Matchmaking and testing for exponentiality in the M/G/∞ queue

Abstract

Customers arrive sequentially at times x₁ < x₂ < · · · < x_n and stay for independent random times Z₁, ..., Z_n > 0. The Z-variables all have the same distribution Q. We are interested in situations where the data are incomplete in the sense that only the order statistics associated with the departure times x_i + Z_i are known, or that the only available information is the order in which the customers arrive and depart. In the former case we explore possibilities for the reconstruction of the correct matching of arrival and departure times. In the latter case we propose a test for exponentiality.