Generalizing Window Flow Control in Bivariate Network Calculus to Enable Leftover Service in the Loop

Network calculus is a methodology to analyse queueing networks in a (probabilistic) worst-case setting. To that end, many results have been presented for different arrival and service processes and under different network topologies. Yet, in the stochastic setting of network calculus it has been found fundamentally hard to deal with feedback systems such as under window flow control constraints. First results on the bivariate feedback equation have been given by Chang et al. in [5]. In this solution, the minimum in a sequence of self-convolutions must be determined. Chang et al. show that under certain conditions the -- theoretically infinite -- sequence of self-convolutions can be cut off after evaluating a finite number of them; the reason being that any further terms are greater than previous terms. As it turns out the required conditions do not allow for leftover service descriptions in the feedback loop. As leftover descriptions are essential in a network analysis, a more general result as in [5] is needed. \ The results of Chang et al. can be translated into the notation of $\sigma$-additive operators (see \cite{chang:performanceguarantees}). This paper's main result is a solution to the feedback equation for such operators. It generalizes present-day solutions and eventually allows to solve the feedback equation for leftover service descriptions inside the feedback loop.