We consider the problem of hypothesis testing in the situation when the first hypothesis is simple and the second one is local one-sided composite. We describe the choice of thresholds and the power functions of Score-function test, General Likelihood Ratio test, Bayesian tests and Wald test in the situation when the intensity function of inhomogeneous Poisson process is smooth with respect to the parameter. It is shown that almost all these tests are asymptotically uniformly most powerful. The results of numerical simulations are presented.