Parulekar, M.Makowski, Armand M.The infinite server model of Cox with arbitrary service time distribution appears to provide a very large class of traffic models - Pareto and log-normal distributions have already been reported in the literature for several applications. Here we begin the analysis of the large buffer asymptotics for a multiplexer driven by this class of inputs. Top do so we rely on recent results by Duffield and O'Connel on overflow probabilities for the general single server queue. In this paper we focus on the key step in this approach which is based on large deviations: The appropriate large deviations scaling is shown to be related to the forward recurrence time for the service time distribution, and a closed form expression is derived for the corresponding generalized limiting log-moment generating function associated with the input process. Tow very different regime are identified. In a companion paper we apply these results to obtain the large buffer asymptotics under a variety of service time distributions.en-USqueueing networksinfinite server queuelarge deviationstail probabilitiesforward recurrence timesscalingsIntelligent Signal ProcessingCommunications SystemsTechnical Report