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 Amacr^thetas (t, s) and - Smacr_i ^-thetas (t, s) be the limits of the cumulant generating functions of Amacr^L (t, s), arrivals to the network, and Smacr_i ^L (t, s), services at node i, during time interval (s, t). Then, for the departures Dmacr^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 Dmacr^thetas (t, s) and Q^thetas (t), respectively, satisfy an inequality Dmacr^thetas (t, s) les Amacr^thetas ominus (Smacr^thetas _{n *} Smacr^thetas _{n-1 *} ... - Smacr^thetas ₁) (t, s) and an equality Q^thetas(t) = Amacr^thetas ominus (Smacr^thetas _{n *} Smacr^thetas _{n-1 *} ... - Smacr^thetas ₁) (t, t), where ominus 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.