Capacity region of large wireless networks Devavrat Shah MIT Urs Niesen Piyush Gupta MIT Bell-Labs

The Problem Given a wireless network of n nodes Determine its dimensional capacity regions That is, determine

Purpose Determining the exact capacity region For large networks Has remained unresolved even for three node network ! For large networks Capacity region serves as guideline to evaluate performance of a given architecture Or, as an ‘oracle’ to determine feasibility of desired performance Reasonable approximate characterization of capacity region Will serve the above stated purposes Likely to bring out key characteristics of a good network architecture

The Approximation Problem Given a wireless network of n nodes Determine its dimensional capacity region up to “scaling” That is, determine that can be “nicely” characterized

The Approximation Problem Given a wireless network of n nodes Determine its dimensional capacity region up to “scaling” That is, determine that can be “nicely” characterized Equivalently, determine approximately for any where

Background The approximation problem Basic problem “parameters”: Does not lend itself to easy solutions Basic problem “parameters”: Node placement: Nodes are placed in a geographic area In general can be arbitrarily placed But, a “nicer” situation is when it is random or regular Arbitrary Random/Regular

Background The approximation problem Basic problem “parameters”: Does not lend itself to easy solutions Basic problem “parameters”: Channel model: Information theoretic Gaussian Fading with power attenuation parameter This allows for possibility of network-wide co-operation Protocol or interference model: transmission do not interfere This implies only inter-neighbor (multihop) transmissions are possible

Background The approximation problem Basic problem “parameters”: Does not lend itself to easy solutions Basic problem “parameters”: Traffic demand: Arbitrary: each node can transmit to all n nodes at varying rates This corresponds to dimensional region (or degree of freedom) Random: each node has only one randomly chosen destination and all nodes wish to transmit at the same rate This corresponds to one-dimensional slice of cap. region, i.e. In summary, we want characterization Ideally, for arbitrary placement, Info. Th. and arbitrary demand

Background Gupta and Kumar (2000) took the key first steps towards this goal Their clever assumptions made it possible to get started Specifically, they considered Random placement (not arbitrary) Protocol model (not info. theory) Random source-destination pairing (not arbitrary traffic) Answer: maximal per node achievable rate scales as Using multi-hop and geographic routing Yields a one-dimensional slice of the capacity region

Background Gupta and Kumar (2000) Random placement (not arbitrary) Protocol model (not info. theory) Random source-destination pairing (not arbitrary traffic) Ozgur, Leveque and Tse (2007) (after a long evolution) considered Information theoretic channel model Obtained complete scaling using “hierarchical” co-operation multi-hop hierarchy

Background Niesen, Gupta and Shah (2007) obtained scaling for Gupta and Kumar (2000) Random placement (not arbitrary) Protocol model (not info. theory) Random source-destination pairing (not arbitrary traffic) Ozgur, Leveque and Tse (2007) (after a long evolution) considered Information theoretic Obtained complete scaling using “Hierarchical” co-operation Niesen, Gupta and Shah (2007) obtained scaling for Arbitrary node placement Information theoretic channel model Random source-destination pairing (not arbitrary traffic) Using our novel interpolation of “multi-hop” and “hierarchical” cooperation multi-hop Interpolation hierarchy

Progress Niesen, Gupta and Shah (2007) obtained scaling for Gupta and Kumar (2000) Random placement (not arbitrary) Protocol model (not info. theory) Random source-destination pairing (not arbitrary traffic) Ozgur, Leveque and Tse (2007) (after a long evolution) considered Information theoretic Obtained complete scaling using “Hierarchical” co-operation Niesen, Gupta and Shah (2007) obtained scaling for Arbitrary placement Information theoretic Random source-destination pairing (not arbitrary traffic) All the above results yield a one-dimensional slice of . Here we consider Random placement (not arbitrary) Arbitrary traffic demand (ie. dimensional region)

Progress Our setup Key challenges Random placement (not arbitrary) Information theoretic Arbitrary traffic demand (ie. dimensional region) Key challenges Random node placement provides “some regularity” But, arbitrary traffic demand requires “co-operative” schemes that depend on traffic demand In most of the previous results, random traffic did not present this challenge Specifically, our “interpolation” scheme did utilize regularity of traffic

Progress Our setup Our solution: somewhat surprisingly, we find that Random placement (not arbitrary) Information theoretic Arbitrary traffic demand (ie. dimensional region) Our solution: somewhat surprisingly, we find that Wireless network capacity region is equal to that of a “wireline” tree networks Tree-construction: “clustering” and use of “multi-hop” or “hierarchical” cooperation Equivalent tree Wireless network

Progress Our setup Our solution: somewhat surprisingly, we find that Random placement (not arbitrary) Information theoretic Arbitrary traffic demand (ie. dimensional region) Our solution: somewhat surprisingly, we find that Wireless network capacity region is equal to that of a “wireline” tree network Tree utilization: given any traffic demand, route it over tree as if it were a capacitated wireline tree with capacity assigned during our construction Equivalent tree Routing

Progress Our setup Our solution: some what surprisingly, we find that Random placement (not arbitrary) Information theoretic Arbitrary traffic demand (ie. dimensional region) Our solution: some what surprisingly, we find that Wireless network capacity region is equal to that of a “wireline” tree networks Therefore, the capacity region is approx. characterized by 2n “weighted cuts” , each corresponding to an “edge” in the tree we created Thus, effectively the dim. capacity region is characterized by 2n out of possible cuts !

Overall Progress REF NODES CHANNEL TRAFFIC GK00 LT01+ OLT07 NGS07 RANDOM PROTOCOL RANDOM ARBITRARY PROTOCOL ARBITRARY RANDOM INFO. TH.(large ) RANDOM RANDOM INFO. TH.(small ) RANDOM ARBITRARY INFO. TH. RANDOM RANDOM INFO. TH. ARBITRARY ARBITRARY INFO. TH. ARBITRARY REF GK00 MSL05,08 + SSG07,08 LT01+ OLT07 NGS07 NGS08 IDEAL INNOVATION Multi-hop and Straight line routing Equivalent to wire-line + Clever routing Random cut evaluation Hierarchical co-op + Interpolation Multi-hop, Hierarchical + Geometry aware scheme + Random cut evaluation Equivalent with “wireline” TREE + Routing over TREE + Separation of PHY and NET layer Get As Close As Possible !

Broad implications We have identified capacity region scaling With random placement Extends to “regular enough” placement as well Optimal architecture and separation principle A “physical layer” or capacitated tree is realized through Combination of multi-hop and hierarchical co-operative schemes A “network layer” is realized by routing demand on this tree Treating it as a wireline network An architecture oblivious to the demands! Lots of exciting details in the poster by Urs Niesen

End of Phase Goals We have made major progress towards Characterizing capacity region of large networks Clearly, the next step is to complete the characterization For arbitrary node placement And, go beyond That is, understand the scaling of the “multicast” region This is a dimensional space and much more complicated We strongly believe that we will be able to resolve it building upon the insights from the unicast case