Stability of covariance matrix for generalized transmission control protocol flow control equations
推广形式传输控制协议流量控制方程的方差稳定性

Given the Transmission Control Protocol(TCP) flow control algorithm in a computer network, how to calculate its stability range is an important problem in the design of computer network. Because the control algorithm is affected by many random factors in the network, solving this problem means doing stability analysis for the system described by stochastic differential/difference equations. Current studies mostly take the expectations on both sides of the system equation directly, and simplify this problem into the stability analysis for the expectation, which simply neglects the random variations in the controlled TCP flows. This paper aims at revealing the un-negligible influence from such random variations to the system stability. Using TCP/RED(TCP flows with random early detection) as an example, based on the stochastic differential equations of the system, the system is converted into a multi-dimensional linear time-invariant system with mixed additive and multiplicative noises by linearization at the equilibrium point. Then, generalized TCP flow control equations for continuous-time and discrete-time cases respectively are given, which are the first-degree timeinvariant stochastic differential or difference equations with multi-noise-inputs. After that, the covariance matrix equation for such generalized system is derived; and based on this matrix equation, the sufficient and necessary condition when the covariance matrix has an asymptotically stable limit is presented, together with the expression of this limit. In engineering design, this condition can be regarded as a substitute criterion for estimating the motion domain. Finally, this general condition is applied to a specific example for demonstrating the change of the stability range when the stability of covariance is considered. Moreover, the results in this paper can be extended to the nonlinear system or time-varying system when treated by similar methods used in deterministic cases.