Statistical analysis of feedback-synchronization signaling delay for multicast flow control

Feedback signaling plays a crucial role in flow control because the traffic source relies on the signaling information to make correct and timely flow-control decisions. Multicast flow-control signaling imposes two additional challenges: scalability and feedback synchronization. We developed a binary-tree deterministic model (Zhang and Shin 1999) and an independent-marking statistical model (Zhang and Shin 2000) to study the delay performance of various multicast feedback-synchronization signaling algorithms. In this paper, we consider the general case in which the congestion markings at different links are dependent. We develop a Markov chain model defined by the link-marking state on each path in a multicast tree. The Markov chain can not only characterize link-marking dependencies, but also yield a tractable analytical model. We also develop a Markov-chain dependency-degree model which can he used to quantify/evaluate all possible Markov-chain dependency degrees without knowing a priori the dependency degree information. Using the Markov-chain and dependency-degree models, we derive the general expressions for the probability distribution of each path bring the multicast-tree bottleneck. Also derived are the closed-form expressions for the first and second moments of multicast signaling delays. The proposed Markov chain is also shown to asymptotically reach an equilibrium, and its limiting state distributions converge to the link-marking marginal probabilities when the Markov chain is irreducible. By applying these two models, we analyze and contrast the feedback-delay scalability of two representative multicast signaling protocols: soft-synchronization protocol and hop-by-hop (HBH) signaling algorithms