Since your input has a delay, then the input can be expressed as, say u(t)=f(t-a), where a is the time delay you mention.
Then, the input in Laplace domain is given by U(S)=exp(-aS)F(s). Then, your output is simply Y(S)=U(S)G(S). Then, you can use Laplace table so to describe the output in time domain. See that G(S) is not changing due to any delay in your input signal. Then, start your simulation at t=0, and if you want to describe the expression mathematically, follow what I wrote in the previous lines. I hope I understood your question correctly. Otherwise, please let me know.