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Differentiated services and other scheduling strategies are now widespread in the traditional, "best effort" Internet. These offer quality of service guarantees for important customers at the same time as supporting less critical applications of lower priority. Since response time, or delay, is a crucial performance metric for delay-sensitive applications, time delays in priority queues have been studied extensively in recent years. We consider a DiffServ node which is modelled as a non-preemptive priority queue with modulated arrivals and derive an expression for the probability distribution of the response time using the generating function method. We consider two service classes: expedited traffic forms the high priority class and is modelled as a Poisson process whereas best effort traffic is in the low priority class and modelled as a Markov modulated Poisson process. The distribution of service time is general. This queue has many real-world applications; in the example considered here, it could model a DiffServ router which provides service differentiation for signalling or management traffic together with standard data streams. Mean delays are derived as explicit expressions and show very close agreement with simulation. Higher moments can be computed in the same way with more routine algebra.