João Luís Sobrinho

Algebra and algorithms for QoS path computation and hop-by-hop routing in the Internet

Prompted by the advent of quality-of-service routing in the Internet, we investigate the properties that path weight functions must have so that hop-by-hop routing is possible and optimal paths can be computed with a generalized Dijkstras algorithm. For this purpose, we define an algebra of weights which contains a binary operation, for the composition of link weights into path weights, and an order relation. Isotonicity is the key property of the algebra. It states that the order relation between the weights of any two paths is preserved if both of them are either prefixed or appended by a common, third, path.We show that isotonicity is both necessary and sufficient for a generalized Dijkstras algorithm to yield optimal paths. Likewise, isotonicity is also both necessary and sufficient for hop-by-hop routing. However, without strict isotonicity, hop-by-hop routing based on optimal paths may produce routing loops. They are prevented if every node computes what we call lexicographic-optimal paths. These paths can be computed with an enhanced Dijkstras algorithm that has the same complexity as the standard one. Our findings are extended to multipath routing as well.As special cases of the general approach, we conclude that shortest-widest paths can neither be computed with a generalized Dijkstras algorithm nor can packets be routed hop-by-hop over those paths. In addition, loop-free hop-by-hop routing over widest and widest-shortest paths requires each node to compute lexicographic-optimal paths, in general.

Network routing with path vector protocols: theory and applications

Path vector protocols are currently in the limelight, mainly because the inter-domain routing protocol of the Internet, BGP (Border Gateway Protocol), belongs to this class. In this paper, we cast the operation of path vector protocols into a broad algebraic framework and relate the convergence of the protocol, and the characteristics of the paths to which it converges, with the monotonicity and isotonicity properties of its path compositional operation. Here, monotonicity means that the weight of a path cannot decrease when it is extended, and isotonicity means that the relationship between the weights of any two paths with the same origin is preserved when both are extended to the same node. We show that path vector protocols can be made to converge for every network if and only if the algebra is monotone, and that the resulting paths selected by the nodes are optimal if and only if the algebra is isotone as well.Many practical conclusions can be drawn from instances of the generic algebra. For performance-oriented routing, typical in intra-domain routing, we conclude that path vector protocols can be made to converge to widest or widest-shortest paths, but that the composite metric of IGRP (Interior Gateway Protocol), for example, does not guarantee convergence to optimal paths. For policy-based routing, typical in inter-domain routing, we formulate existing guidelines as instances of the generic algebra and we propose new ones. We also show how a particular instance of the algebra yields a sufficient condition for signaling correctness of internal BGP.

An algebraic theory of dynamic network routing

We develop a non-classic algebraic theory for the purpose of investigating the convergence properties of dynamic routing protocols. The algebraic theory can be regarded as a generalization of shortest-path routing, where the new concept of free cycle generalizes that of a positive-length cycle. A primary result then states that routing protocols always converge, though not necessarily onto optimal paths, in networks where all cycles are free. Monotonicity and isotonicity are two algebraic properties that strengthen convergence results. Monotonicity implies protocol convergence in every network, and isotonicity assures convergence onto optimal paths. A great many applications arise as particular instances of the algebraic theory. In intra-domain routing, we show that routing protocols can be made to converge to shortest and widest paths, for example, but that the composite metric of Internet Gateway Routing Protocol (IGRP) does not lead to optimal paths. The more interesting applications, however, relate to inter-domain routing and its Border Gateway Protocol (BGP), where the algebraic framework provides a mathematical template for the specification, design, and verification of routing policies. We formulate existing guidelines for inter-domain routing in algebraic terms, propose new guidelines contemplating backup relationships between domains, and derive a sufficient condition for signaling correctness of internal-BGP.

Advances in Modeling of Water Quality in Estuaries

Water quality models are in great demand to complement studies about the status of estuarine waters. However, local models do not perform well when boundary conditions are not properly defined and when biogeochemical processes are not described with adequate detail. This chapter presents advanced modeling applications to perform water quality studies in Portuguese estuaries. Boundary conditions for hydrodynamics and biogeochemistry are provided by the Portuguese Coast Operational Model, downscaled by using nested domains with increasing resolution from the regional to the local scale. The nested models of the estuaries are described, and case studies are presented for specific estuaries to compute sediment transport (Tagus estuary), to calculate residence time of water (Mondego estuary), to forecast quality of bathing waters (Estoril Coast), and to quantify nutrient fluxes between estuaries and the open ocean (Ria de Aveiro). The level of detail used to represent biological processes in water quality models is also addressed, including the description of a case study about modeling of species vulnerable to water quality, such as Zostera noltii in Ria de Aveiro. The need for high level of detail to represent microbial loop and carbon cycle in estuaries is discussed with the application of a complex biological model to the Tagus estuary.

A fresh look at inter-domain route aggregation

We present three route aggregation strategies to scale the Internets inter-domain routing system. These strategies result from a keen understanding on how the customer-provider, peer-peer routing policies propagate routes belonging to long prefixes in relation to how they propagate routes belonging to shorter prefixes that cover the long ones. The first strategy, Coordinated Route Suppression, requires coordination between the Autonomous Systems (ASs) of the Internet, and we present a protocol to perform such coordination. The second strategy, No Import Provider Routes, does not require any coordination between the ASs, but benefits only some of them. The third strategy, Implicit Long Routes, does not rely on any coordination between the ASs either and it is the most efficient strategy. However, it presupposes modifications to the way routers build their forwarding tables. We evaluate the three route aggregation strategies over a publicly available description of the Internet topology and on synthetically generated Internet-like topologies. The results are very promising, with savings in the amount of state information required to sustain inter-domain close to the optimum possible.

Distributed Route Aggregation on the Global Network

The Internet routing system faces serious scalability challenges, due to the growing number of IP prefixes it needs to propagate throughout the network. For example, the Internet suffered significant outages in August 2014 when the number of globally routable prefixes went past 512K, the default size of the forwarding tables in many older routers. Although IP prefixes are assigned hierarchically, and roughly align with geographic regions, todays Border Gateway Protocol (BGP) and operational practices do not exploit opportunities to aggregate routes. We present a distributed route-aggregation technique (called DRAGON) where nodes analyze BGP routes across different prefixes to determine which of them can be filtered while respecting the routing policies for forwarding data-packets. DRAGON works with BGP, can be deployed incrementally, and offers incentives for ASs to upgrade their router software. We present a theoretical model of route-aggregation, and the design and analysis of DRAGON. Our experiments with realistic assignments of IP prefixes, network topologies, and routing policies show that DRAGON reduces the number of prefixes in each AS by about 80% and significantly curtails the number of routes exchanged during transient periods of convergence.

A theory for the connectivity discovered by routing protocols

Route-vector protocols, such as the Border Gateway Protocol (BGP), have nodes elect and exchange routes in order to discover paths over which to send traffic. We ask the following: What is the minimum number of links whose failure prevents a route-vector protocol from finding such paths? The answer is not obvious because routing policies prohibit some paths from carrying traffic and because, on top of that, a route-vector protocol may hide paths the routing policies would allow. We develop an algebraic theory to address the above and related questions. In particular, we characterize a broad class of routing policies for which we can compute in polynomial time the minimum number of links whose failure leaves a route-vector protocol without a communication path from one given node to another. The theory is applied to a publicly available description of the Internet topology to quantify how much of its intrinsic connectivity is lost due to the traditional customer-provider, peer-peer routing policies and how much can be regained with simple alternative policies.

Proposal and performance analysis of a multiple-access protocol for high-speed wireless LANs

Abstract Multiple access schemes are an important component in the development of evolving wireless LANs (WLANs), In this paper, a centralized multiple access protocol is presented for communication among a set of portables and a base station inside a cell of a WLAN. The portables compete for the uplink access to the transmission channel with minipackets in response to controlling commands issued by the base station. The real-time exchange of minipackets and controlling commands allows for the implementation of stable collision resolution algorithms to improve the delay properties of uplink data packets. The proposed protocol is further developed to allow both downlink and uplink traffic directions to time-share a common transmission channel without mutual contention. In the analysis, the operation of the protocol is modeled by a single server queue with two classes of customers having distinct arrival and service characteristics. Stability conditions are derived together with tight upper and lower bounds on the average delays of downlink and uplink data packets. The numerical examples show protocol capacities in excess of 0.85 and 0.70 for nominal channel bit rates of 5 Mbps and 50 Mbps, respectively.

Correctness of Routing Vector Protocols as a Property of Network Cycles

Most analyses of routing vector protocols, such as the Border Gateway Protocol (BGP), are conducted in the context of a single destination in a given network. In that context, for arbitrary routing policies, it is computationally intractable to determine whether or not a routing vector protocol behaves correctly. In this paper, we consider the common scenario where routing policies are specified independently of the destination. In this scenario, we demonstrate that the correctness of a routing vector protocol for all destinations in a given network equates to a property of routing policies around its cycles, designated strict absorbency, similarly to the way that the correctness of a distance vector protocol equates to cycles of positive length. A number of pragmatic conclusions can be derived from this theoretical result. For example, we show that all next-hop routing policies, which are popular in inter-domain routing and in the interconnection of routing instances, cannot fully exploit the physical redundancy of a network. As another example, we show how sibling autonomous systems of the Internet can share all routes between them without introducing oscillations into BGP.

Scaling the Internet Routing System Through Distributed Route Aggregation

The Internet routing system faces serious scalability challenges due to the growing number of IP prefixes that needs to be propagated throughout the network. Although IP prefixes are assigned hierarchically and roughly align with geographic regions, todays Border Gateway Protocol (BGP) and operational practices do not exploit opportunities to aggregate routing information. We present DRAGON, a distributed route-aggregation technique whereby nodes analyze BGP routes across different prefixes to determine which of them can be filtered while respecting the routing policies for forwarding data-packets. DRAGON works with BGP, can be deployed incrementally, and offers incentives for Autonomous Systems (ASs) to upgrade their router software. We illustrate the design of DRAGON through a number of examples, prove its properties while developing a theoretical model of route aggregation, and evaluate its performance. Our experiments with realistic AS-level topologies, assignments of IP prefixes, and routing policies show that DRAGON reduces the number of prefixes in each AS by at least 70% with minimal stretch in the lengths of AS-paths traversed by data packets.