Energy Efficient Transmission Probability and Power Control in Random Access Networks

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1 Eergy Effcet Trasmsso Probablty ad Power Cotrol Radom Access Networks Amrmahd Khodaa Electrcal Egeerg Departmet Sharf Uversty of Techology Tehra, Ira Abstract I ths paper we cosder the ssue of eergy effcecy radom access etworks ad show that optmzg trasmsso probabltes of odes ca ehace etwork performace terms of eergy cosumpto ad faress. Frst, we propose a heurstc power cotrol method that mproves throughput, ad the we model the Utlty Costraed Eergy mzato (UCE) problem whch the utlty costrat takes to accout sgle ad mult ode performace. UCE s modeled as a covex optmzato problem ad Sequetal Quadratc Programmg (SQP) s used to fd optmal trasmsso probabltes. Numercal results show that our method ca acheve faress, reduce eergy cosumpto ad ehace lfetme of such etworks. Keywords-eergy effcecy; slotted Aloha; utlty fucto; covex optmzato I. INTRODUCTION Eergy effcecy of wreless etworks has receved cosderable atteto recet years. Especally, sesor etworks, odes are ot rechargeable ad lfetme of the ode s equal to ts battery lfetme. It s therefore ecessary to avod ay waste of eergy ad esure etwork logevty. O the other had, protocols of these etworks should be smple ad dstrbuted ad thus, radom access protocols are frequetly used such etworks. Slotted Aloha [] s the basc ad most studed radom access whch odes trasmt ther packets wth certa probablty every tme slot. Usually collso model s used to aalyze performace of the protocol []. I ths model, t s assumed that whe two packets arrve at the recever collso occurs ad oe of them ca be decoded wth eglgble error. However, ths model s pessmstc ad provdes a lower boud o system performace, sce may cases the packet wth largest power ca capture the chael ad be receved correctly. Several works (e.g. [2], [3], [4]) have studed the capture effect ad used power cotrol to tesfy t. These schemes usually suppress the weak sgals ad ehace the strog oes order to crease throughput of the etwork. However, t s clear that such a approach wll result ufaress amog odes. For example, a algorthm proposed by etzer[2] reduces trasmtter power of far odes order to ehace throughput but wll smultaeously dmsh success probabltes of such odes. I cotrast, f the power cotrol algorthm tres to acheve faress amog odes by mprovg Babak H. Khalaj Electrcal Egeerg Departmet Sharf Uversty of Techology Tehra, Ira khalaj@sharf.edu weak sgals, capture probablty ad etwork throughput wll be sacrfced. Geeral aalytc framework for faress mult-access wreless chaels was frst proposed [5] where t was show that defg faress these chaels s equvalet to specfyg a utlty fucto ad the logarthmc utlty was used to provde proportoal faress. If we force the utlty fucto to be greater tha a threshold, we ca expect a level of faress amog odes addto to acceptable etwork throughput. Faress has bee also addressed wreless sesor etworks order to esure that data s collected from dfferet regos of the etwork [6]. I ths paper, we use power cotrol ad optmze trasmsso probabltes order to mmze eergy cosumpto of the etwork ad provso faress. Our heurstc power cotrol scheme creases capture probablty wthout affectg faress, ad the computed optmal trasmsso probabltes guaratee faress ad maxmze eergy effcecy of the etwork. ost of the earler studes o radom access have oly focused o etwork throughput ad few of them have cosdered faress. I addto, to the best kowledge of authors, our work s the frst oe that mmzes eergy whle esurg faress amog etwork odes. I [5], [7], [8], ad [9], faress s extesvely studed but oe of them evaluated eergy cosumpto of odes. The mult-group model s used [7] ad a retrasmsso cotrol polcy that ehaces faress s suggested, although the optmalty of the algorthm was ot prove. I [8] ad [9], optmal trasmsso probabltes were foud to acheve faress but oly collso model was used ad eergy cosumpto was gored. Eergy effcecy of the etwork was cosdered [3], [0], ad [], however, they have maly vestgated throughput-eergy tradeoff Aloha etworks wthout takg faress to accout. The structure of ths paper s as follows. I secto II we troduce system model ad ma assumptos of the work. Our power cotrol ad ode classfcato method s descrbed secto III. We formulate our optmzato problem as a fucto of trasmsso probabltes secto IV ad prove ts covexty. I secto V we propose a method for reducg messages set from base stato to odes ad preset the fal algorthm. Numercal results ad coclusos are gve sectos VI ad VII, respectvely.

2 II. SYSTE DISCRIPTION We cosder a system whch a fte umber of odes desre to trasmt ther packets to a Base Stato (BS) (Fg. ). The odes use slotted Aloha whch chael s dvded to tmeslots wth durato T whch s equal to the tme requred to sed a packet. As we wll dscuss secto III, odes are dvded to groups, there are odes group ad ode j of group s deoted by (, j). Each ode trasmts oe slot wth probablty q (called trasmsso probablty) ad the amout of eergy t uses to trasmt a packet s E. Smlar to [0], we suppose that trasmsso ad retrasmsso rates are the same for all odes. It s also assumed that BS estmates chael gas (G ) from receved packets. Thus, the optmzato algorthm recever corporates chael state formato of the curret slot. Ths mples depedecy of the algorthm from dstrbuto of odes ad statstcal characterstcs of chael. A feedback chael s assumed to exst from BS to odes, ad s used for ackowledgmet, sychrozato ad cotrollg trasmsso parameters of odes. It s assumed that the etwork chages very slowly, ad coherece tme of chaels s large. Therefore, updatg the ode parameters does ot take place frequetly ad the effect of ths feedback o total eergy cosumpto ad throughput s eglgble. We have also assumed that whe there s o power cotrol, all odes use the same power, P, to trasmt a packet. I ths work, we do ot take stablty ad delay ssues to accout ad assume that they are cotrolled by settg approprate source rates at hgher etwork layers. III. CAPTRE EFFECT AND POWER CONTROL A. Perfect Capture odel If sgal to terferece ad ose rato (SINR) of a receved packet slotted Aloha etwork s above a certa threshold ad approprate codg s used, relable commucato s possble [2]. If we deote terferece ad ose terms by I ad N respectvely, ad use fxed type of modulato ad codg, the packet wth power P ca be receved successfully f: PG > β () I + N where G s the chael pathga ad β s SINR threshold. It meas that a packet wll capture the chael f ts SINR s above a specfc threshold. Nose term s eglected our aalyss sce terferece amog odes s the most mportat factor mult-access wreless etworks. Also, smlar to [2], we assume that a packet would be receved correctly f ad oly f t has the strogest power amog all of the receved packets. Ths s called perfect capture model ad s close to SINR threshold model for our system because: Our power cotrol (whch we wll subsequetly descrbe ths secto) classfes odes ad esures that receved power from odes of dfferet classes s qute dfferet. Sce we are lookg for a eergy effcet system, trasmsso probabltes should be set small eough 20 0 Nodes order to avod usuccessful trasmssos. Therefore, t s less probable that sum of the terfereces caused by packets wth low power exceeds the strogest packet. I addto, umercal results of secto V cofrm ths clam ad show that performace of our system wth perfect capture model s close to the case that we used SINR threshold model. We should sst here that the assumpto of perfect capture model s hghly related to power cotrol algorthm, ad s ot ecessarly applcable geeral. B. Power Cotrol Our power cotrol algorthm frst classfes odes to groups accordg to ther chael gas. To ths ed, two thresholds are assged for the chael gas of odes each group. For example, the ode (, j) s group f ts ga G satsfes: G < G G + (2) Power cotrol algorthm, (3), esures that packets of odes oe group are receved wth the same power at BS. G d 2 P = P (3) G I order to explot capture effectvely, ad crease probablty of successful trasmsso for odes wth hgher chael ga, we should set proper threshold levels. We set thresholds as (4) to make certa that whe two packets are trasmtted smultaeously from group ad k (<k) the packet from group wll be receved correctly: G G = β (4) As a example, suppose that odes are dstrbuted uformly a crcle ad chael gas deped oly o the dstace of odes from BS (Fg. ). For ths etwork, ga thresholds are equvalet to dstace thresholds ad group cossts of odes that ther dstaces to BS satsfes d <d<d +. BS Fgure. A typcal etwork wth 50 odes ad 20m radus d

3 IV. OPTIAL TRANSISSION PROBABILITIES Utlty Costraed Eergy mzato (UCE) problem ca be formulated as follows: m s. t. = j= E U > U c 0 q where q ad E =P q are trasmsso probablty ad average eergy cosumpto of the ode (,j), U c s the threshold for acceptable value of utlty fucto ad s umber of odes group. I order to solve ths problem, t s assumed that power of each ode s determed earler by the power cotrol algorthm. Frst, we dscuss the utlty fucto ad fd ts maxmum value U max. Smlar to some related work [5] ad [8], we use the logarthmc utlty fucto. I other words f x represets effectve rate of ode (,j), the utlty fucto of a group etwork s gve by: U = = j= (5) log( x ) (6) Sce log(x) goes to egatve fty as x goes toward zero, a fte value for ths utlty fucto makes sure that effectve rates of all odes are above zero. It s also show [5] that logarthmc utlty results proportoal faress amog odes. If we deote throughput of odes by S the: log( x ) = log( L R ) + log( S ) (7) where L s umber of bts a packet ad Rp=/T s packet trasmsso rate. Thus, utlty fucto ca be rewrtte as: U = U + N log( L R ) (8) U = = j= p p log( S ) (9) Sce N log(l.rp) s costat, the rest of paper we use U' stead of U order to ether maxmze utlty or set a threshold for t. Accordg to perfect capture model a packet set by a ode group s successfully receved f ad oly f o other ode groups to has set a packet. Therefore, throughput of ths ode s: S l = q ( qlk ) ( qk ) (0) l= k= k= k j ad U' s gve by: q l = = = = U = log( ) + log( qlk ) () j q l k Rearragg terms we have: q = = = U = log( ) + k log( q ) (2) j q k If we show teral terms of the above summato as φ (.) we have: U = = j= ϕ ( q ) (3) φ (.) s a cocave fucto due to the fact that (4) s ot postve whe there s at least oe ode group. 2 2 q 2 + q k 2q ϕ k= (4) = q 2 ( q ) 2 Thus, U' s cocave because t s the sum of cocave fuctos [3]. Ths result mples covexty of UCE problem sce the objectve of ths problem s a lear fucto of trasmsso probabltes. There exsts a varety of methods to solve covex optmzato problems. Amog them we have chose Sequetal Quadratc Programmg (SQP) [4] for the reaso that our varables are bouded ad the tal guess s ot far from the optmal soluto. I order to fd a rage of U c values that makes UCE problem feasble, we cosder the Utlty axmzato (U) problem: max s. t. U 0 q (5) As we proved earler, U s a cocave fucto ad thus ts maxmum value s reached at ts stable pot. Therefore, trasmsso probabltes that maxmze utlty are gve by: q U = k = ad maxmum utlty ca be calculated as: max U U = U ( q ) k (6) (7) If we set U c <U max, the UCE problem becomes feasble ad optmum values of trasmsso probablty ca be foud. It s terestg to observe that there s o lower boud for U th.

4 V. PROPOSED ALGORITH A. Reducg esseges Accordg to the schemes preseted sectos III ad IV, BS ca calculate both trasmsso powers ad optmal trasmsso probabltes of odes ad broadcast these parameters, however, t s mportat to reduce the overhead of the messages set by BS because they should be broadcasted every tme that etwork topology chages. Equvaletly, reducg messages (or message sze) wll eable us to use ths algorthm etworks that may vary more frequetly. Frst, we show that power cotrol algorthm ca work a dstrbuted maer throughout the etwork. If we assume chaels are symmetrc, odes ca estmate ther chael gas wth beacos trasmtted by BS. Therefore, they ca use threshold levels (whch are set to them at system start-up) order to dcate whch group they belog to, ad cosequetly set ther power levels. I ths case o ole feedback s eeded from BS. We ca also reduce messages set by BS for settg trasmsso probabltes at the cost of makg some calculatos odes. We wll show subsequetly that f BS oly broadcasts the umber of odes each group ad Lagrage multpler the every ode ca calculate ts uque optmal trasmsso probablty. The Lagraga assocated wth UCE problem s: ( Pq ) L( q, λ ) = λu + λϕ (8) c = j= By settg q L = 0 ad usg some algebrac mapulato we wll get the followg quadratc equato whch should be solved at each ode: 2 + q P q P λ k + λ = 0 (9) k= Sce Uc s cocave fucto of q Lagraga s strctly cocave ad thus, t has a uque maxmum. It s also clear that (9) has at least oe root (0, ) because ts rght had sde (r.h.s) has the followg property: r.h.s 0 whe q = ad r.h.s 0 whe q =0. B. Algorthm I summary, our algorthm that specfes power ad trasmsso probabltes cossts of the followg steps: Step. At the system startup, BS specfes chael ga thresholds accordg to SINR threshold ad seds them to the odes. Step 2. Both BS ad odes estmate chael gas ad f t was dfferet wth prevous values they classfy odes wth respect to chael ga thresholds. Step 3. BS evaluates power values of all odes ad tate step 4. Every ode sets ts power. TABLE I. Packet Legth (L) Tme Slot (T ) SINR Threshold (β) Nose Power Spectral Desty SIULATION PARAETERS 000 bt 5 msec Step 4. BS solves UCE problem ad determes lagraga multpler. BS the broadcasts umber of odes each group ad Lagraga multpler to the odes. Step 5. Nodes set ther trasmsso probabltes by solvg (9) ad choosg the root whch s (0,). VI. NUERICAL RESULTS We have appled our algorthm to the sample etwork llustrated Fg., ad explored eergy-utlty tradeoff, comparg our method wth the case that all odes have same trasmsso probabltes (Hereafter we wll call the latter uform polcy). Parameters used for our umercal aalyss are gve table. Fg. 2 shows effectve rate of odes acheved by our algorthm. I ths fgure, odes are sorted accordg to ther dstace to BS. It ca be see that umercal results calculated by the perfect capture model are close to the results of SINR model. Our algorthm s also compared wth the uform polcy ad t s show that whe the same amout of utlty s acheved ear ad far odes our algorthm have less throughput dffereces tha ear ad far odes of the uform polcy. I order to aalyze the tradeoff betwee utlty ad eergy, we have solved UCE problem ad calculated the mmum amout of eergy cosumed for dfferet utlty costrat values. Accordg to Fg.3a, eergy cosumpto s very sestve to U c ad wth small varato of ths threshold mmum requred eergy wll be reduced to half. Eergy effcecy of our algorthm s compared wth uform polcy Fg.3b ad t s observed that our algorthm reduces eergy cosumpto by about 0% for all values of U c. We should ote that accordg to (6) uform polcy ca ever acheve maxmum possble utlty for ay value of eergy. 6 db -4 W/Hz 0 Chael Ga a (G ) 20 d -4 Node tal power (P) Network Radus (R) 200 mwatt 20 m Number of odes(n) 50 Battery Eergy (E B ) 000 Jouls Utlty Costrat (U c ) 29 a. For smplcty we have assumed that chael gas deped oly o the dstace to BS.

5 Effectve datarates (kbt/sec) Uform Polcy Optmal Probablts wth Perfect Capture odel Optmal Probablts wth SINR odel Network Lfetme (hours) Nodes Fgure 2. Effectve data rates of odes a sample etwork (Utlty/axmum Utlty) (a) 4% 0 2% Eergy Cosumpto (dbm) dB (Utlty/axmum Utlty) (a) Lfetme Ehacemet Rato 0% 8 % 6 % 4 % 2 % 0 % (Utlty/ axmum Utlty) (b) Fgure 4. (a) Lfetme-utlty tradeoff (b) lfetme elogato of optmal trasmsso probabltes comparso wth uform polcy Eergy Reducto Rato 6% 4% 2% 0% 8% 6% 4% (Utlty/axmum Utlty) (b) Fgure 3. (a) Eergy-utlty tradeoff (b) Eergy reducto of optmal trasmsso probabltes comparso wth uform polcy It s also of mportace to exame performace of our algorthm terms of lfetme (Fgs 4). I order to do so, we defe lfetme of the etwork as the tme whe 70% of odes ru out of eergy. As we expected, the etwork wll have greater lfetme for smaller values of U c (Fg4a). It s also apparet that our algorthm creases etwork lfetme comparso wth uform polcy (Fg4b). VII. CONCLUDING REARKS We preseted a ovel algorthm to ehace eergy effcecy ad guaratee faress radom access etworks. Based o smulato results, t has bee verfed that the proposed smple algorthm ca reduce eergy cosumpto of the etwork, ehace faress amog odes, ad crease etwork lfetme. Although our algorthm has better lfetme characterstcs tha uform polcy, oe valuable exteso wll be to optmze trasmsso parameters order to drectly maxmze etwork

6 lfetme. I order to do so, oly step 4 of our algorthm wll be chaged ad utlty costraed lfetme elogato wll be solved stead of UCE. Aother approach s to trasform algorthm to a dstrbuted structure. The results of secto IV mply that ay dstrbuted algorthm wth reasoable amout of messages wll be suboptmal sce power of all other odes ad umber of odes each group should be kow order to acheve optmal values. REFERENCES [] Norma Abramso, The throughput of packet broadcastg chaels IEEE Tras. O Comm., Vol 25, No, pp 7-28, Ja [2] J. J etzer, O mprovg utlzato ALOHA etworks, IEEE Trasacto o Comm., Vol. 24, Issue 4, pp , Apr [3] F. Berggre, J. Zader, Throughput ad eergy cosumpto tradeoffs pathga based costraed power cotrol Aloha etworks IEEE Commucato Letters, Vol 4, No 9, pp , Sept [4] X. Q, R. Berry, Explotg ultuser Dversty for edum Access Cotrol I Wreless Networks, Proc. IEEE Ifocom '03, pp , Apr [5] T. Nadagopal, T. E. Km, X. Gao, V. Bharghava, Achevg AC layer faress wreless packet etworks I Proc. AC/IEEE obcom 00, pp , Dallas, USA, Oct. 998 [6] A. Woo, D. Culler. A trasmsso cotrol scheme for meda access sesor etworks. I Proc. obcom 0, Rome, Italy, pp july 200. [7] A.Slvester, T.-K.Lu, A.Polydoros. Retrasmsso cotrol ad faress ssue moble slotted ALOHA etworks wth fadg ad earfar effect. oble Networks ad Applcatos, pp. 0-0, Jue 997. [8] K. Kar, S. Sarkar, ad L. Tassulas, Achevg proportoal faress usg local formato Aloha etworks, IEEE Tras. o Automatc Cotrol, vol. 49, o. 0, pp , Oct [9] J. W. Lee,. Chag, R. A. Calderbak, Jotly optmal cogesto ad coteto cotrol wreless ad hoc etworks, IEEE Comm. Letters, vol. 0, o. 3, pp , arch [0] W. L ad H. Da, Optmal Throughput ad Eergy Effcecy for Wreless Sesor Networks: ultple Access ad ult-packet Recepto, Eurasp Joural o Wreless Commucatos ad Networkg, Vol. 5, Issue 4, pp , September [] A. Chockalgam,. Zorz, Eergy Effcecy of eda Access Protocols for oble Data Networks, IEEE Tras. o Commucatos, vol. 46, o., pp , November 998. [2]. edard, J. Huag, A. Goldsmth, S. ey, ad TP Colema, Capacty of Tme-slotted ALOHA Packetzed ultple-access Systems over the AWGN Chael IEEE Tras. O Wreless Comm. Vol. 3 No. 2, pp , arch [3] S. Boyd, L. Vadeberg, Covex Optmzato, Cambrdge Uversty Press, [4] R. Fletcher, Practcal ethods of Optmzato, Wley, 99