Achieving optimal transmission throughput in data networks in a multi-hop wireless networks is fundamental but hard problem. The situation is aggravated when nodes are mobile. Further, multi-rate system make the analysis of throughput more complicated. In mobile scenario, link may break or be created as nodes are moving within communication range. Route Discovery' which is to find the optimal route and transmission schedule is an important issue. Route discovery entails some cost; so one would not like to initiate discovery too often. On the other hand, not discovering reasonably often entails the risk of being stuck with a suboptimal route and or schedule, which hurts end-to-end throughput. The implementation of the routing decision problem in one dimensional mobile ad hoc network as Markov decision process problem is already is discussed in the paper [1]. A heuristic based on threshold policy is discussed in the same paper without giving a way to find the threshold. In this paper, we suggested a rule for setting the threshold, given the parameters of the system. We also point out that our results remain valid in a slightly different mobility model; this model is a first step towards an open' network in which existing relay nodes can leave and or new relay nodes can join the network.
used linear programming approach to characterize networks with interference, @cite_6 . They used a conflict graph to model constraints on simultaneous transmissions. In the paper @cite_2 , both approximation algorithms that solve both the end-to-end flow routing problem and link scheduling problem near optimal are proposed. In the paper @cite_10 , it is shown that the problem of solving the optimal scheduling given the concurrency constraints to maximize network throughput, is NP-hard.
@cite_10 @cite_6 @cite_2
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We consider the problem of throughput-optimal scheduling in wireless networks subject to interference constraints. We model the interference using a family of K -hop interference models. We define a K-hop interference model as one for which no two links within K hops can successfully transmit at the same time (Note that IEEE 802.11 DCF corresponds to a 2-hop interference model.) .For a given K, a throughput-optimal scheduler needs to solve a maximum weighted matching problem subject to the K-hop interference constraints. For K=1, the resulting problem is the classical Maximum Weighted Matching problem, that can be solved in polynomial time. However, we show that for K>1,the resulting problems are NP-Hard and cannot be approximated within a factor that grows polynomially with the number of nodes. Interestingly, we show that for specific kinds of graphs, that can be used to model the underlying connectivity graph of a wide range of wireless networks, the resulting problems admit polynomial time approximation schemes. We also show that a simple greedy matching algorithm provides a constant factor approximation to the scheduling problem for all K in this case. We then show that under a setting with single-hop traffic and no rate control, the maximal scheduling policy considered in recent related works can achieve a constant fraction of the capacity region for networks whose connectivity graph can be represented using one of the above classes of graphs. These results are encouraging as they suggest that one can develop distributed algorithms to achieve near optimal throughput in case of a wide range of wireless networks. In the interference scheduling problem, one is given a set of n communication requests described by source-destination pairs of nodes from a metric space. The nodes correspond to devices in a wireless network. Each pair must be assigned a power level and a color such that the pairs in each color class can communicate simultaneously at the specified power levels. The feasibility of simultaneous communication within a color class is defined in terms of the Signal to Interference plus Noise Ratio (SINR) that compares the strength of a signal at a receiver to the sum of the strengths of other signals. The objective is to minimize the number of colors as this corresponds to the time needed to schedule all requests. We introduce an instance-based measure of interference, denoted by I, that enables us to improve on previous results for the interference scheduling problem. We prove the upper and lower bounds in terms of I on the number of steps needed for scheduling a set of requests. For general power assignments, we prove a lower bound of @W(I ([email protected])) steps, where @D denotes the aspect ratio of the metric. When restricting to the two-dimensional Euclidean space (as in the previous work) the bound improves to @W(I [email protected]). Alternatively, when restricting to linear power assignments, the lower bound improves even to @W(I). The lower bounds are complemented by an efficient algorithm computing a schedule for linear power assignments using only O(Ilogn) steps. A more sophisticated algorithm computes a schedule using even only O(I+log^2n) steps. For dense instances in the two-dimensional Euclidean space, this gives a constant factor approximation for scheduling under linear power assignments, which shows that the price for using linear (and, hence, energy-efficient) power assignments is bounded by a factor of O([email protected]). In addition, we extend these results for single-hop scheduling to multi-hop scheduling and combined scheduling and routing problems, where our analysis generalizes the previous results towards general metrics and improves on the previous approximation factors. In this work we study the problem of determining the throughput capacity of a wireless network. We propose a scheduling algorithm to achieve this capacity within an approximation factor. Our analysis is performed in the physical interference model, where nodes are arbitrarily distributed in Euclidean space. We consider the problem separately from the routing problem and the power control problem, i.e., all requests are single-hop, and all nodes transmit at a fixed power level. The existing solutions to this problem have either concentrated on special-case topologies, or presented optimality guarantees which become arbitrarily bad (linear in the number of nodes) depending on the network's topology. We propose the first scheduling algorithm with approximation guarantee independent of the topology of the network. The algorithm has a constant approximation guarantee for the problem of maximizing the number of links scheduled in one time-slot. Furthermore, we obtain a O(log n) approximation for the problem of minimizing the number of time slots needed to schedule a given set of requests. Simulation results indicate that our algorithm does not only have an exponentially better approximation ratio in theory, but also achieves superior performance in various practical network scenarios. Furthermore, we prove that the analysis of the algorithm is extendable to higher-dimensional Euclidean spaces, and to more realistic bounded-distortion spaces, induced by non-isotropic signal distortions. Finally, we show that it is NP-hard to approximate the scheduling problem to within n 1-epsiv factor, for any constant epsiv > 0, in the non-geometric SINR model, in which path-loss is independent of the Euclidean coordinates of the nodes. The capacity of an arbitrary ad-hoc network is difficult to estimate due to interference between the links. We use a conflict graph that models this interference relationship to determine if a set of flow rates can be accommodated. Using the cliques (complete subgraphs) of the conflict graph, we derive constraints that are sufficient for a set of flow rates to be feasible, yet are guaranteed to be within a constant bound of the optimal. We also compute an alternate set of sufficient constraints that can be easily derived from the rows of the matrix representation of the conflict graph. These two sets of constraints are particularly useful because their construction and verification may be distributed across the nodes of a network. We also extend the ad-hoc network model to incorporate variations in the interference range, and obstructions in the network. This paper considers a two-user Gaussian interference channel with energy harvesting transmitters. Different than conventional battery powered wireless nodes, energy harvesting transmitters have to adapt transmission to availability of energy at a particular instant. In this setting, the optimal power allocation problem to maximize the sum throughput with a given deadline is formulated. The convergence of the proposed iterative coordinate descent method for the problem is proved and the short-term throughput maximizing offline power allocation policy is found. Examples for interference regions with known sum capacities are given with directional waterfilling interpretations. Next, stochastic data arrivals are addressed. Finally, online and or distributed near-optimal policies are proposed. Performance of the proposed algorithms are demonstrated through simulations. Consider network coded multicast traffic over a wireless network in the bandwidth limited regime. We formulate the joint medium access and subgraph optimization problem by means of a graphical conflict model. The nature of network coded flows is not captured by classical link-based scheduling and therefore requires a novel approach based on conflicting hyperarcs. By means of simulations, we evaluate the performance of our algorithm and conclude that it significantly outperforms existing scheduling techniques. We formulate the problem of delay constrained energy-efficient broadcast in cooperative multihop wireless networks. We show that this important problem is not only NP-complete, but also o(log(n)) inapproximable. We derive approximation results and an analytical lower-bound for this problem. We break this NP hard problem into three parts: ordering, scheduling and power control. We show that when the ordering is given, the joint scheduling and power-control problem can be solved in polynomial time by a novel algorithm that combines dynamic programming and linear programming to yield the minimum energy broadcast for a given delay constraint. We further show empirically that this algorithm used in conjunction with an ordering derived heuristically using the Dijkstra's shortest path algorithm yields near-optimal performance in typical settings. We use our algorithm to study numerically the trade-off between delay and power-efficiency in cooperative broadcast and compare the performance of our cooperative algorithm with a smart non-cooperative algorithm. This paper considers the problem of determining the achievable rates in multi-hop wireless mesh networks with orthogonal channels. We classify wireless networks with orthogonal channels into two types, half duplex and full duplex, and consider the problem of jointly routing the flows and scheduling transmissions to achieve a given rate vector. We develop tight necessary and sufficient conditions for the achievability of the rate vector. We develop efficient and easy to implement Fully Polynomial Time Approximation Schemes for solving the routing problem. The scheduling problem is a solved as a graph edge-coloring problem. We show that this approach guarantees that the solution obtained is within 50 of the optimal solution in the worst case (within 67 of the optimal solution in a common special case) and, in practice, is close to 90 of the optimal solution on the average. The approach that we use is quite flexible and can be extended to handle more sophisticated interference conditions, and routing with diversity requirements. This paper considers jointly optimal design of crosslayer congestion control, routing and scheduling for ad hoc wireless networks. We first formulate the rate constraint and scheduling constraint using multicommodity flow variables, and formulate resource allocation in networks with fixed wireless channels (or single-rate wireless devices that can mask channel variations) as a utility maximization problem with these constraints. By dual decomposition, the resource allocation problem naturally decomposes into three subproblems: congestion control, routing and scheduling that interact through congestion price. The global convergence property of this algorithm is proved. We next extend the dual algorithm to handle networks with timevarying channels and adaptive multi-rate devices. The stability of the resulting system is established, and its performance is characterized with respect to an ideal reference system which has the best feasible rate region at link layer. We then generalize the aforementioned results to a general model of queueing network served by a set of interdependent parallel servers with time-varying service capabilities, which models many design problems in communication networks. We show that for a general convex optimization problem where a subset of variables lie in a polytope and the rest in a convex set, the dual-based algorithm remains stable and optimal when the constraint set is modulated by an irreducible finite-state Markov chain. This paper thus presents a step toward a systematic way to carry out cross-layer design in the framework of “layering as optimization decomposition” for time-varying channel models. In this paper, we consider CSMA policies for scheduling packet transmissions in multihop wireless networks with one-hop traffic. The main contribution of the paper is to propose a novel CSMA policy, called Unlocking CSMA (U-CSMA), that enables to obtain both high throughput and low packet delays in large wireless networks. More precisely, we show that for torus interference graph topologies with one-hop traffic, U-CSMA is throughput optimal and achieves order-optimal delay. For one-hop traffic, the delay performance is defined to be order-optimal if the delay stays bounded as the network-size increases. Simulations that we conducted suggest that (a) U-CSMA is throughput-optimal and achieves order-optimal delay for general geometric interference graphs and (b) that U-CSMA can be combined with congestion control algorithms to maximize the network-wide utility and obtain order-optimal delay. To the best of our knowledge, this is the first time that a simple distributed scheduling policy has been proposed that is both throughput utility optimal and achieves order-optimal delay. In this paper, we address the scheduling problem in wireless ad hoc networks by exploiting the computational advantage that comes when such scheduling problems can be represented by claw-free conflict graphs where we consider a wireless broadcast medium. It is possible to formulate a scheduling problem of network coded flows as finding maximum weighted independent set (MWIS) in the conflict graph of the network. Finding MWIS of a general graph is NP-hard leading to an NP-hard complexity of scheduling. In a claw-free conflict graph, MWIS can be found in polynomial time leading to a throughput-optimal scheduling. We show that the conflict graph of certain wireless ad hoc networks are claw-free. In order to obtain claw-free conflict graphs in general networks, we suggest introducing additional conflicts (edges) while keeping the decrease in MWIS size minimal. To this end, we introduce an iterative optimization problem to decide where to introduce edges and investigate its efficient implementation. Besides, we exemplify some physical modifications to manipulate the conflict graph of a network and also propose a mixed scheduling strategy for specific networks. We conclude that claw breaking method by adding extra edges can perform nearly optimal under the necessary assumptions. We propose a method for global multi-target tracking that can incorporate higher-order track smoothness constraints such as constant velocity. Our problem formulation readily lends itself to path estimation in a trellis graph, but unlike previous methods, each node in our network represents a candidate pair of matching observations between consecutive frames. Extra constraints on binary flow variables in the graph result in a problem that can no longer be solved by min-cost network flow. We therefore propose an iterative solution method that relaxes these extra constraints using Lagrangian relaxation, resulting in a series of problems that ARE solvable by min-cost flow, and that progressively improve towards a high-quality solution to our original optimization problem. We present experimental results showing that our method outperforms the standard network-flow formulation as well as other recent algorithms that attempt to incorporate higher-order smoothness constraints. We consider the problem of designing an online scheduling scheme for a multi-hop wireless packet network with arbitrary topology and operating under arbitrary scheduling constraints. The objective is to design a scheme that achieves high throughput and low delay simultaneously. We propose a scheduling scheme that - for networks operating under primary interference constraints - guarantees a per-flow end-to-end packet delay bound of 5dj (1-ρj), at a factor 5 loss of throughput, where dj is the path length (number of hops) of flow j and ρj is the effective loading along the route of flow j. Clearly, dj is a universal lower bound on end-to-end packet delay for flow j. Thus, our result is essentially optimal. To the best of our knowledge, our result is the first one to show that it is possible to achieve a per-flow end-to-end delay bound of O(# of hops) in a constrained network. Designing such a scheme comprises two related subproblems: Global Scheduling and Local Scheduling. Global Scheduling involves determining the set of links that will be simultaneously active, without violating the scheduling constraints. While local scheduling involves determining the packets that will be transferred across active edges. We design a local scheduling scheme by adapting the Preemptive Last-In-First-Out (PL) scheme, applied for quasi-reversible continuous time networks, to an unconstrained discrete-time network. A global scheduling scheme will be obtained by using stable marriage algorithms to emulate the unconstrained network with the constrained wireless network. Our scheme can be easily extended to a network operating under general scheduling constraints, such as secondary interference constraints, with the same delay bound and a loss of throughput that depends on scheduling constraints through an intriguing "sub-graph covering" property. In this paper, we propose a cross-layer scheduling algorithm that achieves a throughput “ e-close” to the optimal throughput in multihop wireless networks with a tradeoff of O([1 (e)]) in average end-to-end delay guarantees. The algorithm guarantees finite buffer sizes and aims to solve a joint congestion control, routing, and scheduling problem in a multihop wireless network while satisfying per-flow average end-to-end delay constraints and minimum data rate requirements. This problem has been solved for both backlogged as well as arbitrary arrival rate systems. Moreover, we discuss the design of a class of low-complexity suboptimal algorithms, effects of delayed feedback on the optimal algorithm, and extensions of the proposed algorithm to different interference models with arbitrary link capacities. Multichannel communication in a wireless mesh network with routers having multiple radio interfaces significantly enhances the network capacity. Efficient channel assignment and routing is critical for realization of optimal throughput in such networks. In this paper, we investigate the problem of finding the largest number of links that can be activated simultaneously in a wireless mesh network subject to interference, radio and connectivity constraints. Our goal is to activate all such links and we present an interference aware channel assignment algorithm that realizes this goal. We show that the Link Interference Graph created by utilizing a frequently used interference model gives rise to a special class of graphs, known as overlapping double-disk (ODD) graphs. We prove that the Maximum Independent Set computation problem is NP-complete for this special class of graphs. We provide a Polynomial Time Approximation Scheme (PTAS) for computation of the Maximum Independent Set of an ODD graph. We use this PTAS to develop a channel assignment algorithm for a multiradio multichannel Wireless Mesh Network. We evaluate the performance of our channel assignment algorithm by comparing it with the optimal solution obtained by solving an integer linear program. Experimental results demonstrate that our channel assignment algorithm produces near optimal solution in almost all instances of the problem. This paper considers the problem of determining the achievable rates in multi-hop wireless networks. We consider the problem of jointly routing the flows and scheduling transmissions to achieve a given rate vector. We develop tight necessary and sufficient conditions for the achievability of the rate vector. We develop efficient and easy to implement Fully Polynomial Time Approximation Schemes for solving the routing problem. The scheduling problem is a solved as a graph edge-coloring problem. We show that this approach guarantees that the solution obtained is within 67 of the optimal solution in the worst case and, in practice, is typically within about 80 of the optimal solution. The approach that we use is quite flexible and is a promising method to handle more sophisticated interference conditions, multiple channels, multiple antennas, and routing with diversity requirements. In this paper, we revisit the problem of determining the minimum-length schedule that satisfies certain traffic demands in a wireless network. Traditional approaches for the determination of minimum-length schedules are based on a collision channel model, in which neighboring transmissions cause destructive interference if and only if they are within the "interference region" of the receiving nodes. By contrast, we adopt here a more realistic model for the physical layer by requiring that a threshold be exceeded by the signal-to-interference-plus-noise ratio (SINR) for a transmission to be successful. We present a novel formulation of the problem that incorporates various power and rate adaptation schemes while seamlessly integrating the generation of "matchings" (i.e., sets of links that can be activated simultaneously) by taking into consideration the SINR constraints at the receivers. For the formulated problem, we propose a column-generation-based solution method and show that it theoretically converges to a globally optimal solution, with a potential advantage of not having to enumerate all the feasible matchings a priori. We also discuss the influence of power control, spatial reuse, and variable transmission rates on network performance. Furthermore, we include aspects of the routing problem and provide computational results for our proposed column-generation-based solution procedure. We present a framework to study linear deterministic interference networks over finite fields. Unlike the popular linear deterministic models introduced to study Gaussian networks, we consider networks where the channel coefficients are general scalars over some extension field Fpm (scalar m-th extensionfield models), m×m diagonal matrices over Fp (m-symbol extension ground-field models), and m×m general non-singular matrices (MIMO ground field models). We use the companion matrix representation of the extension field to convert m-th extension scalar models into MIMO ground-field models where the channel matrices have special algebraic structure. For such models, we consider the 2× 2× 2 topology (two-hops two-flow) and the 3-user interference network topology. We derive achievability results and feasibility conditions for certain schemes based on the Precoding-Based Network Alignment (PBNA) approach, where intermediate nodes use random linear network coding (i.e., propagate random linear combinations of their incoming messages) and non-trivial precoding decoding is performed only at the network edges, at the sources and destinations. Furthermore, we apply this approach to the scalar 2×2×2 complex Gaussian IC with fixed channel coefficients, and show two competitive schemes outperforming other known approaches at any SNR, where we combine finite-field linear precoding decoding with lattice coding and the Compute and Forward approach at the signal level. As a side result, we also show significant advantages of vector linear network coding both in terms of feasibility probability (with random coding coefficients) and in terms of coding latency, with respect to standard scalar linear network coding, in PBNA schemes. In this paper, we address the scheduling problem in wireless ad hoc networks by exploiting the computational advantage that comes when such scheduling problems can be represented by claw-free conflict graphs. It is possible to formulate a scheduling problem of network coded flows as finding maximum weighted independent set (MWIS) in the conflict graph of the network. We consider activation of hyperedges in a hypergraph to model a wireless broadcast medium. We show that the conflict graph of certain wireless ad hoc networks are claw-free. It is known that finding MWIS of a general graph is NP-hard, but in a claw-free conflict graph, it is possible to apply Minty's or 's algorithms in polynomial time. We discuss our approach on some sample networks. We consider the scheduling of arbitrary wireless links in the physical model of interference to minimize the time for satisfying all requests. We study here the combined problem of scheduling and power control, where we seek both an assignment of power settings and a partition of the links so that each set satisfies the signal-to-interference-plus-noise (SINR) constraints. We give an algorithm that attains an approximation ratio of O(log n ċ log log Δ), where n is the number of links and Δ is the ratio between the longest and the shortest link length. Under the natural assumption that lengths are represented in binary, this gives the first approximation ratio that is polylogarithmic in the size of the input. The algorithm has the desirable property of using an oblivious power assignment, where the power assigned to a sender depends only on the length of the link. We give evidence that this dependence on Δ is unavoidable, showing that any reasonably behaving oblivious power assignment results in a Ω(log log Δ)-approximation. These results hold also for the (weighted) capacity problem of finding a maximum (weighted) subset of links that can be scheduled in a single time slot. In addition, we obtain improved approximation for a bidirectional variant of the scheduling problem, give partial answers to questions about the utility of graphs for modeling physical interference, and generalize the setting from the standard 2-dimensional Euclidean plane to doubling metrics. Finally, we explore the utility of graph models in capturing wireless interference.