Performance Modeling of Ad Hoc Networks with Time-Varying Carrier Sense Range and Physical Capture Capability

Transcription

1 Performance Modeling of 802. Ad Hoc Networks with Time-Varying Carrier Sense Range and Physical Capture Capability Jin Sheng and Kenneth S. Vastola Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute Abstract In a slow fading environment, the carrier sense range is not constant, so there is not a constant set of hidden terminals for a mobile station. The probability of capture with a set of interferers is not a fixed value either, and it fundamentally affects the loss rate and throughput of the whole network. We estimate the expectation of the capture probability in a single hop ad hoc network and incorporate it with our previously proposed model for 802. DCF that considers the time-varying nature of carrier sensing. The system throughput is then derived from an individual station s point of view. The model is verified against simulations, and extensive numerical experiments are performed to demonstrate its application. I. INTRODUCTION The wireless channel is complicated in nature. Any movement of wireless stations or slight changes in the environment could cause the signal strength to vary around a local mean that is determined by path loss and shadowing. As a result, the reliability to detect the carrier between senders varies over time even if a network s topology remains constant. We studied this issue in a symmetric single hop ad hoc network [] assuming the basic access mechanism of 802. DCF and a block-fading Rayleigh channel. The result is interesting yet not entirely convincing without considering the physical capture effect. We demonstrate the complications induced by a timevarying channel from a sender-receiver (S-R) pair s view as in Fig., in which A, A and A are stations competing with S. We draw two disks (not necessarily the same size) around S and R, and we denote PS A as the power received by S from A. The distance between A and S is inversely proportional to PS A, while the distance between A and R is inversely proportional to PR S/P R A, i.e. the signal-to-interference ratio. A competing station is within the disk centered at S if PS A > C, the carrier sense threshold, and it is within the disk centered at R if PR S/P R A > T, the physical capture threshold. Thus if the channel is constant, a competing station stays in one of the regions around an S-R pair, i.e. the hidden, exposed, deferred and don t-care regions. A station in the hidden region can not be detected by S and it would destroy the transmission between the S-R pair should it collide with S. An exposed station unnecessarily inhibits S from transmitting as it cannot prevent the signal of S from capturing R. A station in the deferred region is effectively controlled by the 802. DCF so as to have the minimal chance to collide with S. A station cannot neither be detected nor interfere with the S-R pair Exposed t Exposed S S t A Deferred A A A Deferred R Hidden Hidden A A R Fig.. S-R Centric View of Competing Stations. In a constant channel, all stations remain in a region. In a time-varying channel, a station can be in a region at time t while jumping to another region at time. Observed over a large time window, the position of a competing station follows a probability distribution (not necessarily a bell-shaped one). resides in the don t-care region, i.e. outside of both disks. In a time-varying channel, as all stations are separated by many times the wavelength, the variations of the channel gains between A, S and R are independent, so that a station jumps across the boundaries all the time. Thus as perceived by the S-R pair, the positions of the competing senders are random variables, which, to the best of our knowledge, has not been described mathematically with the physical carrier sense and backoff mechanism of 802. DCF in an ad hoc network. The concept of time-varying carrier sense and capture is briefly mentioned in [2], but the discussion is confined in a wireless LAN, assuming that all interfering frames have a fixed and identical mean power at the receiver, which is not suitable t

2 for ad hoc networks. In other discussions about capture effect in Rayleigh fading channels of IEEE 802. [3] [4] and ALOHA [5] protocols, perfect carrier sense is always assumed, hence the analysis is valid only within a limited area. Previous research on IEEE 802.-based ad hoc networks beyond a carrier sense range is conducted assuming the constant carrier sense capability. A numerical model of network performance in non-fading channel has been thoroughly developed in [6]. In this paper, we provide the first attempt to model the performance of an ad hoc network running on IEEE 802. DCF basic access scheme with considerations of both time-varying carrier sense and capture induced by a block-fading Rayleigh channel. We assume a saturated MAC queue in order to make use of the previous model [], and confine our analysis within a fixed homogenous topology where all senders have the same probability to transmit and the frame error is the same for each S-R pair, so as to avoid further complications in solving nonlinear equations. An iteration algorithm was developed in [7] to find the solutions for heterogenous topologies, but it cannot be readily extended beyond a carrier sense range to accommodate our system equations. Our paper is organized as follows. We present the modifications of system equations from our previous model in Section II and derive system throughput in Section III. In Section IV, we estimate the overall capture probability under a slow Rayleigh fading channel in a symmetrical network. Our model is verified against the simulation results, and we present the related content in Section V along with its application and limitation. Concluding remarks and a brief introduction to our on-going work are summarized in Section VI. II. SYSTEM EQUATIONS We have extended Bianchi s two-dimensional Markov chain [8] to study the probabilistic Carrier Sensing Outage (CSO) [] without capture. We were interested in the combined saturated frame error rate and throughput that can be achieved in a homogeneous single hop network consisting of n slowly moving contending stations with a quasi-stationary topology, given imperfect carrier sensing due to slow Rayleigh fading. For a tagged station, we have the knowledge of carrier sensing outage probabilities α = [α, α 2,..., α n ] between this station and its n contenders. The individual outage probability is calculated as the probability that the sum of the received mean power X i from station i (independent of the path loss and shadowing models) and the power of white noise N is less than the carrier sense threshold C []: α i = P[X i < C N] = e C N X i. Our analysis can be found in []. Here we focus on the results. For a network consisting of n pairs of sender and receiver running 802. DCF with the maximum retry limit m and minimum backoff window W min, given the conditional collision probability p and correct backoff probability q, the channel access probability τ is given by: τ = ( q) 2 /[( q) 2 + ( q) A B W min ], () where A = [ p + (p/2) m+ ]/( p/2), m B = ( p) ( p 2 )i q 2i W min + ( p 2 )m q 2m W min. i=0 If the physical layer capture is not considered, p and q can be determined by τ and the mean carrier sensing outage probability ᾱ respectively: n p = p coll = ( τ) (n ) ( α i ), (2) q = ᾱ[ ( τ) (n ) ]. (3) Equations (), (2) and (3) form a nonlinear system that can be numerically solved to determine the frame loss probability and system throughput given the knowledge of carrier sensing outage probability between the tagged station and all the neighbors α. The physical layer capture effect invalidates Equation (2). We redefine p as the conditional frame error rate and assume that the physical layer capture probability p cap is known. Equation (2) of the nonlinear system shall be replaced by: p = ( p cap ) p coll. (4) Now we have constructed a new nonlinear system to depict the behavior of the basic access mechanism of 802. DCF by Equations (), (4) and (3) with consideration of capture effect and carrier sensing outage. In Section IV, we will estimate the values of p cap, which is dependent on τ. III. SYSTEM THROUGHPUT When the capture effect is considered within one single CS range [7], i.e. every transmission can be detected by all other stations, the whole network is perfectly synchronized and the throughput of the entire network can be calculated once τ and p are determined by the nonlinear system. When hidden terminals are present, an alternative method was proposed to study the throughput of a single station [6] [9]. Our assumptions, however, invalidate both assumptions on time slot synchronization and constant set of hidden terminals. Inspired by the concept of average transmission time [6], we study the throughput from a station s point of view. As we assume a homogenous topology and a fixed frame length, all stations achieve the same numbers of successful and failed transmissions in the long run. Here we define average transmission time E[D] as the average duration between the end of two consecutive transmissions of a station, i.e. the expected time it takes to transmit one frame, no matter if it fails or not. This definition leads to a simple form of the normalized system throughput of a network with n stations whose payload in each frame takes the amount of time T L to transmit: S = n ( p) T L, (5) E[D] where the only unknown is E[D] which can be further decomposed into transmission, backoff and busy-waiting stages. Let

3 us omit the difference between the duration of a successful transmission T s and a failure, because it is not significant unless the data rate is high. Then the duration of transmission stage is always equal to T s. The average slot time spent in backoff is directly related to the frame loss rate p, by which we calculate the probability that the backoff counter is zero τ as follows [8]: τ = 2( 2p) ( 2p)(W + ) + pw( (2p) m ), (6) and the expected number of backoff slots is τ. The complications lie in the duration of the busy-waiting stage, in which the received signal strength is above the CS threshold so that the station refrains from transmitting. Undetected transmissions do not take a part here. The expected number of transmissions that can be sensed is given by i ( α i). The probability that a transmission collides with at least one other transmission with the knowledge of p and p cap can be easily deduced from Equation (4) as p coll = p/( p cap ). We omit collisions that involve more than two stations, so that the number of sensed busy slots are estimated as i ( α i)( p coll 2 ). At last, as we observed that when the carrier sensing is working perfectly, all collisions are perfectly aligned and the alignment is gradually violated as the CSO grows, we multiply + ᾱ to the number of sensed busy slots to accommodate this issue. Thus the average transmission time is calculated as follows: E[D] = ( τ )σ+t s [+( p coll 2 )(+ᾱ) i α i ], (7) where T s = PHY hdr + MAC hdr + T L + SIFS + ACK + DIFS, in which σ is the slot time of 802., PHY hdr and MAC hdr are the length of headers of physical and MAC layers respectively, ACK is the time spent on an ACK frame, SIFS and DIFS are the inter-frame spacings. IV. CAPTURE PROBABILITY ESTIMATION Since it was experimentally justified that several available 802. cards can capture stronger frames even if they are detected later than weaker frames [0], the capture conditions for our analysis and Qualnet implementation are: ) A frame that is weaker than the sensitivity threshold T cap, equal to the carrier sensing threshold as implemented in Qualnet, is not considered as signal. Instead, its power is combined with other types of noise and interference; 2) The strongest frame, whose power is above the sensitivity threshold, involved in a collision is treated as signal. The sum of the power of other frames, no matter how strong it might be, is treated as interference; The power of interference from the interferer i has a local mean X i and it is exponentially distributed due to Rayleigh fading. The combined power of the interference and the constant thermal noise N has a shifted version of exponential distribution: fi+n i (x) = X i e x N X i. (8) The received signal power follows an exponential distribution f S (y) whose mean is Y and the PDF of SINR is calculated as follows: f i Z(z) = 0 xf i I+N(x)f S (xz)dx = [ γ i γ + Y(γ i+z) γi (γ i+z) ]e z γ y, 2 (9) in which γ i = Y X i and γ Y = Y N. Since a frame must be stronger than the carrier sensing threshold to be detected, the effective SINR threshold to capture is T = max(t cap, C/N), and the capture probability is given by: T p i cap = fz i (z)dz = e γ Y + T. (0) γ i T For a strong wireless link, the capture probability can be γ further approximated as i γ. i+t Given a set of K interferers colliding with a tagged station, the PDF of the power of interference has a closed form if all the local mean powers at the intended receiver are distinct []. Even if there are multiple interferers with the same power, we can introduce some minor corrections and apply the following formula: f K I+N(x) = K j=,j i X K 2 i (X i X j) e x N X i. () The distribution of SINR with K interferers is: K fz K (z) = fi+n K (x)dx = X K i fz i (z). (2) X i X j N j=,j i The combined capture probability can be proved to follow a product form [2]: p K cap = K j=,j i T e γ Y X i X j X K i + T γ i = e T γy γ i γ i+t. (3) It is computationally expensive to quantify the probability that each combination of potential interferers collide with the tagged station, so as to calculate the corresponding capture probability of each case. This also requires us to consider the CSO involving multiple stations. Thus we resort to a first-order approximation to cope with the challenge. Let us look at the probability that a specific competing station i is involved in a collision with the tagged station. Note that we do not exclude the cases in which other stations are also involved in that collision. Based on the assumptions about the independence of transmission and CSO, given the channel access probability τ, the probability that the competing station i collides with the tagged station is estimated as: w i = ( τ i )( α i ) = τ i ( α i ) + α i, (4) which is chosen as the weight factor to calculate the overall capture probability. Several important properties are preserved

4 TABLE I PARAMETERS IN QUALNET SIMULATIONS 802. PHY 802.b Data Rate Mbps TX Power 5dBm CS Threshold -89 to -95dBm Retry Limit m 6 W min 32 RTS/CTS Disabled Slot Time σ 20µs SIFS 0µs DIFS 50µs PHY hdr 92µs MAC hdr 224µs ACK 304µs Payload 024bytes Network Radius +d Receiver -d Network Radius P cap m Sim +50m Model m Sim +00m Model m Sim 50m Model 0. 00m Sim 00m Model (a) Capture Probability Sender Fig. 2. Sample Homogeneous Topologies for n = 9 with two transmission directions. in it. First of all, the station that transmits faster has more influence on the capture probability when carrier sensing works. On the other hand, the station that senses the carrier with lower reliability has more chance to collide. With such an approximation, the overall unconditional capture probability in a network with n competing stations is given below and used in our numerical experiments: p cap = e T n γ Y γ w i i γ. (5) i+t Although this estimation does not consider multiple concurrent interferers explicitly, it does produce accurate computational results against simulations. In Section V, we validate our model based on the estimation, before discussing our observations and its application. V. VALIDATION AND APPLICATION A. Topologies and Settings Two sample topologies consisting of 9 contending stations are shown in Fig. 2. All wireless stations are placed on two concentric circles, in which all competing senders occupy one and receivers have the other. The radius of the circle that hosts senders is referred to as the radius of the network. A saturated CBR flow is set up between each S-R pair, which is collinear with the center of the circles. The distance between each S-R pair d is thus the difference of the concentric circle s radii. When the direction of transmission is towards to the center, then d < 0 and vice versa. With the same network radius, the CSO experienced by senders are similar, while the capture capability of the receivers vary with both d and the transmission direction. The topology is designed to enforce the same amount of competition on all the senders and same interference on all the receivers in the long run. We use P loss m Sim m Model m Sim 50m Model m Sim +00m Model m Sim +50m Model Normalized Throughput m Sim +50m Model +00m Sim +00m Model 50m Sim 50m Model 00m Sim 00m Model (b) Frame Loss Rate Fig. 3. (c) Normalized Throughput Model vs. Simulation n = 5, C = static routes to prevent routing oscillation and short routing packets. All nodes move at m/s towards the same direction and maintain the relative distances. We then vary the radius of the network while maintaining a constant d. Five rounds of simulations are conducted for each set of parameters and each round runs over 00 seconds. All shared parameters on IEEE 802. MAC are presented in Table I. B. Numerical vs. Simulation Results The effectiveness of the previous model without capture has been shown in []. This extended version relies on the estimation of the overall capture probability discussed in Section IV. In Qualnet, we tagged each frame with a boolean variable that is set to true only if the overall interference

5 power is greater than the thermal noise power. It allows us to record how many frames are received or lost with and without interference from other stations, so that we can calculate p cap for each round of simulation. It is straightforward to obtain the loss rate of DATA frames, as we observed that the loss rate of ACK frames is less than 2% even in the worst case. The normalized throughput within a specified duration is the ratio between the amount of delivered payload and the raw bit rate of the wireless channel. If there is no capture effect, the normalized throughput is strictly less than. But it can certainly exceed as we enable the capture effect to allow a portion of frames to survive collisions and gradually enlarge the network radius. Its value can be used as a factor of spatial reuse. Here we present the comparison of the overall capture rate, frame loss probability and normalized throughput for 5 competing stations obtained from simulations and our numerical model in Fig. 3a, 3b and 3c respectively. Both transmission directions are included and the distance between a S-R pair is 50 or 00 meters. The results of our model are closely matched with the simulation data, except the 00m case which is still acceptable. Varying the CS threshold from 89dBm to does not alter the statement, which are not shown due to the limited space. C. Observations As the network radius increases, the throughput experiences a three-phase change. In the first phase, all stations can hear each other perfectly and the channel access probability is well controlled to stay low. The boundary of this phase is related to the CS threshold. The higher the threshold, the earlier the network loses synchronization. The performance is dominated by capture effect in this phase, which falls into the same region as the work of Chang et al. [7] except the computation of capture probability. If the capture probability increases significant during the stage, the throughput of the network would obviously benefits. The next stage starts as the stations are losing perfect awareness of the whole network, and the collision probability rises rapidly as the effect of CSO cannot be ignored. If the capture effect has not grown strong enough to compensate the rocketing collision rates, the network is going to suffer significant frame losses. This phenomenon is clearly shown as the network radius ranges from 00 to 50 meters for the 00m cases in Fig. 3c. A high CS threshold leads to fast increasing collision rate that is hard to be kept up with, thus we can expect high loss rate and low throughput. This stage is a missing link out of previous related works and to the best of our knowledge, this paper is the first work that is able to describe it mathematically. Finally, the capture effect goes so strong that most frames survive collisions and the throughput is limited by sensed transmissions. In this stage, the throughput grows linearly and the rate of growth is almost solely determined by the CS threshold. But this time a high threshold leads to more throughput, because senders spend less time to withhold transmissions that should get through even with the interference. A fixed CS threshold is two-bladed. We have to trade the worst case throughput in a dense topology with the spatial reuse factor in a sparse one. This fact holds even if the Rayleigh fading is not considered. One may adjust the transmit power of each station and/or their CS thresholds to balance the two effects. Assuming a static wireless channel, an generic analytical model on the optimal ratio between transmit power and CS threshold was proposed in [3], which was further supplemented by [4], [5] and [6] that considered MAC overhead and multiple data rates. However, the effectiveness of these models and associated algorithms have not been verified in fading channels. We are looking forward to applying our model to the related topics. D. Application A fixed carrier sense threshold might not achieve the best performance at different network radii. However, our model enables us to quickly calculate the normalized throughput of a network with different size and MAC parameters. For instance, we set n = 5, and for each value of C between 89dBm and, we compute the normalized throughput as the network radius increases from 20 to 300 meters. Then at each step of the network radius, the throughput achieved at a different carrier sense threshold is again normalized to the maximum value. Finally, we can look for a threshold that remains competitive throughout the whole range. The results for transmission directions at [ 50, 75, 00] meters are shown in Fig. 4a, 4b and 4c respectively. Simply by inspection, we can identify the carrier sense threshold with the most stable performance for every case. For d = 50m, C = 90dBm or 9dBm guarantees that the throughput never drops under 90% of the best case. For d = 75m, a threshold between 9 and 92dBm can do that. We can t make the same statement for d = 00m, but C = 92dBm guarantees 85% of the maximum throughput. On the other hand, for d = +50, +75, +00m, C = 90dBm is the obvious choice as the throughput remains above 95% of the maximum. Due to the limited space, we only show the graph for d = +00m in Fig. 4d. E. Computational Expense and Limitations The numerical calculation routings are implemented using Matlab and we run both the numerical experiments and Qualnet simulations on a laptop with a Intel Core 2 Duo T5500 (.67GHz) processor. It takes over 7,000 seconds to produce the simulation results in a line in Fig. 4 but less than seconds to compute the capture probability, loss rate and throughput at the same points using our model and have them plotted. That is 0 4 times of speedup. The accuracy of our model is limited by the estimation of capture probability which is not easy when the network begins to lose synchronization due to CSO. The first-order approximation that we used in this paper exhibits good performance unless the wireless link is weak and the interference is strong.

6 A computationally efficient technique has to be developed to generate precise results for wireless links under any conditions. VI. CONCLUSION We considered both slow Rayleigh channel induced issues, i.e. time-varying carrier sensing outage and physical layer capture, in a numerical model of system throughput for a static homogenous wireless network consisting of multiple saturated single hop flows. We gave first order estimations of physical capture probability with multiple Rayleigh interferers. The model is extremely computational efficient and accurate against simulation results, so that we can compute the performance with various settings and find the optimal value of MAC parameters. We are currently working on further enhancement of our estimation of physical capture probability and extending our results to an arbitrary topology. REFERENCES [] J. Sheng and K. S. Vastola, Physical Carrier Sensing Outage in Single Hop IEEE 802. Ad Hoc Networks with Slowly Moving Stations, in Proc. IEEE WCNC 08, [2] Z. Hadzi-Velkov and B. Spasenovski, The influence of flat rayleigh fading channel with hidden terminals and capture over the ieee 802. wlans, in IEEE VTC 0, 200, pp vol.2. [3] J. H. Kim and J. K. Lee, Capture effects of wireless csma/ca protocols in rayleigh and shadow fading channels, IEEE Transactions on Vehicular Technology, vol. 48, no. 4, pp , 999. [4] X. Li and Q.-A. Zeng, Capture effect in the ieee 802. wlans with rayleigh fading, shadowing, and path loss, in IEEE WiMob 06, June 9-2, 2006, pp [5] J. Arnbak and W. van Blitterswijk, Capacity of slotted aloha in rayleighfading channels, IEEE Journal on Selected Areas in Communications, vol. 5, no. 2, pp , Feb 987. [6] K. Medepalli and F. A. Tobagi, Towards performance modeling of ieee 802. based wireless networks: A unified framework and its applications, in Proc. IEEE INFOCOM 06, [7] H. Chang, V. Misra, and D. Rubenstein, A general model and analysis of physical layer capture in 802. networks, in Proc. IEEE INFOCOM 06, [8] G. Bianchi, Performance analysis of the IEEE 802. distributed coordination function, IEEE Journal on Selected Areas in Communications, vol. 8, no. 3, pp , [9] T.-C. Hou, L.-F. Tsao, and H.-C. Liu, Analyzing the throughput of IEEE 802. DCF scheme with hidden nodes, in Proc. IEEE VTC 03, vol. 5, 2003, pp [0] A. Kochut, A. Vasan, A. U. Shankar, and A. Agrawala, Sniffing out the correct physical layer capture model in 802.b, in Proc. IEEE ICNP 04, 2004, pp [] Y. D. Yao and A. U. H. Sheikh, Outage probability analysis for microcell mobile radio systems with cochannel interferers in rician/rayleigh fading environment, Electronics Letters, vol. 26, no. 3, pp , 990. [2] M. Zorzi and S. Pupolin, Optimum transmission ranges in multihop packet radio networks in the presence of fading, IEEE Transactions on Communications, vol. 43, no. 7, pp , Jul 995. [3] J. Zhu, X. Guo, L. L. Yang, W. S. Conner, S. Roy, and M. M. Hazra, Adapting physical carrier sensing to maximize spatial reuse in 802. mesh networks: Research articles, Wirel. Commun. Mob. Comput., vol. 4, no. 8, pp , [4] X. Yang and N. Vaidya, On physical carrier sensing in wireless ad hoc networks, in Proc. IEEE INFOCOM 05, vol. 4, 2005, pp [5] T.-S. Kim, J. C. Hou, and H. Lim, Improving spatial reuse through tuning transmit power, carrier sense threshold, and data rate in multihop wireless networks, in Proc. ACM MobiCom 06, 2006, pp [6] T. Y. Lin and J. C. Hou, Interplay of spatial reuse and SINR-determined data rates in CSMA/CA-based, multi-hop, multi-rate wireless networks, in Proc. IEEE INFOCOM 07, 2007, pp dBm dBm 9dBm dBm 5 94dBm (a) d = 50m 89dBm dBm 9dBm dBm 5 94dBm (b) d = 75m 89dBm dBm 9dBm dBm 5 94dBm (c) d = 00m 89dBm 90dBm 9dBm 5 92dBm 94dBm (d) d = 00m Fig. 4. n = 5