I have several test results of server response delay. According to our theory analysis, the delay distribution (The probability distribution function of response delay) should have heavy-tail behavior. But how could I prove that the test result does follow heavy-tail distribution?
I'm not sure if I'm interpreting your question correctly, so let me know, and I could adapt or delete this answer. First, we don't prove things regarding our data, we just show that something isn't unreasonable. That can be done several ways, one of which is through statistical tests. In my opinion, however, if you have a pre-specified theoretical distribution, the best approach is just to make a qq-plot. Most people think of qq-plots as only being used to assess normality, but you can plot empirical quantiles against any theoretical distribution that can be specified. If you use R, the car package has an augmented function qq.plot() with a lot of nice features; two that I like are that you can specify a number of different theoretical distributions beyond just the Gaussian (e.g., you could to