This paper studies alternative price-service mechanisms for a provider that serves customers whose delay cost depends on their service valuations. We propose a generalized delay cost structure that augments the standard additive model with a multiplicative component, capturing the interdependence between delay cost and values. We derive and compare the revenue-maximizing and socially optimal equilibria under uniform pricing, preemptive, and nonpreemptive priority auctions with an admission price. We find that the delay cost structure has a paramount effect on system behavior. The classical result that the revenue-maximizing admission price is higher and the utilization lower than is socially optimal can be reversed under our generalized structure, and we identify the conditions driving this reversal under each mechanism. We show that the conditional bid equilibria are unique and induce the socially optimal allocations. The auctions yield gains in system net value and provider profit over uniform pricing, which are dramatically larger for the preemptive mechanism. Both auctions perform better under multiplicative compared to additive delay costs. The highest-value customers always gain under the preemptive, but may lose under the nonpreemptive auction. The lowest-value customers always gain in either auction.