A queueing analysis of max-min fairness, proportional fairness and balanced fairness

We compare the performance of three usual allocations, namely max-min fairness, proportional fairness and balanced fairness, in a communication network whose resources are shared by a random number of data flows. The model consists of a network of processor-sharing queues. The vector of service rates, which is constrained by some compact, convex capacity set representing the network resources, is a function of the number of customers in each queue. This function determines the way network resources are allocated. We show that this model is representative of a rich class of wired and wireless networks. We give in this general framework the stability condition of max-min fairness, proportional fairness and balanced fairness and compare their performance on a number of toy networks.     

Poly-symmetry in processor-sharing systems

We consider a system of processor-sharing queues with state-dependent service rates. These are allocated according to balanced fairness within a polymatroid capacity set. Balanced fairness is known to be both insensitive and Pareto-efficient in such systems, which ensures that the performance metrics, when computable, will provide robust insights into the real performance of the system considered. We first show that these performance metrics can be evaluated with a complexity that is polynomial in the system size if the system is partitioned into a finite number of parts, so that queues are exchangeable within each part and asymmetric across different parts. This in turn allows us to derive stochastic bounds for a larger class of systems which satisfy less restrictive symmetry assumptions. These results are applied to practical examples of tree data networks, such as backhaul networks of Internet service providers, and computer clusters.     

Window flow control in FIFO networks with cross traffic

We focus on window flow control as used in packet-switched communication networks. The approach consists in studying the stability of a system where each node on the path followed by the packets of the controlled connection is modeled by a FIFO (First-In-First-Out) queue of infinite capacity which receives in addition some cross traffic represented by an exogenous flow. Under general stochastic assumptions, namely for stationary and ergodic input processes, we show the existence of a maximum throughput allowed by the flow control. Then we establish bounds on the value of this maximum throughput. These bounds, which do not coincide in general, are reached by time-space scalings of the exogenous flows. Therefore, the performance of the window flow control depends not only on the traffic intensity of the cross flows, but also on fine statistical characteristics such as the burstiness of these flows. These results are illustrated by several examples, including the case of a nonmonotone, nonconvex and fractal stability region. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Kelly networks with shuffling

We consider Kelly networks with shuffling of customers within each queue. Specifically, each arrival, departure or movement of a customer from one queue to another triggers a shuffle of the other customers at each queue. The shuffle distribution may depend on the network state and on the customer that triggers the shuffle. We prove that the stationary distribution of the network state remains the same as without shuffling. In particular, Kelly networks with shuffling have the product form. Moreover, the insensitivity property is preserved for symmetric queues.      

Flow-level performance and capacity of wireless networks with user mobility

The performance evaluation of wireless networks is severely complicated by the specific features of radio communication, such as highly variable channel conditions, interference issues, and possible hand-offs among base stations. The latter elements have no natural counterparts in wireline scenarios, and create a need for novel performance models that account for the impact of these characteristics on the service rates of users. Motivated by the above issues, we review several models for characterizing the capacity and evaluating the flow-level performance of wireless networks carrying elastic data transfers. We first examine the flow-level performance and stability of a wide family of so-called α-fair channel-aware scheduling strategies. We establish that these disciplines provide maximum stability, and describe how the special case of the Proportional Fair policy gives rise to a Processor-Sharing model with a state-dependent service rate. Next we turn attention to a network of several base stations with inter-cell interference. We derive both necessary and sufficient stability conditions and construct lower and upper bounds for the flow-level performance measures. Lastly we investigate the impact of user mobility that occurs on a slow timescale and causes possible hand-offs of active sessions. We show that the mobility tends to increase the capacity region, both in the case of globally optimal scheduling and local α-fair scheduling. It is additionally demonstrated that the capacity and user throughput improve with lower values of the fairness index α.     

Insensitive Bandwidth Sharing in Data Networks

We represent a data network as a set of links shared by a dynamic number of competing flows. These flows are generated within sessions and correspond to the transfer of a random volume of data on a pre-defined network route. The evolution of the stochastic process describing the number of flows on all routes, which determines the performance of the data transfers, depends on how link capacity is allocated between competing flows. We use some key properties of Whittle queueing networks to characterize the class of allocations which are insensitive in the sense that the stationary distribution of this stochastic process does not depend on any traffic characteristics (session structure, data volume distribution) except the traffic intensity on each route. We show in particular that this insensitivity property does not hold in general for well-known allocations such as max-min fairness or proportional fairness. These results are ilustrated by several examples on a number of network topologies.     

On Stochastic Bounds for Monotonic Processor Sharing Networks

We consider a network of processor sharing nodes with independent Poisson arrival processes. Nodes are coupled through their service capacity in that the speed of each node depends on the number of customers present at this and any other node. We assume the network is monotonic in the sense that removing a customer from any node increases the service rate of all customers. Under this assumption, we give stochastic bounds on the number of customers present at any node. We also identify limiting regimes that allow to test the tightness of these bounds. The bounds and the limiting regimes are insensitive to the service time distribution. We apply these results to a number of practically interesting systems, including the discriminatory processor sharing queue, the generalized processor sharing queue, and data networks whose resources are shared according to max–min fairness.     

Performance of CSMA in multi-channel wireless networks

We analyze the performance of CSMA in multi-channel wireless networks, accounting for the random nature of traffic. Specifically, we assess the ability of CSMA to fully utilize the radio resources and in turn to stabilize the network in a dynamic setting with flow arrivals and departures. We prove that CSMA is optimal in the ad-hoc mode, when each flow goes through a unique dedicated wireless link from a transmitter to a receiver. It is generally suboptimal in infrastructure mode, when all data flows originate from or are destined to the same set of access points, due to the inherent bias of CSMA against downlink traffic. We propose a slight modification of CSMA that we refer to as flow-aware CSMA, which corrects this bias and makes the algorithm optimal in all cases. The analysis is based on some time-scale separation assumption which is proved valid in the limit of large flow sizes.