Almost Disjunct Codes in Large Scale Multihop Wireless Network Media Access Control

Transcription

1 Almost Disjunct Coes in Large Scale Multihop Wireless Network Meia Access Control D. Charles Engelhart Anan Sivasubramaniam Penn. State University University Park PA 682 Abstract We iscuss the application of coing constructs to the problem of raio transmission scheuling in a hoc wireless networks. We are particularly intereste in the ability to scale such a hoc wireless networks to thousans of noes. Current meia access protocols o not scale well to larger networks than a few ozen noes. We examine the possible ways to apply superimpose coes an almost isjunct coes to these networks meia access layer. We analyze how well one can buil coes that are not as powerful as superimpose coes but have better parameters for the case of a hoc networks. We look at the local network performance that a particular noe can expect using these coes. We also examine the system wie performance using an approximate simulation of thousans of a hoc wireless noes. Introuction We woul like to improve the performance of meia access control (MAC) in wireless a hoc networks. We are intereste in performance both in terms of quality of service measures like throughput an latency but also in terms of measures that will affect battery life, such as collision rate. In orer to transmit ata a wireless noe nees to be sure that no other noes in its raio range are transmitting at the same time so the transmission can be hear by its neighbors. Traitionally MAC protocols can be classifie as either centralize or istribute algorithms. Centralize protocols like TDMA suffer from low spatial utilization as generally only one noe in the system is allowe to transmit at a time, but guarantee no conflicts. Distribute algorithms offer the possibility of spatial reuse but suffer from significant contention problems even in moerately loae systems, an usually cannot guarantee reliable elivery. We are particularly intereste in how these systems will scale to thousans of noes or more. Since these noes are usually mobile evices an thus are resource constraine, especially with regar to battery life, we will be more intereste in the efficiency of the system than throughput or latency. Recently avance protocols base on the ieas of TDMA, reliable elivery an little contention, have starte to appear. These new protocols use avance coing structures to reuse the space an still guarantee reliable elivery. These systems were first iscusse in [2] an refine in [3]. Both protocols use constructions of orthogonal coes to ensure that a conflict free scheule can be obtaine without knowlege of the unerlying network topology. Both stuies analyze the theoretical throughput of these systems an the latter more closely analyzes guarantee throughput as well as scheuling elays. A recent paper by Syrotiuk, et al. [6] generalizes the use of orthogonal arrays for these systems an analyzes both the probability of successful transmission an throughput of such systems built from general orthogonal arrays of strength 2 an 3. The remainer of this paper is organize as follows. Section 2 goes over some basic coing constructs. The ways they can be applie to wireless networks is iscusse in Section 3. Section 4 examines the localize network performance when using superimpose coes. An approximate simulation of a large scale wireless network is use to compare an almost isjunct coe to an a hybri protocol in Section 5. 2 Orthogonal arrays an Superimpose coes In general we are going to use a binary matrix to represent each noes transmission scheule. The rows of the ma-

2 trix correspon to the given time-slots that noes can transmit in. Each column of the matrix represents a single noe. Thus each time a appears in a row i column j then noe j is allowe to transmit uring time-slot i. The scheule is then repeate inefinitely. The number of columns of such a matrix will limit the number of noes in a system. In general we will let n be the number of noes in the system. 2. Orthogonal arrays Previous work [6] has use orthogonal arrays to construct these scheules. Def. OA(t, k, v) is a k v t matrix using the symbols {,, 2...v } is a strength t orthogonal array if for any set of t of its rows each t-tuple of column elements appears exactly once, for that row set. One important property of such an orthogonal array is that each column, or coewor, of OA(t, k, v) intersects every other column in fewer than t positions. Using this property we can buil a transmission scheule. First no two columns intersect in more than t positions, so given a set of D other columns the given column iffers from them in at least k D(t ) positions. As long as this ifference is positive we can say that there is at least one row where the given column is ifferent from all of any set of D other columns. If we assume this D is boun for the number of neighbors any given noe can have we can then assure each noe has a time-slot in which it is allowe to talk when none of its neighbors are. The symbols {,, 2...v } are mappe to the set of ientity vectors. Using this binary vectorization we can see that as long as 2 symbols are ifferent then uring that row the 2 noes will not interfere with each other. So using OA(t, k, v) we can hanle up to v t noes an a scheule inclues k v time-slots. 2.2 Superimpose coes We can also construct viable scheules using superimpose coes [4]. We say one binary vector a covers another b iff for every in b there is a in the same position in a. Def. A binary matrix is -isjunct iff it has the property that the union of any coewors oes not cover any coewor not in that set of coewors. We can then construct a scheule using such a -isjunct superimpose coe, because we know there is at least one row where there exists a corresponing to a given noe an there are all s for all of its neighbors. Each noe is guarantee a time-slot where it can talk an none of its neighbors may. We can then see that what we constructe with the binary vectorize orthogonal arrays was a D-isjunct superimpose coe. 2.3 Almost isjunct coes We woul like to relax the property of having an absolute guarantee that there exists a time-slot in which every noe is allowe to talk with no interference. To o so we wish to consier α-almost -isjunct coes []. Def. A binary matrix is α-almost -isjunct if for any ranomlyselecte set of columns, the probability that they cover no other column is at least α. So we can say that a -almost-isjunct matrix is exactly a -isjunct superimpose coe. These matrices have the property that each noe has probability α that it has a time-slot where it can have interference free communication The Matrix * Operator One way of increasing the number of noes that a coe can hanle without increasing its scheule length much is by using the matrix * operator. The * operator applie to a single matrix prouces another matrix that has every orere pair of columns from the first matrix. Given a matrix, M, the matrix M then has twice as many rows, but the number of columns is square. It is worth noting that given any matrix M we cannot guarantee that M is even 2-isjunct, but it may very well be highly α-almost -isjunct for a given. 3 Superimpose coes in wireless networks The scheules in previous work have been built on OA(2,v,v) in [2] an enhance to OA(2,v +,v) in [6] for an increase in throughput. The resulting scheules will then have v 2 coewors of length either v 2 or v 2 + v an can hanle up to v neighbors, that is it will be a v-isjunct superimpose coe. In [6] they also iscuss the aition of strength 3 orthogonal coes an show that for small numbers of neighbors that they can significantly increase throughput. They again focus on orthogonal arrays where k = v +, in oing so they increase the number of coewors to v 3.However,by not increasing the length of each coewor they can have a maximum of (v +)/2 neighbors, which roughly cuts in half the power of the superimpose coe. In orer to not reuce the power of the resulting superimpose coe, an thus the maximum number of neighbors allowe in the system they woul have to fin an orthogonal array with k approximately twice v. This woul yiel a scheule with coewors of length approximately 4v 2. It can be seen that in the initial constructions using orthogonal arrays that the length of each coewor an the number of coewors is asymptotically the same. This means that a scheule must be as long as the number of 2

3 noes in the system. So in regar to scheule length these scheules are no better than traitional TDMA. They o, however, significantly increase the amount of ata that it is possible to transmit uring that time perio. If we plan to use these scheules for very large systems, they are also going to be require to have a very large scheule length, approximately the size of the number of noes. For example in a system with one million noes we woul nee to construct OA(2,, ) (is this easy to o?). We coul then hanle neighbors among any of the one million noes, this may seem a bit too powerful for what our nees may be, perhaps we woul nee to only be able to hanle neighbors. More importantly each scheule has one million time-slots. It may be the case that certain noes are only allowe to transmit uring one small section of that scheule, because they are only guarantee one time-slot in which none of their neighbors are allowe to transmit. This woul possibly lea to large latencies, an if we want to increase the number of noes further it coul lea to near starvation. 3. Incience matrix construction of superimpose coes One well known [5] construction of superimpose coes is base on the iea of incience matrices of sets. Each row an column in such a matrix represents a given subset of m objects. We then say there is a in position (i.j) iff the set corresponing to row i is completely containe in the set corresponing to column j. A-isjunct superimpose coe can be constructe with ( ) ( m rows (time-slots) an m ) s columns (noes) where m<s<. Consequently we can have superimpose coes that asymptotically grow much faster in the number of coewors than the scheule length. We can see that even with a relatively small these coes are only going to be suite for large systems. The other main factor that will contribute to their performance will be the ensity of the coes, that is the percentage of ones that appear in the coe, that is the percentage of time the noes are allowe to transmit. 3.2 Constructing Almost isjunct coes Our initial metho for constructing almost isjunct coes is to simply remove some percentage of the rows. By oing this we are cutting the scheule length but we are also potentially reucing the power of the coe. We nee to be careful that we o not remove rows in a manner which overly egraes the power of the coe, or one that penalizes some noes significantly. We will remove rows uniformly at ranom in orer to try an minimize these cases. This is not guarantee to minimize conflicts, but on average shoul penalize no coewor(noe) more than any other. With this reuction in scheule length we are not going to be able to make the same guarantees as before but hopefully these scheuling conflicts can be minimize to the point that they only occur with very low probability. The way in which the superimpose were initially constructe will make the ensity of the coe remain constant when removing rows. alpha m=2, s=6, =3 Incience Matrix Coe Frame Length Figure. The α- almost isjunct property as rows are remove from the superimpose coe. Figure shows how alpha changes as we remove rows in the superimpose coes constructe with the incience matrix construction. There were 22 rows in the initial coe, notice that with only /4 of the original rows that the coe is still.5-almost 3 isjunct, an with half the rows remove it is still about.85-almost 3 isjunct. We can also use a less powerful superimpose coe than the number of neighbors. For example with a power 3 superimpose coe we can only guarantee success in the case of 3 or less neighbors, but it is very likely that we will still have some success with more than 3 neighbors. Figure 2 shows the alpha values in the case of a =3 superimpose coe with up to 3 neighbors. Using a coe that is only guarantee to be 3 isjunct it is.95-almost 5 isjunct, an.6-almost 3 isjunct. It appears there are great gains to be ha if we are willing to eal with a chance that our coe oes not strictly meet the isjunct property of a superimpose coe. In a wireless network this woul mean that on a given scheule there is a α chance that a noe woul not be guarantee a time slot where only it was talking amongst its neighbors. 3.3 Parameter Comparison Figure 3 compares the scheule length of 7-isjunct superimpose coes constructe with orthogonal arrays an the incience matrix construction, using completely isjunct coes. The OA(2,v,v) coe has the longest scheule 3

4 alpha m=2, s=6, =3 Incience Matrix noes scheule length neighbors TDMA n n n orthogonal coes v t k v k t OA(2,v,v) coes v 2 ) v ( 2 v m ) incience matrix almost isjunct ( m s ( m s ) β ( ) m?? Table. Parameters of various MAC protocol coes Desire Figure 2. The α- almost isjunct property with a lower power superimpose coe. lengths of the three over all values of network sizes. The OA(2, 8,v) orthogonal coe performs better at lower numbers of noes, but when the network size is increase significantly the growth rate of the incience matrix superimpose coe has shorter scheule lengths. Table 3.3 shows the various parameter constraints for the coes iscusse. Scheule Length OA(2,8,v) Incience matrix 7 isjunct TDMA an OA(2,v,v) Noes the coe. So in orer to be able to talk about how it shoul perform we will want expressions for the ensity (the fraction of ones) of each coe. The simplest way to look at the ensity of a superimpose coe constructe via an incience matrix is not to look at the ones, but instea the situations that will lea to zeros in the coe. It is clear that a superimpose coe constructe via the incience matrix metho will have the same number of ones or zeros in each column, an similarly in each row. So to fin the ensity of the coe we only nee to look at one row or column. Recall that a row in such a superimpose coe correspons to a set of objects chosen from m objects, similarly columns correspon to a set of s objects chosen from those same m objects. Looking at any column we can see that the only time a one will appear in any position is when the set corresponing to the row is completely containe in the set corresponing to that column. There are exactly ( s ) sets of size that are completely containe in the s-set corresponing to a given column. Since there are exactly elements in a given column the coe ensity is then ) (s. This ensity ( m ) remains constant across rows of the matrix as well as the columns. Figure 3. The scheule length of various 7- isjunct superimpose coes. =3 =5 =7 4 Localize Network Performance Coe Density 2 In orer to unerstan how a wireless network will perform we will first look at the theoretical network performance in the neighborhoo aroun a given noe. 4. Coe Density For superimpose coes one of the primary characteristics that will effect network performance is the ensity of Noes (m=[2,5], s=m/2) Figure 4. Coe ensity as the number of noes increases. 4

5 Figure 4 shows how the ensity of the coe changes as the number of noes (coewors) in the system increases for 3, 5 an 7-isjunct superimpose coes, constructe with the incience matrix construction. The values for m are in the interval [2,5], an the s values are always chosen to be half of a given m, remember that m an s etermine the maximum number of noes for a given superimpose coe. Notice that as the power of the superimpose is increase the ensity of the coe is lower. Also as the number of noes increases the ensity increases slightly. A successful transmission then occurs when the noe in question is allowe to transmit an all of its neighbors are silent. Thus the probability of successful reception for a noe that wishes to communicate is simply P r = P a ( P a ) P b. Figure 6 shows how the expecte reception ratio changes as the ensity of the coe increases, with a fixe number of neighbors. Notice that when all noes wish to communicate all the time the esire coe ensity is higher than in a more lightly loae system..5 m=4, s=2 5.8 % transmission probability 2% transmission probability.6.35 Coe Density.3 5 Reception Probability Coe Density Figure 5. Coe ensity as the power,, ofthe superimpose coe increases. Figure 6. The reception probability as the ensity of the coe changes. Figure 5 shows how the ensity changes with the power,, of the superimpose coe. As we saw before as the power of the coe increases its ensity ecreases. This ensity remains constant as we remove rows to from an almost isjunct coe. The ensity for stanar TDMA is clearly /n an for orthogonal arrays is /v which in the case of OA(2,v,v) coes is / n. These coes will all be somewhat inflexible if we wish to change the ensity of the coe to suit a particular network configuration, but we have some choice in coe ensity when using an incience matrix superimpose coe. 4.2 Throughput Throughput at the MAC layer is a simple function of the reception success rate for a given noe that wishes to transmit. We erive a simplifie moel of reception success that ignores complex propagation etails. Let P r be the probability of a successfully receive transmission. P r will epen on several factors. The probability that the noe is allowe to transmit uring that times lot, P a, is the first factor. Secon, this success rate will also epen on the number of neighbors,, that noe has as well as the likelihoo that a given noe will wish to transmit, P b. Figure 7 shows how the reception probability changes with varying the number of neighbors, for various coe ensities. As expecte increasing the number of neighbors reuces the probability of a successful communication, but notice how the lower ensity coes become more esirable as the number of noes increases. Reception Probability ensity=.5 ensity= ensity=.3 ensity= ensity= Figure 7. The reception probability as more neighbors are ae to the system. Figure 8 shows how varying both the number of neigh- 5

6 bors an transmission probability effects the reception probability. Reception Probability Transmission Probability Figure 8. The reception probability varying with both the number of neighbors an their transmission probability. 5 Simulation Although looking at the performance localize aroun a group of noes may give us some insight into how to optimize such networks we must look at the whole network working together in orer fully analyze any MAC protocol. To o this we use an approximate simulation that moels a complete, large scale, wireless network. We use Matlab [7] to simulate these networks. The noes in the networks we simulate are all static, that is they o not move over the course of the simulation. They are istribute uniformly at ranom over a square area. Raio propagation is hanle by simply calculating the istance between pairs of noes an if they are within some threshol istance of each other they can communicate, an otherwise they cannot. Messages are generate using intervals etermine by using an exponential istribution, governe by a parameter λ. Asλ increases the time between message generation shortens, thus there is more loa in the system. Messages are sent from any ranom noe to another ranom noe. Routes for the messages are mae using a geographic algorithm for simplicity, the next hop of a message is sent to the noe in its communication range that is geographically closest to its final estination. The communication ranges were set so that there woul be 6 neighbors per noe on average. We use a mbps raio moel, using 52 byte packets. We ignore all packets elivere uring an initial start up winow, an most of the simulations ran for a few minutes of simulate time. We ran two experiments for each raio moel. First we varie the size of the network an kept the loa (etermine by the parameter λ in the exponential traffic generator) constant. Secon we kept the network size constant an varie the loa in the system generate by the exponential traffic generator. For each experiment we measure the total number of routing layer packets elivere, the collision rate an the average latency for routing layer packets. Note that the collision rate is going to have a irect effect on the power use by noes in the system, as collisions are simply transmissions where power was consume an no work was one. 5. MAC Protocols We compare three ifferent MAC protocols for each of the simulations. First we use a protocol. We use time intervals that were smaller than the time to transmit a packet (/25th the time to sen a packet) for raio contention. In this moel a given transceiver only tries to communicate if no one is currently using the channel. If there is a collision, or anytime that the raio wants to talk after it has hear another transceiver talking it must wait a ranom number of the smaller time intervals before it can try to transmit. That ranom amount of timer intervals is base on an exponential back off timer. The back off winow has a minimum size of 8 time intervals an a maximum of 256, with it oubling every time the noe cannot communicate, an resetting to 8 every time it communicates successfully. We also use a coe base MAC protocol. For simplicity sake we use a very simple coe that has goo α -almost isjunct properties. We use the minimum In matrix that coul support the number of noes use in the simulation, for example if there were simulate noes we woul have use I, which has a scheule length of 2 an a coe ensity of.. We i not use any smaller time intervals for this simulation, an the noes ability to communicate uring a transmission winow is completely base on the column of the coe corresponing to that noe. In orer to try to reuce the latency problems associate with the coe base MAC protocol we use a hybri approach that use the same coe use for the coe base protocol, but also incorporates the smaller time intervals. So a given noe can start transmitting uring a time slot where it has a an continues transmitting until it elivers the whole packet (which takes 25 of the smaller time units). There is also a carrier sense mechanism use so that no noe will try to start communicating when another noe is alreay transmitting. 6

7 25 9 coe base hybri coe base hybri Messages passe Average Latency Noes Noes Figure 9. Routing layer messages passe as the size of the network grows. Figure. Average latency of routing layer packets as the size of the network grows. 5.2 Network size scaling First we examine how each of the protocols scales as the network grows in size. The parameter for the traffic generator, λ, was kept constant at.. Figure 9 shows the number of messages passe for each protocol. The coe base protocol starts out the worst of the 3 when there are fewer noes in the system, but it overtook the approach aroun 5 noes. It is worth noting the hybri approach completely faile (near throughput, tocols perform almost ientically until about 3 noes are reache, where the coe base protocol starts to perform significantly better than. Overall the coe base protocol seems to outperform the base an hybri protocols as the size of the network grows. 5.3 Network traffic scaling We fixe the number of noes in the system at an varie the rate at which traffic was generate by varying λ..5 5 coe base hybri 4 coe base hybri 2.35 Collision Rate.3 5 Messages passe Noes λ 6 8 x Figure. The collision rate as the size of the network grows. Figure 2. The number of routing layer messages passe as more traffic is generate. Figure shows the collision rate for each protocol. Notice that both the coe base an protocols have a significantly lower collision rate than the hybri system. The coe base protocol has a lower collision probability over the range of network sizes. Figure shows the average latency of routing level packets for each protocol. The coe base an pro- Figure 2 shows the number of messages passe for each protocol as network traffic increases. As we woul expect as more traffic is generate more messages are passe overall. The hybri protocol seems to pass the most messages with the an coe base protocols performing similarly. Figure 3 shows collision rate for each protocol as network traffic increases. The hybri protocol reaches satura- 7

8 .9 coe base hybri 6 Summary an Conclusions Collision Rate λ 6 8 x Figure 3. The collision rate as more traffic is ae to the system. tion with the lowest traffic rate of the three protocols. The protocol also reaches saturation soon after, but the coe base system seems to taper off with a collision rate aroun 3% coe base hybri We examine several coing constructs for use in multihop wireless networks. Of the currently use coes the incience matrix constructe superimpose coes ha better growth rates when scaling to large numbers of noes, an offere some flexibility in terms of coe ensity. There appear to be great gains possible if we are willing to relax the isjunct property of superimpose coes an use almost isjunct coes. We also examine how various properties of the coes use woul effect localize network performance. As we increase the number of noes in the system a very low ensity coe appears to be esirable. Increasing the reception probability locally woul also have a two-fol effect, it not only woul increase the network performance in terms of latency an throughput, but will also reuce the power costs of the system by avoiing collisions. In our simulations there was not a clear winner in terms of throughput or latency, but in terms of collision rate it appeare the almost isjunct coe performe the best. The hybri approach showe some potential, but also showe its limitations when scaling the number of noes in the system. References Average Latency λ 6 8 x Figure 4. The average routing layer packet latencyasmoretrafficisgenerate. Figure 4 shows average routing layer packet latency for each protocol as network traffic increases. As expecte, in general when we pump more traffic into the system the average latency increases. The hybri protocol seems to perform best of the 3 protocols with the an coe base protocols performing somewhat similarly, until the protocol is near saturation. Although the other protocols may seem to have favorable throughput numbers it seems the coe base system is better because it oes not saturate as quickly as the network traffic is increase. Throughout the situations teste as network characteristics scale up the coe base system, using approximate superimpose coes, outperforms the other two methos. [] A.J.MACULA, V.V.RYKOV, S. Y. Trivial two-stage group testing for complexes using almost isjunct matrices. Discrete an Applie Mathematics 37 (24), [2] CHLAMTAC, I.,AND FARAGÓ, A. Makingtransmission scheules immune to topology changes in multihop packet raio networks. IEEE/ACM Transactions on Networking 2, (February 994), [3] JU, J.-H., AND LI, V. O. K. An optimal topologytransparent scheuling metho in multihop packet raio networks. IEEE/ACM Transactions on Networking 6,3 (June 998), [4] KAUTZ, W.H.,AND SINGLETON, R. Nonranom binary superimpose coes. IEEE Transaction on Information Theory (October 964), [5] MACULA, A. J. Simple construction of -isjunct matrices with certain constant weights. Discrete Mathematics 62 (996), [6] SYROTIUK, V.R.,COLBOURN, C.J.,AND LING, A. C. H. Topology-transparent scheuling for manets using orthogonal arrays. Preprint, May 23. [7] THE MATHWORKS, I. Matlab. 8