DESIGNING TRANSMISSION SCHEDULES FOR WIRELESS AD HOC NETWORKS TO MAXIMIZE NETWORK THROUGHPUT

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1 DESIGNING TRANSMISSION SCHEDULES FOR WIRELESS AD HOC NETWORKS TO MAXIMIZE NETWORK THROUGHPUT Bran J. Wolf, Joseph L. Hammond, and Harlan B. Russell Dept. of Electrcal and Computer Engneerng, Clemson Unversty, Clemson SC ABSTRACT We present a method for maxmzng the throughput of moble ad hoc packet rado networks usng broadcast transmsson schedulng. In such networks, a termnal may become a bottleneck f t s not allocated enough transmsson slots n the current transmsson schedule to handle the traffc flowng through t. Topology nduced bottlenecks may arse frequently n ad hoc networks due to uneven dstrbutons of termnals. Termnals n sparse areas of a network may be requred to forward a large amount of traffc to facltate communcaton between dense areas of the network. We address ths problem by modfyng the broadcast transmsson schedule so that termnals handlng more traffc have more opportuntes to transmt. Frst, we descrbe a theoretcal framework for analyzng the performance of a gven schedule n terms of end-to-end stable throughput; we also defne the upper bound for ths performance. Next, we ntroduce a centralzed algorthm that uses a process smlar to smulated annealng to generate schedules wth near optmal performance. We conduct smulaton studes to show that transmsson schedules produced by the centralzed algorthm offer greatly mproved performance over a smple, collson-free transmsson schedule n terms of end-to-end packet delay, throughput, and completon rate. These studes are performed on a varety of test networks to generalze results and demonstrate the wde applcablty of these prncples. INTRODUCTION Schedulng transmssons n ad hoc networks s an effectve way of provdng network termnals regular access to the channel. Ths s mportant when the network s supportng applcatons whch have qualty-of-servce requrements or when the network has a heavy traffc load. In both cases, random channel-access schemes may perform poorly. Schedulng transmssons n ad hoc networks does not come wthout a cost n overhead, however. Coordnaton s requred among termnals to set up an ntal transmsson schedule. Also, spatal reuse allows some network termnals to share transmsson slots, resultng n a hgh degree of channel utlzaton; however Ths work was supported by the Multdscplnary Unversty Research Intatve (MURI) program admnstered by the Offce of Naval Research under Grant N and by NSF under Grant ANI termnal moblty may result n new nterference as termnals assgned the same slots come nto proxmty. In ths case termnals may have to coordnate n alterng the transmsson schedule to avod the new source of nterference. The basc problem of assgnng non-nterferng transmsson slots to termnals so that the length of the schedule, and hence channel access delay, s mnmzed and the number of transmssons s maxmzed s referred to as the Broadcast Schedulng Problem [9]. Dffcultes of ad hoc networkng envronments, such as termnal moblty, rregular topology, and scalablty have generated a varety of dstrbuted schedulng protocols []. These nclude topology transparent protocols [2][5], protocols n whch termnals contend for slot reservatons [8], [], and dynamc schedulng protocols [], [0]. In certan stuatons, some network termnals may be requred to forward a much larger share of traffc than other termnals. Ths can be thought of n terms of a bottlenecked flow. If a large flow of traffc s drected through a sngle termnal t may begn experencng packet overflow, even when surroundng termnals are not usng all of ther assgned capacty. Such a termnal may be sad to be unstable n the sense that ts packet arrval rate exceeds ts forwardng rate. Ths stuaton may arse from any combnaton of uneven traffc flows, poor routng decsons, and network topology. In these stuatons, the stable network capacty can be greatly ncreased by assgnng transmsson slots to the termnals that need them most. In ths paper we demonstrate how transmsson schedulng can postvely or negatvely affect network capacty, whch we assocate wth a bound on stable endto-end throughput as defned below. The research has three prmary contrbutons. Frst, for a gven network and traffc model, we derve a theoretcal upper bound on stable end-to-end throughput. Second, we present a centralzed algorthm for generatng near-optmum transmsson schedules for a network. These transmsson schedules are near-optmum n the sense they can support traffc rates close to the theoretcal upper bound on stable end-to-end throughput. Fnally, we smulate traffc flows n several networks, and compare results usng a basc transmsson schedule wth those usng the near-optmum schedule. These smulatons demonstrate the advantages of usng schedules that are senstve to traffc load. The centralzed schedulng algorthm also provdes a pont of of 7

2 reference for evaluatng the performance of dstrbuted schedulng protocols. NETWORK MODEL We model the ad hoc network as a unt-dsk graph. In ths model, the network s represented as a graph G consstng of termnals that represent the vertces of G, so that the set of all termnals n the network s V(G). Every network termnal has a unform communcatons range. Two termnals whch are wthn communcatons range share a half-duplex, bdrectonal lnk represented as an edge n the graph, and may transmt and receve data packets wthout error as long as no packet collsons occur. We say two termnals are -neghbors f they are wthn communcatons range. They are 2-neghbors f they are not -neghbors and there exsts a termnal whch s a - neghbor to both. The neghborhood of a termnal conssts of the termnal tself and ts and 2-neghbors. Termnals use omndrectonal antennas to broadcast to all other termnals n range, and we assume the termnals are synchronzed to the slot boundares. A broadcast transmsson s successful f t s receved by all other termnals n range. A broadcast transmsson s not successful f a collson occurs at any elgble recever. Packet collsons may occur n two ways. In the frst nstance, two termnals whch are -neghbors transmt at the same tme, resultng n nterference and preventng the recepton of ether packet at the transmttng termnals. In the second nstance, two termnals whch are 2-neghbors transmt smultaneously, causng a packet collson at all - neghbors they have n common. A collson-free broadcast transmsson schedule can be constructed by assgnng each termnal a color that s unque among other termnals n ts neghborhood, and then assgnng to each color a unque transmsson slot n a transmsson frame. We use a more sophstcated method for assgnng transmsson slots to colors that was developed by Lyu [7]. Each termnal has a transmsson frame wth a length equal to the smallest power of two greater than or equal to the maxmum color number n that termnal s neghborhood. Termnals transmt at least once n each frame, and addtonal slots n the frame are assgned to termnals based on the colorng of the neghborhood. A concse descrpton of Lyu s algorthm and dscusson of ts propertes s gven n [4]. In ths paper, we demonstrate how modfyng slot assgnments can greatly mprove several aspects of network performance. We defne end-to-end throughput, γ, for a network as the total steady-state rate at whch traffc reaches ts destnatons, assumng each termnal generates traffc for every other termnal at an equal generaton rate for each destnaton. The largest value of γ for whch the steadystate traffc arrval rate to each termnal s less than or equal to ts forwardng rate s the maxmum stable end-toend throughput, denoted Γ. The value of Γ s a functon of the forwardng rate of each termnal, the network topology, and the method of routng. We lmt the networks studed to those whch are connected, so a route exsts between each source-destnaton par. Routes are selected that mnmze the number of hops from source to destnaton. If there are multple mnmum-hop routes one s selected at random. Ths ncreases the dversty of routes chosen n the network, and thereby reduces the effects of routngnduced bottlenecks. ANALYTIC RESULTS To derve an expresson for Γ, we proceed as follows. Gven a network topology and routng, for each termnal we defne Λ as the number of source-destnaton paths for whch forwards packets. We defne λ as the average traffc rate on each source-destnaton path. We set the traffc rate for each source-destnaton par to be equal n order to place the focus on network bottlenecks caused by topology, as opposed to partcular traffc patterns. Let N be the number of termnals n the network so that the number of source-destnaton pars s N(N ). The average traffc rate on each source-destnaton path can then be related to γ through the equaton γ λ =. () N( N ) The rate termnal forwards traffc can be expressed as Λ γ λ Λ =. (2) N( N ) Let C equal the number of slots n termnal s transmsson frame, let S represent the number of slots assgned to that termnal n ts frame, and normalze the transmsson rate at each termnal to packet per slot. The effectve transmsson rate assgned to termnal s then S / C packets per slot. A termnal s stable f the average rate of ncomng traffc s less than the termnal s effectve transmsson rate. Thus λλ C <. () S The left hand sde of the nequalty n () s a measure of utlzaton. It s convenent to normalze the utlzaton as follows, Λ C LF( ) =. (4) S te that LF() s equal to the utlzaton of termnal dvded by λ. We refer to LF() as the load factor of termnal, or smply the load factor. Combnng (), (), and (4) gves 2 of 7

3 N( N ) γ <. (5) LF( ) The maxmum stable end-to-end throughput s the largest value of γ such that (5) s satsfed for every termnal. Thus, N( N ) Γ =. (6) Max[ LF( )] From (6) t s evdent that a transmsson schedule whch mnmzes the maxmum load factor wll maxmze the bound on stable end-to-end throughput. It s possble to determne a lower bound on the maxmum load factor of (6) for a partcular topology. To do ths we assume the effectve transmsson rate s any real number, nstead of restrctng t the rato of the number of assgned slots to slots per frame. Wth the effectve transmsson rate defned n ths manner, we can normalze the frame sze C to and express the slots S assgned to each termnal as a fracton of the total frame. We defne a neghborhood clque as a set of termnals for whch every termnal n the set s wthn 2 hops of every other termnal n the set. As a result, no two termnals n a neghborhood clque may transmt at the same tme. We also defne the traffc load sum for a clque as the sum of the Λ over all clque members. w we make two clams. Frst, f we schedule any neghborhood clque ndependently of the rest of the network, the maxmum load factor among members of the clque s mnmzed by assgnng portons of the frame so that all members of the clque have equal load factors. Ths s true snce the termnal n the clque wth the largest load factor can take portons of the frame from other termnals and reduce ts load factor untl t s no longer largest. However, the termnal whch now fnds tself wth the largest load factor can proceed n the same manner to reduce ts load factor and the process can be contnued, resultng n an equal load factor for all termnals n the clque. Second, the largest traffc load sum, taken over all neghborhood clques n the network, s the lower bound on the maxmum load factor n the network. To see ths, frst defne the termnals n a neghborhood clque as set A, and note that the load factors n A are mnmzed by forcng them to be equal. Equatng load factors usng (4), wth C set to, produces the followng relaton Λ Λ j =,, j A. (7) S S j Snce termnals n a clque cannot transmt smultaneously, the sum of the portons of the frame assgned to the ndvdual termnals must be no greater than the total frame length. We utlze the entre frame for assgnments n order to mnmze the load factors, thus, A S =. (8) We solve (7) for S, then substtute the resultng expresson nto (8) whch, wth C set to, yelds LF( j) = Λ, j A. (9) A Snce the load factors for termnals n A are equal, (9) can be wrtten as Max LF( ) = Λ. (0) A [ ] A For any schedule, a neghborhood clque cannot have a smaller maxmum load factor than that equal to the traffc load sum of the clque. By enumeratng all neghborhood clques n the network and fndng the traffc load sum for each clque, the lower bound on the maxmum load factor for the network, for any schedulng, can be dentfed as the largest traffc load sum. The lower bound on the maxmum load factor can be used n (6) to determne the upper bound on Γ. It should be noted that the lower bound on the load factor determned above s not tght for all graphs; for example, the slot assgnments, S, n (8) should account for all of the termnals n s neghborhood whch transmt n separate slots, and all of these termnals may not be ncluded n a neghborhood clque. EXAMPLES OF CREATING SCHEDULES WHICH OPTIMIZE THROUGHPUT To maxmze stable throughput, t s suffcent to mnmze the maxmum load factor over all termnals n the network. In practce load factors can be reduced by assgnng termnals addtonal colors, resultng n addtonal transmsson slots. Usng Lyu s algorthm for slot assgnment, the frame sze s determned by the largest color number n a gven neghborhood. Thus, n order to bound access delay as colors are added, we mpose a constrant on the sze of the largest added color and hence on the frame sze. The basc frame sze s the frame sze resultng from a basc colorng whch mnmzes the number of colors used. It s convenent to specfy the frame bound as a multple of the basc frame sze, for example by requrng the actual frame sze to be no greater than twce the basc frame sze. Although another multple such as 4 or 8 could be used, we set the constrant at twce the basc frame sze unless otherwse stated. Consder the sx node network shown n Fgure along wth a basc colorng and the resultng schedule. Routng s trval snce only one route exsts between each source-destnaton par. Termnal only forwards packets that t generates, and the destnatons of these packets are equally dstrbuted among the other 5 termnals, so Λ = 5; by symmetry, Λ 6 = 5. Termnal 2 must handle the 5 unts of traffc t generates, plus 4 unts representng pars (,), (,4), (,5), and (,6), and 4 more unts representng pars of 7

4 (,), (4,), (5,), (6,), hence Λ 2 = ; also by symmetry, Λ 5 =. In a smlar fashon, Λ and Λ 4 may be calculated as 7. Only colors are requred to color ths network, thus the frame sze for the network s 4. Slot assgnments resultng from use of Lyu s algorthm are shown n the fgure. te that Lyu s algorthm assgns termnals and 4 an extra slot n each frame, thus these termnals have half the load factor of ther symmetrc counterparts. The maxmum load factor s 68, hence the maxmum expected stable throughput, Γ, s 0.44 packets per slot Termnal Color 2 2 Slots 2, 4 2, 4 Λ LF() Fgure. The sx termnal network wth a smple colorng, and the resultng calculatons for Λ and LF(). We next consder how modfyng slot assgnments can reduce the load factors of the termnals. In an extended frame of 8 slots, Lyu s algorthm assgns 2 and 4 transmsson slots to termnals and 4 respectvely. If these termnals are both assgned slots out of 8, ther resultng load factors are 45., and the maxmum load factor n the network s 52 (from termnals 2 and 5). The resultng value of Γ s then packets per slot, an mprovement of more than 0%. One way to mplement the modfed slot assgnment s to assgn termnal an addtonal color number 6. Ths forces the frame sze to be 8 and assgns termnal slots, 5, and 6, whle termnal 4 s assgned slots 2, 4, and 8. te that ncreasng the frame sze provdes a larger set of colors from whch termnals can select an addtonal color number. If the frame sze s not doubled, termnal has only color 4 as an avalable color. If termnal takes on color 4, termnal 4 s load factor ncreases to 68, so there s no overall mprovement. Increasng the frame sze comes at a cost, however, snce some termnals wll experence a greater access delay wth a larger frame. The example llustrates the addtonal concept that by assgnng termnals more than one color we can fne-tune the schedule wth respect to traffc load and reduce the load factors of some termnals, at some cost to neghborng termnals. CENTRALIZED OPTIMIZATION ALGORITHM For a gven network, routng, and frame sze constrant, we wsh to fnd a transmsson schedule that performs as close to the optmum schedule as possble. We defne the optmum schedule as the schedule wth the smallest maxmum load factor over all possble schedules made wth the same constrants. We conjecture that fndng the optmum schedule s NP-complete. An actual proof of NP-completeness s beyond the scope of ths paper. Our strategy for optmzng a transmsson schedule nvolves assgnng termnals addtonal color numbers, whch n turn allows the schedule to be fne-tuned to the network. We use a process smlar to smulated annealng to assgn addtonal colors to termnals. Smulated annealng s a probablstc approach to fndng a globally optmum soluton, and has been used to fnd good solutons to the travelng salesman problem, among others [6]. To set up a smulated annealng process, four components must be defned:. An ntal system confguraton 2. A random process for makng changes or modfcatons to the system. A quanttatve objectve functon to be optmzed 4. An annealng schedule, whch specfes the magntudes of allowed changes and the number of changes allowed at each magntude For our problem, the ntal system confguraton conssts of a network topology, routng, and an ntal colorng. These elements allow us to determne a basc frame sze and the load factor for each termnal. The functon to be optmzed s the maxmum load factor over all termnals; by mnmzng ths value, we maxmze the value of Γ. The random process for makng changes to the system conssts of a lst of termnals ordered by load factor from hghest to lowest. A termnal s randomly chosen from the lst accordng to a geometrc dstrbuton, thus the termnals wth hgher load factors are chosen more frequently. The chosen termnal attempts to mprove ts load factor by addng a color wthn the frame sze lmt, startng wth the lowest avalable color. If a termnal adds a color that mproves ts own load factor wthout ncreasng the maxmum load factor n the network, t keeps that color, otherwse t contnues to try hgher color numbers untl t reaches the frame sze lmt. The whole process repeats when a termnal adds a color or runs out of colors to try. Fnally, the annealng schedule results naturally from the way we add colors to termnals. The core concept of an annealng schedule s that large changes occur at hgh temperatures, or magntudes, and smaller changes occur as the temperature, or magntude of allowed changes, decreases. Termnals attempt to add colors startng from the smallest color avalable. As more 4 of 7

5 and more termnals add colors, only the hgher colors are left. Under Lyu s schedulng algorthm, colors 5 through 8 transmt once every 8 slots, colors 9-6 transmt once every 6 slots, colors 7-2 transmt once every 2 slots, and so on. As termnals add hgher colors, they affect the schedule less frequently, resultng n a smaller mpact on the load factors. Thus, the hgh temperature to low temperature transton occurs naturally as the termnals use up avalable colors. Start N = Number of teratons to execute Sort termnals accordng to load factor (hgh to low), Choose a target termnal, X, accordng to a Geometrc(0.) dstrbuton. Determne smallest color, C, avalable to X good choce for N depends on the number of termnals n the network, the frame sze constrant, and tme requred to run the algorthm. SIMULATION RESULTS We smulate two types of networks to llustrate the performance gans possble wth a load-based schedule over a schedule whch does not account for traffc. Network conssts of 4 termnals arranged to create a severe traffc bottleneck. It s antcpated that large gans n performance are possble wth an optmzed schedule for ths network. The topology of Network s shown n Fgure. We also smulate a set of 00 randomly generated networks, each consstng of 00 termnals randomly placed n a 000m x 000m square. In these networks, each termnal has a transmsson radus of 200m, the average degree of each termnal s approxmately, and the network dameter s approxmately 9. Add color C to X, Update load factors Decreased LF(X)? Is C greater than Frame sze lmt? Increased Max[LF(G)]? Remove C from X, Update load factors, Determne next smallest avalable color, C. N teratons Completed? de X keeps C Fgure 2. Outlne of algorthm used to add colors to termnals to approxmate optmal schedules. End The centralzed algorthm for generatng a nearoptmal schedule for a network s shown n Fgure 2. For a gven network, ths algorthm s run wth hundreds of dfferent random number seeds to generate a large number of near-optmal schedules. Each random number seed results n a dfferent sequence of termnals chosen to add colors, thus the colors taken by termnals and the resultng schedule wll be dfferent for each seed. Out of these schedules, we keep the one that results n the smallest maxmum network load factor and use t n smulatons. The number of teratons executed, N, for each random number seed s analogous to the annealng schedule. A Fgure. Network, desgned to have a severe topologynduced bottleneck. Each smulaton run conssts of a network topology and a collson-free transmsson schedule. Mnmum-hop routes are calculated for each source-destnaton par; f multple mnmum-hop routes exst, one s chosen at random wth equal probablty. One consequence of ths s that dfferent random number seeds result n dfferent routes, hence the route loads and the maxmum stable throughput are dfferent. A Posson process s used to generate packets at a gven rate. The source and destnaton of each packet are chosen at random wth equal probablty. Each termnal has a queue sze of 0; f a packet s generated at or transmtted to a termnal wth a full queue t s dropped. The duraton of each smulaton s 0000 slots. Durng the frst 000 slots the networks experence a transent phase, durng whch traffc s generated and flows through the network, allowng queues to reach steady state condtons. Only packets generated after ths startup phase contrbute to the performance measurements. 5 of 7

6 End-to-End Delay End-to-End Throughput Packet Completon Rate Traffc Indep Traffc Indep Traffc Indep Fgure 4. Performance measurements of Network for traffc-ndependent and load-based schedulng. Results for Network are averaged over 00 random seeds, whle results for the random networks are averaged over the 00 random topologes. For both Network and the random networks, two sets of results are generated at dfferent packet generaton rates. The frst set of results s generated usng a traffc-ndependent schedule, whch corresponds to a smple colorng of the network graph. The second set of results s generated usng a load-based schedule generated for that network by the centralzed optmzaton algorthm. Fgure 4 shows the end-to-end delay, throughput, and completon rate for Network. Usng the traffcndependent schedule, the network s stable up to a generaton rate of about 0. packets per slot. The average load factor over these 00 runs s 488, whch ndcates a maxmum stable throughput of about 0.2 packets per slot usng (6). Usng load-based schedules, generated by runnng the centralzed optmzaton algorthm wth the frame sze lmt set to 2, the network s stable up to a generaton rate of about 0.48 packets per slot. For these schedules, the average maxmum load factor s 45, and the correspondng predcted maxmum stable throughput s 0.5 packets per slot. Ths more than four-fold ncrease n performance demonstrates the large performance gans possble usng ths schedulng approach n a network wth a severe bottleneck. The throughput of the traffcndependent smulatons contnues to ncrease, even past the stable regon of operaton, due to the ncreasng amount of successful -hop traffc n the group of 8 fully connected termnals. We also ran the centralzed optmzaton algorthm on these networks wth a frame sze lmt of 28, and the resultng average maxmum load factor for these schedules s 287. The theoretcal lower bound on the maxmum load factor, found by enumeratng all neghborhood clques, s 28. Fgure 5 shows end-to-end delay, throughput, and completon rate for the random networks. The maxmum load factor for these networks usng traffc-ndependent schedules, averaged over the 00 topologes, s 408; ths ndcates a predcted maxmum stable throughput of packets per slot. Usng load-based schedules reduces the average maxmum load factor to 050; ths ndcates a predcted maxmum stable throughput of packets per slot. The completon rate curves show ths ncrease, wth the traffc-ndependent schedules begnnng to drop packets at generaton rates around 0.2 packets per slot, and the load-based schedules begnnng to drop packets at generaton rates around packets per slot. These values are slghtly lower than predcted snce, n the smulaton results, the completon rate begns droppng off when the network wth the largest maxmum load factor frst begns droppng packets. As n Network, the throughput curves contnue to ncrease past the stable pont snce more short range traffc s beng successfully delvered. Allowng the frame sze to double does ncrease the channel access delay for termnals. The delay curves of Fgure 5 exhbt ths phenomenon, as the load-based schedules result n slghtly hgher packet delay than the traffc-ndependent schedules at very low traffc levels. However, ths delay ncrease s very slght, allowng us to conclude that queung delay has a much greater mpact on end-to-end packet delay than does channel access delay. CONCLUSIONS AND FUTURE WORK We have shown that the stable end-to-end throughput of networks usng transmsson schedulng can be greatly 6 of 7

7 mproved by assgnng termnals effectve transmsson rates that are proportonal to ther traffc load. We have shown how to analyze the traffc carryng ablty of networks n our model, culmnatng wth an expresson for calculatng the predcted maxmum stable throughput. We have also explaned how to arrve at a bound on the maxmum possble stable throughput by enumeratng the neghborhood clques. Algorthms based on the concepts of annealng have been used to generate near-optmal solutons to the Broadcast Schedulng Problem, as n [9]. We have shown that smlar algorthms are applcable to our problem as well. End-to-End Delay End-to-End Throughput Packet Completon Rate Traffc Independent Traffc Independent Traffc Independent Fgure 5. Performance measurements of random networks for traffc-ndependent and load-based schedulng. In a network desgned to beneft from ths type of schedulng, we have shown that load-based schedules result n a fve-fold ncrease n stable throughput capacty over traffc-ndependent schedules. In randomly generated networks, load-based schedules, on average, result n a trplng of stable throughput capacty. Ths performance ncrease comes at the cost of an almost neglgble ncrease n access delay due to the longer frame szes used to facltate optmzaton. Actual mplementaton of load-based schedules n an ad hoc network s dffcult. Estmatng load factors n a dstrbuted envronment s a sgnfcant problem whch s compounded by the task of dstrbutng ths nformaton over a neghborhood of termnals. Future research wll focus on developng ways to duplcate the performance gans of the load-based schedules n a dstrbuted envronment. REFERENCES [] P. K. Appan, J. L. Hammond, D. L. neaker, and H. B. Russell, Operaton of a dynamc transmsson-schedulng protocol for moble ad hoc networks, Proc. IEEE Mltary Communcatons Conf., October 200. [2] I. Chlamtac and A. Farago, Makng transmsson schedules mmune to topology changes n mult-hop packet rado networks, IEEE/ACM Trans. Networkng, vol. 2. pp. 2-29, Feb [] A. Myers and S. Basagn, Wreless Meda Access Control, n Handbook of Wreless Networks and Moble Computng, I. Stojmenovć, Ed. New York: John Wley & Sons, 2002, pp [4] J. L. Hammond and H. B. Russell, Propertes of a transmsson assgnment algorthm for multple-hop packet rado networks, IEEE Trans. on Wreless Communcatons, vol., no. 4, July [5] J. H. Ju and V. O. K. L, An optmal topology-transparent schedulng method n multhop packet rado networks, IEEE/ACM Trans. Networkng, vol. 6, pp , June 998. [6] S. Krkpatrck, C. D. Gelatt and M. P. Vecch, Optmzaton by Smulated Annealng, Scence, vol. 220, Number 4598, pp , 98. [7] W.-P. Lyu, Desgn of a new operatonal structure for moble rado networks, Ph.D. dssertaton, Clemson Unversty, August 99. [8] A. D. Myers, G. V. Záruba, and V. R. Syrotuk, An adaptve generalzed transmsson protocol for ad hoc networks, Moble Networks and Applcatons, vol. 7, no. 6, pp , December [9] G. Wang and N. Ansar, "Optmal broadcast schedulng n packet rado networks usng mean feld annealng, IEEE Journal on Selected Areas n Communcatons, vol. 5, no. 2, pp , Feb [0] B. J. Wolf, J. L. Hammond, D. L. neaker, and H. B. Russell, Dstrbuted formaton of broadcast transmsson schedules for moble ad hoc networks, Proc. IEEE Mltary Communcatons Conf., vember [] C. Zhu and M. S. Corson, A fve-phase reservaton protocol (FPRP) for moble ad hoc networks, Wreless Networks, vol. 7, no. 4, pp. 7-84, July of 7