Adaptation of TDMA Parameters Based on Network Conditions

Transcription

1 Adaptation of TDMA Paametes Based on Netwok Conditions Boa Kaaoglu Dept. of Elect. and Compute Eng. Univesity of Rocheste Rocheste, NY Tolga Numanoglu Dept. of Elect. and Compute Eng. Univesity of Rocheste Rocheste, NY Wendi Heinzelman Dept. of Elect. and Compute Eng. Univesity of Rocheste Rocheste, NY Abstact Soft clusteing of the nodes combined with time division multiple access (TDMA) channel access within a cluste has been shown to povide an enegy-efficient solution fo Mobile Ad-Hoc Netwoks (MANET). Such channel access schemes use a paamete that is citical in detemining netwok pefomance: the numbe of fames pe supefame, which detemines the amount of spatial euse possible, simila to the fequency euse facto in cellula netwoks. When a smalle numbe of fames pe supefame is used, each fame will consist of a lage numbe of slots, enabling the fame (i.e., cluste) to suppot moe nodes, but also limiting the choices of fames fo clusteheads to select, causing highe co-channel intefeence and collisions. Convesely, when a lage numbe of fames pe supefame is used, the clusteheads will only be able to gant channel access to a limited numbe of nodes, which in tun inceases the numbe of dopped packets (i.e., blocked channel access). The optimum value of the numbe of fames is the one that minimizes the combined effect of both collisions and dopped packets. By analytically detemining the effects of dopped packets and collisions, we can find the optimal value fo any given scenaio. This pape develops a model to detemine the optimal TDMA stuctue unde vaious settings, showing the advantages that can be obtained by adapting potocol paametes as netwok conditions change. I. INTRODUCTION The MAC potocol is the key element in the potocol stack that detemines the ability of a wieless netwok to meet application equiements, since the MAC potocol has a diect impact on thoughput, Quality of Sevice (QoS), enegy dissipation, fainess, stability, and obustness [1], [2]. In paticula, coodinated channel access schemes povide suppot fo QoS, educe enegy dissipation and incease thoughput fo low-to-mid noise levels and fo dense netwoks [3]. MH-TRACE [4] and IEEE [5] ae examples of such coodinated potocols. The IEEE potocol is specifically designed fo high-ate and shot ange WPAN netwoks [6]. MH-TRACE is designed fo mid-ange mediumate tansmissions [4]. Both of those algoithms use a TDMA stuctue togethe with soft clusteing of the nodes fo channel access, as this appoach has been shown to povide satisfactoy pefomance in tems of QoS and enegy dissipation. Many of the potocol paametes in cluste-based potocols ae set a-pioi based on estimates of netwok conditions This wok was suppoted in pat by the Univesity of Rocheste Cente fo Electonic Imaging Systems and in pat by Hais Copoation, RF Communications Division. and based on a specific physical laye. TDMA paametes, which detemine the amount of spatial euse and intefeence, distibute the available bandwidth among clustes so as to educe the intefeence thoughout the netwok. Reducing the intefeence is a desiable goal since high intefeence leads to high eo ates, deceasing the thoughput as studied in [7]. Howeve, educing the available bandwidth pe cluste also deceases the capacity pe cluste. The non-unifom distibution and node mobility may incease the local load above the cluste capacity, esulting in dopped packets and deceasing the thoughput fo eal-time taffic [4]. The decision of how to set these paametes should thus be based on this tade-off and would be effected by vaious conditions such as node density and physical laye paametes. The elationship between the paametes can only be detemined by analytical analysis. Although simulation studies eveal the pefomance of a potocol fo a cetain set of conditions, the statistical accuacy of the simulation esults is questionable unless epeated extensively. Fo lage and dense netwoks, this appoach equies excessive amounts of pocessing powe. Moeove, esults obtained fom those studies ae only valid fo the selected paametes and do not eveal thei impact on the pefomance of the potocol. Theefoe, we have developed an analytical model that eflects the elationships between potocol paametes and the oveall pefomance of the potocol unde diffeent netwok conditions fo the TDMA-based clusteed potocol MH-TRACE. Specifically, we develop a model that elates the TDMA fame paametes (numbe of slots pe fame and numbe of fames pe supefame) and the node density to the expected numbe of dopped packets and the expected numbe of collisions. This model enables us to find the set of paametes that maximize oveall thoughput fo MH-TRACE, and, by extension, fo TDMA-based clusteed potocols such as IEEE The oganization of this pape is as follows. In section II, a bief summay of MH-TRACE is pesented. An analytical model that helps in identifying the pefomance of MH- TRACE fo a given set of paametes is pesented in section III. Section IV discusses the validity of the analytical model and potocol paamete decisions based on that model by compaing the analytical esults with those found via simulation. Section V concludes the discussion with final comments.

2 Fame 1 Fame 2 Fame 3 Fame 4 Fame 5 Fame 6 CA Slot Beacon Contention Slots Contol sub-fame Heade IS Slots Data sub-fame Data Slots Supefame N - 1 Supefame N Supefame N + 1 Fig. 1. A snapshot of MH-TRACE clusteing and medium access. Diamonds epesent selected clusteheads and dots epesent the nodes in the netwok. CH-fame matching togethe with the contents of each fame is depicted. II. BACKGROUND: MH-TRACE In ode to undestand the esults of the study, it is impotant to undestand how the MH-TRACE potocol woks. This section biefly explains the potocol s cucial mechanisms and aims to give the eade a bief undestanding of the potocol. The complete potocol desciption is available in [4]. In MH-TRACE, time is divided into supefames of equal length, as shown in Fig. 1, whee the supefame is epeated in time and futhe divided into fames. Each cluste opeates in one of the fames, and the clustehead (CH) manages the channel access scheme of that fame. Thee is a spatial euse mechanism that allows moe than one cluste to opeate in the same time fame povided that the intefeence is low enough. Each fame in the supefame is futhe divided into subfames. The contol sub-fame is used fo management ovehead. The CH selection and maintenance as well as intefeence minimization algoithms use the CH announcement and beacon slots of the contol sub-fame. The est of the contol sub-fame is devoted to the pope collection of channel access equests and announcement of channel access gants as well as data summaization. The data subfame contains data slots whee nodes that ae successful in secuing channel access tansmit thei data in the ode announced by the CH. III. ANALYTICAL ANALYSIS In TDMA-based clusteed potocols such as MH-TRACE, thee ae two main factos limiting the pefomance (in tems of thoughput); dopped packets and collisions. Thus, ou goal is to model the effect of the potocol paametes on both the expected numbe of dopped packets and the expected numbe of collisions in ode to estimate the thoughput. MH-TRACE is an ideal potocol fo suppoting eal-time taffic, such as voice steaming, whee a data steam is divided into seveal packets of fixed length. In ode to maintain continuity of the steam, data slots should be assigned peiodically to nodes that have data to send, and this peiod must be matched to the length of each packet and the data fequency (e.g., voice sampling ate). This povides bounds on some of the TDMA paametes, such as slot size and supefame time, in ode to guaantee that the taffic meets its delay bounds. In MH-TRACE, the nodes within a cluste equest channel access fom the CH, and the CH gants the channel access by assigning available data slots to the nodes that equest channel access. Howeve, the numbe of data slots is limited and mainly detemined by the length of the fame. If the CH gants channel access and assigns a data slot to a node, it continues to eseve the same data slot in consecutive supefames as long as the node s tansmission continues. Thus, the time inteval between the data slots in two consecutive supefames, which is equal to the supefame duation, is fixed and detemined by the length of each packet and the sampling ate. Given that the length of the supefame must be matched to the eal-time data fequency, as the numbe of fames in the supefame (N f ) is inceased, each fame must be shote and thus has fewe data slots. This inceases the pobability of having no available data slot when a node equests channel access fom the CH. The packet must be dopped due to eal time taffic equiements. On the othe hand, as N f is deceased, spatial euse is inceased which inceases capacity pe cluste but also inceases the intefeence and collisions. Thus, thee is a tade-off in tems of thoughput between dopped packets and collisions, and ou goal is to model both of these souces of thoughput loss in ode to analytically examine this tade-off. Fo a fixed physical laye and voice model, ou model povides an estimate of the expected numbe of dopped packets and the expected numbe of collisions based on the values of the following paametes: N f, the netwok aea, and the numbe of nodes in the netwok. Thoughput is estimated based on the numbe of dopped packets and the expected numbe of collisions, as descibed in the following sections. A. Dopped Packets The pobability of dopped packets can be calculated as ( ) Load Capacity P dp = max, 0, (1) Load whee Load epesents the numbe of nodes that equie channel access within a cluste pe supefame and Capacity is the total numbe of data slots available fo the cluste (i.e., the numbe of data slots pe fame). Each cluste is capable of supplying channel access only to a limited numbe of nodes pe supefame that is equal to the numbe of data slots in the fame. The load in the cluste is mainly detemined by thee pobabilities: p s : Pobability of a node to be in sput duation, p A : Pobability of a node to be in the communication ange of a CH.

3 p d : Pobability of a node that is in the communication ange of a CH to choose that CH as its channel access povide. We assume a voice model whee nodes ae eithe in sput o gap, with voice data being geneated in the sput duation and no data being geneated in the gap duation. Hence, p s accounts fo the fact that not all nodes will have data to send at a given time. Futhemoe, MH-TRACE allows nodes to access the channel fom any CH within ange. Theefoe, if one CH has no data slots available, a node can equest channel access fom anothe CH in ange. Hence, p d takes into account the fact that a node s expected load to a cluste is distibuted among those CHs in the communication ange of that node. Given those distibutions, the load pe cluste pe supefame fo a given CH (and its location, CH loc ) is a binomial distibution with a success pobability of p dn = p s p A p d as given in (2). ( ) Nnodes P(Load = k CH loc )= p k dn (1 p dn ) (N nodes k) k (2) 1) Pobability of a node to be in sput duation(p s ): The voice model we use incopoates exponentially distibuted sput and gap duations (T sput and T gap, espectively) followed by each othe. The pobability of finding a node in sput state can be calculated as given in (3). E[T sput ] P(Node i in sput) = E[T sput + T gap ] = p s (3) 2) Pobability of a node to be in the communication ange of a CH (p A ): Nodes ae assumed to be i.i.d. distibuted accoding to a unifom distibution. Thus, the pobability of a node to be in any given egion is the aea of that egion divided by the whole simulation aea. Aea of Region A P(Node in Region A) = Simulation Aea = p A (4) The coveage egion of a cluste is a cicle centeed at the CH s location having a adius equal to the maximum eliable communication distance at the selected powe level. Pat of a CH s coveage egion may lie outside the netwok aea depending on the CH s location, CH loc. The netwok aea can be divided into 3 egions, as in Fig. 2. The coveage aea that lies within the simulation aea, α 1, α 2, and α 3 fo a CH in egion 1, 2 and 3, espectively, ae calculated in [7] and epeated hee fo efeence. α 1 = π 2, α 2 = π 2 2I(d 0 ), α 3 = 3 4 π2 I(x 0 ) I(y 0 )+x 0 y 0, (5a) (5b) (5c) whee x 0 and y 0 ae the distances to the closest vetical (fo x 0 ) and hoizontal (fo y 0 ) limits of the netwok aea, d 0 = min(x 0,y 0 ) and the function I(x) is as given in I(x) = π 4 2 x 2 x acsin(x/). (6) 2 H n 3 L-2 n 2 H-2 H-2 H n 1 L L-2 L Region 1 Region 2 Region 3 Fig. 2. Patitioning of the simulation egion. Pat of the coveage egion lies outside the netwok egion fo CHs located in egions 2 and 3. Fig. 3. Alignment of CHs unde maximum packing conditions. Replacing RegionA with the appopiate tem fom (5), leads to desied p A fo a given CH. 3) Pobability of a node to be in the communication ange of a CH (p d ): The entie simulation aea will be fully coveed fo sufficiently lage node densities. Afte that point, vaiations in the node density do not alte the expected numbe of cluste head nodes in the netwok and thei positions. Thus, the numbe of CHs in the communication ange of each node is independent of node density fo sufficiently lage node densities. Simulations indicate that 35% of the nodes ae in the communication ange of only one CH wheeas 48% and 17% of the nodes ae in the communication ange of two and thee CHs, espectively fo a communication ange of 250m. Nodes can select thei channel access povide, unifomly among the CHs that ae in thei communication ange. Using the afoementioned values p d can be calculated as Substituting (5a)-(5c) into (4) and (2), p A and the statistical distibution of load can be obtained fo a given CH. Taking a weighted aveage, using the pobability of CH s locations as the weights, gives the distibution of load fo a geneic CH as P(Load = k) = P(Load = k CH loc )P(CH loc )ds, S (7) whee S is the simulation aea and CH loc is a CH s location. The integal in (7) can be taken numeically fo all k values. 4) Effect of dopped packets on thoughput: Using the distibutions of Load, the distibution of P dp is calculated by (1). If a dopped packet was tansmitted successfully, that packet would have been eceived by all the nodes that ae in the tansmission ange of the tansmitting node. Thus, each dopped packet deceases the thoughput by the numbe of neighboing nodes. The total decease in thoughput pe supefame due to dopped packets, f dop, is anothe andom vaiable and is elated to P dp by f dop = P dp (P s N nodes ) E[N neighbos ]. (8) The expected numbe of neighbos, E[N neighbos ], can be calculated as E[N neighbos ]=p a N nodes, (9) whee p a, the pobability of a node to be in the communication ange of anothe node, is calculated in a manne simila to the calculation of p A.

4 y d- d 2 d x (a) (b) Fig. 4. (a) Intesection egion fo two cicles with adius sepaated by d. (b) A typical alignment of co-fame clustes togethe with thei vulneable egion (lightly shaded egion) and a node in vulneable egion epesented by the disk. The dak shaded egion is the egion of membe nodes whose packets could collide at the node epesented by the disk. B. Collisions CHs manage the channel access within the cluste so as to pevent simultaneous tansmissions of the nodes in the cluste. Hence, collisions within the cluste ae completely eliminated. Clustes opeating in sepaate fames access the channel in distinct time instances, and hence collisions among nodes in these clustes ae also not possible. The only emaining souce of collisions is co-fame clustes. Ou appoach fo calculating the numbe of collisions pe supefame includes two steps. The fist step is to elate the numbe of fames pe supefame (N f ) to the co-fame CH sepaation, and the second step is to elate the co-fame CH sepaation to the numbe of collisions pe supefame. 1) Relation between N f and co-fame CH sepaation: The numbe of collisions inceases as the sepaation between cofame CHs deceases. Due to the CH ceation mechanism, the minimum sepaation between two CHs is equal to the communication distance,, in the wost case scenaio. Unde maximum packing conditions, a thid CH is also sepaated by fom both of the pevious CHs as in Fig. 3. The lines connecting the 3 CHs foms an equilateal tiangle. At most 6 CHs can be located aound the fist CH as in Fig. 3. Consideing an imaginay cell bounday passing pependicula to the lines connecting the neighboing CHs, a hexagonal cellula stuctue can be constucted in the fom of the standad stuctue used in cellula systems [8]. The labeling stuctue that is used in cellula systems is applicable fo the clusteing stuctue unde concen. Distinct labels coespond to clustes opeating in distinct fames in the supefame stuctue. The sepaation between co-labeled cells and hence co-fame CHs is given as d ch = N f. (10) We use this sepaation as a basis to calculate the collisions, although a egula stuctue that maximizes the distance between co-fame CHs exists only fo a cetain set of N f values. 2) Relation between CH sepaation and numbe of collisions: The fist step in detemining the elation between the numbe of collisions and the CH sepaation, d ch, is to find a function, f(d, ), that calculates the intesection aea of two cicles having adii, and whose cente points ae sepaated by d. This function will be used fequently late on. Fig. 4a shows two such cicles. Fom symmety, the total intesection y d ch x aea is fou times the shaded aea and can be calculated as f(d, ) =4 d 2 d 2 (x d) 2 0 dydx ( sin(2θ) =4 2 + θ ) θ= acsin( d 2 ). (11) 4 2 θ= π 2 Nodes vulneable to collisions should be located at most 2 times the communication adius,, fom a CH. Futhemoe, those nodes should also be located within 2 times the communication adius of a co-fame CH. Thus, the vulneable egion fo two co-fame CHs is the aea of intesection of two cicles with adii 2 and centes located at the co-fame CH positions. The lightly shaded aea in Fig. 4b depicts the vulneable egion fo two CHs epesented by two diamonds. The nodes in the vulneable egion ae called vulneable nodes. As the co-fame CH sepaation is deceased, the vulneable egion and hence the numbe of vulneable nodes inceases. The nodes whose packets may collide at a node in the vulneable egion should be in the communication ange of that node. In Fig. 4b, the little disk located at (x, y) epesents an abitaily selected node among the vulneable nodes. Packets tansmitted by the nodes in the dak shaded egions will collide at the selected node when these packets ae scheduled at the same data slot. As the co-fame CH sepaation is deceased, the aea of the dak shaded egions and hence the numbe of nodes in those egions inceases, which in tun leads to an inceased pobability of collision at the selected node. The numbe of nodes whose packets may collide at a selected node in the vulneable egion, N nodes2coll, is a function of the elative position of the selected node with espect to the CH locations and the distance between the CHs. The sum of N nodes2coll fo all the nodes in the vulneable aea leads to TN nodes2coll, which is a function of CH sepaation. N nodes2coll, fo the disk located at (x, y) in Fig. 4b, is the sum of the numbe of nodes in the dak shaded egions and can be calculated by multiplying the node density with the aea of these egions as in (12) since nodes ae i.i.d. and unifomly distibuted. N nodes2coll = d n [ f( x 2 + y 2,)+f( ] (d ch x) 2 + y 2,), (12) whee d n is the node density and d ch is the CH sepaation. Integating (12) fo all nodes in the vulneable egion, V, (lightly shaded aea in Fig. 4b) yields TN nodes2coll = [ f( x 2 + y 2,)+f( ] (d ch x) 2 + y 2,) dxdy. d 2 n V (13) Assuming the CHs ae well sepaated (making the egion of integation small), (13) can be appoximated by [ TN nodes2coll = d 2 n f (d ch, 2) 2 f( d ] ch 2,). (14)

5 TABLE I SUPERFRAME PARAMETERS Numbe of Fames Numbe of Data Numbe of Contention Supefame N f Slots,N d Slots,N c Time T sf , , , , ,992 Numbe of Packets Lost pe Supefame f dop Simulated (Stationay) f Simulated (Stationay) coll f Theoetical dop f Theoetical coll f dop Simulated (Mobile) f Simulated (Mobile) coll The exact elationship between TN nodes2coll and the numbe of collisions pe supefame, f coll, is complex in natue. We use a fist ode simplification, and assume that they ae linealy elated. Unde this assumption, the numbe of collisions can be obtained by scaling TN nodes2coll with a constant, C, as in (15). The constant is independent of N nodes and N f since those effects ae aleady taken into account in TN nodes2coll though density and co-fame CH sepaation. The constant can be detemined by compaing the analytical esults with simulation esults fo spase netwoks whee the simulation is possible. Using the same constant, f coll can be detemined fo any node density and any N f value. f coll = C TN nodes2coll (15) IV. PROOF OF CONCEPT The aim of the analytical study is to ceate a model that will enable us to examine the pefomance of the MH-TRACE potocol fo any given values of N f, netwok size and numbe of nodes. Using this model, fo example, we can detemine a good N f value that maximizes the thoughput in the system fo any given node density. The analytical model given above is a fist ode appoximation fo a complex system. Thus, the appoximation eos may effect the decision about the optimal N f value and may lead to suboptimal solutions. In this section we evaluate the pefomance of ou analytical model in detemining the optimal N f value. A. Simulation Envionment Fo compaison puposes, we conduct ns-2 simulations of MH-TRACE unde diffeent netwok scenaios and fo vaious N f values. Voice communications is simulated with sputs and gaps whose lengths ae exponentially distibuted and statistically independent with mean duations of 1.0s and 1.35s, espectively[9][10]. The souce application is the convesational voice, coded at 32Kbps. 100 byte long packets ae geneated fom that application fo all nodes. The channel ate is set to 2Mbps. Given these values, in ode to maintain data packet timing, the supefame time is adjusted to be as close as possible to 25ms, which is the voice packet geneation peiod. Table I shows details fo each value of N f. Beacon, CA, contention, and IS packets ae all 4 bytes long. The heade packet has a vaiable length of (4 +2N d ) bytes depending on the numbe of data slots. Thee is a 4 byte heade and 100 byte data packet associated with each data slot. Details of the vaious heade packet lengths can be found in [4]. Numbe of Packets Lost pe Supefame Numbe of Nodes f dop Simulated (Stationay) f coll Simulated (Stationay) f dop Theoetical f coll Theoetical f dop Simulated (Mobile) f coll Simulated (Mobile) (a) N f = Numbe of Nodes (b) N f =8 Fig. 5. Packet loss functions fo dopped packets and collisions fo both stationay and mobile nodes. The andom way-point mobility model [11][12] is used in mobile scenaios whee the node speeds ae chosen fom a unifom andom distibution between 0.0 m/s and 5.0 m/s with zeo pause time. The enegy model discussed in [13] and the default popagation model that is available in ns-2 [14] ae used. We used a constant tansmit powe that esults in a constant eceiving ange of 250m. All simulations ae un in an aea of 1km x 1km fo 100 seconds. The simulations ae epeated with the same paametes 5 times. All esults ae based on sample aveage of those 5 iteations. B. Results and Compaison The loss in thoughput due to collisions and dopped packets ae depicted in Fig. 5 fo N f equal to 6 and 8, espectively. Fo stationay case, the analytical estimations that ae obtained with the method descibed in the pevious section tightly follow the simulation esults fo all node densities and all N f values, veifying the model. The simulation esults fo mobile case follows the same tend as the theoetical esults but ae not as close as the stationay case. The fist eason fo this is that the andom way-point model esults in a non-unifom node distibution by binging the nodes togethe[11][12]. The assumption of unifom node density used thoughout the theoetical analysis fails as time evolves unde the andom way-point model. The second eason is that mobility causes CHs that come nea each othe to esign and let othe nodes fom new clustes. Duing the CH fomation pocess, the local capacity aound esigned CHs deceases, which in tun leads to a highe numbe of dopped packets. Anothe obsevation is that as N f is inceased, the collisions decease, wheeas the dopped packets incease. Inceasing

6 N f inceases the sepaation between co-fame clustes which in tun deceases collisions. On the othe hand, since the supefame length is fixed, inceasing N f deceases the fame length and hence the numbe of data slots pe fame and the capacity pe CH. Thus, thee is a highe numbe of dopped packets as N f is inceased. Theoetically, the maximum thoughput should be ealizable at the N f value that minimizes the combined effects of collisions and dopped packets. Thus, the optimal N f paamete of the analytical model is the value that minimizes the analytical estimate of collisions and dopped packets fo a given node density. In ode to detemine the applicability of using the analytical model fo selecting the optimal N f value, we measue the simulated thoughput fo all N f values as a pecentage of the maximum ealizable thoughput at a given node density. Fig. 6 depicts the pefomance of the decisions obtained though the analytical estimations togethe with the wost decision fo N f {4, 5, 6, 7, 8} fo all node densities and fo both the mobile and stationay cases. All decisions obtained though analytical analysis lie within 4% and 9% of the maximum ealizable thoughput fo the stationay and mobile cases, espectively. Analytical analysis is successful in detemining the optimal N f value fo dense netwoks, in both the mobile and stationay cases, although in the mobile case, individual estimations fo dopped packets and collisions obtained fom the analytical analysis ae not as close to the simulated values as they ae in the stationay case. Fo low node densities, the demand fo channel access is faily low and the capacity is sufficient. The dopped packets ae meely the esults of non unifomities in node distibution. Thus, inceasing capacity pe cluste has elatively less impact on thoughput. Also the impact of collisions on thoughput is less fo spase netwoks since collisions ae popotional to the squae of the node density. Hence, the decision on which N f value to use is elatively less impotant fo spase netwoks as can be obseved in Fig. 6 by the deceasing pefomance diffeence between the wost and best decisions as node density is deceased. On the othe hand, the decision about which N f value to use is quite impotant fo dense netwoks. Fo 300 node stationay and mobile netwoks, choosing the wost N f value achieves only 65% and 64% of the optimum thoughput, espectively. Thus, appopiate setting of the N f paamete is extemely impotant, and the analytical model povided in this pape can be used to optimally set this value fo abitay netwoks. V. CONCLUSION The analysis made in this pape eveals the impotance of pope selection of TDMA paametes accoding to netwok conditions. It has been shown that impope selection of these paametes may esult in seious thoughput loses (up to 35% fo stationay and 36% fo mobile scenaios) in dense netwoks, while the decision has smalle impact fo spase netwoks. The pimay detemining facto in the decision is the node density. Othe factos such as tansmitting powe and noise levels as well as mobility pattens also have an impact % Est. N f =7 Est. N f =6 Est. N f =5 Est. N f =4 70 Pefomance of Theoetical Analysis (Stationay) Pefomance of Wost Decision (Stationay) Pefomance of Theoetical Analysis (Mobile) 65 Pefomance of Wost Decision (Mobile) Numbe of Nodes Fig. 6. Pefomance of the N f decision of the analytical analysis togethe with the wost decision as a pecentage of the best decision fo all node densities. Selected N f values though analytical analysis is displayed on the top of the gaph. but they ae out of the scope of this pape and ae the subjects fo futue wok. The analytical analysis pesented in this pape not only makes the analysis possible fo vey dense netwoks whee the simulation of the netwok is impactical but also eveals the impact of paametes on system pefomance. It should be emphasized that, although the analytical model developed in this pape is specifically designed fo MH-TRACE, it can be easily adapted to any potocol whee the bandwidth is shaed among clustes. REFERENCES [1] A. Chanda, V. Gummalla, and J. O. Limb. Wieless medium access contol potocols. IEEE Communications Suveys and Tutoials, 3:2 15, [2] P. Mohapata, J. Li, and C. Gui. Qos in mobile ad hoc netwoks. IEEE Wieless Communications Magazine, 10:44 52, [3] T. Numanoglu, B. Tavli, and W. Heinzelman. An analysis of coodinated and non-coodinated medium access contol potocols unde channel noise. Militay Communications Confeence, MILCOM IEEE, pages Vol. 4, Oct [4] B. Tavli and W. B. Heinzelman. MH-TRACE: Multi hop time esevation using adaptive contol fo enegy efficiency. 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