[show abstract][hide abstract] ABSTRACT: A key problem in the control of packet-switched data networks is to schedule the data so that the queue sizes remain bounded over time. Scheduling algorithms have been developed in a number of different models that ensure network stability as long as no queue is inherently overloaded. However, this literature typically assumes that each server runs at a fixed maximum speed. Although this is optimal for clearing queue backlogs as fast as possible, it may be suboptimal in terms of energy consumption. Indeed, a lightly loaded server could operate at a lower rate, at least temporarily, to save energy. Within an energy-aware framework, a natural question arises: "What is the minimum energy that is required to keep the network stable?" In this paper, we demonstrate the following results towards answering that question. Starting with the simplest case of a single server in isolation, we consider three types of rate adaptation policies: a heuristic policy, which sets server speed depending on queue size only, and two more complex ones that exhibit a tradeoff between queue size and energy usage. We also present a lower bound on the best such tradeoff that can possibly be achieved. Next, we study a general network environment and investigate two scenarios. In a temporary sessions scenario, where connection paths can rapidly change over time, we propose a combination of the above rate adaptation policies with the standard Farthest-to Go scheduling algorithm. This approach provides stability in the network setting, while using an amount of energy that is within a bounded factor of the optimum. In a permanent sessions scenario, where connection paths are fixed, we examine an analogue of the well-known Weighted Fair Queueing scheduling policy and show how delay bounds are affected under rate adaptation.
[show abstract][hide abstract] ABSTRACT: Current Internet service-provider networks are typically over-provisioned, with the actual traffic through a network element often being much less than the capacity of the network element. However, current network element power consumption is largely independent of actual traffic. This presents an opportunity to reduce network power usage. Such an opportunity may be exploited locally, by redesigning individual network elements to make them rate-adaptive, or globally, by power-aware traffic routing. Instantiating either approach requires significant engineering effort. We attempt to quantify, as realistically as possible, the power-savings opportunity that can be obtained using these two approaches, in isolation or together. In particular, we investigate whether power-aware routing provides any additional benefit if network elements are rate-adaptive. We adopt a fairly simple model of network power use, where power consumption is attributed to links. A link may be turned off, in which case its power usage is zero. Otherwise, a link that is turned on consumes no less than a certain amount of power, called base power, and power use increases further with traffic. A significant parameter of the model is the ratio of base power over full-capacity power. Since it is difficult to estimate feasible values for this ratio, we investigate multiple scenarios in which its value ranges from 0 to 100%. We demonstrate that the combination of rate-adaptivity and power-aware routing saves a significant fraction of network power consumption, for a wide variety of network topologies, traffic loads, and base power ratios. More specifically, if the base power ratio is 50% or more, then power-aware routing appears to be of significant additional benefit over rate adaptivity alone; if the ratio is 25% or less, then power-aware routing offers a relatively small additional benefit.
[show abstract][hide abstract] ABSTRACT: We study traffic grooming in optical network design. The goal is to aggregate low-bandwidth traffic streams to efficiently utilize high-bandwidth media such as wavelength channels. More precisely, given traffic demands to be routed in a network, the design problem is to define a collection of light paths such that each demand can follow a sequence of consecutive light paths. Each light path has unit-wavelength bandwidth, and multiple sub-wavelength demands may share a common light path. Traffic must enter and depart from a light path at its two endpoints only. Most previous work on grooming focused on the ring topology and typically dealt only with uniform bandwidth demands, whereas we consider more general settings. We study two objectives. One is to minimize total equipment cost necessary to support the light paths; the other is simply to minimize the number of light paths. Even for the extremely restricted special case of a line topology and traffic demands that request half-wavelength bandwidth, we show that both objectives are APX-hard to optimize, which means we cannot approximate the optimum arbitrarily closely. On the other extreme of generality, for arbitrary network topologies and traffic demands that request arbitrary amounts of bandwidth, we show a logarithmic approximation for cost minimization and a 2-approximation for minimizing light path counts. Furthermore, we discover that the special case of half-wavelength demands has rich combinatorial properties, closely related to graph problems such as cycle packing and pattern matching problems such as interchange distance in strings. We show how to approximate both objectives up to small constant factors in this case, while similarly improving the approximation and hardness of the interchange distance problem as well.
[show abstract][hide abstract] ABSTRACT: We consider approximation algorithms for buy-at-bulk network design, with the additional constraint that demand pairs be protected against edge or node failures in the network. In practice, the most popular model used in high speed telecommunication networks for protection against failures, is the so-called 1+1 model. In this model, two edge or node-disjoint paths are provisioned for each demand pair. We obtain the first non-trivial approximation algorithms for buy-at-bulk network design in the 1+1 model for both edge and node-disjoint protection requirements. Our results are for the single-cable cost model, which is prevalent in optical networks. More specifically, we present a constant-factor approximation for the single-sink case, and an O(log<sup>3</sup> n) approximation for the multi-commodity case. These results are of interest for practical applications and also suggest several new challenging theoretical problems.