2 research outputs found

    Gaussian queues in light and heavy traffic

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    In this paper we investigate Gaussian queues in the light-traffic and in the heavy-traffic regime. The setting considered is that of a centered Gaussian process X{X(t):tR}X\equiv\{X(t):t\in\mathbb R\} with stationary increments and variance function σX2()\sigma^2_X(\cdot), equipped with a deterministic drift c>0c>0, reflected at 0: QX(c)(t)=sup<st(X(t)X(s)c(ts)).Q_X^{(c)}(t)=\sup_{-\infty<s\le t}(X(t)-X(s)-c(t-s)). We study the resulting stationary workload process QX(c){QX(c)(t):t0}Q^{(c)}_X\equiv\{Q_X^{(c)}(t):t\ge0\} in the limiting regimes c0c\to 0 (heavy traffic) and cc\to\infty (light traffic). The primary contribution is that we show for both limiting regimes that, under mild regularity conditions on the variance function, there exists a normalizing function δ(c)\delta(c) such that QX(c)(δ(c))/σX(δ(c))Q^{(c)}_X(\delta(c)\cdot)/\sigma_X(\delta(c)) converges to a non-trivial limit in C[0,)C[0,\infty)

    Convergence of the all-time supremum of a L\'evy process in the heavy-traffic regime

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    In this paper we derive a technique of obtaining limit theorems for suprema of L\'evy processes from their random walk counterparts. For each a>0a>0, let {Yn(a):n1}\{Y^{(a)}_n:n\ge 1\} be a sequence of independent and identically distributed random variables and {Xt(a):t0}\{X^{(a)}_t:t\ge 0\} be a L\'evy processes such that X1(a)=dY1(a)X_1^{(a)}\stackrel{d}{=} Y_1^{(a)}, EX1(a)<0\mathbb E X_1^{(a)}<0 and EX1(a)0\mathbb E X_1^{(a)}\uparrow0 as a0a\downarrow0. Let Sn(a)=k=1nYk(a)S^{(a)}_n=\sum_{k=1}^n Y^{(a)}_k. Then, under some mild assumptions, Δ(a)maxn0Sn(a)dR    Δ(a)supt0Xt(a)dR\Delta(a)\max_{n\ge 0} S_n^{(a)}\stackrel{d}{\to} R\iff\Delta(a)\sup_{t\ge 0} X^{(a)}_t\stackrel{d}{\to} R, for some random variable RR and some function Δ()\Delta(\cdot). We utilize this result to present a number of limit theorems for suprema of L\'evy processes in the heavy-traffic regime
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