On asymptotic accuracy in queueing theory

Recently results have shown that a single server queue Poisson Pareto Burst Process input has a tail which is bounded by hyperbolic functions. We show that the hyperbolic upper and lower bounds for this system can be very misleading, that this hyperbolic tail result is relevant only from a certain threshold onwards, and the magnitude of this threshold may be very large. We also show that any hyperbolic upper and lower bounds for a tail of the stationary waiting time complementary distribution necessarily become further apart as the rate of the process increases.