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. However, it is difficult to design an efficient signaling algorithm since each 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 design and implementation of algorithms without analyzing their delay performances. To remedy this deficiency, we developed a binary-tree deterministic model [1] and an independent-marking statistical model [2] to study the delay performance of various multicast feedbacksynchronization signaling algorithms. In this paper, we consider the general case in which the congestion markings at different links are dependent. Including congestion-marking dependencies in the analysis is usually much harder than that under the independence assumption. However, the analysis without assuming independent markings can more accurately capture statistical characteristics for many practical cases where the congestion markings are not independent. Specifically, 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 be used to quantify/evaluate all possible Markov-chain dependency degrees without knowing a priori the dependencydegree information. Using the Markov-chain and dependency-degree models, we derive the general expressions for the probability distribution of each path being the multicast-tree bottleneck. Also derived are the closed-form expressions for the first and second moments of multicast signaling delays. The modeling accuracy and analytical findings have been confirmed by simulations. 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 feedbackdelay scalability of two representative multicast signaling protocols: SoftSynchronization Protocol (SSP) [1], [3] and Hop-By-Hop (HBH) signaling algorithms [4], [5], [6]. The proposed modeling techniques are generic, and thus can be applied not only to the multicast signaling-delay analysis, but also to other Markov-chain-based analyses abstracted from other applications.