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.