Summary: A stochastic model originating in chemical physics is interpreted as a randomized random walk on the integers with an unusual exponential pattern for the inter-step time intervals. Some features of this are analyzed, and then attention is turned to the queue counterpart, where the walk is confined to the non-negative integers. The system state and busy period processes are analyzed in some detail, yielding modified Bessel function formulae. A continued fraction expansion technique is used, but not described, to give specimen values of the queueing probabilities. An interesting operational feature is that, whatever the parameters, the queue has no finite steady state.