Background & Motivation. Layering as Optimization Decomposition: A Mathematical Theory of Network Architectures. Layered Architecture

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1 Background & Motivation Layering as Optimization Decomposition: A Mathematica Theory o Network Architectures Network Architecture? Convex Optimization max U(x) Ax C x 0 Francesco Lo Presti Universita di Roma Tor Vergata opresti@ino.uniroma2.it U(x) concave unction x, C vectors A matrix 1 2 Layered Architecture Layered Architecture Communication system as composed o ordered set o ayers Each ayer provides a service (to the ayer above) Service: set o unctions oered to the service users Error Contro, Fow Contro reying on services provided by ayer beow via its own interna-ayer actions Each ayer provides a vaue added (communication) service with respect to the (communication) service provided by the ayer beow appication transport appication transport TCP/P Architecture Layer 4, Tranport: Congestion Contro end-to-end Layer 3, Network: Routing Layer 2, Data Link, MAC: Media Access Layer 1, Physica: Coding 3 4

2 Layered Architecture Key Observation [Key98] Probem Today Protocos mosty based on engineering intuitions and ad-hoc heuristics TCP Congestion Contro and its additive increase and mutipicative decrease MAC Layer Coision Detection and Binary Backo Agoritms Exceptions Coding: Maximize Data Rate subject to error probabiity constraints Routing: Minimum Path Cost Routing How we do they work as a whoe? 5 Reverse Engineering o Protocos Behaviour TCP and its variants sove in a distributed way a Network Utiity Maximization Probem (convex) Contention based MAC as a game theoretic seish utiity maximization 6 Forward Engineering 1. Reverse Engineering discovers the underying mathematica probems soved by protocos 2. Forward Engineering based on insights obtained rom 1 above then systematicay improves the protocos Wireess Protocos ssues Wireess s chaenges MAC ag, routing, mobiity, power contro, cong. contro, etc. Cross-Layer Design/Optimization Do away with the strict ayers separation in the protoco stack design Take advantage o (possiby strict) coordination among dierent ayers We consider the congestion contro probem as a joint optimization with routing and MAC Formuated as a utiity based optimization probem 7 8

3 Outine o the Tak Mode Mode Probem Formuation Convex Optimization Probem with inear constraints Probem Soution Via Dua Decomposition and (sub)-gradient agorithm Distributed Agorithm or Congestion Contro, Routing and MAC Numerica Exampes Concusions 9 Static Muti-Hop Wireess Network Transmission range R Logica channe capacity C Link (i,j) ii distance(i,j) R Direct Graph G=(N,E) (i,j) E (j,i) E 10 Mode Mode Links contention Transmission over dierent inks may interere Link contention graph to describe ink intererence Nodes represent inks Edge between nodes when inks interere Contention Graph: Exampe Two inks interere when either ends o one o the two inks as within carrier sense range o the other CSMA/CA-ike contention with RTS/CTS mechanism Assume Carrier sense range~transmission range 11 12

4 Mode Mode Links contention Ony inks orming an independent set in the contention graph can transmit at the same time without causing intererence Transmission Schedue Sequence o independent set 1,2,3,,n, Contention imits the average ink rate c c= C a =C ar : a raction o time independent set appears in the transmission schedue r independent set characteristic vector r=1 i, otherwise r=0 13 Traic: Set o ows F Fow F Source, destination x ow rate y ow rate aong ink Utiity unction U(x) ncreasing Concave Continuousy Dierentiabe e.g.,u(x)=og x 14 Probem Formuation Probem Formuation: Constraints Maximize aggregate utiity Given the constraints max U(x)= U(x) 1. Fow conservation constraints 2. Link capacity constraints 3. Links contention constraints 1. Fow Conservation Constraints For each ow, traic which enters a node n, must eave it 1. n source, destination y= y n enters node n eaves node n n 15 16

5 Probem Formuation: Constraints Probem Formuation: Constraints 1. Fow Conservation Constraints For each ow, traic which enter a node n, must eave it 2. n source node x= y eaves node n n 1. Fow Conservation Constraints For each ow, traic which enter a node n, must eave it 3. n destination node enters node n y = x n Probem Formuation: Constraints Probem Formuation: Constraints 2. Link capacity constraints For each ink, ows traversing it must not exceed its average transmission rate c 3. Links contention constraints Ony inks orming an independent set in the contention graph can transmit at the same time Contention imits the achievabe ink rate c y c c= C ar,, a=1, a

6 Probem Formuation Dua Approach max U(x)= U(x) Consider the dua probem min D(p) subject to p 0 y= y, y 0 n y c c= C ar,, a=1, a 0 21 D(p) (partia) dua unction p=(p) L vector o Lagrangian Mutipiers (ink prices) Reax the ink capacity constraints Sove the dua Strong Duaity hods: the soution o the dua yieds a soution o the prima t is (reativey) easy to compute in a distributed way - D(p) 22 and its derivatives Dua Approach Probem Formuation Separabe wrt x Soution via (Sub)-Gradient agorithm D(p) is a convex (non-dierentiabe) unction 1. Choose initia soution p(0) 2. terate p(k+1)=p(k)+γ [ D(p(k)) ] + p A ows rates y aong ink + c max U(x)= U(x) Fow rate variabes x y 3. Unti convergence criteria are satisied Computation can be distributed among nodes. ts impementation deines the congestion contro, the routing and the MAC contro agorithms 23 y= y, y 0 n y c c= C ar,, a=1, a 0 Link rates c 24

7 Dua Function: Lagrangian Lagrangian Lagrangian Mutipiers Dua Function D(p) L(x,y,c,a,p)= U(x) - p( y-c) y= y, y 0 n y c c=c ar,, a=1, a 0 one mut. per ink p as ink price 25 D(p)=max L(x,y,c,a,p)= x,y,c,a =max U(x) - p( y-c) x,y,c,a y= y, y 0 n c= C ar,, a=1, a 0 x(p),y(p),c(p)=argmax L(x,y,c,a,p) D(p)= U(x(p))- p( y(p)-c(p)) x,y,c,a 26 Dua Probem Dua Probem Decomposition min D(p)=ΣU(x(p))-Σp(Σy(p)-c(p)) p 0 We sove via the (sub)-gradient ag. Compute Communicate Comm. new prices x,y,c new x,y,c p(k+1) Sub gradient update Compute p(k+1) [ ] + p(k+1)=p(k)-γ D(p(k)) p Compute x1(p),y1(p) Compute xn(p),yn(p) Compute c(p) 27 28

8 Network mpementation Network mpementation Compute Communicate Comm. new prices x,y,c new x,y,c p(k+1) Network Update inks price p(k+1) teration 1. Compute x, y, c 1. Sources adjust their rate to paths price (queueing deay) 2. Source seect the cheapest path or routing traic 3. MAC ayer schedue transmission aong competing inks Fow comp. x1(p),y1(p) Fow comp. xn(p),yn(p) MAC ayer Compute c(p) Communicate x, y, c 1. Sources, Routing and MAC behave according to 1 above 3. Compute p(k+1) 1. Link queues enght changes as consequence o the new traic rate/routes 4. Communicate p(k+1) 1. Sources measures the new path prices deays 30 Dua Function D(p) Dua Function D(p) D(p)=max L(x,y,c,a,p)= x,y,c,a =max (U(x) - py)+ pc x,y,c,a Fow rate variabes x y Link rates c y= y, y 0 n c= C ar,, a=1, a 0 31 D(p)=max L(x,y,c,a,p)= x,y,c,a = D(p) + DMAC(p) Fow probem D(p)=max U(x) - py x y y= y, y 0 n MAC probem DMAC(p)=max pc c a c= C ar,, a=1, a 0 32

9 Fow Probem max U(x) - py y= y, y 0 n Fow Net Utiity Net utiity maximized when traic is sent aong minimum cost paths pπ minimum cost x(p)=argmax U(x) -pπx=u -1 (pπ) x Y(p) are set accordingy 33 MAC Probem max pc c= C ar,, a=1, a 0 Convex hu o independent sets characteristic vectors soution is the independent set o maxima price c(p)=c i o maxima price; otherwise c(p)=0 34 (Sub)-Gradient Agorithm (Sub)-Gradient Agorithm mpementation terate p(k+1)=p(k)-γ D(p(k)) p [ ] + D(p)= U(x(p))- p( y(p)-c(p)) D(p) p =- y(p)-c(p)=queue i rate [ ] + p(k+1)= p(k)+γ( y(p(k))-c(p(k))) proportiona to ink queue enght 35 Sotted time Sot duration mutipe o packet transmission time Sot k: Given p(k) Fow sets its rate to x(p(k)), F Routing agorithm adjusts routes according to y(p(k)) Link adjust its transmission rate according to c(p(k)) Link updates its price according to p(k+1)= p(k)+γ( y(p(k))-c(p(k))) [ ] + [ ] + p(k+1)= p(k)+ γ(traic rate traversing -ink transmission rate) p measures ink congestion p ~ ink queue ength Network converges to (socia) optima operating point 36

10 MAC agorithm Routing Agorithm Compute Link Transmission Rate during sot k ndependent set price: p= p Schedue or transmission inks which beong to the independent set o maxima price c=c i o maxima price; otherwise c=0 No exact distributed agorithm exists or genera graph Distributed agorithm exists or perect graphs Requires knowedge o nodes within 2-hops Becomes an approximation or genera graphs More Research is needed Traic is routed aong minimum cost paths!!! Link price p as the ink metric Mutipe paths can be determined by means o a suitabe mutipe path routing protocos Congestion Contro Agorithm To Summarize Fow adjusts its rate according to the end-to-end path price minimum cost path π=1,2,n End-to-end path price pπ=σpj x=u -1 (pπ) og utiity -> x=1/pπ Path prices proportiona to end-to-end deay 39 Price p associated to ink Evoves according to aw o suppy and demand ncreases when traic exceeds ink rate, decrease otherwise Measures ink congestion MAC ayer: schedue the (non interering) inks with maxima aggregate price Routing: minimum cost path routing with ink price as metric Congestion Contro: Sources adjust rates according to end-to-end price (congestion) Price Rate Price Rate 40

11 Exampe 6 nodes CSMA/CA-ike contention ony (1,2) and (5,6) can transmit at the same time C=10Mbps 1 ow: 1->6, og utiity Source Rate over Time Topoogy Average Link Rate Exampe MAC agorithm Link Scheduing Topoogy C/ Concusions Layers as Optimization Decomposition Mode or joint congestion contro, routing and MAC or ad hoc wireess s Utiity maximization probem with routing and ink access constraints Soution via dua decomposition which transates into a distributed cross-ayer scheme Much work yet to be done to make this practica!! 43