TY - GEN A1 - Harks, Tobias T1 - Nash Equilibria in Online Sequential Routing Games N2 - In this paper, we study the efficiency of Nash equilibria for a sequence of nonatomic routing games. We assume that the games are played consecutively in time in an online fashion: by the time of playing game $i$, future games $i+1,\dots,n$ are not known, and, once players of game $i$ are in equilibrium, their corresponding strategies and costs remain fixed. Given a sequence of games, the cost for the sequence of Nash equilibria is defined as the sum of the cost of each game. We analyze the efficiency of a sequence of Nash equilibria in terms of competitive analysis arising in the online optimization field. Our main result states that the online algorithm $\sl {SeqNash}$ consisting of the sequence of Nash equilibria is $\frac{4n}{2+n}$-competitive for affine linear latency functions. For $n=1$, this result contains the bound on the price of anarchy of $\frac{4}{3}$ for affine linear latency functions of Roughgarden and Tardos [2002] as a special case. Furthermore, we analyze a problem variant with a modified cost function that reflects the total congestion cost, when all games have been played. In this case, we prove an upper bound of $\frac{4n}{2+n}$ on the competitive ratio of $\sl {SeqNash}$. We further prove a lower bound of $\frac{3n-2}{n}$ of $\sl {SeqNash}$ showing that for $n=2$ our upper bound is tight. T3 - ZIB-Report - 06-43 KW - Congestion Game KW - Online Optimization KW - Nash equilibria Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9376 ER - TY - GEN A1 - Harks, Tobias T1 - Utility Proportional Fair Bandwidth Allocation - An Optimization Oriented Approach N2 - In this paper, we present a novel approach to the congestion control and resource allocation problem of elastic and real-time traffic in telecommunication networks. With the concept of utility functions, where each source uses a utility function to evaluate the benefit from achieving a transmission rate, we interpret the resource allocation problem as a global optimization problem. The solution to this problem is characterized by a new fairness criterion, \e{utility proportional fairness}. We argue that it is an application level performance measure, i.e. the utility that should be shared fairly among users. As a result of our analysis, we obtain congestion control laws at links and sources that are globally stable and provide a utility proportional fair resource allocation in equilibrium. We show that a utility proportional fair resource allocation also ensures utility max-min fairness for all users sharing a single path in the network. As a special case of our framework, we incorporate utility max-min fairness for the entire network. To implement our approach, neither per-flow state at the routers nor explicit feedback beside ECN (Explicit Congestion Notification) from the routers to the end-systems is required. T3 - ZIB-Report - 04-32 KW - Utility proportional fairness KW - resource allocation KW - congestion control KW - real-time applications Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8079 ER - TY - GEN A1 - Harks, Tobias A1 - Heinz, Stefan A1 - Pfetsch, Marc T1 - Competitive Online Multicommodity Routing N2 - We study online multicommodity minimum cost routing problems in networks, where commodities have to be routed sequentially. Arcs are equipped with load dependent price functions defining the routing weights. We discuss an online algorithm that routes each commodity by minimizing a convex cost function that depends on the demands that are previously routed. We present a competitive analysis of this algorithm showing that for affine linear price functions this algorithm is $4K/2+K$-competitive, where $K$ is the number of commodities. For the parallel arc case this algorithm is optimal. Without restrictions on the price functions and network, no algorithm is competitive. Finally, we investigate a variant in which the demands have to be routed unsplittably. T3 - ZIB-Report - 06-27 KW - Online Optimization KW - Routing KW - Telecommunications Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9212 ER - TY - GEN A1 - Harks, Tobias A1 - Heinz, Stefan A1 - Pfetsch, Marc T1 - Competitive Online Multicommodity Routing N2 - In this paper we study online multicommodity routing problems in networks, in which commodities have to be routed sequentially. The flow of each commodity can be split on several paths. Arcs are equipped with load dependent price functions defining routing costs, which have to be minimized. We discuss a greedy online algorithm that routes each commodity by minimizing a convex cost function that only depends on the demands previously routed. We present a competitive analysis of this algorithm showing that for affine linear price functions this algorithm is 4K2 (1+K)2 -competitive, where K is the number of commodities. For the single-source single-destination case, this algorithm is optimal. Without restrictions on the price functions and network, no algorithm is competitive. Finally, we investigate a variant in which the demands have to be routed unsplittably. T3 - ZIB-Report - 07-16 KW - Online Optimization KW - Routing KW - Telecommunications Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9599 ER -