Abstract

In this paper we consider the generalized processor sharing (GPS) mechanism serving two traffic classes. These classes consist of a large number of independent identically distributed Gaussian flows with stationary increments. We are interested in the logarithmic asymptotics or exponential decay rates of the overflow probabilities. We first derive both an upper and a lower bound on the overflow probability. Scaling both the buffer sizes of the queues and the service rate with the number of sources, we apply Schilder’s sample-path large deviations theorem to calculate the logarithmic asymptotics of the upper and lower bound. We discuss in detail the conditions under which the upper and lower bound match. Finally we show that our results can be used to choose the values of the GPS weights. The results are illustrated by numerical examples.

Keywords: Sample-path large deviations; Gaussian traffic; Schilder’s theorem; Generalized processor sharing; Many-sources asymptotics; Communication networks; Differentiated services; Weight setting

Article Outline

1. Introduction

1.1. Large deviations

1.2. Contribution

2. Model and preliminaries

2.1. Generalized processor sharing

2.2. Gaussian processes

2.3. Sample-path large deviations

3. Generic upper and lower bound on the probability

4. Lower bound on the decay rate: class 2 in underload

5. Upper bound on the decay rate: class 2 in underload

5.1. Lower bound on J^U(x)

5.2. Conditions for exactness

6. Analysis of the decay rate: class 2 in overload

7. Discussion of the results

7.1. Structure of the solution

7.1.1. Ad Case (i): Class 2 in overload

7.1.2. Ad Case (ii): Class 2 in underload, with ₂ small

7.1.3. Ad Case (iii): Class 2 in underload, with ₂ large

7.2. Numerical results

7.3. Brownian motion input

8. Weight setting

8.1. Approximation of the overflow probabilities

8.2. Weight setting algorithm

8.3. Admissible region

9. Concluding remarks

Acknowledgements

Appendix A. Analysis of underload regime with large ₂

Appendix B. Weight setting algorithm: partial derivatives

References