Some reflections on the renewal-theory paradox in queueing theory

The classical renewal-theory (waiting time, or inspection) paradox states that the length of the renewal interval that covers a randomly-selected time epoch tends to be longer than an ordinary renewal interval. This paradox manifests itself in numerous interesting ways in queueing theory, a prime example being the celebrated Pollaczek-Khintchine formula for the mean waiting time in the M/G/1 queue. In this expository paper, we give intuitive arguments that ‘explain’ why the renewal-theory paradox is ubiquitous in queueing theory, and why it sometimes produces anomalous results. In particular, we use these intuitive arguments to explain decomposition in vacation models, and to derive formulas that describe some recently-discovered counterintuitive results for polling models, such as the reduction of waiting times as a consequence of forcing the server to set up even when no work is waiting.