Markov-chain modeling for multicast signaling delay analysis

Feedback signaling plays a key role in flow control because the traffic source relies on the signaling information to make correct and timely flow-control decisions. However, it is difficult to design an efficient signaling algorithm since a signaling message can tolerate neither error nor latency. Multicast flow-control signaling imposes two additional challenges: scalability and feedback synchronization. Previous research on multicast signaling has mainly focused on the development of algorithms without analyzing their delay performance. To remedy this deficiency, we have previously developed a binary-tree model and an independent-marking statistical model for multicast-signaling delay analysis. This paper considers a general scenario where the congestion markings at different links are dependent - a more accurate but complex case. Specifically, we develop a Markov-chain model defined by the link-marking state on each path in the multicast tree. The Markov chain can not only capture link-marking dependencies, but also yield a tractable analytical model. We also develop a Markov-chain dependency-degree model to evaluate all possible Markov-chain dependency degrees without any prior knowledge of them. Using the above two models, we derive the general probability distributions of each path becoming the multicast-tree bottleneck. Also derived are the first and second moments of multicast signaling delays. The proposed Markov chain is also shown to asymptotically reach an equilibrium, and its limiting distribution converges to the marginal link-marking probabilities when the Markov chain is irreducible. Applying the two models, we analyze and contrast the delay scalability of two representative multicast signaling protocols: Soft-Synchronization Protocol (SSP) and Hop-By-Hop (HBH) algorithms.