Analysis and Implementation of Scalable Clock Synchronization Protocols in IEEE Ad Hoc Networks

Transcription

1 Analysis and Implementation of Scalable Clock Synchronization Protocols in IEEE Ad Hoc Networks Dong Zhou Ten-Hwang Lai Department of Computing and Information Science The Ohio State University {zhoudo, Abstract This paper studies a fundamental problem, clock synchronization, in IEEE ad hoc networks. Clock synchronization is important for frequent hopping spread spectrum (FHSS) to ensure that all stations hop at the same time; it is also necessary for FHSS, direct sequence spread spectrum (DSSS) and Orthogonal Frequency-Division Multiplexing (OFDM) to perform power management. The synchronization mechanism specified in the IEEE standards has severe scalability problem. Remedies have been proposed to solve the scalability problem, but these solutions either can not handle mobility very well or the protocol is too experimental without solid analysis. In this paper, we analyze the root cause of the scalability problem and propose two protocols with analytical guidance for implementation. The new solutions are distributed, scalable and very adaptive to station mobility. The maximum clock drift is improved from over 4000 µs to under 125µs and 50µs respectively for our new protocols. Keywords: IEEE , ad hoc networks, scalability, clock synchronization 1 Introduction The IEEE standards support the peer-topeer mode Independent Basic Service Set (IBSS), which is an ad hoc network with all its stations within each other s transmission range. An ad hoc network can be constructed quickly without much planning and any infrastructure support. Clock synchronization is important for frequent hopping and power management. IEEE standards [1, 2, 3, 4] specify a Timing Synchronization Function (TSF) for the ad hoc mode in the MAC layer s MAC protocol has been extensively studied [5, 6, 7] TSF scalability problems and solutions have been presented only recently [8],[9]. The TSF works well if there are only a couple of stations in an IBSS. With the availability of a and g, an IEEE IBSS can easily support a few hundred nodes as far as bandwidth is concerned (54 Mbps). Huang and Lai [8] show that TSF can be out of synchronization frequently when the number of stations is not very small. The problem is more severe the larger the size of an IBSS. A distributed algorithm called adaptive timing synchronization procedure (ATSP) was proposed in [8] to address the scalability problem. It scales very well compared with TSF. But the protocol is slow to be stabilized. It could be out of synchronization before the protocol is stabilized and it is also slow to be re-stabilized after the fastest station leaves the IBSS. Station mobility handling is another drawback for ATSP. An improved time synchronization procedure was proposed in [9] to better handle station mobility. However, it is too experimental and does not give guidance on how to set the values of various parameters. In this paper, we will analyze the root cause of the scalability problem in detail and propose two new protocols with analytical guidance for implementation. The new solutions are fully distributed, scalable and very adaptive to station mobility. The maximum clock drift can be controlled within 50 to 125 µs. It is a huge improvement compared with over 4000 µs for TSF and over 500 µs for ATSP. The rest of the paper is organized as follows. Section 2 reviews TSF and ATSP. Root causes /04/$20.00 '2004 IEEE 255

2 of the scalability problem are analyzed. Section 3 proposed two solutions to solve the problem. Simulation results are discussed in section 4. Section 5 concludes the paper. 2 Overview This section reviews the TSF and ATSP. The root cause of the scalability problem is also analyzed. The analysis results of the severity of the asynchronism are summarized here to help the analysis of our new protocols in the next section. 2.1 IEEE TSF and ATSP In TSF, clock synchronization is achieved by stations periodically exchanging timing information through beacon frames, which contains a timestamp among other parameters. All stations in the IBSS competes for beacon transmission every abeaconperiod second. The typical value is 0.1s. This time period is called a beacon period (BP). At the beginning of each beacon period, there is a beacon generation window consisting of w + 1 slots each of length aslottime, where w = 2 acwmin. The value of acwmin is 15 for FHSS and OFDM, while the value of acwmin is 31 for DSSS. Each station calculates a random delay uniformly distributed in [0, w] and is scheduled to transmit a beacon when the delay timer expires. If a beacon arrives before the random delay timer has expired, the station cancels the pending beacon transmission and the remaining random delay. Upon receiving a beacon, a station sets its TSF timer to the timestamp of the beacon if the value of the timestamp is later than the station s TSF timer. (It is important to note that clocks only move forward and never backward) The number of slots needed to send a beacon is called beacon length. It differs from 4 to 34 for various standards and bit-rates. ATSP was proposed in [8] to solve the scalability problem. The basic idea is to let the fastest station to compete for beacon transmission every beacon period and let other stations to compete only occasionally. Each station i keeps a variable I(i), which means how many BPs it waits before competing for beacon contension. It is initialized to a random number between 1 and I max. Whenever station i receives a beacon with a timing value later than its own, the station sets its timer to this value, increases I(i) by 1. If station i does not receive any beacon frame with a timing value later than its own for I max consecutive BPs, it decrements I(i) by 1 In ATSP, the fastest station will not receive a timing value greater than its own; its I-value will gradually decrease to 1 and stay there. The other stations will gradually increase their I-values until they reach I max. So the fastest station will compete every beacon period while other stations competes every I max beacon periods. 2.2 Metrics for asynchronism Before we start the root cause analysis on the TSF scalability problem, let s define the metrics of asynchronism. If the clocks of two stations differ so much that either power management or frequency hopping can not work properly, the two stations are said to be out of synchronization. Let be the maximum clock difference tolerable by power management and FHSS. Assume that the clocks in an IBSS are all different in speed (or accuracy), which is typically the case in real systems. Thus, there is a unique fastest station, whose clock is fastest in the system. We are particularly interested in three situations: Fastest-station asynchronism This refers to a situation where the fastest station s clock is ahead of all other stations clocks by more than. Global asynchronism Given a value k between 1 and 100, k percent global asynchronism (or simply k percent asynchronism) refers to the situation that at least k percent of the n(n 1)/2 pairs of stations are out of synchronization. Simple asynchronism This refers to a situation where the fastest station s clock is ahead of the slowest station s clock by more than. The difference of the fastest and slowest clock is called maximum clock drift. 2.3 Root Causes of Asynchronism and Some Possible Remedies All three types of Asynchronism are due to the fact that the fastest or all the faster stations repeatedly 256

3 lose beacon contentions. How can stations repeatedly lose beacon contentions? There are several reasons due to s design of the TSF. Beacon collision: Stations contend with one another for beacon generation. Equal opportunity: All stations are equal in their opportunity to win a beacon contention. There is at most one successful beacon transmission in each beacon period. Unidirectional clocks: The ad-hoc-mode TSF let clocks move only forward and never backward. A station adjusts its clock only if the clock s time is behind (smaller than) the timestamp on an arriving beacon. In order to fix the TSF s scalability problem, we need to remove one or more of its causes. There may be various ways to get around each of the above causes, but, unfortunately, not all of them work well. Before describing our proposed solution, we wish to outline several unsatisfactory solutions, which would help the reader understand why we have designed the proposed protocols as they are. Reducing the beacon contention rate. Instead of contending every BP, a random delay is introduced. It only contends for beacon transmission when the delay timer is up. This scheme does not work well because the fastest station may delay its contention for long time and cause the fastest clock to drift further away from the pack. Let the fastest clock to contend every BP and other stations contend not frequently (i.e. ATSP). It takes long time for ATSP to figure out who is the fastest station, which may result in large maximum clock drift. This scheme does not react well when the fastest station leaves the IBSS. It will take IBSS quite a while to recognize the departure of the fastest station and select the next fastest station. Allowing multiple beacon transmission in the same BP. This can help in the situation that a slow station sends the beacon first and blocks a potential fast station from sending out beacon. But this solution alone can not solve the beacon collision problem when the number of stations is large. Allow stations to set their timers forward or backward as appropriate; i.e., no matter which station wins the beacon contention, let all other stations synchronize their timers with the beacon s timing. This approach is undesirable for two reasons. First, the collisions still happen quite often with large number of stations. We will show in simulation that the improvement is limited. It still suffers the scalability problem. Second, it is incompatible with the current TSF. To see this, suppose the fastest station in the IBSS happen to implement the current TSF, while the rest of the IBSS happen to implement the modified TSF. Also, suppose the fastest station and the slowest station are currently out of synchronization, and the slowest station happens to win the beacon contention, then the fastest station s timer will stay intact, while the others all synchronize with the slowest station. This would result in more out-of-sync links. 2.4 Analysis of Beacon Contention We will only summarize the results of the analysis due to the space limitation. Please refer to [10] for detail. Suppose there are n stations, and denote the beacon generation window by [0, w], which consists of w + 1 slots numbered 0 through w. Let b be the beacon length. Let p(n, w) be the probability of the event E that at least one of the n stations succeeds in beacon transmission in window [0, w]. + ( ) n W p(n, W ) = p(n, W 1) W + 1 ( ) ( ) n 1 1 W + n W + 1 W + 1 { n n i ( ) i ( ) j 1 b 1 Ci n C n i j W + 1 W + 1 i=2 j=0 ( ) n i j W b + 1 p(n i j, W b)} W + 1 (1) The boundary condition for p(n, w) is p(0, w) = p(n, 0) = 0, provided n > 1. Let p (n, w) be the probability that a particular station, say A, successfully sends a beacon in a given BP. 257

4 p (n, w) = 1 w + 1 w p (n, w, k) (2) k=0 C n 1 x π(i, x, y) = Cx n 1 y ( ) x ( ) y ( ) n 1 x y 1 b 1 w i b + 1 (3) w + 1 w + 1 w + 1 For n 3 and k b, the function p (n, w, k) satisfies the following recurrence: p (n, w, k) = ( ) n 1 w k + w + 1 k b n 1 i=0 x=2 n 1 x y=0 {π(i, x, y) p (n x y, w i b, k i b)} (4) The boundary condition for p (n, w, k) is p (0, w, k) = 0 and p (n, 0, k) = 0. Let E(L ) be the expected length of time between two consecutive incidents of fastest-node asynchronism. d is the difference in clock accuracy between the fastest station and the second fastest station. T is the length of each beacon period. τ = /(d T ). E(L ) = 1 p (n, w)(1 p (n, w)) τ 1 p (n, w) 3 Our Solutions 3.1 Adaptive bi-directional TSF (5) If we do not need to consider the compatibility issue and design the TSF from scratch, the bi-directional TSF can become a very good scheme with minor changes. We call the new protocol ABTSF. From our simulation, we can see collision is the major reason for the clock drift. We need to reduce the collision by controlling the frequency of beacon contention. Each station competes for beacon transmission every I max BPs. If I max is large enough, the possibility of collision is very slim. But a large I max increase the chances of idle when no station is competing for beacon transmission. So we introduce the concept of token. The token is assigned to the initiator of the IBSS first and the token is rotated among the stations periodically. The token holder will compete for beacon transmission every beacon period and other stations competes for beacon transmission every I max BPs. The token a Beacon Frame Extension Time Stamp Beacon Interval... ~ 550 bits Token Bits 2 bits Figure 1: Beacon Frame Format for ABTSF token bits role 00 none token holder 10 token holder 11 token holder releasing token 01 starting token contention Table 1: Token bits explanation holder consumes more power due to its frequency of beacon contention, rotation of token holder will distribute the power consumption. Each station maintains a list of station IDs of the IBSS. These IDs are collected by deriving the source address of the beacon. When the token holder wants to release the token to another station, it randomly picks one station from the list and announces the new token holder. To implement the token, we extend the beacon frame format, adding two token bits. Fig. 1 shows the frame format. The token holder will set the token-bit to 10 and other stations set the token-bit to 00. The token holder will set it to 11 when it starts the token rotation and the ID of the new token holder is also added in the beacon frame. When the old token holder sees a new beacon with token-bit 10, it resets the token-bit back to 00. When a station sees an idle beacon period, it implies the beacon holder leaves the IBSS or crashes, each station sets its token-bit to 01 and contends to be the beacon holder. The station of the first successful beacon transmission with token-bit 01 will become the new beacon holder. Other stations will reset their token-bit to 00 after receiving a beacon with token-bit 01. There is only one parameter to pick for ABTSF, I max. Eq. 6 is used to select I max. p(m, w) 0.95, m = 1 + n/i max (6) Eq. 1 can be used to calculate the value of p(m, w), combing Eq. 6, the value of I max can be 258

5 p(n,w) Analysis w=30 b=4 Analysis w=30 b=13 generation? We now propose a three-tier protocol called TATSF 1 to utilize this capability and handle station mobility more gracefully. TATSF: Tiered Adaptive Timing Synchronization Function Number of Stations Figure 2: p(n, w) determined. This solution is very scalable and adaptive to station mobility. The overhead is only 2-bit plus the new token ID at rotation time. The 2-bit addition can be considered free since this addition does not increase beacon length. The rotation happens once in a while. The overhead is also minor. We will show later in the simulation that the maximum clock drift can be controlled to be less than 50µs for number of stations from 100 to 600. The movement of token holder has very slim impact since the new token holder can be selected very fast. With ATSP, it may take up to Imax 2 BPs before the new leader is selected. For ABTSF, the first successful beacon transmission with token bit 10 marks the end of leader election. The average BPs to elect a new initiator is: E(leader) = 1/p(n 1, w) (7) From Fig. 2 we can see, E(leader) < 2 if 2 n 70 for b (w = 30, b = 4) and E(leader) < 2 if 2 n 115 for a (w = 30, b = 13). Remember that there is no need to elect a leader unless the token holder crashes. Even when the token holder does crash, it only takes a few beacon periods to elect a new leader. 3.2 Tiered Adaptive Timing Synchronization Function Now, suppose we must keep the IEEE TSF s provision of moving forward only. Since faster stations can synchronize slower stations but not the other way around, why not give faster stations a higher priority in the process of beacon 1. At the beginning of each BP, if C(i) I(i) then resets C(i) to zero and let station i participate in beacon contention. 2. [Initialization] If station i initiates a new IBSS, let I(i) := 1; otherwise, when it joins the (existing) IBSS, let I(i) := I max2. In any case, let C(i) := random(0..i(i)). 3. Each station keeps an observation counter, its initial value is 0 when it joins the IBSS. The counter is incremented by one for each BP. When the counter reaches a pre-set value, the station is at the end of observation period and the counter is reset to At the end of each observation period, update I(i) as follows. 1 if est rank(i) = 1 I(i) := I max2 if 1 < est rank(i) THD if est rank(i) > THD I max4 If there is an actual change in the value of I(i), let C(i) := random(0..i(i)). 5. At the end of each beacon contention window, update I(i) as follows: I max4 if est rank(i) > THD I(i) 2 I(i) := if there are collisions min{i max3, I(i)/2 } if the medium was idle In any case, let C(i) := random(0..i(i)). 6. At the end of each BP, increment C(i) by 1. In order to classify the stations into different tiers, envision that the stations are ranked according to their frequencies, with the fastest station ranked 1 The time synchronization procedure proposed in [9] is a two-tier protocol, mobility handling is more refined by adding a tier. 259

6 number 1, the second fastest station ranked number 2, and so on. With such ranks, we can put those stations with rank 1 in S 1, and those with ranks between 1 and some threshold in S 2, and put the rest in S 3. Unfortunately, precise ranks are not easy to obtain, and so we satisfy ourselves with an estimate. Each station divides the time line into observation periods each consisting of OBS BPs, where OBS is a pre-specified integer. In each observation period, each station counts the number of stations different stations from which it has received at least one beacon with a timing later than its own timer, and uses this number as an estimate of its rank. More precisely, at the end of each contention window, station i estimates/updates its rank as: est rank(i) = 1 + the number of different stations with faster beacons received by i (8) I(i) is an integer that indicates how often station i will participate in beacon contention. Its values is increased if collision happens or decreased if the medium is idle. Analysis of TATSF There are 5 prameters in the TATSF protocol: THD, OBS, I max2, I max3, I max4. The protocol s performance hinges on the values of these parameters. Should THD = OBS, I max2 = 1, I max3 = 1, and I max4 = 1, the protocol degenerates to the current TSF. This section discusses how to set these parameters. Lemma 1 The fastest station is always in S 1 except possibly for the first several BPs after it joins or initiates the IBSS. Proof. Let f be the fastest station in the IBSS. When f joined the IBSS, it synchronized with some station, from which it heard a beacon. At that moment, f s timer might or might not be the largest (latest). Let δ be the difference between f s timer and the largest (latest) timer in the IBSS, and let d be the difference in accuracy between f s timer and the second fastest timer in the IBSS. Then, at most a number δ/(d T ) of BPs after f joined the IBSS, f s timer would be the latest and est rank(f) would always be 1. Thus, As long as f stays in the IBSS, it will stay in S 1 until a new fastest station joins the IBSS. The case where f initiated the IBSS is similar. In normal situation, tier 1 stations contend for beacon generation in every BP, and on average there are S 2 /I max2 tier 2 stations participating in each contention. The number of beacon contenders from tier 3 can be ignored if I max4 is set large enough. On average, there are m = S 1 + S 2 /I max2 contending stations in each BP. From Eq. 1, the probability that at least one of the m stations wins the contention is more than 0.95 if m 20. Whenever a station wins the contention, its beacon synchronizes all timers with a smaller timing value. Let be the desired accuracy of timing synchronization for the IBSS and L be the time interval between two consecutive events of simple asynchronism. Then E(L ), the expected length of L (in terms of the number of BPs), can be approximated using Eq. 5, E(L ) 1 p (m, w)(1 p (m, w)) τ 1 p (m, w) (9) where τ = /(d T ) and d is the difference in accuracy between the fastest and the slowest stations in S 1 S 2. Note that E(L ) approaches infinity as p approaches 1. In order to have a sufficiently large E(L ), it s important that p be as large as possible. For instance, if = 100µs, w = 30, and d = 0.01%, then E(L ) 352 years, 141 days, 3 days, for p = 0.9, 0.8, 0.7, respectively. The p (m, w) in the above formula is the fastest station s probability of winning a contention where m = S 1 + S 2 /I max2, and w is the size of the beacon generation window. From Eq. 2, in order to obtain a value as large as 0.7 for p (m, w), m must be less than 2. That is, S 1 + S 2 /I max2 < 2, which implies S 2 /I max2 < 1 (since S 1 1 by Lemma 1). On the other hand, it is desired that S 2 /I max2 0. To see this, suppose S 2 /I max2 = 0. Then, S 2 = 0 or I max2 =. If I max2 =, then S 3 = ; should all stations in S 1 leave the IBSS, all stations in S 2 (i.e., all stations in the IBSS) will enter beacon contention and force many collisions. If S 2 = 0, then should all stations in S 1 leave the IBSS, all stations in S 3 will enter beacon contention, which is too slow. Thus, we wish S 2 and I max2 to statisfy 0 < S 2 /I max2 < 1. I max2 should be small enough so that I(i) will reduce to 1 quickly in case the medium was idle. Desirably, log I max2 4. I max3 should be bigger than I max2 but small enough so that I(i) will reduce to 1 quickly in case 260

7 the all the stations in S 1 and S 2 leave the IBSS. Desirably, log I max3 5. If all stations in S 1 leave the IBSS, we want to quickly identify the new members of S 1. In the worst case this process may take OBS BPs. Thus, OBS should not be too large, enen though the larger the value of OBS, the more accurate is est rank(i). Tha value of I max4 is more flexible than the other parameters as long as it is sufficiently large. Idealy, I max4 > 2n. We intended to let the fastest station to win the contention with a probability no less than 0.7. With such an odd for the fastest station, the stations in S 2 would win no more than 0.3 OBS contentions in an observation period. Therefore, THD should be no more than 0.3 OBS. 4 Simulation Study on Maximum Clock Drift of TSF, bi-directional TSF and TATSF In the simulation, we let the clock speed uniformly distributed in the range of [ d, d]. We pick d = 0.01%. This is a very inaccurate clock. It is the worst clock accuracy allowed by the standard. Due to the low cost nature of devices, inaccurate clock with d = 0.01% is often used in these devices. We also study the impact of packet transmission error and mobility of stations. We run the simulation for OFDM system: w = 30, b = 4. The desired accuracy of timing synchronization ( ) is 100µs. We use the result of our analysis as the guideline to pick the parameters of TATSF: T HD = 3, OBS = 10, I max2 = 16, I max3 = 32 and I max4 = For ABTSF, we set I max = 16 when n 200; 32 otherwise according to Eq. 6. We set up our simulation as follows: We run the simulation for beacon periods. We set the packet error rate to be 0.01%. The accuracy of receive timestamp is to within ±5µs of the reported clock time. The local clock is updated only when receiving a faster timestamp that is more than 5µs. binum max drift average max drift µs 1770 µs µs 1470 µs µs 2047 µs µs 2053 µs µs 226 µs Table 2: Maximum clock drift of bi-directional TSF for n = 200 We let 5% of the fastest stations leave at beacon period k 2000 ( k > 1 ). They return after 500 beacon periods. We measure the performance by the largest clock drift between the fastest clock and slowest clock in the IBSS during the whole simulation period. The performance metric we use is simple asynchronism. The reason is that fastest-station asynchronism and global asynchronism rarely happen for all the proposed protocols except TSF, only simple asynchronism can distinguish the best protocols among the improved schemes TSF/ bi-directional TSF Fig. 4 shows the maximum clock drift for TSF when n = 200. We can see the maximum is more than 4000µs. For ATSP, it is difficult to pick the value of I max. If I max is small, then it is not scalable. If the I max is large, it takes long time to find the fastest station. Either way, the maximum clock drift can be over 500 µs. Due to space limitation, please refer to [9] for simulation details. Bi-directional TSF is the same as TSF except that the clock will reset its value whenever a beacon is received regardless which clock is faster. We compare the maximum clock drift in a IBSS where some stations run TSF and some run bi-directional TSF. Let binum be the number stations running the bi-directional TSF. When binum = 0, it becomes TSF. When binum = n, then all the stations run bi-directional TSF. It is a mixed mode when binum takes other value. Table 2 shows the maximum and average clock drift for bi-directional TSF when n = 200. The maximum drift is still over 1500µs when running in complete bi-directional mode. It is a good improvement from the TSF where the maximum drift is over 4000µs, but still not good enough. When n = 300, the max- 261

8 max clock drift(us) TATSF n= 600 n TATSF max drift ABTSF max drift µs 39 µs µs 41 µs µs 41 µs µs 42 µs µs 41 µs µs 41 µs Table 3: ABTSF Maximum clock drift of TATSF and Number of BP Figure 3: Maximum clock drift for TATSF imum drift will be over 20000µs. In mixed mode, the performance is more or less at the level of TSF. The scalability issue is still not resolved. TATSF/ ABTSF Table 3 shows the maximum clock drift for TATSF and ABTSF with n ranging from 100 to 600. Fig. 3 shows the maximum clock drift during each beacon period for TATSF when n = 600. The improvement over TSF is quite large for both TATSF and ABTSF. We can see the maximum drift does not increase with the number of stations. It is very scalable. To our surprise, the maximum drift cross the 100µs mark. When we rerun the same simulation with I max2 = 4, the maximum drift drop to 82 for 100 stations and 99 for 400 stations. It seems not consistent with the analysis. According to the analysis, S 1 should be close to 1 with a larger I max2. The fastest station is in contention every beacon period, the clock should be very synchronized. That is true when the timestamp can be trusted at face value and the air interface is error-free. In the simulation, when a station receives a timestamp that is faster than its local clock but less than 5µs faster, the receiving station will not increase est rank. So it is possible, there are multiple stations in S 1. The average size of S 1 is 6.2 and the average size of S 2 is When all the stations in S 1 leave the IBSS, the stations in S 2 can quickly become member of S 1 with a small I max2. We can avoid multiple rounds of idle under these situation. Packet errors and movement of satiations have bad impact in term of maximum clock drift. The impact is much limited due to the adaptive nature of TATSF. For example, the maximum drift for 400, 600 stations can be reduce to 72,74 µs with stationary IBSS and error-free radio channels. Our assumptions are very conservative. In the real life, maybe only one station in S 1 leaves during one beacon period. We let all the stations leave at the same beacon period and the impact is still very limited. 5 Conclusion In this paper, we analyze the root cause of the scalability problem and propose new protocols with analytical guidance for implementation. The maximum clock drift can be higher than 4000 µs for TSF. ATSP can reduce the maximum clock drift to around 500 µs but it does not handle mobility well. For the two protocols we proposed in this paper, TATSF is compatible with the TSF and can control the maximum clock drift under 125µs. ABTSF is deigned from scratch and it can achieve even better synchronization (the maximum clock drift is under 50µs). The new solutions are fully distributed, scalable and very adaptive to station mobility. References [1] IEEE Std Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specification, 1999 edition. [2] IEEE Std a. Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specification, 1999 edition. 262

9 TSP n= 200 [10] Ten-Hwang Lai, Dong Zhou, L. Huang On the Scalability of IEEE Ad-Hoc-Mode Timing Synchronization Function. In preparation for journal submission max clock drift(us) Number of BPs Figure 4: Maximum clock drift for TSP [3] IEEE Std b. Higher-Speed Physical Layer Extension in the 2.4 GHz Band, 1999 edition. [4] IEEE Std g. Amendment 4: Further Higher Data Rate Extension in the 2.4 GHz Band, 2003 edition. [5] G. Bianchi, L. Fratta, and M. Oliveri. Performance evaluation and enhancement of the CSNA/CA MAC protocol for wireless LANs. In Proc. PIMRC 1996, pp [6] G. Bianchi. Performance analysis of the IEEE distributed coordination function. IEEE Journal on Selected Areas in Communications, 18(3): , [7] Y. C. Tay and K. C. Chua. A capacity analysis for the IEEE MAC protocol. ACM/Baltzer Wireless Networks, 7(2): , [8] Lifei Huang, Ten-Hwang Lai. On the Scalability of IEEE Ad Hoc Networks In Proceedings of MobiHoc 2002, pp [9] Ten-Hwang Lai, Dong Zhou, Efficient and scalable IEEE ad-hoc-mode timing synchronization function, 17th International Conference on Advanced Information Networking and Applications, 2003, pp