Since callers encountering busy signals often want to redial, modern communication networks have been designed to provide automatic redialling. Redialling services commonly have two parameters: a maximum number n of retries and a total duration τ over which retries are to be made. Typically, retries are made at evenly spaced time intervals of length τ/n until either the call succeeds or n retries have failed. This rule has an obvious intuitive appeal; indeed, among the main results of this paper are proofs that τ/n-spacing is optimal in certain basic models of called-number behaviour. However, it is easy to find situations where τ/n-spacing is not optimal, as the paper verifies. All of our models assume Poisson arrivals, but different assumptions are studied for the call durations; for a given mean, these are allowed to have the relatively high-variance exponential distribution or the zero-variance distribution concentrated at a point. We approximate the probability of success for the Erlang loss model with c > 1 trunks and we calculate exact probabilities of success for the c = 1 Erlang model and the model with one trunk and constant call durations. For the latter model, we present two intriguing conjectures, one about the optimal choice of τ when n = 1 and one about the optimality of the τ/n-spacing policy. In spite of their apparent simplicity, these conjectures seem difficult to resolve. Finally, we study policies that continue redialling until they succeed, balancing a short mean wait against a small mean number of retries to success.
