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The use of passive optical networks (PONs) enables access rates of multi-Gbit/sec bandwidth and provision of quality of service for high definition multimedia services. In this paper, we consider and analyse a generic multi-priority dynamic bandwidth allocation (DBA) algorithm for TDM PONs serving multimedia traffic in an upstream link. PON traffic is served strictly according to its priority. We consider this DBA algorithm using two approaches: (i) the algorithm assigns a fixed service quantum to each priority service and (ii) different service quanta are assigned to different priority services. The mean message delay is evaluated using a multiqueue processor sharing (MPS) model and an MPS with Heterogeneous Traffic (MPS-HT)model for the two approaches respectively. The MPS model is a classical processor sharing model limited by the critical assumption that there is egalitarian service sharing among all users, which is inefficient for multimedia applications in PONs. We extend the MPS model to a general MPS-HT model that enables the analysis of message delay performance in the case where the service quanta may be different for different services.