Wei Kang Tsai

Complexity of gradient projection method for optimal routing in data networks

In this paper, we derive a time-complexity bound for the gradient projection method for optimal routing in data networks. This result shows that the gradient projection algorithm of Goldstein-Levitin-Poljak type formulated by Bertsekas (1982), Bertsekas and Gallager (1987) and Bertsekas et al. (1984) converges to within /spl epsi/ in relative accuracy in O(/spl epsi//sup -2/h/sub min/N/sub max/) number of iterations, where N/sub max/ is the number of paths sharing the maximally shared link, and h/sub min/ is the diameter of the network. Based on this complexity result, we also show that the one-source-at-a-time update policy has a complexity bound which is O(n) times smaller than that of the all-at-a-time update policy, where n is the number of nodes in the network. The result of this paper argues for constructing networks with low diameter for the purpose of reducing complexity of the network control algorithms. The result also implies that parallelizing the optimal routing algorithm over the network nodes is beneficial.

A theory of convergence order of maxmin rate allocation and an optimal protocol

The problem of allocating maxmin rates with minimum rate constraints for connection-oriented networks is considered. This paper proves that the convergence of maxmin rate allocation satisfies a partial ordering in the bottleneck links. This partial ordering leads to a tighter lower bound for the convergence time for any maxmin protocol. An optimally fast maxmin rate allocation protocol called the distributed constraint precedence graph (CPG) protocol is designed based on this ordering theory. The new protocol employs bi-directional minimization and does not induce transient oscillations. The distributed CPG protocol is compared against ERICA, showing far superior performance.

A highly parallel algorithm for multistage optimization problems and shortest path problems

Abstract It appears that all of the known algorithms for solving multistage optimization problems are based explicitly on standard dynamic programming concepts. Such algorithms are inherently serial in the sense that computation must be completed at the current stage before meaningful computation can begin at the next stage. In this paper we present a technique which recursively divides the original problem into a set of smaller problems which can be solved in parallel. This technique is based on the recursive application of a simple aggregation procedure. For a multistage process with n stages, we show that our new algorithm can achieve a time complexity of O(log n). In contrast, competing algorithms based exclusively on the standard dynamic programming technique can only achieve a time complexity of Φ(n). The new algorithm is designed to operate on a tightly coupled parallel computer. As some important applications, it is shown that our algorithm can serve as a fast and efficient means of decoding convolutional codes, solving shortest path problems, and determining minimum-fuel flight paths.

Time-optimal network queue control: the case of multiple congested nodes

The fundamental problem of time-optimal queue control in packet-switched networks is how to adjust source rates in time after network disturbances so that the network queue sizes converge to desired values in the minimum time, while ensuring that always at least one link remains fully utilized in every flows path. This nonlinear feedback control problem had been solved in a previous paper for a single queue in a single congested node and the solution proven robust to queue size and bandwidth estimation errors. In this paper, we generalize that solution to a general network of flows crisscrossing queues, with link delays being arbitrary. The solution, derived for desired queue sizes of 0, turns out to be simple: two computationally simple conditions on the source rate control, viz. QRE-feasibility and maximally utilizing property are sufficient to ensure time-optimality, regardless of the packet scheduling scheme used inside the network nodes.

A Lexicographic Optimization Framework to the Flow Control Problem

In this paper, a theory of lexicographic optimization for convex and compact feasible sets is presented. Existence, globality, unimodality, and uniqueness of the solution to the problem are proved. Also, necessary and sufficient conditions are derived that establish the relationship between the lexicographic problem and the maxmin problem. This framework is shown to be useful in the problem of designing flow control protocols. Towards this objective, a theory of bottleneck ordering is introduced, which unveils the convergence properties of the flow control problem.

Time-optimal network queue control: the case of a single congested node

The problem of time-optimal network queue control is solved: what are the input data rates that make network queue sizes converge to their ideal size in the least possible time after a disturbance while still maintaining maximum link utilization at all times, even in the transient? The problem is nontrivial especially because of the vast possible heterogeneity in packet propagation delays in the network. In this paper, we derive the time-optimal queue control for a single congested network node with a single finite queue shared by flows with arbitrary network delays. We neatly separate the derivation of the optimal arrival rate sequence from that of the feedback control protocol to achieve it. The time-optimal control is robust to bandwidth and queue size estimation errors. Its complexity is only a function of the size of the network delays and no per-flow computation is needed. The time-optimality and robustness properties are proven to hold under all queue operating regimes with no need for linearizing approximations.

Temporal flow control for ABR traffic management in integrated networks

In the integrated networks where real-time traffic coexists with non-real-time traffic, ABR flow control (FC) has to deal with time-varying background traffic. Dynamic changes in available bandwidth (AB) cause rate and queue oscillations, poor utilization, cell loss, extra delay, and jitters. The oscillations are due to slow source response to FC and aggravated by disparity in feedback latencies. We use the multi-time-scale (MTS) approach to differentiate between short and long latency connections so that AB windows are allocated according to latency: dynamically-available bandwidth should be allocated only to those connections that can use it timely. In this paper, we introduce an improved classification scheme to differentiate between short and long latency connections. Extensive simulations show that the MTS protocol significantly improves network performance in terms of queue size, throughput, fairness, and sensitivity to various parameters in the background traffic.

Stability analysis of intelligent marking EPRCA for ABR congestion control in ATM

The EPRCA (enhanced proportional rate control algorithm) is a leading congestion control protocol for available bit rate (ABR) service in ATM networks. Even though EPRCA has been adopted by the ATM Forum as the standard for ABR congestion control, we show that this protocol is only marginally stable. The stability analysis presented in this paper does not consider interactions among network components and should be considered as a bottle-neck stability result. We show that, if a queue is the bottleneck of a congested path, then its queue size will eventually decrease. However, if a queue is not congested, its queue size can actually increase and the congestion can be self-induced.

Least Square Approach to Multi-Path Maxmin Rate Allocation

The maxmin fair bandwidth allocation has been proposed as a flow control mechanism for managing the data traffic in connection-oriented networks. This paper devises a fast convergence algorithm based on the least square approach to solve multi-level multi-path maxmin problems. We prove that the least square result obtained by our method converges to the optimal solution which is multi-level multi-path maxmin. Our approach is the only known algorithm that can achieve lexicographical separation among the many maxmin levels

Thresholds: Revisit the Strings Versus Clouds Debate for the Internet Architecture, Part II: QoS, Control, Management, and TCP

The clouds (IP) versus strings (connection-oriented) debate over the Internet architecture is reexamined. Controllability and observability are shown to be the key to the performance (QoS) of the networks. The clouds architecture treats the network as a black box, making it uncontrollable and unobservable; in contrast, the strings architecture was designed to be a controllable and observable structure. In network management, the need for centralized management and control to obtain efficiency and optimal performance argues for strings architecture. Finally, TCP is shown to be unscalable in performance because of its poor observability and controllability.