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Many measurement studies have shown that telecommunication traffic usually exhibits self-similar nature and the service times of packets often follow heavy-tailed distributions. However, due to the high complexity of modelling the fractal self-similar properties and heavy-tailed distributions, most existing studies on analytical modelling of queuing systems have been confined to investigate the effects of either traffic self-similarity or heavy-tailed service times only. To fill this gap, in this paper we develop a new analytical model for a single server queuing system in the presence of self-similar inputs and heavy-tailed service time distributions. Specifically, we derive the closed-form expressions for three important quality-of-service (QoS) metrics, namely, the tail distributions of queue length, packet loss, and packet delay of the queuing systems where traffic arrivals follow the fractional Brownian motion (fBm) model and the packet sizes have lognormal or Pareto distributions. We validate the accuracy of the developed model through comparing analytical results to those obtained from experimental simulations.