Abstract
We consider a buffered queueing system that is fed by a Gaussian source and drained at a constant rate. The fluid offered to the system in a time interval $(0,t]$ is given by a separable continuous Gaussian process $Y$ with stationary increments. The variance function $\sigma^2: t \mapsto \mathbb{V}\mbox{ar} Y_t$ of $Y$ is assumed to be regularly varying with index $2H,$ for some $0<H<1.$
By proving conditional limit theorems, we investigate how a high buffer level is typically achieved. The underlying large deviation analysis also enables us to establish the logarithmic asymptotics for the probability that the buffer content exceeds $u$ as $u\to\infty.$ In addition, we study how a busy period longer than $T$ typically occurs as $T\to\infty,$ and we find the logarithmic asymptotics for the probability of such a long busy period.
The study relies on the weak convergence in an appropriate space of $\{Y_{\alpha t}/\sigma(\alpha): t\in\mathbb{R}\}$ to a fractional Brownian motion with Hurst parameter $H$ as $\alpha\to\infty.$ We prove this weak convergence under a fairly general condition on $\sigma^2,$ sharpening recent results of Kozachenko et al. (Queueing Systems Theory Appl. 42 (2002) 113). The core of the proof consists of a new type of uniform convergence theorem for regularly varying functions with positive index.
Original language | English |
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Pages (from-to) | 849-873 |
Number of pages | 25 |
Journal | Stochastic processes and their applications |
Volume | 115 |
Issue number | 5 |
DOIs | |
Publication status | Published - 2005 |
Keywords
- EWI-17607
- Busy period
- Gaussian processes
- Metric entropy
- IR-70161
- Large deviations
- Weak convergence
- Overflow probability
- METIS-224182
- Regular variation