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Steady-state asymptotics for tandem, split-match and other feedforward queues with heavy tailed service. (English)
Queueing Syst. 33, No.1-3, 233-259 (1999).
Summary: Consider a stable FIFO GI/GI/$1\rightarrow/$GI/1 tandem queue in which the equilibrium distribution of service time at the second node $S(2)$ is subexponential. It is shown that when the service time at the first node has a lighter tail, the tail of steady-state delay at the second node, $D(2)$, has the same asymptotics as if it were a GI/GI/1 queue: $$P(D(2) > x) \sim \frac{ρ_{2}}{1-ρ_{2}} P(S_{e}(2) >x), \quad x\rightarrow\infty,$$ where $S_{e}(2)$ has equilibrium (integrated tail) density $P(S(2) >x)/ E[S(2)]$, and $ρ_{2} = λE[S(2)]$ ($λ$ is the arrival rate of customers). The same result holds for tandem queues with more than two stations. For split-match (fork-join) queues with subexponential service times, we derive the asymptotics for both the sojourn time and the queue length. Finally, more generally, we consider feedforward generalized Jackson networks and obtain similar results.