Maximum remaining service time in infinite-server queues

We study the maximum remaining service time in infinite-server queues of type $M|G|\infty$ (at a given time and in a stationary regime). The following cases for the arrival flow rate are considered: (1) time-independent, (2) given by a function of time, (3) given by a random process. As examples of service time distributions, we consider exponential, hyperexponential, Pareto, and uniform distributions. In the case of a constant rate, we study effects that arise when the average service time is infinite (for power-law distribution tails). We find the extremal index of the sequence of maximum remaining service times. The results are extended to queues of type $M^X|G|\infty$, including those with dependent service times within a batch.