Max-min Fair Allocation Research Articles

Max-Min Fair Scheduling in Input-Queued Switches

Fairness in traffic management can improve the isolation between traffic streams, offer a more predictable performance, eliminate transient bottlenecks, mitigate the effect of certain kinds of denial-of-service attacks, and serve as a critical component of a quality-of-service strategy to achieve certain guaranteed services such as delay bounds and minimum bandwidths. In this paper, we choose a popular notion of fairness called max-min fairness and provide a rigorous definition in the context of input-queued switches. We show that being fair at the output ports alone or at the input ports alone or even at both input and output ports does not actually achieve an overall max-min fair allocation of bandwidth in a switch. Instead, we propose a new algorithm that can be readily implemented in a distributed fashion at the input and output ports to determine the exact max-min fair rate allocations for the flows through the switch. In addition to proving the correctness of the algorithm, we propose a practical scheduling strategy based on our algorithm. We present simulation results, using both real traffic traces and synthetic traffic, to evaluate the fairness of a variety of popular scheduling algorithms for input-queued switches. The results show that our scheduling strategy achieves better fairness than other known algorithms for input-queued switches and, in addition, achieves throughput performance very close to that of the best schedulers.

Efficient resource allocation in bandwidth-sharing networks

Document transfer in the Internet is regulated by distributed packet-based congestion control mechanisms, usually relying on TCP. By dividing a document into packets, parts of one file reside at different nodes along the transmission path. The "instantaneous transfer rate" of the entire document can be thought of as being equal to the minimum transfer rate along the entire path. Bandwidth-sharing networks as considered by Massoulié & Roberts [2] provide a natural modeling framework for the dynamic flow-level interaction among document transfers. The class α-fair policies for such networks, as introduced by Mo & Walrand [3], captures a wide range of distributed allocation mechanisms such as TCP, the proportional fair allocation and the max-min fair allocation.

A Unified Framework for Max-Min and Min-Max Fairness With Applications

Max-min fairness is widely used in various areas of networking. In every case where it is used, there is a proof of existence and one or several algorithms for computing it; in most, but not all cases, they are based on the notion of bottlenecks. In spite of this wide applicability, there are still examples, arising in the context of wireless or peer-to-peer networks, where the existing theories do not seem to apply directly. In this paper, we give a unifying treatment of max-min fairness, which encompasses all existing results in a simplifying framework, and extend its applicability to new examples. First, we observe that the existence of max-min fairness is actually a geometric property of the set of feasible allocations. There exist sets on which max-min fairness does not exist, and we describe a large class of sets on which a max-min fair allo cation does exist. This class contains, but is not limited to the compact, convex sets of RN. Second, we give a general purpose centralized algorithm, called Max-min Programming, for computing the max-min fair allocation in all cases where it exists (whether the set of feasible allocations is in our class or not). Its complexity is of the order of N linear programming steps in RN, in the case where the feasible set is defined by linear constraints. We show that, if the set of feasible allocations has the free disposal property, then Max-min Programming reduces to a simpler algorithm, called Water Filling, whose complexity is much lower. Free disposal corresponds to the cases where a bottleneck argument can be made, andWater Filling is the general form of all previously known centralized algorithms for such cases. All our results apply mutatis mutandis to min-max fairness. Our results apply to weighted, unweighted and util-max-min and min-max fairness. Distributed algorithms for the computation of max-min fair allocations are outside the scope of this paper.

Fairness and Load Balancing in Wireless LANs Using Association Control

The traffic load of wireless LANs is often unevenly distributed among the access points (APs), which results in unfair bandwidth allocation among users. We argue that the load imbalance and consequent unfair bandwidth allocation can be greatly reduced by intelligent association control. In this paper, we present an efficient solution to determine the user-AP associations for max-min fair bandwidth allocation. We show the strong correlation between fairness and load balancing, which enables us to use load balancing techniques for obtaining optimal max-min fair bandwidth allocation. As this problem is NP-hard, we devise algorithms that achieve constant-factor approximation. In our algorithms, we first compute a fractional association solution, in which users can be associated with multiple APs simultaneously. This solution guarantees the fairest bandwidth allocation in terms of max-min fairness. Then, by utilizing a rounding method, we obtain the integral solution from the fractional solution. We also consider time fairness and present a polynomial-time algorithm for optimal integral solution. We further extend our schemes for the on-line case where users may join and leave dynamically. Our simulations demonstrate that the proposed algorithms achieve close to optimal load balancing (i.e., max-min fairness) and they outperform commonly used heuristics.

Utility Max-Min Flow Control Using Slope-Restricted Utility Functions

We present a network architecture for the distributed utility max-min flow control of elastic and nonelastic flows where utility values of users (rather than data rates of users) are enforced to achieve max-min fairness. The proposed link algorithm converges to utility max-min fair bandwidth allocation in the presence of round-trip delays without using the information of users' utility functions. To show that the proposed algorithm can be stabilized not locally but globally, we found that the use of nonlinear control theory is inevitable. Even though we use a distributed flow-control algorithm, it is shown that any kind of utility function can be used as long as the minimum slopes of the functions are greater than a certain positive value. Though our analysis is limited to the single-bottleneck and homogeneous-delay case, we believe that the proposed algorithm is the first to achieve utility max-min fairness with guaranteed stability in a distributed manner

Fairness comparison of FAST TCP and TCP Reno

In networking, it is often required to quantify by how much one protocol is fairer than another and how certain parameter setting and/or protocol enhancements improve fairness. This paper provides a framework to evaluate the fairness of various protocols in a general telecommunications network. Within this framework, there are two key components: (1) a benchmark and (2) a single dimension metric. We suggest to use the max–min fairness bandwidth allocation as the benchmark and the Euclidean distance between any bandwidth allocation under any protocol and the max–min bandwidth allocation as the metric. Explicitly, we provide a method to compare the fairness of two sets of bandwidth allocation under two different protocols for a given network by using this metric. On the basis of this new framework, we evaluate the fairness of FAST TCP and TCP Reno relative to the max–min fairness criteria. The distance between the max–min fair allocation and allocations based on each of the two protocols is measured using the Euclidian norm. We derive explicit expressions for these distances for a general network and compare the fairness of these two protocols by using their corresponding utility functions. Finally, we numerically demonstrate how this method can be applied to compare the fairness of FAST TCP and TCP Reno for a “Parking Lot” linear network and for the NSFNET Backbone network. In addition to merely a comparison between protocols, such numerical results can provide guidelines for better choice of parameters to make a protocol fairer in a given scenario.

Computing Optimal Max-Min Fair Resource Allocation for Elastic Flows

In this paper, we consider the max-min fair resource allocation problem as applied to elastic flows. We are interested in computing the optimal max-min fair rate allocation. The proposed approach is a linear programming based one and allows the computation of optimal routing paths with regard to max-min fairness, in stable and known traffic conditions. We consider non-bounded access rates, but we show how the proposed approach can handle the case of upper-bounded access rates. A proof of optimality and some computational results are also presented

Max-Min Fair Rate Allocation Scheme in Multirate Multicast Networks Using Supervisory Control

This paper describes a max-min fair rate allocation scheme in multicast networks based on the supervisory control framework for discrete event systems. Based on the discrete event model of multicast networks, we design a supervisor to guarantee the max-min fair allocation of bandwidth. Analysis and simulations are used to show that the controlled networks guarantee the max-min fair sharing with a low message exchange overhead and fast convergence time.

Auction-Based Spectrum Sharing

We study auction mechanisms for sharing spectrum among a group of users, subject to a constraint on the interference temperature at a measurement point. The users access the channel using spread spectrum signaling and so interfere with each other. Each user receives a utility that is a function of the received signal-to-interference plus noise ratio. We propose two auction mechanisms for allocating the received power. The first is an auction in which users are charged for received SINR, which, when combined with logarithmic utilities, leads to a weighted max-min fair SINR allocation. The second is an auction in which users are charged for power, which maximizes the total utility when the bandwidth is large enough and the receivers are co-located. Both auction mechanisms are shown to be socially optimal for a limiting "large system" with co-located receivers, where bandwidth, power and the number of users are increased in fixed proportion. We also formulate an iterative and distributed bid updating algorithm, and specify conditions under which this algorithm converges globally to the Nash equilibrium of the auction.

Price-based max-min fair rate allocation in wireless multi-hop networks

The interaction between links in wireless multihop networks introduces extra constraints on the combinations of achievable flow rates. Algorithms have been proposed to achieve max-min fairness under these additional constraints. This letter provides a simple price-based max-min fair rate allocation scheme, building on a utility maximization scheme recently proposed for such networks.

Intra-Session Fairness in Multicast Communications

Multicast communication achieves scalability by sending data to multiple receivers at the same time. Receivers in a multicast session usually share the fate with each other, even though their processing speed and the capacity of the path they use can be quite different. A conventional multicast session usually consists of a single multicast group and the problem is how to set the group rate so that it is fair to both fast and slow receivers, to some extent. In a replicated multicast service, receivers are divided into groups based on their capacities and a multicast session can consist of multiple multicast groups. The question is how to divide receivers into groups exactly and set appropriate group rates so that it is fair to all the receivers. Most of current work focuses on optimizing the social welfare represented as a sum of some performance measures of receivers [Kar et al., 2002; Stoenescu et al., 2003]. In this paper, we define a new concept called intra-session fairness and give an optimal solution that can achieve fairness among receivers in the same session. The goal is to maximize the minimum fairness value of the receivers. The novelty of the framework is that it is independent of the specific definition of the fairness function on individual receivers. We illustrate a layering method to implement the max-min intra-session fair allocation and demonstrate the significant difference in fairness achieved by the maximal social welfare algorithm and the max-min intra-session fairness algorithm.

Achieving fair rates with ingress policing

We study a simple ingress policing scheme for a stochastic queuing network that uses a round-robin service discipline, and derive conditions under which the flow rates approach a max-min fair share allocation. The scheme works as follows: Whenever any of a flow's queues exceeds a policing threshold, the network discards that flow's arriving packets at the network ingress, and does so until all of that flow's queues fall below their thresholds. To prove our results, we use previously known results relating the stability of a queuing system to the stability of its fluid limit and extend these results to relate the flow rates of the stochastic system to those of a corresponding fluid model. In particular, we consider the fluid limit of a sequence of queuing networks with increasing thresholds. Using a Lyapunov function derived from the fluid limits, we find that as the policing thresholds are increased the state of the stochastic system is attracted to a relatively smaller and smaller neighborhood surrounding the equilibrium of the fluid model. We then show how this property implies that the achieved flow rates approach the max-min rates predicted by the fluid model.

Fair distributed congestion control in multirate multicast networks

We study fairness of resource allocation in multirate, multicast networks. In multirate networks, different receivers of the same multicast session can receive service at different rates. We develop a mathematical framework to model the maxmin fair allocation of bandwidth with minimum and maximum rate constraints. We present a necessary and sufficient condition for a rate allocation to be maxmin fair in a multirate, multicast network. We propose a distributed algorithm for computing the maxmin fair rates allocated to various source-destination pairs. This algorithm has a low message exchange overhead, and is guaranteed to converge to the maxmin fair rates in finite time.

Maxmin fair scheduling in wireless ad hoc networks

We investigate from an algorithmic perspective the maxmin fair allocation of bandwidth in wireless ad hoc networks. We formalize the maxmin fair objective under wireless scheduling constraints, and present a necessary and sufficient condition for maxmin fairness of a bandwidth allocation. We propose an algorithm that assigns weights to the sessions dynamically such that the weights depend on the congestion in the neighborhood, and schedules the sessions that constitute a maximum weighted matching. We prove that this algorithm attains the maxmin fair rates, even though it does not use any information about the statistics of the packet arrival process.

Maxmin fair scheduling in wireless ad hoc networks

We investigate from an algorithmic perspective the maxmin fair allocation of bandwidth in wireless ad hoc networks. We formalize the maxmin fair objective under wireless scheduling constraints, and present a necessary and sufficient condition for maxmin fairness of a bandwidth allocation. We propose an algorithm that assigns weights to the sessions dynamically such that the weights depend on the congestion in the neighborhood, and schedules the sessions that constitute a maximum weighted matching. We prove that this algorithm attains the maxmin fair rates, even though it does not use any information about the statistics of the packet arrival process.

Fair bandwidth allocation for multicasting in networks with discrete feasible set

We study fairness in allocating bandwidth for loss-tolerant real-time multicast applications. We assume that the traffic is encoded in several layers so that the network can adapt to the available bandwidth and receiver processing capabilities by varying the number of layers delivered. We consider the case where receivers cannot subscribe to fractional layers. Therefore, the network can allocate only a discrete set of bandwidth to a receiver, whereas a continuous set of rates can be allocated when receivers can subscribe to fractional layers. Fairness issues differ vastly in these two different cases. Computation of lexicographic optimal rate allocation becomes NP-hard in this case, while lexicographic optimal rate allocation is polynomial complexity computable when fractional layers can be allocated. Furthermore, maxmin fair rate vector may not exist in this case. We introduce a new notion of fairness, maximal fairness. Even though maximal fairness is a weaker notion of fairness, it has many intuitively appealing fairness properties. For example, it coincides with lexicographic optimally and maxmin fairness, when maxmin fair rate allocation exists. We propose a polynomial complexity algorithm for computation of maximally fair rates allocated to various source-destination pairs, which incidentally computes the maxmin fair rate allocation, when the latter exists.

Network Border Patrol: Preventing Congestion Collapse and Promoting Fairness in the Internet

The Internet's excellent scalability and robustness result in part from the end-to-end nature of Internet congestion control. End-to-end congestion control algorithms alone, however, are unable to prevent the congestion collapse and unfairness created by applications that are unresponsive to network congestion. To address these maladies, we propose and investigate a novel congestion-avoidance mechanism called network border patrol (NBP). NBP entails the exchange of feedback between routers at the borders of a network in order to detect and restrict unresponsive traffic flows before they enter the network, thereby preventing congestion within the network. Moreover, NBP is complemented with the proposed enhanced core-stateless fair queueing (ECSFQ) mechanism, which provides fair bandwidth allocations to competing flows. Both NBP and ECSFQ are compliant with the Internet philosophy of pushing complexity toward the edges of the network whenever possible. Simulation results show that NBP effectively eliminates congestion collapse and that, when combined with ECSFQ, approximately max-min fair bandwidth allocations can be achieved for competing flows.

A fair resource allocation protocol for multimedia wireless networks

Wireless networks are expected to support real-time interactive multimedia traffic and must be able, therefore, to provide their users with quality-of-service (QoS) guarantees. Although the QoS provisioning problem arises in wireline networks as well, mobility of hosts and scarcity of bandwidth makes QoS provisioning a challenging task in wireless networks. It has been noticed that multimedia applications can tolerate and gracefully adapt to transient fluctuations in the QoS that they receive from the network. The additional flexibility afforded by the ability of multimedia applications to tolerate and adapt to transient changes in QoS can be exploited by protocol designers to significantly improve the overall performance of wireless systems. This paper presents a fair resource allocation protocol for multimedia wireless networks that uses a combination of bandwidth reservation and bandwidth borrowing to provide network users with QoS in terms of guaranteed bandwidth, call blocking, and call dropping probabilities. Our view of fairness was inspired by the well-known max-min fairness allocation protocol for wireline networks. Simulation results are presented that compare our protocol to similar schemes.

The impact of multicast layering on network fairness

Many definitions of fairness for multicast networks assume that sessions are single rate, requiring that each multicast session transmits data to all of its receivers at the same rate. These definitions do not account for multirate approaches, such as layering, that permit receiving rates within a session to be chosen independently. We identify four desirable fairness properties for multicast networks, derived from properties that hold within the max-min fair allocations of unicast networks. We extend the definition of multicast max-min fairness to networks that contain multirate sessions, and show that all four fairness properties hold in a multirate max-min fair allocation, but need not hold in a single-rate max-min fair allocation. We then show that multirate max-min fair rate allocations can be achieved via intra-session coordinated joins and leaves of multicast groups. However, in the absence of coordination, the resulting max-min fair rate allocation uses link bandwidth inefficiently, and does not exhibit some of the desirable fairness properties. We evaluate this inefficiency for several layered multirate congestion control schemes, and find that, in a protocol where the sender coordinates joins, this inefficiency has minimal impact on desirable fairness properties. Our results indicate that sender-coordinated layered protocols show promise for achieving desirable fairness properties for allocations in large-scale multicast networks.

Fast, Fair and Frugal Bandwidth Allocation in ATM Networks

Abstract. ATM networks are used to carry a variety of types of traffic. For some types of traffic, in particular Available Bit Rate (ABR) traffic, the bandwidth of a network is typically insufficient to satisfy the requests of all the sessions, and so some fair allocation scheme must be devised. The ATM Forum, the standards setting body for ATM networks, has specified that the fairness criterion for ABR traffic should be Max-Min Fairness , which intuitively means that raising the bandwidth of any session comes at the expense of some other session of no greater bandwidth. Protocols to allocate bandwidth to sessions in a max-min fair manner are an important part of a network design. For a protocol to be realistic, it must conform to the Resource Management (RM) cell mechanism specified by the ATM Forum. Such RM cells get sent as a constant fraction of all cells sent by the source; however, they have only a few fields. RM cells are the only means of communication allowed from link to link so any reasonable protocol is totally distributed and asynchronous, since the RM cell mechanism does not easily lend itself to synchronization. Finally, RM Cells must be handled very quickly at each link. We call a protocol frugal if at each link it performs O(1) computation per RM cell and uses O(1) space per session. Recently, several frugal RM cell protocols have been proposed for ABR traffic, but none have been shown to converge to the max-min fair state. Protocols that are known to converge in a linear number of maximum round-trip times require RM cell processing that is linear in the number of sessions that go through a link. In this paper we give a frugal RM cell protocol for ABR that matches the convergence time of the fastest known non-frugal protocol. A second type of ABR traffic is the Minimum Cell Rate (MCR) type, where every session can specify a minimum amount of bandwidth. The max-min fair allocation should then respect these MCR requests. We extend our results to give the first frugal RM cell protocol for MCR and achieve a quadratic convergence rate.