A heavy-traffic theorem for the GI/G/1 queue with a Pareto-type service time distribution. (1998) - Cohen, J. W.
Abstract : For the G I / G / 1 queueing model with traffic load a < 1, service time distribution B ( t ) and interarrival time distribution A ( t ), whenever for t → ∞ 1 − B ( t ) ∼ c ( t / β ) ν + O ( e − δ t ), c > 0, 1 < ν < 2, δ > 0, and ∫ 0 ∞ t μ d A ( t ) < ∞ for μ > ν, ( 1 − a ) 1 ν − 1 w converges in distribution for a ↑ 1 . Here w is distributed as the stationary waiting time distribution. The L.-S. transform of the limiting distribution is derived and an asymptotic series for its tail probabilities is obtained. The theorem actually proved in the text concerns a slightly more general asymptotic behavior of 1 − B ( t ), t → ∞, than mentioned above.