Some analytical results concerning PI controller tuning based on integrator plus time delay models are worked out and presented. A method for obtaining PI controller parameters, K p = α kτ , and, T i = βτ , which ensures a given prescribed maximum time delay error, dτ max , to time delay, τ , ratio parameter δ = dτ max τ , is presented. The corner stone in this method, is a method product parameter,c = αβ. Analytical relations between the PI controller parameters, T i , and, K p , and the time delay error parameter, δ, is presented, and we propose the setting, β =c a (δ + 1), and, α = a δ+1 , which gives, T i =c a (δ + 1)τ , and, K p = a (δ+1)kτ , where the parameter, a, is constant in the method product parameter,c = αβ. It also turns out that the integral time, T i , is linear in, δ, and the proportional gain, K p , inversely proportional to, δ + 1. For the original Ziegler Nichols (ZN) method this parameter is approximately,c = 2.38, and the presented method may e.g., be used to obtain new modified ZN parameters with increased robustness margins, also documented in the paper.