733 research outputs found
Robust distributed routing in dynamical networks with cascading failures
We consider a dynamical formulation of network flows, whereby the network is modeled as a switched system of ordinary differential equations derived from mass conservation laws on directed graphs with a single origin-destination pair and a constant inflow at the origin. The rate of change of the density on each link of the network equals the difference between the inflow and the outflow on that link. The inflow to a link is determined by the total flow arriving to the tail node of that link and the routing policy at that tail node. The outflow from a link is modeled to depend on the current density on that link through a flow function. Every link is assumed to have finite capacity for density and the flow function is modeled to be strictly increasing up to the maximum density. A link becomes inactive when the density on it reaches the capacity. A node fails if all its outgoing links become inactive, and such node failures can propagate through the network due to rerouting of flow. We prove some properties of these dynamical networks and study the resilience of such networks under distributed routing policies with respect to perturbations that reduce link-wise flow functions. In particular, we propose an algorithm to compute upper bounds on the maximum resilience over all distributed routing policies, and discuss examples that highlight the role of cascading failures on the resilience of the network.National Science Foundation (U.S.). Office of Emerging Frontiers in Research and Innovation (ARES Grant 0735956)United States. Air Force Office of Scientific Research (Grant FA9550-09-1-0538
Robust Network Routing under Cascading Failures
We propose a dynamical model for cascading failures in single-commodity
network flows. In the proposed model, the network state consists of flows and
activation status of the links. Network dynamics is determined by a, possibly
state-dependent and adversarial, disturbance process that reduces flow capacity
on the links, and routing policies at the nodes that have access to the network
state, but are oblivious to the presence of disturbance. Under the proposed
dynamics, a link becomes irreversibly inactive either due to overload condition
on itself or on all of its immediate downstream links. The coupling between
link activation and flow dynamics implies that links to become inactive
successively are not necessarily adjacent to each other, and hence the pattern
of cascading failure under our model is qualitatively different than standard
cascade models. The magnitude of a disturbance process is defined as the sum of
cumulative capacity reductions across time and links of the network, and the
margin of resilience of the network is defined as the infimum over the
magnitude of all disturbance processes under which the links at the origin node
become inactive. We propose an algorithm to compute an upper bound on the
margin of resilience for the setting where the routing policy only has access
to information about the local state of the network. For the limiting case when
the routing policies update their action as fast as network dynamics, we
identify sufficient conditions on network parameters under which the upper
bound is tight under an appropriate routing policy. Our analysis relies on
making connections between network parameters and monotonicity in network state
evolution under proposed dynamics
Resilience of Locally Routed Network Flows: More Capacity is Not Always Better
In this paper, we are concerned with the resilience of locally routed network
flows with finite link capacities. In this setting, an external inflow is
injected to the so-called origin nodes. The total inflow arriving at each node
is routed locally such that none of the outgoing links are overloaded unless
the node receives an inflow greater than its total outgoing capacity. A link
irreversibly fails if it is overloaded or if there is no operational link in
its immediate downstream to carry its flow. For such systems, resilience is
defined as the minimum amount of reduction in the link capacities that would
result in the failure of all the outgoing links of an origin node. We show that
such networks do not necessarily become more resilient as additional capacity
is built in the network. Moreover, when the external inflow does not exceed the
network capacity, selective reductions of capacity at certain links can
actually help averting the cascading failures, without requiring any change in
the local routing policies. This is an attractive feature as it is often easier
in practice to reduce the available capacity of some critical links than to add
physical capacity or to alter routing policies, e.g., when such policies are
determined by social behavior, as in the case of road traffic networks. The
results can thus be used for real-time monitoring of distance-to-failure in
such networks and devising a feasible course of actions to avert systemic
failures.Comment: Accepted to the IEEE Conference on Decision and Control (CDC), 201
Integrating fluctuations into distribution of resources in transportation networks
We propose a resource distribution strategy to reduce the average travel time
in a transportation network given a fixed generation rate. Suppose that there
are essential resources to avoid congestion in the network as well as some
extra resources. The strategy distributes the essential resources by the
average loads on the vertices and integrates the fluctuations of the
instantaneous loads into the distribution of the extra resources. The
fluctuations are calculated with the assumption of unlimited resources, where
the calculation is incorporated into the calculation of the average loads
without adding to the time complexity. Simulation results show that the
fluctuation-integrated strategy provides shorter average travel time than a
previous distribution strategy while keeping similar robustness. The strategy
is especially beneficial when the extra resources are scarce and the network is
heterogeneous and lowly loaded.Comment: 14 pages, 4 figure
Robust distributed routing in dynamical networks-part II: Strong resilience, equilibrium selection and cascaded failures
Original manuscript: March 25, 2011Strong resilience properties of dynamical networks are analyzed for distributed routing policies. The latter are characterized by the property that the way the outflow at a non-destination node gets split among its outgoing links is allowed to depend only on local information about the current particle densities on the outgoing links. The strong resilience of the network is defined as the infimum sum of link-wise flow capacity reductions making the asymptotic total inflow to the destination node strictly less than the total outflow at the origin. A class of distributed routing policies that are responsive to local information is shown to yield the maximum possible strong resilience under such local information constraints for an acyclic dynamical network with a single origin-destination pair. The maximal achievable strong resilience is shown to be equal to the minimum node residual capacity of the network. The latter depends on the limit flow of the unperturbed network and is defined as the minimum, among all the non-destination nodes, of the sum, over all the links outgoing from the node, of the differences between the maximum flow capacity and the limit flow of the unperturbed network. We propose a simple convex optimization problem to solve for equilibrium flows of the unperturbed network that minimize average delay subject to strong resilience guarantees, and discuss the use of tolls to induce such an equilibrium flow in traffic networks. Finally, we present illustrative simulations to discuss the connection between cascaded failures and the resilience properties of the network.National Science Foundation (U.S.). Office of Emerging Frontiers in Research and Innovation (Grant 0735956)United States. Air Force Office of Scientific Research (Grant FA9550-09-1-0538
On resilient control of dynamical flow networks
Resilience has become a key aspect in the design of contemporary
infrastructure networks. This comes as a result of ever-increasing loads,
limited physical capacity, and fast-growing levels of interconnectedness and
complexity due to the recent technological advancements. The problem has
motivated a considerable amount of research within the last few years,
particularly focused on the dynamical aspects of network flows, complementing
more classical static network flow optimization approaches. In this tutorial
paper, a class of single-commodity first-order models of dynamical flow
networks is considered. A few results recently appeared in the literature and
dealing with stability and robustness of dynamical flow networks are gathered
and originally presented in a unified framework. In particular, (differential)
stability properties of monotone dynamical flow networks are treated in some
detail, and the notion of margin of resilience is introduced as a quantitative
measure of their robustness. While emphasizing methodological aspects --
including structural properties, such as monotonicity, that enable tractability
and scalability -- over the specific applications, connections to
well-established road traffic flow models are made.Comment: accepted for publication in Annual Reviews in Control, 201
Robust Distributed Routing in Dynamical Networks - Part II: Strong Resilience, Equilibrium Selection and Cascaded Failures
Original manuscript: March 25, 2011Strong resilience properties of dynamical networks are analyzed for distributed routing policies. The latter are characterized by the property that the way the outflow at a non-destination node gets split among its outgoing links is allowed to depend only on local information about the current particle densities on the outgoing links. The strong resilience of the network is defined as the infimum sum of link-wise flow capacity reductions making the asymptotic total inflow to the destination node strictly less than the total outflow at the origin. A class of distributed routing policies that are responsive to local information is shown to yield the maximum possible strong resilience under such local information constraints for an acyclic dynamical network with a single origin-destination pair. The maximal achievable strong resilience is shown to be equal to the minimum node residual capacity of the network. The latter depends on the limit flow of the unperturbed network and is defined as the minimum, among all the non-destination nodes, of the sum, over all the links outgoing from the node, of the differences between the maximum flow capacity and the limit flow of the unperturbed network. We propose a simple convex optimization problem to solve for equilibrium flows of the unperturbed network that minimize average delay subject to strong resilience guarantees, and discuss the use of tolls to induce such an equilibrium flow in traffic networks. Finally, we present illustrative simulations to discuss the connection between cascaded failures and the resilience properties of the network.National Science Foundation (U.S.). Office of Emerging Frontiers in Research and Innovation (Grant 0735956)United States. Air Force Office of Scientific Research (Grant FA9550-09-1-0538
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