A stochastic network calculus for many flows

The stochastic network calculus receives much attention as a new methodology for end-to-end performance evaluation of networks, taking account of the effect of statistical multiplexing. In this paper, we present a new stochastic network calculus for many flows from an approach like large deviations techniques. In an n-node discrete-time tandem network with L flows, let Ā^θ (t, s) and — S¯^−θ (t, s) be the limits of the cumulant generating functions of Ā^L (t, s), arrivals to the network, and S¯^L<inf>i</inf> (t, s), services at node i, during time interval (s, t). Then, for the departures D¯^L (t, s) from the network during time interval (s, t) and the backlog Q^L(t) in the network at time t, we prove that the limits of the cumulant generating functions of them denoted by D¯^θ (t, s) and Q^θ (t), respectively, satisfy an inequality D¯^θ (t, s) ≤ Ā^θ ⊘ (S¯^θ<inf>n</inf> ∗ S¯^θ<inf>n−1</inf> ∗ … − S¯^θ<inf>1</inf>) (t, s) and an equality Q^θ(t) = Ā^θ ⊘ (S¯^θ<inf>n</inf> ∗ S¯^θ<inf>n−1</inf> ∗ … − S¯^θ<inf>1</inf>) (t, t), where ⊘ and ∗ are deconvolution and convolution operators. By using these results, we propose approximation formulas for the end-to-end evaluation of output burstness and backlog, and we apply the formula on backlog to a tandem network with cross traffic as an example.