In a mobile telecommunications network, the period when a mobile station (MS) resides in a cell (the radio coverage of a base station) is called the cell residence time of that MS. The period between when a call arrives at the MS and when the MS moves out the cell is called the excess life of the cell residence time for that MS. In performance evaluation of a mobile telecommunications network, it is important to derive the excess life distribution from the cell residence times. This distribution determines if a connected call will be handed over to a new cell, and therefore significantly affects the call dropping probability of the network. In mobile-telecommunications-network simulation, generating the excess-life random numbers is not a trivial task, which has not been addressed in the literature. This paper shows how to generate the random numbers from the excess life distribution, and develop the excess-life random number generation procedures for cell residence times with gamma, Pareto, lognormal, and Weibull distributions. This paper indicates that the generated random numbers closely match the true excess-life distributions.