In this paper a new class of test-statistics is proposed. This class of statistical tests is particularly useful to test the null hypothesis that an arrival process is a Poisson process, and widely applicable to teletraffic problems, including telephone and packet-switched networks. After an introduction where a wide set of possible applications is examined (Section 1), in Sections 2 and 3 the class of test-statistics under examination is introduced, and its main theoretical properties are studied. In Section 4 a numerical evaluation (based on Monte Carlo simulation methods) of the power of the test is performed. Finally, in Section 5 an application to real data is considered.