Balaji Prabhakar

Throughput-delay trade-off in wireless networks

Gupta and Kumar (2000) introduced a random network model for studying the way throughput scales in a wireless network when the nodes are fixed, and showed that the throughput per source-destination pair is /spl otimes/(1//spl radic/nlogn). Grossglauser and Tse (2001) showed that when nodes are mobile it is possible to have a constant or /spl otimes/(1) throughput scaling per source-destination pair. The focus of this paper is on characterizing the delay and determining the throughput-delay trade-off in such fixed and mobile ad hoc networks. For the Gupta-Kumar fixed network model, we show that the optimal throughput-delay trade-off is given by D(n) = /spl otimes/(nT(n)), where T(n) and D(n) are the throughput and delay respectively. For the Grossglauser-Tse mobile network model, we show that the delay scales as /spl otimes/(n/sup 1/2//v(n)), where v(n) is the velocity of the mobile nodes. We then describe a scheme that achieves the optimal order of delay for any given throughput. The scheme varies (i) the number of hops, (ii) the transmission range and (iii) the degree of node mobility to achieve the optimal throughput-delay trade-off. The scheme produces a range of models that capture the Gupta-Kumar model at one extreme and the Grossglauser-Tse model at the other. In the course of our work, we recover previous results of Gupta and Kumar, and Grossglauser and Tse using simpler techniques, which might be of a separate interest.

Energy-efficient packet transmission over a wireless link

The paper considers the problem of minimizing the energy used to transmit packets over a wireless link via lazy schedules that judiciously vary packet transmission times. The problem is motivated by the following observation. With many channel coding schemes, the energy required to transmit a packet can be significantly reduced by lowering transmission power and code rate, and therefore transmitting the packet over a longer period of time. However, information is often time-critical or delay-sensitive and transmission times cannot be made arbitrarily long. We therefore consider packet transmission schedules that minimize energy subject to a deadline or a delay constraint. Specifically, we obtain an optimal offline schedule for a node operating under a deadline constraint. An inspection of the form of this schedule naturally leads us to an online schedule which is shown, through simulations, to perform closely to the optimal offline schedule. Taking the deadline to infinity, we provide an exact probabilistic analysis of our offline scheduling algorithm. The results of this analysis enable us to devise a lazy online algorithm that varies transmission times according to backlog. We show that this lazy schedule is significantly more energy-efficient compared to a deterministic (fixed transmission time) schedule that guarantees queue stability for the same range of arrival rates.

The throughput of data switches with and without speedup

In this paper we use fluid model techniques to establish two results concerning the throughput of data switches. For an input-queued switch (with no speedup) we show that a maximum weight algorithm for connecting inputs and outputs delivers a throughput of 100%, and for combined input- and output-queued switches that run at a speedup of 2 we show that any maximal matching algorithm delivers a throughput of 100%. The only assumptions on the input traffic are that it satisfies the strong law of large numbers and that it does not oversubscribe any input or any output.

CHOKe - a stateless active queue management scheme for approximating fair bandwidth allocation

We investigate the problem of providing a fair bandwidth allocation to each of n flows that share the outgoing link of a congested router. The buffer at the outgoing link is a simple FIFO, shared by packets belonging to the n flows. We devise a simple packet dropping scheme, called CHOKe, that discriminates against the flows which submit more packets per second than is allowed by their fair share. By doing this, the scheme aims to approximate the fair queueing policy. Since it is stateless and easy to implement, CHOKe controls unresponsive or misbehaving flows with a minimum overhead.

Energy-efficient transmission over a wireless link via lazy packet scheduling

The paper considers the problem of minimizing the energy used to transmit packets over a wireless link via lazy schedules that judiciously vary packet transmission times. The problem is motivated by the following key observation: in many channel coding schemes, the energy required to transmit a packet can be significantly reduced by lowering the transmission power and transmitting the packet over a longer period of time. However, information is often time-critical or delay-sensitive and transmission times cannot be made arbitrarily long. We therefore consider packet transmission schedules that minimize energy subject to a deadline or a delay constraint. Specifically, we obtain an optimal offline schedule for a node operating under a deadline constraint. An inspection of the form of this schedule naturally leads us to an online schedule which is shown, through simulations, to be energy-efficient. Finally, we relax the deadline constraint and provide an exact probabilistic analysis of our offline scheduling algorithm. We then devise a lazy online algorithm that varies transmission times according to backlog and show that it is more energy efficient than a deterministic schedule that guarantees stability for the same range of arrival rates.

Optimal throughput-delay scaling in wireless networks: part I: the fluid model

Gupta and Kumar (2000) introduced a random model to study throughput scaling in a wireless network with static nodes, and showed that the throughput per source-destination pair is Theta(1/radic(nlogn)). Grossglauser and Tse (2001) showed that when nodes are mobile it is possible to have a constant throughput scaling per source-destination pair. In most applications, delay is also a key metric of network performance. It is expected that high throughput is achieved at the cost of high delay and that one can be improved at the cost of the other. The focus of this paper is on studying this tradeoff for wireless networks in a general framework. Optimal throughput-delay scaling laws for static and mobile wireless networks are established. For static networks, it is shown that the optimal throughput-delay tradeoff is given by D(n)=Theta(nT(n)), where T(n) and D(n) are the throughput and delay scaling, respectively. For mobile networks, a simple proof of the throughput scaling of Theta(1) for the Grossglauser-Tse scheme is given and the associated delay scaling is shown to be Theta(nlogn). The optimal throughput-delay tradeoff for mobile networks is also established. To capture physical movement in the real world, a random-walk (RW) model for node mobility is assumed. It is shown that for throughput of Oscr(1/radic(nlogn)), which can also be achieved in static networks, the throughput-delay tradeoff is the same as in static networks, i.e., D(n)=Theta(nT(n)). Surprisingly, for almost any throughput of a higher order, the delay is shown to be Theta(nlogn), which is the delay for throughput of Theta(1). Our result, thus, suggests that the use of mobility to increase throughput, even slightly, in real-world networks would necessitate an abrupt and very large increase in delay.

Energy-efficient scheduling of packet transmissions over wireless networks

The paper develops algorithms for minimizing the energy required to transmit packets in a wireless environment. It is motivated by the following observation: In many channel coding schemes it is possible to significantly lower the transmission energy by transmitting packets over a long period of time. Based on this observation, we show that for a variety of scenarios the offline energy-efficient transmission scheduling problem reduces to a convex optimization problem. Unlike for the special case of a single transmitter-receiver pair studied by (see Prabhakar, Uysal-Biyikoglu and El Gamal. Proc. IEEE Infocom 2001), the problem does not, in general, admit a closed-form solution when there are multiple users. By exploiting the special structure of the problem, however, we are able to devise energy-efficient transmission schedules. For the downlink channel, with a single transmitter and multiple receivers, we devise an iterative algorithm, called MoveRight, that yields the optimal offline schedule. The MoveRight algorithm also optimally solves the downlink problem with additional constraints imposed by packet deadlines and finite transmit buffers. For the uplink (or multiaccess) problem MoveRight optimally determines the offline time-sharing schedule. A very efficient online algorithm, called MoveRightExpress, that uses a surprisingly small look-ahead buffer is proposed and is shown to perform competitively with the optimal offline schedule in terms of energy efficiency and delay.

pFabric: minimal near-optimal datacenter transport

In this paper we present pFabric, a minimalistic datacenter transport design that provides near theoretically optimal flow completion times even at the 99th percentile for short flows, while still minimizing average flow completion time for long flows. Moreover, pFabric delivers this performance with a very simple design that is based on a key conceptual insight: datacenter transport should decouple flow scheduling from rate control. For flow scheduling, packets carry a single priority number set independently by each flow; switches have very small buffers and implement a very simple priority-based scheduling/dropping mechanism. Rate control is also correspondingly simpler; flows start at line rate and throttle back only under high and persistent packet loss. We provide theoretical intuition and show via extensive simulations that the combination of these two simple mechanisms is sufficient to provide near-optimal performance.

Multicast scheduling for input-queued switches

We design a scheduler for an M/spl times/N input-queued multicast switch. It is assumed that: 1) each input maintains a single queue for arriving multicast cells and 2) only the cell at the head of line (HOL) can be observed and scheduled at one time. The scheduler needs to be: 1) work-conserving (no output port may be idle as long as there is an input cell destined to it) and 2) fair (which means that no input cell may be held at HOL for more than a fixed number of cell times). The aim is to find a work-conserving, fair policy that delivers maximum throughput and minimizes input queue latency, and yet is simple to implement. When a scheduling policy decides which cells to schedule, contention may require that it leave a residue of cells to be scheduled in the next cell time. The selection of where to place the residue uniquely defines the scheduling policy. Subject to a fairness constraint, we argue that a policy which always concentrates the residue on as few inputs as possible generally outperforms all other policies. We find that there is a tradeoff among concentration of residue (for high throughput), strictness of fairness (to prevent starvation), and implementational simplicity (for the design of high-speed switches). By mapping the general multicast switching problem onto a variation of the popular block-packing game Tetris, we are able to analyze various scheduling policies which possess these attributes in different proportions. We present a novel scheduling policy, called TATRA, which performs extremely well and is strict in fairness. We also present a simple weight-based algorithm, called WBA.

Approximate fairness through differential dropping

Many researchers have argued that the Internet architecture would be more robust and more accommodating of heterogeneity if routers allocated bandwidth fairly. However, most of the mechanisms proposed to accomplish this, such as Fair Queueing [16, 6] and its many variants [2, 23, 15], involve complicated packet scheduling algorithms. These algorithms, while increasingly common in router designs, may not be inexpensively implementable at extremely high speeds; thus, finding more easily implementable variants of such algorithms may be of significant practical value. This paper proposes an algorithm called Approximate Fair Dropping (AFD), which bases its dropping decisions on the recent history of packet arrivals. AFD retains a simple forwarding path and requires an amount of additional state that is small compared to current packet buffers. Simulation results, which we describe here, suggest that the design provides a reasonable degree of fairness in a wide variety of operating conditions. The performance of our approach is aided by the fact that the vast majority of Internet flows are slow but the fast flows send the bulk of the bits. This allows a small sample of recent history to provide accurate rate estimates of the fast flows.