Equilibrium of Heterogeneous Congestion Control: Optimality and Stability

When heterogeneous congestion control protocols that react to different pricing signals share the same network, the current theory based on utility maximization fails to predict the network behavior. The pricing signals can be different types of signals such as packet loss, queueing delay, etc, or different values of the same type of signal such as different ECN marking values based on the same actual link congestion level. Unlike in a homogeneous network, the bandwidth allocation now depends on router parameters and flow arrival patterns. It can be non-unique, suboptimal and unstable. In Tang et al. (“Equilibrium of heterogeneous congestion control: Existence and uniqueness,” IEEE/ACM Trans. Netw., vol. 15, no. 4, pp. 824–837, Aug. 2007), existence and uniqueness of equilibrium of heterogeneous protocols are investigated. This paper extends the study with two objectives: analyzing the optimality and stability of such networks and designing control schemes to improve those properties. First, we demonstrate the intricate behavior of a heterogeneous network through simulations and present a framework to help understand its equilibrium properties. Second, we propose a simple source-based algorithm to decouple bandwidth allocation from router parameters and flow arrival patterns by only updating a linear parameter in the sources’ algorithms on a slow timescale. It steers a network to the unique optimal equilibrium. The scheme can be deployed incrementally as the existing protocol needs no change and only new protocols need to adopt the slow timescale adaptation

On the flow of non-axisymmetric perturbations of cylinders via surface diffusion

We study the surface diffusion flow acting on a class of general (non--axisymmetric) perturbations of cylinders $\mathcal{C}_r$ in ${\rm I \! R}^3$. Using tools from parabolic theory on uniformly regular manifolds, and maximal regularity, we establish existence and uniqueness of solutions to surface diffusion flow starting from (spatially--unbounded) surfaces defined over $\mathcal{C}_r$ via scalar height functions which are uniformly bounded away from the central cylindrical axis. Additionally, we show that $\mathcal{C}_r$ is normally stable with respect to $2 \pi$--axially--periodic perturbations if the radius $r > 1$,and unstable if $0 < r < 1$. Stability is also shown to hold in settings with axial Neumann boundary conditions.Comment: 20 pages, 1 figure, submitted for publicatio

Network equilibrium of heterogeneous congestion control protocols

When heterogeneous congestion control protocols that react to different pricing signals share the same network, the resulting equilibrium may no longer be interpreted as a solution to the standard utility maximization problem. We prove the existence of equilibrium under mild assumptions. Then we show that multi-protocol networks whose equilibria are locally non-unique or infinite in number can only form a set of measure zero. Multiple locally unique equilibria can arise in two ways. First, unlike in the single-protocol case, the set of bottleneck links can be non-unique with heterogeneous protocols even when the routing matrix has full row rank. The equilibria associated with different sets of bottleneck links are necessarily distinct. Second, even when there is a unique set of bottleneck links, network equilibrium can still be non-unique, but is always finite and odd in number. They cannot all be locally stable unless it is globally unique. Finally, we provide various sufficient conditions for global uniqueness. Numerical examples are used throughout the paper to illustrate these results

Heterogeneous Congestion Control: Efficiency, Fairness and Design

When heterogeneous congestion control protocols that react to different pricing signals (e.g. packet loss, queueing delay, ECN marking etc.) share the same network, the current theory based on utility maximization fails to predict the network behavior. Unlike in a homogeneous network, the bandwidth allocation now depends on router parameters and flow arrival patterns. It can be non-unique, inefficient and unfair. This paper has two objectives. First, we demonstrate the intricate behaviors of a heterogeneous network through simulations and present a rigorous framework to help understand its equilibrium efficiency and fairness properties. By identifying an optimization problem associated with every equilibrium, we show that every equilibrium is Pareto efficient and provide an upper bound on efficiency loss due to pricing heterogeneity. On fairness, we show that intra-protocol fairness is still decided by a utility maximization problem while inter-protocol fairness is the part over which we don¿t have control. However it is shown that we can achieve any desirable inter-protocol fairness by properly choosing protocol parameters. Second, we propose a simple slow timescale source-based algorithm to decouple bandwidth allocation from router parameters and flow arrival patterns and prove its feasibility. The scheme needs only local information

Stochastic user equilibrium assignment with traffic-responsive signal control

This paper considers the Stochastic User Equilibrium (SUE) assignment problem for a signal-controlled network in which intersection control is flow-responsive. The problem is addressed within a Combined Traffic Assignment and Control (CTAC) modeling framework, in which the calculation of user equilibrium link flows is integrated with the calculation of consistent signal settings [1]. It is assumed that network equilibrium is dispersed due to user misperceptions of travel times, and that the intersection control system is designed to allow the persistent adjustment of signal settings in response to traffic flow variations. Thus, the model simulates real- world situations in which network users have limited information and signal control is traffic-actuated. The SUE- based CTAC model is solved algorithmically by means of the so- called Iterative Optimization and Assignment (IOA) procedure, a widely used heuristic which relies on the alternate execution of a control step (signal setting calculation for fixed link flows) and an assignment step (network equilibration under fixed signal settings). The main objective of the study is to define a methodological framework for the evaluation of the performance of various traffic-responsive signal control strategies in interaction with different levels of user information, as represented by the spread parameter of the perceived travel time distribution assumed in the SUE assignment submodel. The results are of practical relevance in a policy context, as they provide a basis for assessing the potential integration of Advanced Traveler Information Systems (ATIS) and signal control systems. Several computational experiments are carried out on a small, contrived network and using realistic intersection delay functions, in order to test the behavior of the model under a wide range of conditions; in particular, convergence pattern and network performance measures at equilibrium are analyzed under alternative information/control scenarios and for various demand levels. The issue of uniqueness of the model solution is addressed as well. Reference: [1] Meneguzzer C. (1997). Review of models combining traffic assignment and signal control. ASCE Journal of Transportation Engineering, vol. 123, no. 2, pp. 148-155.