Adaptive Graph Partitioning Wireless Protocol S. L. Ng 1, P. M. Geethakumari 1, S. Zhou 2, and W. J. Dewar 1 1

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1 Adaptive Graph Partitioig Wireless Protocol S. L. Ng 1, P. M. Geethakumari 1, S. Zhou 2, ad W. J. Dewar 1 1 School of Electrical Egieerig Uiversity of New South Wales, Australia 2 Divisio of Radiophysics CSIRO, Australia ABSTRACT: We propose a ew wireless protocol, called the Adaptive Graph Partitioig Wireless Protocol, that dyamically partitios odes ito the available logical chaels i such a maer so as to improve performace by balacig the itra-logical chael traffic betwee the available logical chaels, ad miimisig the iter-logical chael traffic. We have simulated the performace of a simplified model of the system, at a istat i time, based o the followig parameters: odal load, umber of chaels, ode buffer size, cetral etity buffer size, chael locality factor, ad packet trasmissio probability. The simulatios show the beefits of graph partitioig: whe the chael loads are balaced ad the chael locality factor is high, performace is sigificatly improved. We have also developed aalysis for this simplified model of the system which serves as a upper boud calculatio for packet loss i the system. 1. INTRODUCTION Wireless Local Area Networks (LANs) will exted existig wired arragemets, providig uparalleled coveiece ad mobility i the future [1]. A wireless LAN typically supports a limited umber of users i a well-defied idoor area [2]. These wireless data systems are desiged for packetswitched ( asychroous ) rather tha circuitswitched ( isochroous ) operatios, ad provide high-data rate commuicatios over a small area. Wireless LANs are primarily targeted for data rates geerally above 1 Mbps ad i-buildig applicatios, ad are preferable to their wired couterparts i situatios where wirig is difficult or impractical, or some degree of mobility is required [3]. The existig techologies for wireless LAN s iclude licesed cellular systems operatig at GHz, ulicesed spread-spectrum systems operatig i ISM bads, ad diffused ad directed-beam ifrared (IR) systems [2]. There are curretly a umber of products available such as FreePort, AT&T s WaveLAN, ad Motorola s ALTAIR. FreePort provides a wireless Etheret (IEEE 802.3) hub ad uses direct sequece spreadig, WaveLAN provides peer-to-peer commuicatio ad uses direct sequece spreadig with a Carrier Sese Multiple Access/Collisio Avoidace (CSMA/CA) protocol, ad ALTAIR uses the Etheret protocol [3]. Stadards are beig developed for wireless LANs, uder IEEE i the Uited States ad ETSI/RES10 i Europe (kow as HIPERLAN for High Performace Radio LAN). There are a umber of similarities betwee the IEEE ad the HIPERLAN work. Both stadards are iteded for rates exceedig 1 Mbps, ad will support architectures with ifrastructure as well as other ad hoc architectures, whereby termials commuicate directly with each other (peer-to-peer) without the mediatio of a fixed base statio. Poit-to-poit, poit-to-multipoit, ad broadcast services will be supported. As it is also aticipated that may termials will be battery-powered, the stadards will icorporate a sleep mode for power maagemet. Furthermore, the HIPERLAN stadard is plaed to iclude provisios for a flexible forwardig mechaism, to exted the effective rage of termials. The odes are subdivided ito forwardig ad o-forwardig odes [3]. The medium i a wireless eviromet is shared amogst all mobile hosts. Therefore a most importat issue i wireless commuicatios is how to cotrol access to the shared medium. There are two mai techiques for cotrollig access to the medium: prioritisatio ad reservatio. The HIPERLAN medium access cotrol (MAC) [4] uses prioritisatio, ad IEEE [5] ad ALTAIR [6] use reservatio. The performace of the MAC protocol depeds greatly o the traffic load i the chael. Whe the chael is heavily loaded, performace deteriorates. Noe of the above wireless MAC protocols implemets a scheme to optimise performace. I this paper, we propose a wireless protocol called the adaptive graph partitioig wireless protocol to optimise performace i a wireless LAN by usig adaptive graph partitioig. 2. ADAPTIVE GRAPH PARTITIONING WIRELESS PROTOCOL This protocol defies ode (termial) partitioig ad commuicatio suitable for a wireless LAN eviromet, usig frequecies i the microwave or

2 radio wave rage. The wireless eviromet cosists of multiple odes, each capable of trasmittig ad receivig iformatio o multiple logical chaels, however oly o oe assiged logical chael at ay oe time. Whe the umber of odes is greater tha the umber of logical chaels, multiple odes will share a commo logical chael to trasmit ad receive iformatio. Allocatio of the logical chael over which each ode operates is performed by a cetral etity, which also relays the iformatio oto the correct logical chael whe the trasmittig ode operates at a differet logical chael to the receivig ode. Thus the cetral etity is capable of trasmittig ad receivig iformatio over all available logical chaels simultaeously, ad has the dual roles of determiig odal logical chael assigmet at regular itervals ad of switchig iformatio to the appropriate logical chael wheever ecessary. Nodal logical chael assigmet is dyamic ad reflects the optimal odal assigmet for the curret traffic patter. Oe beefit of usig the cetral etity is that it shifts the complexity ad processig itelligece from the odes oto the cetral etity. The cetral etity also serves as a commo iterface (gateway) of the wireless eviromet to the exteral world. Whe the source (trasmittig) ode shares the same assiged logical chael as the destiatio (receivig) ode, termed itra-logical chael traffic, the source ode trasmits the iformatio o its assiged logical chael, which the destiatio ode will immediately accept. Whe the source ode has a differet assiged logical chael to the destiatio ode, termed iter-logical chael traffic, the source ode trasmits the iformatio o its assiged logical chael, which the cetral etity will receive ad subsequetly trasmit o the logical chael of the destiatio ode, which the destiatio ode will immediately accept. Logical chael ode allocatio (the partitioig of the odes ito the available logical chaels) is determied i such a maer so as to improve performace by balacig the itra-logical chael traffic betwee the available logical chaels (load balacig), ad miimisig the iter-logical chael traffic. 3. GRAPH PARTITIONING I graph partitioig problems, the aim is to partitio the odes for a optimised cost fuctio i a give graph. Graph partitioig has bee applied successfully i umerous applicatios such as i the desig of VLSI chips to fid the optimum umber of compoets i a module ad the umber of coectios betwee modules, routig i distributed systems to subdivide the computer etwork ito smaller clusters to optimise the cotrol overhead, image processig i computer visio to represet the image segmets to optimise the weighted edge betwee two segmets represetig the relatioship betwee the segmets of image, virtual memory pagig to optimise the umber of refereces betwee objects distributed i memory pages, ad mappig parallel algorithms o parallel architectures. Hill climbig ad simulated aealig algorithms are the two most widely used optimisatio techiques. The disadvatage of hill climbig is that it is suitable oly for fidig the global miimum i covex spaces. Simulated aealig overcomes this disadvatage at the expese of valuable computatio time. Graph partitioig problems ca also be solved usig heuristic methods. However oly approximate solutios may be obtaied, although the solutios are always obtaied i a reasoable amout of time. There are two differet approaches. Oe ca either use geeral purpose optimisatio algorithms idepedet of a give applicatio or adopt a heuristic approach desiged specifically for a uique problem. The mathematical formulatio of the graph partitioig problem is outlied i [7]. Give : - a udirected graph G = (V,E); - a applicatio Ω 1 from V ito Z +, such that Ω 1 (v i ) = w 1i is the weight of vertex v i ; - a applicatio Ω 2 from E ito Z +, such that Ω 2 (e j ) = w 2j is the weight of edge e j ; - ad a set of umerical costraits Φ = {φ 1, φ 2,..., φ m } o these weights; The graph partitioig problem has to fid a partitio Π of V (Π = {π 1, π 2,...,π } ) satisfyig the costraits Φ. A classical ad well studied set Φ1 of costraits expresses that: - for each sub-set π i of V belogig to the partitio Π, the sum of the weights of its vertices must be iferior to a give value B π Π, Ω ( v) B i v πi 1 - the sum of the weights of edges goig from oe ode of π i to oe ode of some other π j must be iferior to some give value C

3 e ε Ω 2 ( e) C with ε = x, y x, y E& x π & y π & i j {( ) ( ) i j } chael iter-chael traffic switch chael The graph partitioig problem uder costraits Φ1 has bee proved NP-complete. Most applicatios correspod to the followig set Φ2 of costraits where the weights of all odes are set to 1: - for each sub-set π i of V belogig to the partitio Π, the umber of odes i π i is equal to a give value B i π Π, Ω ( v) = i v πi with v V, Ω 1 ( v) = 1 - the total cost of the edges goig from oe π i to aother π j should be miimum MIN e ε Ω 2 1 ( e) The graph partitioig problem uder costraits Φ2 has also bee proved NP-complete. Applicatios such as the mappig of parallel programs o parallel architectures follow the set Φ3 of costraits: -miimise the sum of the total commuicatio costs betwee processors (total cost of the edges goig from oe π i to aother π j ) ad of the variace of the loads of the differet processors (variace of cost of vertices belogig to a give π i ): MIN Ω ( e) 2 + ( K ( e ε πi π v B i 2 Ω ( v) 1 πi Π v V 2 Ω1( v) )) Π With K = 0 the set of costraits Φ3 reduces to Φ2. This proves that the partitioig problem uder costraits Φ3 is NP-complete. For the mappig problem, K is the weight of the cotributio of the commuicatio cost relative to the computatioal load balace across the system. Choosig a suitable value for K depeds o kowledge about characteristics of the applicatio. itra-chael traffic Figure 1. Itra- ad Iter-Chael Traffic. For the wireless applicatio proposed i this paper, graph partitioig is used to miimise the iterchael traffic ad balace the loads betwee the chaels, i order to optimise performace. The graph partitioig cocept has also bee successfully used i persoal commuicatio etworks [8]. Here, graph partitioig is used to assig cells to the switches of PCS etwork i a optimum maer. I this problem the space is divided ito cells ad graph partitioig is used to assig these cells to the switches of PCS etwork i a optimum way. The two types of costs addressed i this problem are the cost of hadoffs betwee cells ad the cost of cablig betwee the cell site ad its associated switch. The costrait is the cell volume that ca be hadled by a switch. Three heuristic algorithms are discussed to fid the optimum solutio. Graph partitioig has also bee employed i [9] to obtai balaced schedulig of air space i a air traffic cotrol system. A geetic algorithm approach is used to solve this problem. 4. NETWORK MODEL I our simulatios ad aalysis we assume that there are c chaels ad x odes i each chael. Each ode has a trasmissio buffer of b packets. All iter-chael traffic successfully trasmitted is trasfered ito the appropriate chael queue of a output buffered switch with a queue size of b s packets for each chael. I a practical implemetatio, a shared buffer would be employed with memory maagemet of the virtual chael buffers. Cotetio i each chael is based o the slotted ALOHA protocol [2]. I slotted ALOHA, users sychroise their trasmissio times to uiformly spaced timeslots. If more tha oe buffer attempts to trasmit i a time slot the o packets are successfully set. We defie parameter L to be the locality factor, the probability that a packet geerated by a ode is destied for aother ode i the same chael. All packet destiatios for itrachael traffic withi the same chael, ad itrachael traffic are radomly distributed.

4 5. ANALYSIS We defie the followig parameters: p as the load ito each ode. p as the load out of each ode. p s as the load out of each switch buffer. p c as the probability that a chael seds a packet to a particular switch buffer. l as the packet loss of each ode. l s as the packet loss of each switch buffer. t as the probability that a packet tries to trasmit a packet i the chael. s as the probability that a ode successfully trasmits a packet. s s as the probability that a switch buffer successfully trasmits a packet. Uiform Load Algorithm: Iterate util parameters reach their steady state values s = ( 1- Pr( b = 0) ) t (( 1- Pr( b = )) ( t) ( b Pr = 0) ) (( 1- Pr( b ))( t) s = Pr( bs = 0) ) [ l, p] = queue ( p,1, b, s ) ss = ( 1- Pr( bs = 0) ) t (( 1- Pr( b = 0) ) ( 1 t) + Pr( b = 0 )) xp s ( 1 L) p = c c 1 l, p = queue p, c - 1, b, s [ ] ( ) s s c s s ed Total probability of packet loss = cxl + cl s Total probability of packet loss i switch = cl s The fuctio [ l, o] ( p i b, s) x 1 = queue,, represets the shared buffer of a multiplexer with i iputs which is modelled usig a discrete time oe dimesioal Markov chai. Based o the kowledge of the load (p), the umber of simultaeous iputs (i), the umber of buffers i the queue (b), ad the probability that the buffer trasmits a packet successfully (s), the probability of packet loss (l) ad offered load (o) parameters ca be calculated. The uiform load algorithm ca be easily exteded to model o-uiform loads withi each of the chaels. x 6. RESULTS We observe that aalysis achieves good correlatio with simulatio at high loads. Results i Figure 2 show that, icreasig the locality decreases packet loss, icreasig the umber of odes per chael icreases packet loss, icreasig the ode buffer size decreases overall packet loss but icreases packet loss i the switch, ad icreasig the switch buffer size decreases packet loss i the switch. Other results show i Table 1 idicate that icreasig the umber of chaels decreases packet loss, ubalaced loads betwee the chaels icreases packet loss, ad the effect buffer size ad the trasmissio probability have o packet loss. 7. CONCLUSION We have simulated the performace of a simplified model of the system, at a istat i time, based o the followig parameters: odal load, umber of chaels, ode buffer size, cetral etity buffer size, chael locality factor, ad packet trasmissio probability. As oe would expect, the simulatios show that the packet loss icreases with the load offered ad the umber of odes per logical chael. We also observe that the packet loss decreases as the ode buffer size ad the cetral etity buffer size icreases. Most importatly, the simulatios show the beefits of graph partitioig, that is whe the chael loads are balaced ad the chael locality factor is high, performace is sigificatly improved. We have also developed aalysis for this simplified model of the system, however we have foud that it is accurate oly i certai situatios. Nevertheless the aalysis provides a useful upper boud calculatio for packet loss i the system. 8. REFERENCES [1] Wireless Network Commuicatios: Cocepts, Techology, ad Implemetatio. Bud Bates, McGraw Hill, [2] K. Pahlava, A. H. Levesque, Wireless Data Commuicatios, Proceedigs of the IEEE, Vol. 82, No. 9, pp , Sept [3] J. E. Padgett, C. G. Guthuer, ad T. Hattori, Overview of Wireless Persoal Commuicatio, IEEE Commuicatios Magazie, pp.28-41, Ja [4] ETSI, Radio Equipmet ad Systems (RES); High Performace Radio Local Area Network (HIPERLAN); Fuctioal Specificatio, PT 41 & RES 10, Ja [5] K. S. Nataraja, C. C. Huag, D. F. Batz, Medium Access Cotrol Protocol for Wireless LANs (A Update), IEEE P802.11/92-39, Mar

5 p = 0.05, b = 10, t = 0.15, x = 4 p = 0.05, b = 10, L = 0.8, t = Locality Nodes p = 0.1, b = 10, t = 0.15, x = p = 0.1, b = 10, L = 0.8, t = 0.15, Locality Nodes p = 0.1, b = 10, L = 0.8, x = b = 10, L = 0.8, t = 0.15, x = Trasmissio Probability Load p = 0.1, b s = 10, L = 0.8, t = 0.15, x = 4 p = 0.1, b = 10, L = 0.8, t = 0.15, x = Node Buffers Switch Buffers Figure 2 Aalysis ad Simulatio Results for a 3 Chael System.

6 Simulatio Total Probability Total Probability of p b b s L t x c of Packet Loss Switch Packet Loss E-3 3.4E E-5 1.6E E-7 4.0E E-3 7.0E E-6 4.0E E-5 2.4E E-5 2.0E E-4 2.0E-6 [6] D. Buchholz, P. Odlyzko, M. Taylor, R. White, Wireless I-Buildig Network Architecture ad Protocols, IEEE Network Magazie, pp.31-38, Nov [7] E. G. Talbi ad P. Bessiere, A Parallel Geetic Algorithm for the Graph Partitioig Problem, ftp://archive.cis.ohio-state.edu/pub/ europrose/bessiere.acm-ics91.ps.z [8] A. Merchat ad B. Segupta, Multiway Graph Partitioig With Applicatio to PCS Networks, Proc. IEEE INFOCOM 94, pp [9] D. Delahaya et. al., Geetic Algorithms for Partitioig Air Space, Proc. IEEE 10th Coferece o Artificial Itelligece for Applicatios 94, pp Table 1. Simulatio Results Ubalaced load simulatio. The three chaels have a load of 0.04, 0.05, ad 0.06 respectively.