10 ms, p₂ = 100 ms, p₃ = 500 ms, and p₄ = 1000 ms. The execution times are e₁ = 2 ms, e₂ = 15 ms, e₃ = 100 ms, and e₄ = 10 ms, respectively.

However, task τ 1 is a critical control loop, whose execution period affects directly to the achievable control performance. Hence, in principle, that execution period should be as short as possible. What is the minimum execution period for τ₁( = p_1,min), if the maximum allowed CPU utilization factor is 0.91 ( “ dangerous ” )?