2.1 The Alternative Kolmogorov-Smirnov Statistical Tests.................... 3 2.2 Proof of the Log Test...................................... 5

Service systems such as call centers and hospital emergency rooms typically have strongly time-varying arrival rates. Thus, a nonhomogeneous Poisson process (NHPP) is a natural model for the arrival process in a queueing model for performance analysis. Nevertheless, it is important to perform statistical tests with service system data to confirm that an NHPP is actually appropriate, as emphasized by Brown et al. (2005). They suggested a specific statistical test based on the Kolmogorov-Smirnov (KS) statistic after exploiting the conditional-uniform (CU) property to transform the NHPP into a sequence of i.i.d. random variables uniformly distributed on [0, 1] and then performing a logarithmic transformation of the data. We investigate why it is important to perform the final data transformation and consider what form it should take. We conduct extensive simulation experiments to study the power of these alternative statistical tests. We conclude that the general approach of Brown et al. (2005) is excellent, but that an alternative data transformation proposed by Lewis (1965), drawing upon Durbin (1961), produces a test of an NHPP test with consistently greater power. We also conclude that the KS test after the CU transformation, without any additional data transformation, tends to be best to test against alternative hypotheses that primarily differ from an NHPP only through stochastic and time dependence. Thi