arXiv subjects

P. R. Kumar

Publications and source records attributed to P. R. Kumar.

84 records · Page 5Linked to original sources

arXiv2011

A digital interface for Gaussian relay and interference networks: Lifting codes from the discrete superposition model

For every Gaussian network, there exists a corresponding deterministic network called the discrete superposition network. We show that this discrete superposition network provides a near-optimal digital interface for operating a class consisting of many Gaussian networks in the sense that any code for the discrete superposition network can be naturally lifted to a corresponding code for the Gaussian network, while achieving a rate that is no more than a constant number of bits lesser than the rate it achieves for the discrete superposition network. This constant depends only on the number of nodes in the network and not on the channel gains or SNR. Moreover the capacities of the two networks are within a constant of each other, again independent of channel gains and SNR. We show that the class of Gaussian networks for which this interface property holds includes relay networks with a single source-destination pair, interference networks, multicast networks, and the counterparts of these networks with multiple transmit and receive antennas. The code for the Gaussian relay network can be obtained from any code for the discrete superposition network simply by pruning it. This lifting scheme establishes that the superposition model can indeed potentially serve as a strong surrogate for designing codes for Gaussian relay networks. We present similar results for the K x K Gaussian interference network, MIMO Gaussian interference networks, MIMO Gaussian relay networks, and multicast networks, with the constant gap depending additionally on the number of antennas in case of MIMO networks.

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arXiv2011

Optimal Computation of Symmetric Boolean Functions in Collocated Networks

We consider collocated wireless sensor networks, where each node has a Boolean measurement and the goal is to compute a given Boolean function of these measurements. We first consider the worst case setting and study optimal block computation strategies for computing symmetric Boolean functions. We study three classes of functions: threshold functions, delta functions and interval functions. We provide exactly optimal strategies for the first two classes, and a scaling law order-optimal strategy with optimal preconstant for interval functions. We also extend the results to the case of integer measurements and certain integer-valued functions. We use lower bounds from communication complexity theory, and provide an achievable scheme using information theoretic tools. Next, we consider the case where nodes measurements are random and drawn from independent Bernoulli distributions. We address the problem of optimal function computation so as to minimize the expected total number of bits that are transmitted. In the case of computing a single instance of a Boolean threshold function, we show the surprising result that the optimal order of transmissions depends in an extremely simple way on the values of previously transmitted bits, and the ordering of the marginal probabilities of the Boolean variables. The approach presented can be generalized to the case where each node has a block of measurements, though the resulting problem is somewhat harder, and we conjecture the optimal strategy. We further show how to generalize to a pulse model of communication. One can also consider the related problem of approximate computation given a fixed number of bits. In this case, the optimal strategy is significantly different, and lacks an elegant characterization. However, for the special case of the parity function, we show that the greedy strategy is optimal.

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arXiv2010

Scheduling Periodic Real-Time Tasks with Heterogeneous Reward Requirements

We study the problem of scheduling periodic real-time tasks so as to meet their individual minimum reward requirements. A task generates jobs that can be given arbitrary service times before their deadlines. A task then obtains rewards based on the service times received by its jobs. We show that this model is compatible to the imprecise computation models and the increasing reward with increasing service models. In contrast to previous work on these models, which mainly focus on maximize the total reward in the system, we aim to fulfill different reward requirements by different tasks, which offers better fairness and allows fine-grained tradeoff between tasks. We first derive a necessary and sufficient condition for a system, along with reward requirements of tasks, to be feasible. We also obtain an off-line feasibility optimal scheduling policy. We then studies a sufficient condition for a policy to be feasibility optimal or achieves some approximation bound. This condition can serve as a guideline for designing on-line scheduling policy and we obtains a greedy policy based on it. We prove that the on-line policy is feasibility optimal when all tasks have the same periods and also obtain an approximation bound for the policy under general cases.

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arXiv2010

Optimal computation of symmetric Boolean functions in Tree networks

In this paper, we address the scenario where nodes with sensor data are connected in a tree network, and every node wants to compute a given symmetric Boolean function of the sensor data. We first consider the problem of computing a function of two nodes with integer measurements. We allow for block computation to enhance data fusion efficiency, and determine the minimum worst-case total number of bits to be exchanged to perform the desired computation. We establish lower bounds using fooling sets, and provide a novel scheme which attains the lower bounds, using information theoretic tools. For a class of functions called sum-threshold functions, this scheme is shown to be optimal. We then turn to tree networks and derive a lower bound for the number of bits exchanged on each link by viewing it as a two node problem. We show that the protocol of recursive innetwork aggregation achieves this lower bound in the case of sumthreshold functions. Thus we have provided a communication and in-network computation strategy that is optimal for each link. All the results can be extended to the case of non-binary alphabets. In the case of general graphs, we present a cut-set lower bound, and an achievable scheme based on aggregation along trees. For complete graphs, the complexity of this scheme is no more than twice that of the optimal scheme.

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arXiv2010

Optimal ordering of transmissions for computing Boolean threhold functions

We address a sequential decision problem that arises in the computation of symmetric Boolean functions of distributed data. We consider a collocated network, where each node's transmissions can be heard by every other node. Each node has a Boolean measurement and we wish to compute a given Boolean function of these measurements. We suppose that the measurements are independent and Bernoulli distributed. Thus, the problem of optimal computation becomes the problem of optimally ordering node's transmissions so as to minimize the total expected number of bits. We solve the ordering problem for the class of Boolean threshold functions. The optimal ordering is dynamic, i.e., it could potentially depend on the values of previously transmitted bits. Further, it depends only on the ordering of the marginal probabilites, but not on their exact values. This provides an elegant structure for the optimal strategy. For the case where each node has a block of measurements, the problem is significantly harder, and we conjecture the optimal strategy.

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arXiv2010

A digital interface for Gaussian relay networks: lifting codes from the discrete superposition model to Gaussian relay networks

For every Gaussian relay network with a single source-destination pair, it is known that there exists a corresponding deterministic network called the discrete superposition network that approximates its capacity uniformly over all SNR's to within a bounded number of bits. The next step in this program of rigorous approximation is to determine whether coding schemes for discrete superposition models can be lifted to Gaussian relay networks with a bounded rate loss independent of SNR. We establish precisely this property and show that the superposition model can thus serve as a strong surrogate for designing codes for Gaussian relay networks. We show that a code for a Gaussian relay network, with a single source-destination pair and multiple relay nodes, can be designed from any code for the corresponding discrete superposition network simply by pruning it. In comparison to the rate of the discrete superposition network's code, the rate of the Gaussian network's code only reduces at most by a constant that is a function only of the number of nodes in the network and independent of channel gains. This result is also applicable for coding schemes for MIMO Gaussian relay networks, with the reduction depending additionally on the number of antennas. Hence, the discrete superposition model can serve as a digital interface for operating Gaussian relay networks.

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arXiv2009

The Importance, Design and Implementation of a Middleware for Networked Control Systems

Due to the advancement of computing and communication technology, networked control systems may soon become prevalent in many control applications. While the capability of employing the communication network in the control loop certainly provides many benefits, it also raises several challenges which need to be overcome to utilize the benefits. In this chapter, we focus on one major challenge: a middleware framework that enables a networked control system to be implemented. Indeed our thesis is that a middleware for networked control sys important for the future of networked control systems. We discuss the fundamental issues which need to be considered in the design and development of an appropriate middleware for networked control systems. We describe \emph{Etherware}, a middleware for networked control system which has been developed at the University of Illinois, as an example of such a middleware framework, to illustrate how these issues can be addressed in the design of a middleware. Using a networked inverted pendulum control system as an example, we demonstrate the powerful capabilities provided by Etherware for a networked control system.

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arXiv2009

Optimal strategies for computing symmetric Boolean functions in collocated networks

We address the problem of finding optimal strategies for computing Boolean symmetric functions. We consider a collocated network, where each node's transmissions can be heard by every other node. Each node has a Boolean measurement and we wish to compute a given Boolean function of these measurements with zero error. We allow for block computation to enhance data fusion efficiency, and determine the minimum worst-case total bits to be communicated to perform the desired computation. We restrict attention to the class of symmetric Boolean functions, which only depend on the number of 1s among the n measurements. We define three classes of functions, namely threshold functions, delta functions and interval functions. We provide exactly optimal strategies for the first two classes, and an order-optimal strategy with optimal preconstant for interval functions. Using these results, we can characterize the complexity of computing percentile type functions, which is of great interest. In our analysis, we use lower bounds from communication complexity theory, and provide an achievable scheme using information theoretic tools.

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arXiv2009

Fundamentals of Large Sensor Networks: Connectivity, Capacity, Clocks and Computation

Sensor networks potentially feature large numbers of nodes that can sense their environment over time, communicate with each other over a wireless network, and process information. They differ from data networks in that the network as a whole may be designed for a specific application. We study the theoretical foundations of such large scale sensor networks, addressing four fundamental issues- connectivity, capacity, clocks and function computation. To begin with, a sensor network must be connected so that information can indeed be exchanged between nodes. The connectivity graph of an ad-hoc network is modeled as a random graph and the critical range for asymptotic connectivity is determined, as well as the critical number of neighbors that a node needs to connect to. Next, given connectivity, we address the issue of how much data can be transported over the sensor network. We present fundamental bounds on capacity under several models, as well as architectural implications for how wireless communication should be organized. Temporal information is important both for the applications of sensor networks as well as their operation.We present fundamental bounds on the synchronizability of clocks in networks, and also present and analyze algorithms for clock synchronization. Finally we turn to the issue of gathering relevant information, that sensor networks are designed to do. One needs to study optimal strategies for in-network aggregation of data, in order to reliably compute a composite function of sensor measurements, as well as the complexity of doing so. We address the issue of how such computation can be performed efficiently in a sensor network and the algorithms for doing so, for some classes of functions.

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arXiv2009

On approximating Gaussian relay networks by deterministic networks

We examine the extent to which Gaussian relay networks can be approximated by deterministic networks, and present two results, one negative and one positive. The gap between the capacities of a Gaussian relay network and a corresponding linear deterministic network can be unbounded. The key reasons are that the linear deterministic model fails to capture the phase of received signals, and there is a loss in signal strength in the reduction to a linear deterministic network. On the positive side, Gaussian relay networks are indeed well approximated by certain discrete superposition networks, where the inputs and outputs to the channels are discrete, and channel gains are signed integers. As a corollary, MIMO channels cannot be approximated by the linear deterministic model but can be by the discrete superposition model.

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arXiv2009

Scheduling Heterogeneous Real-Time Traffic over Fading Wireless Channels

We develop a general approach for designing scheduling policies for real-time traffic over wireless channels. We extend prior work, which characterizes a real-time flow by its traffic pattern, delay bound, timely-throughput requirement, and channel reliability, to allow time-varying channels, allow clients to have different deadlines, and allow for the optional employment of rate adaptation. Thus, our model allow the treatment of more realistic fading channels as well as scenarios with mobile nodes, and the usage of more general transmission strategies. We derive a sufficient condition for a scheduling policy to be feasibility optimal, and thereby establish a class of feasibility optimal policies. We demonstrate the utility of the identified class by deriving a feasibility optimal policy for the scenario with rate adaptation, time-varying channels, and heterogeneous delay bounds. When rate adaptation is not available, we also derive a feasibility optimal policy for time-varying channels. For the scenario where rate adaptation is not available but clients have different delay bounds, we describe a heuristic. Simulation results are also presented which indicate the usefulness of the scheduling policies for more realistic and complex scenarios.

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arXiv2009

Utility Maximization for Delay Constrained QoS in Wireless

This paper studies the problem of utility maximization for clients with delay based QoS requirements in wireless networks. We adopt a model used in a previous work that characterizes the QoS requirements of clients by their delay constraints, channel reliabilities, and delivery ratio requirements. In this work, we assume that the utility of a client is a function of the delivery ratio it obtains. We treat the delivery ratio for a client as a tunable parameter by the access point (AP), instead of a given value as in the previous work. We then study how the AP should assign delivery ratios to clients so that the total utility of all clients is maximized. We apply the techniques introduced in two previous papers to decompose the utility maximization problem into two simpler problems, a CLIENT problem and an ACCESS-POINT problem. We show that this decomposition actually describes a bidding game, where clients bid for the service time from the AP. We prove that although all clients behave selfishly in this game, the resulting equilibrium point of the game maximizes the total utility. In addition, we also establish an efficient scheduling policy for the AP to reach the optimal point of the ACCESS-POINT problem. We prove that the policy not only approaches the optimal point but also achieves some forms of fairness among clients. Finally, simulation results show that our proposed policy does achieve higher utility than all other compared policies.

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