Throughput-Optimal Configuration of Fixed Multi-Hop Wireless Networks

1. Throughput-Optimal Configuration of Fixed Multi-Hop Wireless Networks C. Rosenberg This work was done in collaboration with A. Karnik, A. Iyer, and S. Muthaiah.

2. Context: Scheduled Wireless Mesh Networks • Mesh routers (e.g., subscriber stations, access points, etc.) form a multi-hopwireless network. • Network is managed. • No mobility. • Wireless  • Understanding & modelling physical layer is key (fading, interference, etc.). • Access scheme is key. • WLAN traffic is aggregated at the router and the traffic flows are (mostly) to or from the gateway. WLAN Primarily interested in engineering fixed and managed wireless mesh networks (WMN).

3. Modeling Wireless Communications • Designing wireless networks is quite challenging, owing to the complexity underlying wireless communications. • A particular source of complexity: interference. • Wireless medium characteristics: • Signal power radiated into space; • Signal power attenuates over distance; • Signal decoded by treating sum of unwanted signals as noise; decoding is probabilistic. • Unlike the wireline case, the physical layer cannot be abstracted by few simple parameters.

4. Managing Access to the Medium • Owing to the interference wireless links cause one another, it is necessary to arbitrate access to the wireless medium. • Medium access could be based on scheduling or random access. • Conflict-free scheduled networks: the focus of this work. • Results for networks based on IEEE 802.16 • Bounds on performance of IEEE 802.11-networks

5. Focus of Our Work • Interplay between physical layer, network mechanisms, and performance.

6. Aim of Our Work • We seek answers to the following two questions: Q1: Capacity: Given a set of nodes with arbitrary locations, and a set of data flows specified as source-destination pairs, what is the maximumachievable throughput, under certain constraints on the radio parameters (in particular, regulatory constraints on transmit power)? Q2: Optimal Configuration: Further, how should the network be configured to achieve this maximum? By configuration, we mean the complete choice of the set of links, flow routes, link schedules, and transmit power and modulation scheme for each link. • Maximum throughput: • max-min flow rate – maximize the minimum end-to-end flow throughput achievable in the network • Classical notion of capacity a la Gupta-Kumar • Cumulative interferences from multiple links are considered, SINR (signal-to-interference-and-noise ratio) model. • Link rate is a step function of the transmit power. • Our model allows for multiple modulations and multiple power levels

7. An Important Preamble When we compared simplistic physical channel models used in the literature, with one based on SINR, we found: • Models such as the capture threshold model (used in ns-2), or the protocol model, predict different qualitative behavior while studying performance of MAC protocols. • The capture threshold model, the protocol model and a model based on fixed ranges for communication and interference, also predict throughput performance which is in contrast with the predictions of the SINR based model for scheduled networks. • In particular, for a grid mesh network, the maximum throughput at very low power, has different orders of growth in the number of nodes, for different models.

8. Prior Work • Asymptotic Approach – Capacity Scaling: • Characterize capacity in an order sense, Gupta-Kumar [1] • Decode and forward architecture is order optimal, Xie-Kumar [2] • More work in scaling laws under different models, Mhatre-Rosenberg [3] • Mainly addresses capacity (Q1) in an order sense; proofs construct optimal configuration (Q2) again in an order sense • Algorithmic Approach – Joint Optimization: • Dynamic Routing and Power Control, Neely et. al. [4] • Minimize power subject to link data-rate constraints, Cruz and Santhanam [5] • Algorithms/policies implicitly construct optimal configuration (Q2); does not address capacity (Q1)

9. Motivation • Despite works on capacity scaling and joint optimization • No feel for actual numbers for capacity. • No insights into overall trends of capacity with transmit power, modulation schemes, etc. • No structural insights. • Our aim is to address Q1 and Q2 for arbitrary networks. • Important from two perspectives: • Structural Properties: • Does optimal configuration have any structural implications? • Is power used to improve range or data-rate? • Is min-hop routing optimal? When? … and so on • Engineering of networks: upcoming standards like IEEE 802.16 provide: • Multi-rate capability via adaptive modulation and coding • Message-passing mechanisms • Framework for coordinated network operation • A complete answer to Q1 and Q2: • not only provides capacity, • but also optimal design in terms of the sophisticated features of upcoming standards, • Engineering guidelines.

10. Assumptions and Problem Setup • Traffic requirements are static or quasi-static: • Appropriate for large backbone/back-haul type of networks • Channel gains are almost time-invariant: • Realistic in urban/suburban areas with roof-top antennas [6] • Radio parameters and physical layer • Allowable power vectors: • Feasible modulation and coding schemes: • In wireless communication, a successful transmission is specified in terms of an acceptable (bit) error rate (BER). • A modulation scheme z provides a specific transmission rate c(z) • To transmit data from i to j using z while maintaining a specified BER, the Signal to Noise Interference Ratio (SINR) at the receiver must be greater than some threshold β(z).

11. Assumptions and Model (contd.) • Hence the SINR requirement for transmission on link l from i to j is specified as • Plis the transmitting power of i, N0is the average thermal noise power. • The sum in the denominator is over the links transmitting simultaneously with l. • Gll and Gl’l resp. denote the channel gains on link l, and the channel gain from the transmitter of link l’ to j. • Thus for transmission success not all wireless links can be active simultaneously • Link conflict relationships • Given a setting of power and a modulation scheme, each link has a conflict set: • Conflict set consists of subsets; each subset member consists of links which “together disturb” the given link

12. Assumptions and Model (contd.) • Conflict Set • Each v represents a conflict set member for link l • Realistic scheduling constraints as compared to protocol model or k-hop interference model • Conflict graphs only model pairwise or “binary” conflict relationships • Conflict set idea is more general • Conflict sets specify multiple conflict graphs • If v is a unit vector, conflict relationships can be expressed as a conflict graph • Independent Sets • An independent set I is a set of links which can be activated simultaneously, without conflicts • Denote the set of independent sets by

13. Assumptions and Model (contd.) • Link Scheduling: • Set of independent sets: • Set of conflict-free schedules: • Link capacities under a conflict-free schedule: • Data flows and routing: • Flow traffic split along routes such that:

14. Formal Problem Statement • First set of constraints – link capacity constraints • LHS – Traffic imposed on link • RHS – Link capacity under conflict-free schedule a • Third set – to maximize the minimum • Optimal solution exists under certain conditions

15. A Note on Problem Formulation • Our formulation is very powerful and allows for numerous scenarios • Can be reduced to optimal routing problem if the powers and modulations are chosen from discrete sets • introduce an artificial link for every feasible combination of power and modulation • Can be generalized to achieve weighted max-min throughput • flow f has weight wf depending on its priority, willingness-to-pay, etc. • can yield throughputs proportional to per node traffic demands • Conflict set structure can be seen as specifying multiple contention graphs • can include any interference model as special case

16. Scaling Transmit Power Improves Capacity$ • Here ll(z, P) denotes the optimal solution for fixed PHY parameters • If transmit power of each node is scaled up by same factor, then optimum throughput cannot decrease • Why? • SINR improves: • New links could become feasible • In contrast with protocols like COMPOW – common minimum power [7] • $Also, proved by Behzad and Rubin [8]

17. Addressing Computational Complexity • Hardness result: • The problem of determining the max-min throughput of a network, given any conflict structure specified in terms of the conflict sets, is NP-hard • Independent set problem can be reduced to our problem • Smart enumerative technique: • Under the assumptions: • Channel gain is isotropic path loss • Minimum node separation of dmin • Bounded network area of LxL • Maximum number of links that can be scheduled simultaneously is upper bounded by B depending only on dmin, L, the path loss exponent and minimum SINR requirement • Enumerate all subsets of links, of size B or less, and check if they are independent • We develop a computational tool • input: node/gateway locations, available power levels and modulation schemes at each node, flows • output: routing, scheduling and PHY parameters on each link

18. Set-up for Computational Results • For grids and arbitrary networks. • For grids: • nodes placed on a unit grid with gateway in the bottom left corner (except when stated otherwise); NxN grid has 1 gateway and n = N2- 1 nodes • grid side=8m, far-field crossover distance=0:1m, noise power=-100dBm • The flows are all assumed identical and going towards the gateway (unidirectional) or from and to the gateway (bi-directional) • All nodes use same transmission power and same modulation (except when stated otherwise) • Base modulation has rate 1 with SINR threshold 10 dB • Higher modulations: rate 4, threshold 20 dB, and rate 8, threshold 25 dB • Channel experiences only isotropic path loss (can be generalized), path loss exponent = 4 (except when stated otherwise)

19. 5x5 Grid Network: 1 Power, 1 Modulation • The optimum throughput is non decreasing with power (bottom right) • Let Pmin be the minimum power for which the grid is connected (-13.8 dBm) • The maximum throughput is low • The optimal routing is min hop (bottom left) • Let PSH be the minimum power for which single hop is possible (+16.2 dBm) • Routing and scheduling are very simple • Maximum throughput is R(m)/n where R(m) is the link rate when modulation m is used. • For intermediate powers, routing can be quite complex (bottom center: uses -1.85 dBm)

20. 5x5 Grid: Impact of Modulation Schemes • Normalized throughput (left) and size of largest independent set used in optimal configuration (right) as a function of transmit power for different (single) modulation schemes. • A higher rate modulation reduces spatial reuse, but may lead to high throughput gains • A lower rate modulation provides connectivity at lower transmit powers • Pmin= -13.8 dBm, -3.8dBm, +1.2 dBm and PSH= +16.2 dBm, +26.2dBm, +31.2 dBm (for modulation 1,2,3 resp.)

21. Multiple Power and Modulation Levels • Top right: Normalized throughput vs. transmit power for the 5x5 grid of previous slide • With modulation 1 • With modulation 2 • With both modulations 1 & 2 • Using both modulations provides • Connectivity at lower powers • High throughput at higher powers • Bottom right: Normalized throughput vs. transmit power for a 4x4 grid network • With 1 or 2 power levels and modulation 2 • We compare 1 power level P with 2 power levels P and P - 5dBm. • Using 2 power levels provides better flexibility at the physical layer

22. Interference-avoiding Routing • Two flows: red and blue; source and destination apparent from RHS figure • Optimal routing (left) – throughput 2/7 – solid links carry 85% of the traffic • Minimum hop routing (right) – throughput 1/7 • Illustration of interference-avoiding routing

23. No Obvious Trade-offs • Mesh network: 15 nodes, 1 gateway on 4x4 grid • Optimal routing with 2 power levels and 1 modulation scheme (left); and 2 power levels and 2 modulation schemes (right)

24. Hexagonal Topology and Many Flows • IEEE 802.16-like mesh network • 36 subscriber stations and 1 base station (center) • Many-to-one (upload) traffic (30% of traffic) and one-to-many (download) traffic (70% of traffic) • Transmit power of -13 dBm: uplink throughput = 0.00144; downlink = 0.00433 • Routes unlike the tree-based structures considered in literature

25. Single Gateway Placement Grid Topology Throughput Curves • Motivation: • Gateway placement is necessary.

26. Single Gateway Placement Optimal Throughput Curves Arbitrary Network Topology Gateway placement is necessary in arbitrary networks and no placement is optimal for all Ps.

27. Beyond Omni-Directional Antennas • Limitations of omni-directional (om-d) antennas. • high power is required for high throughput. • Interference reduces spatial reuse. • Advantages of smart antennas • Low interference. • Less power to reach the same distance as compared to om-d antennas. • Designed to transmit or receive power in a particular direction • Characterized by • Directivity: concentration of radiated power in a particular direction • Beamwidth (θ). : maximum spread of the gain defined at 3dB points • Radiation pattern: depiction of the relative field strength • Gain (г)

28. 5x5 Grid with Smart Antennas Variation of Spatial-Reuse with power Variation of λ∗ with transmit power (in dBm) • Using smart antennas reduces the transmitter power requirements considerably. • Better spatial reuse is enabled by the use of smart antennas. • Network throughput at low powers is considerably enhanced. • However the maximum achievable network throughput remains identical to omni- directional antennas since the bottleneck is the gateway.

29. 5x5 Grid with Smart Antennas Omni-directional antenna with transmit power −7.75 dBm Smart antennas with beam-width 52o and transmit power −22.48 dBm

30. Extensions and Future Work • Two main assumptions: • Static channel gains • Traffic is static • For uncertain channel gains: • Possible to use robust design; consider independent sets which always remain independent • For stochastically time-varying channel gains: • If channel process iid across slots, can use average statistics to yield sufficient conditions on outage • Static traffic: • Makes sense for a managed network • Further studies: • Lifetime • Other objective functions • Multiple gateway placement • Computational tool

31. References [1] P. Gupta and P. R. Kumar. The Capacity of Wireless Networks. IEEE Transactions on Information Theory, Vol. IT-46, no. 2, pp. 388-404, March 2000. [2] L.-L. Xie and P. R. Kumar. A Network Information Theory for Wireless Communication: Scaling Laws and Optimal Operation. IEEE Transactions on Information Theory, Vol. 50, no. 5, pp. 748-767, May 2004. [3] V. Mhatre and C. Rosenberg. The Capacity of Random Ad Hoc Networks under a Realistic Link Layer Model. submitted to IEEE Transactions on Information Theory, 2005. [4] M. J. Neely, E. Modiano, and C. E. Rohrs. Dynamic power allocation and routing for time-varying wireless networks. IEEE Journal of Selected Areas in Communications, 23(1):89–103, January 2005. Special Issue on Wireless Ad Hoc Networks. [5] R. L. Cruz and A. V. Santhanam. Optimal routing, link scheduling and power control in multi-hop wireless networks. In Proceedings of the IEEE INFOCOM 2003, April 2003. [6] V. Erceg, L. Greenstein, Y. Tjandra, S. Parkoff, A. Gupta, B. Kulic, A. Julius and R. Bianchi. An Empirically-based Path Loss Model for Wireless Channels in Suburban Environments. IEEE J. Sel. Areas Commun., vol. 17, pp 1205-1211, July 1999. [7] S. Narayanaswamy, V. Kawadia, R. S. Sreenivas, and P. R. Kumar, Power Control in Ad Hoc Networks: Theory, Architecture, Algorithm and Implementation of the COMPOW Protocol. in Proceedings of the European Wireless Conference 2002, Florence, Italy, Feb. 2002, pp. 156- 162 [8] A. Behzad and I. Rubin, High Transmission Power Increases the Capacity of Ad Hoc Wireless Networks. IEEE Transactions on Wireless Communications, Vol. 5, No. 1, January 2006.

32. Our References • A. Karnik, A. Iyer and C. Rosenberg; “Throughput-optimal Configuration of Fixed Wireless Networks”, accepted in IEEE/ACM Transaction in Networking, July 07. • S. N. Muthaiah and C. Rosenberg; “ Single Gateway Placement in Wireless Mesh Networks” submitted to ICC’08. • S. N. Muthaiah, A. Iyer, A. Karnik and C. Rosenberg; “Design of High Throughput Scheduled Mesh Networks: A Case For Directional Antennas”, in proceedings of IEEE Globecom 2007, Washington, November 2007. • A. Iyer, C. Rosenberg and A. Karnik, “What is the Right Model for Wireless Channel Interference? in the proceedings of The Third International Conference on Quality of Service in Heterogeneous Wired/Wireless Networks (QShine 2006), Waterloo, Canada, August 2006. Invited paper.