Convexity and Robustness of Dynamic Traffic Assignment and Freeway Network Control

We study the use of the System Optimum (SO) Dynamic Traffic Assignment (DTA) problem to design optimal traffic flow controls for freeway networks as modeled by the Cell Transmission Model, using variable speed limit, ramp metering, and routing. We consider two optimal control problems: the DTA problem, where turning ratios are part of the control inputs, and the Freeway Network Control (FNC), where turning ratios are instead assigned exogenous parameters. It is known that relaxation of the supply and demand constraints in the cell-based formulations of the DTA problem results in a linear program. However, solutions to the relaxed problem can be infeasible with respect to traffic dynamics. Previous work has shown that such solutions can be made feasible by proper choice of ramp metering and variable speed limit control for specific traffic networks. We extend this procedure to arbitrary networks and provide insight into the structure and robustness of the proposed optimal controllers. For a network consisting only of ordinary, merge, and diverge junctions, where the cells have linear demand functions and affine supply functions with identical slopes, and the cost is the total traffic volume, we show, using the maximum principle, that variable speed limits are not needed in order to achieve optimality in the FNC problem, and ramp metering is sufficient. We also prove bounds on perturbation of the controlled system trajectory in terms of perturbations in initial traffic volume and exogenous inflows. These bounds, which leverage monotonicity properties of the controlled trajectory, are shown to be in close agreement with numerical simulation results

Design Issues of Reserved Delivery Subnetworks, Doctoral Dissertation, May 2006

The lack of per-flow bandwidth reservation in today\u27s Internet limits the quality of service that an information service provider can provide. This dissertation introduces the reserved delivery subnetwork (RDS), a mechanism that provides consistent quality of service by implementing aggregate bandwidth reservation. A number of design and deployment issues of RDSs are studied. First, the configuration problem of a single-server RDS is formulated as a minimum concave cost network flow problem, which properly reflects the economy of bandwidth aggregation, but is also an NP-hard problem. To make the RDS configuration problem tractable, an efficient approximation heuristic, largest demands first (LDF), is presented and studied. In addition, performance improvements with local search heuristic is investigated. A traditional negative cycle reduction and a new negative bicycle reduction algorithms are applied and evaluated. The study of RDS configuration problems is then extended to multi-server RDSs. The configuration problem can be similarly formulated as the single-server RDS configuration problem; however, the major challenge of multi-server RDS configuration is the optimal server locations. A number of server placement algorithms are evaluated using simulations. The simulation results show that a class of greedy algorithms provide the best solutions. In addition to configuration problem, the dynamic load redistribution mechanism is studied to improve the tolerance to server failures. A configuration algorithm to build redistribution subnetworks is proposed and evaluated to deal with single server failures in a group of servers. Besides the exclusive bandwidth access, there are potentials to further improve end-to-end performance in an RDS because end hosts can utilize the knowledge about the underlying networks to achieve better performance than in the ordinary Internet. These improvements are illustrated with a source traffic regulation technique to resolve the unbalanced bandwidth utilization problem in an RDS. A per-connection and an aggregated regulation algorithm for single-server and multi-server RDSs are presented and studied

Improved Local Search Algorithms with Multi-Cycle Reduction for Minimum Concave Cost Network Flow Problems

The minimum concave cost network flow problem (MCCNFP) has many applications in areas such as telecommunication network design, facility location, production and inventory planning, and traffic scheduling and control. However, it is a well known NP-hard problem, and all existing search based exact algorithms are not practical for networks with even moderate numbers of vertices. Therefore, the research community also focuses on approximation algorithms to tackle the problems in practice. In this paper, we present an improved local search algorithm for the minimum concave cost network flow problem based on multi-cycle reduction. The original cycle reduction local search algorithm as proposed by Gallo and Sodini considers only negative cost single cycles; however, we find that such cycle reduction is not complete. We show that negative cost multi-cycles may exist in a network with concave edge costs that has no negative cost cycles, and an existing flow can be reduced to an adjacent neighboring flow with lower cost by redirecting flows along these negative multi-cycles. In this paper, we present an improved local search algorithm based on multi-cycle reduction. We evaluate our proposed algorithm in networks with a simple concave edge cost in different topologies and sizes. The experimental results show that the original cycle reduction algorithms can improve the quality of solutions obtained from a simple minimum cost augmentation approximation heuristic (LDF), and that a multi-cycle reduction yields more improvements; however, it reaches a point of diminished returns when we attempt to reduce more than bicycles

Dynamic vs Oblivious Routing in Network Design

Consider the robust network design problem of finding a minimum cost network with enough capacity to route all traffic demand matrices in a given polytope. We investigate the impact of different routing models in this robust setting: in particular, we compare \emph{oblivious} routing, where the routing between each terminal pair must be fixed in advance, to \emph{dynamic} routing, where routings may depend arbitrarily on the current demand. Our main result is a construction that shows that the optimal cost of such a network based on oblivious routing (fractional or integral) may be a factor of \BigOmega(\log{n}) more than the cost required when using dynamic routing. This is true even in the important special case of the asymmetric hose model. This answers a question in \cite{chekurisurvey07}, and is tight up to constant factors. Our proof technique builds on a connection between expander graphs and robust design for single-sink traffic patterns \cite{ChekuriHardness07}

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology