Subexponential loss rates in a GI/GI/1 queue with applications

Abstract

Consider a single server queue with i.i.d. arrival and service processes, $\{ A,A_n ,n \geqslant 0\}$ and $\{ C,\;C_n ,n\;\; \geqslant \;\;0\}$, respectively, and a finite buffer B. The queue content process $\{ Q_n^B ,n \geqslant 0\}$ is recursively defined as $Q_{n + 1}^B = \min ((Q_n^B + A_{n + 1} - C_{n + 1} )^ + ,B),\;\;q^ + = \max (0,q)$. When $\mathbb{E}(A - C) < 0$, and A has a subexponential distribution, we show that the stationary expected loss rate for this queue $E(Q_n^B + A_{n + 1} - C_{n + 1} - B)^ +$ has the following explicit asymptotic characterization:

$${\mathbb{E}}\left( {Q_n^B + A_{n + 1} - C_{n + 1} - B} \right)^ + ~{\mathbb{E}}\left( {A - B} \right)^ + {as} B \to \infty ,$$

independently of the server process C _n . For a fluid queue with capacity c, M/G/∞ arrival process A _t , characterized by intermediately regularly varying on periods σ^on, which arrive with Poisson rate Λ, the average loss rate $\lambda _{{loss}}^B$ satisfies λ ^B_loss ∼ Λ E(τ^onη — B)⁺ as B → ∞, where $\eta = r + \rho - c,\;\rho \; = \mathbb{E}A_t < \;\;c;r\;\;(c \leqslant r)$ is the rate at which the fluid is arriving during an on period. Accuracy of the above asymptotic relations is verified with extensive numerical and simulation experiments. These explicit formulas have potential application in designing communication networks that will carry traffic with long-tailed characteristics, e.g., Internet data services.