We study routing and scheduling in packet-switched networks. We assume an adversary that controls the injection time, source, and destination for each packet injected. A set of paths for these packets is admissible if no link in the network is overloaded. We present the first on-line routing algorithm that finds a set of admissible paths whenever this is feasible. Our algorithm calculates a path for each packet as soon as it is injected at its source using a simple shortest path computation. The length of a link reflects its current congestion. We also show how our algorithm can be implemented under today's Internet routing paradigms. When the paths are known (either given by the adversary or computed as above) our goal is to schedule the packets along the given paths so that the packets experience small end-to-end delays. The best previous delay bounds for deterministic and distributed scheduling protocols were exponential in the path length. In this paper we present the first deterministic and distributed scheduling protocol that guarantees a polynomial end-to-end delay for every packet. Finally, we discuss the effects of combining routing with scheduling. We first show that some unstable scheduling protocols remain unstable no matter how the paths are chosen. However, the freedom to choose paths can make a difference. For example, we show that a ring with parallel links is stable for all greedy scheduling protocols if paths are chosen intelligently, whereas this is not the case if the adversary specifies the paths.
The problem of choosing routes for a fixed set of packets was studied by Srinivasan and Teo @cite_5 and Bertsimas and Gamarnik @cite_13 . For example, @cite_5 presents an algorithm that minimizes the congestion and dilation of the routes up to a constant factor. This result complemented the paper of Leighton, Maggs and Rao @cite_1 which showed that packets could be scheduled along a set of paths in time @math congestion @math dilation @math .
@cite_1 @cite_5 @cite_13
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We study routing and scheduling in packet-switched networks. We assume an adversary that controls the injection time, source, and destination for each packet injected. A set of paths for these packets is admissible if no link in the network is overloaded. We present the first on-line routing algorithm that finds a set of admissible paths whenever this is feasible. Our algorithm calculates a path for each packet as soon as it is injected at its source using a simple shortest path computation. The length of a link reflects its current congestion. We also show how our algorithm can be implemented under today's Internet routing paradigms.When the paths are known (either given by the adversary or computed as above), our goal is to schedule the packets along the given paths so that the packets experience small end-to-end delays. The best previous delay bounds for deterministic and distributed scheduling protocols were exponential in the path length. In this article, we present the first deterministic and distributed scheduling protocol that guarantees a polynomial end-to-end delay for every packet.Finally, we discuss the effects of combining routing with scheduling. We first show that some unstable scheduling protocols remain unstable no matter how the paths are chosen. However, the freedom to choose paths can make a difference. For example, we show that a ring with parallel links is stable for all greedy scheduling protocols if paths are chosen intelligently, whereas this is not the case if the adversary specifies the paths. We present polylogarithmic approximations for the R|prec|Cmax and R|prec|∑jwjCj problems, when the precedence constraints are “treelike” – i.e., when the undirected graph underlying the precedences is a forest. We also obtain improved bounds for the weighted completion time and flow time for the case of chains with restricted assignment – this generalizes the job shop problem to these objective functions. We use the same lower bound of “congestion+dilation”, as in other job shop scheduling approaches. The first step in our algorithm for the R|prec|Cmax problem with treelike precedences involves using the algorithm of Lenstra, Shmoys and Tardos to obtain a processor assignment with the congestion + dilation value within a constant factor of the optimal. We then show how to generalize the random delays technique of Leighton, Maggs and Rao to the case of trees. For the weighted completion time, we show a certain type of reduction to the makespan problem, which dovetails well with the lower bound we employ for the makespan problem. For the special case of chains, we show a dependent rounding technique which leads to improved bounds on the weighted completion time and new bicriteria bounds for the flow time. This paper considers two inter-related questions: (i) Given a wireless ad-hoc network and a collection of source-destination pairs (s i ,t i ) , what is the maximum throughput capacity of the network, i.e. the rate at which data from the sources to their corresponding destinations can be transferred in the network? (ii) Can network protocols be designed that jointly route the packets and schedule transmissions at rates close to the maximum throughput capacity? Much of the earlier work focused on random instances and proved analytical lower and upper bounds on the maximum throughput capacity. Here, in contrast, we consider arbitrary wireless networks. Further, we study the algorithmic aspects of the above questions: the goal is to design provably good algorithms for arbitrary instances. We develop analytical performance evaluation models and distributed algorithms for routing and scheduling which incorporate fairness, energy and dilation (path-length) requirements and provide a unified framework for utilizing the network close to its maximum throughput capacity.Motivated by certain popular wireless protocols used in practice, we also explore "shortest-path like" path selection strategies which maximize the network throughput. The theoretical results naturally suggest an interesting class of congestion aware link metrics which can be directly plugged into several existing routing protocols such as AODV, DSR, etc. We complement the theoretical analysis with extensive simulations. The results indicate that routes obtained using our congestion aware link metrics consistently yield higher throughput than hop-count based shortest path metrics. We consider the problem of routing traffic to optimize the performance of a congested network. We are given a network, a rate of traffic between each pair of nodes, and a latency function for each edge specifying the time needed to traverse the edge given its congestion; the objective is to route traffic such that the sum of all travel times---the total latency---is minimized.In many settings, it may be expensive or impossible to regulate network traffic so as to implement an optimal assignment of routes. In the absence of regulation by some central authority, we assume that each network user routes its traffic on the minimum-latency path available to it, given the network congestion caused by the other users. In general such a "selfishly motivated" assignment of traffic to paths will not minimize the total latency; hence, this lack of regulation carries the cost of decreased network performance.In this article, we quantify the degradation in network performance due to unregulated traffic. We prove that if the latency of each edge is a linear function of its congestion, then the total latency of the routes chosen by selfish network users is at most 4 3 times the minimum possible total latency (subject to the condition that all traffic must be routed). We also consider the more general setting in which edge latency functions are assumed only to be continuous and nondecreasing in the edge congestion. Here, the total latency of the routes chosen by unregulated selfish network users may be arbitrarily larger than the minimum possible total latency; however, we prove that it is no more than the total latency incurred by optimally routing twice as much traffic.