The free space and two-ray ground reflection models were widely used in prior work. According to such channel models, the transmission range of a node is circular, and all of the nodes within the disk are one-hop neighbors. In typical wireless network environments, however, the variations in the received powers as measured by different receivers with the same distance from a same transmitter are random and independent [32], and can be characterized by the log-normal slow fading [16]. Consider a pair of transmitter and receiver where the distance between them is d. According to the log-normal shadowing propagation model, the received power of the intended signal at the receiver in dB, denoted as P_r,dB, is represented by [16]: (1) P r , d B = P 0 , d B − 10 β log ( d d 0 ) + X rwhere P₀,_dB is the reference power measured at a distance of d₀ in dB, d₀ is the reference distance, β is the path-loss exponent, and X_r represents a normal random variable with zero mean and standard deviation σ_dB. That is, X_r is a random variable with the following normal probability density function, (2) f X r ( x ) = 1 2 π σ d B exp ( − x 2 2 σ d B 2 )

In (1), the received power consists of two parts: the deterministic attenuation depending on the distance between the transmitter and the receiver, and the probabilistic part X_r. The variation of the received power is determined by the standard deviation σ_dB. Note that typically σ_dB ranges from 4 dB to 12 dB in outdoor environments [33]. Due to the probabilistic part X_r, the transmission range of a sending node is not circular anymore, as shown in Figure 1. The received power can be expressed in unit of Watts as follows: (3) P r = P 0 10 − β log ( d d 0 ) + 0.1 X r = C ( Y r d β )where P₀ is the reference power measured at d₀ in unit of Watts, C = P 0 d 0 β is a constant, and Y_r = 10^{(0.1 X}_r⁾ represents a log-normal random variable with zero mean and standard deviation σ = ( ln 10 10 ) σ d B.

We next derive the success probability of a transmission in the presence of other interfering transmissions. Suppose that there are N interfering transmitters around a tagged receiver, and that the distance between interfering Node i and the receiver is r_i. The distance between the tagged transmitter and receiver is d. Similar to (3), the interference by interfering Node i, P_i, is represented by: (4) P i = P 0 10 − β log ( r i d 0 ) + 0.1 X i = C ( Y i r i β )where Y_i is a log-normal random variable with zero mean and standard deviation σ. From (4), the cumulative interference P_I measured at the receiver amounts to: (5) P I = ∑ i = 1 N P i = ∑ i = 1 N C ( Y i r i β )

From (3) and (5), we obtain the signal-to-interference ratio (SIR) at the tagged receiver as: (6) 1 SIR = P I P r = ∑ i = 1 N C ( Y i r i β ) C ( Y r d β ) = ∑ i = 1 N ( d r i ) β ( Y i Y r )

In order to receive and decode the packet successfully, the SIR at the tagged receiver should be greater than a threshold T_SIR. The probability of such an event is: (7) P succ = Pr { SIR > T SIR } = Pr { 1 Y r ∑ i = 1 N ( d r i ) β Y i < 1 T SIR }

Let U_t = (d/r_s)^β Y_i, which is a log-normal random variable with mean μ_i = β ln (d/r_s) and variance

σ². Furthermore, let V = ∑ i = 1 N U i. Since U_i's are independent, V can be approximated by a log-normal random variable W with the following mean μ_w and variance σ w 2 according to the Fenton-Wilkinson approximation method [34]: (8) { μ w = log ( ∑ i = 1 N e μ i ) + σ 2 − σ w 2 2 σ w 2 = log [ ( e 2 σ 2 − 1 ) ∑ i = 1 N e 2 μ i ( ∑ i = 1 N e μ i ) 2 + 1 ]

Finally, (W/Y_r) is a log-normal random variable with mean μ_w and variance ( σ w 2 + σ 2 ).

By utilizing the logistic distribution for the cumulative distribution function (CDF) of the log-normal distribution [35], F ( x ; μ , σ ) = [ ( e μ x ) π / ( σ 3 ) + 1 ] − 1, we finally obtain the success probability of the transmission as: (9) P succ = [ ( T SIR e μ w ) π 3 ( σ w 2 + σ 2 ) + 1 ] − 1

In networks that are not very dense, the number of interfering nodes around a receiver is usually small. Consider the case of a single interfering node, e.g., the one that is the closest to the tagged receiver among all interfering notes. The interfering power of this node may dominate the overall interfering power from all other interfering nodes. Letting the distance between the single interfering transmitter and the target receiver be r, the success probability in (7) is reduced to: (10) P succ = Pr { Y i Y r < 1 T SIR ( r d ) β } = Pr { Z < 1 T SIR ( r d ) β }where Z = Y_i/Y_r represents a log-normal random variable with zero mean and variance 2σ². Applying the logistic distribution, we find the success probability to be: (11) P succ = { [ T SIR ( d r ) β ] π / ( σ 6 ) + 1 } − 1

As an example, we plot the success probability P_succ for different σ's by evaluating (11) in Figure 2. The parameters are selected as T_SIR = 10 and β = 4 to emulate a typical wireless sensor network scenario. The mean interference range R I = d T SIR β = 35.6 m. We find that P_succ is a decreasing function of σ when r > R_I, but an increasing function of σ when r < R_I. For example, consider the two extreme cases when σ = 0 and σ = ∞. When σ = 0, the channel is a deterministic one; the received power depends on the distance only and does not have any randomness. Equation (11) becomes a step function as: P succ = { 0 , r < R I 1 , r > R I. That is, there is a perfect channel with the distance is within R_I, and there is no connection atall when the distance is larger than R_I. This is the deterministic disk model used in many prior studies. On the other hand, when σ = ∞, the success probability tends to be 1/2 in all cases. It means that the success probability is no longer dependent on the distances d and r.

When a node first overhears an RTS and then a DATA frame from the same transmitter, it can be identified as an exposed terminal with regard to the overheard transmission. In practice, exposed terminals can be identified before receiving the complete frame as follows.

The IEEE 802.11 PHY frame structure when direct spread spectrum sequencing (DSSS) is used is shown in Figure 3. As soon as the physical layer convergence protocol (PLCP) header, which precedes the current MAC DATA frame, is completely received and verified (by checking HEC, a 16-bit CRC code), the exposed node will know the MPDU (MAC protocol data unit) length. The exposed node can determine if the MAC frame is DATA from the length information since DATA frames are always longer than control frames (i.e., RTS, CTS, and ACK). By comparing the Length value with that in the preceding RTS, the exposed node can infer if the RTS and the DATA frame are from the same sender. It can further validate its inference if the interval between the DATA frame and the RTS is (SIFS + CTS + SIFS) plus some propagation delay.

Suppose that a node (called “free transmitter”) wins the channel and is transmitting to a one-hop neighbor (called “free receiver”). When a node near the free transmitter identifies itself as an exposed terminal, it can validate the feasibility of its concurrent transmission with a probabilistic approach based on location information and the shadowing channel model. We call such a node a “scheduled transmitter” and its target receiver “scheduled receiver” if the concurrent transmission is successfully scheduled.

When testing the feasibility of concurrent transmission, we focus on the case of two (free and scheduled) transmitters and two (free and scheduled) receivers. This simplification can be justified by the fact that the chance of having multiple interfering transmitters in the neighborhood of a tagged receiver is usually small in many wireless networks. When the path loss exponent is large, usually the interference power from the nearest one dominates the overall interference power. If an identified exposed node has a frame to transmit, it will evaluate the following four probabilities:

P_DATA1 – the success probability of the free DATA frame;

Under log-normal shadowing channels and with location information available, each of the probabilities can be calculated using (11). Table 1 shows the validation procedure validate schdTx(), where P_th is a prescribed threshold. In order to schedule the concurrent transmission, each of the above probabilities should be greater than P_th. As P_th is increased, there will be fewer scheduled transmissions allowed.

The operation of the proposed location-assisted MAC protocol is illustrated in Figure 4. First, an exposed node is identified by examining the PHY PLCP frame header, as described in Section 4.1. Once an exposed node is identified, it will try to validate its concurrent transmission, by executing the validation procedure shown in Table 1.

If this test is passed, the exposed node will attempt its scheduled transmission as follows. First, schdTx_margin is calculated: (12) schdTx _ margin = ( free _ duration ) − SIFS − CTSSIFS − ( PLCP _ reading _ time ) − ( schd _ data _ duration ) − SIFS − ACK − ( round − trip _ prop _ delay )where free_duration is the value in the duration field of the preceding RTS frame. If p = chdTx_margin is negative, the scheduled transmission will be canceled. That is, the concurrent DATA transmission cannot be aligned with the free DATA transmission. Otherwise, in order to avoid the case of multiple scheduled transmissions in a small neighborhood, a random delay t_d is introduced as: (13) t d = [ ( rando m _ integer ) % t d max ] × a SlotTimewhere t d max = ⌈ ( schdT x _ m arg i n ) / aSlotTime ⌉.

During the back-off period t_d, the exposed node will keep on detecting if there is a single transmission (i.e., the free transmission) or multiple transmissions (e.g., a scheduled transmission from another exposed node with a smaller t_d value, in addition to the free transmission). Such events can be detected by a sudden increase in received power or bit error rate when decoding the free transmission. If the latter case is detected, the exposed node will not attempt the concurrent transmission; otherwise, it will start sending its DATA frame, which carries the information T_info, defined as: (14) T info = t d max − t d aSlotTime

When a scheduled transmission is received, the scheduled receiver will return an ACK after a delay of SIFS + T_info x aSlotTime, in order to align its ACK with the ACK from the free transmission.

The proposed scheme assumes location information is available for neighboring nodes. Such location information can be distributed by defining a specific control message. When a node's location is changed, it broadcasts the newly defined control message to all neighbors to update its location information stored at these nodes. Alternatively, we can modify the RTS frame format to piggyback the locations of the sender and its target receiver. When a node overhears an RTS, it extracts the location information, and stores them in a location table. Nodes will also exchange their location table with neighbors so that the location of two-hop neighbors are all known after some initialization phase.

So far we have assumed that the channel parameters, i.e., both β and σ_dB, are known. In practice, these parameters may not be given a priori. After the initial deployment, the nodes need to measure received signal powers and estimate these parameters, assuming the location information of neighboring nodes are known. We describe how to estimate β and σ_dB in the following.

According to the shadowing channel model, the received power P_r,dB is governed by (1), and X_r is normally distributed. Thus, (1) can be categorized as the ANOVA (Analysis of Variance) model I (also known as a fixed-effects model ANOVA) [36]. During network operation, each node computes the distance to other transmitters based on location information, and measures the received powers. Assume that a node has made a set of n_T observations, which consists of n_i observations P r , d B i j, j = 1,…,n_i, for each corresponding distance d_i,i = 1, 2, …, N. Note that ∑ i = 1 N n i = n T. According to the least squares criterion, the sum of the squared deviations of the observations around the expected values, denoted by Q, should be minimized: Q = ∑ i = 1 N Q i = ∑ i = 1 N ∑ j = 1 n i ( P r , d B i j − μ i ) 2where Q i = ∑ j = 1 n i ( P r , d B i j − μ i ) 2.

Differentiating Q with respect to μ_i, we obtain that: d Q d μ i = d Q i d μ i = ∑ j = 1 n i [ − 2 ( P r , d B i j − μ i ) ]

Setting the derivatives to zero and replacing μ_i with the least squares estimator μ̂_i, we obtain that: (15) μ ^ i = 1 n i ∑ j = 1 n i P r , d B i j

Since μ_i = P₀,_dB − 10β log ( d i d 0 ) according to the attenuation model (i.e, X_r is zero mean), the estimator of the path loss exponent β, denoted as β̂, can be obtained as follows: (16) β ^ = μ ^ i − P 0 , d B − 10 log ( d i d 0 ) = ( 1 n i ) ∑ j = 1 n i P 0 , d B − P r , d B i j 10 log ( d i d 0 )

With (16), we next derive σ ^ d B 2, the estimator of σ ^ d B 2. Since the expected value of mean square error (MSE) is equal to σ ^ d B 2, we derive σ ^ d B 2 as: (17) σ ^ d B 2 = MSE = ∑ i = 1 N ∑ j = 1 n i [ P r , d B i j − μ ^ i ] 2 n T − N

We first examine the scheme for estimating channel parameters as described in Section 4.4. During each simulation, the channel parameters β and σ_dB are specified in the TCL script file and used in modeling log-normal shadowing channels for each transmission [33]. We assume that the proposed MAC protocol is not aware of the values, and uses the scheme described in Section 4.4 to estimate the parameters based on received signal powers.

We consider a network with a chain topology, where eight nodes are placed in a row with 20 m separation between adjacent nodes. The average transmission range is 26.9 m. The 8-node chain network has two CBR sessions: one from Node 1 to Node 8 with 1,000-byte frames and the other from Node 8 to Node 1 with 700-byte frames. We trace the estimated value of β and σ_dB at an intermediate node every 0.01 second for 45 seconds after the sessions are started.

The simulation results are plotted in Figure 5. We observe that in general the estimation schemes are quite effective. The estimated values quickly converge to the actual values set in the TCL scripts. Specifically, the estimation of path exponents β is more accurate than that of the standard deviation σ_dB for a given number of samples of received power. Furthermore, the convergence of β estimation is much faster than that of σ_dB. The convergence is also slower when σ_dB becomes larger. This is because when the channel exhibits high randomness (as indicated by the larger σ_dB value), more samples are needed to get an accurate estimate. We define the convergence ratio for σ_dB estimation as: (18) R σ = 1 − | σ d B − σ ^ d B σ d B |

For the worst case when σ_dB = 6, the convergence ratio at 30 seconds is 95.31%. After 45 seconds, the convergence ratio becomes 97.83%. On the other hand, the convergence ratios for the cases of σ_dB = 2 dB and σ_dB = 4 dB become very close to 1 after a few seconds of estimation, due to the lower randomness of the channels.

Similar results are observed when other topologies are used, but are omitted for brevity. The estimation algorithms can be continuously executed due to its low complexity. The estimated values can be updated for every frame received. We conjecture that the proposed approach is also applicable to the case of time-varying channels where the parameters vary over time, as long as they conform to the log-normal slow fading channel model.

We next examine the throughput and delay performance of the proposed scheme using the chain topology, as shown in Figure 7(a). The distance between two adjacent nodes is set to 20 m. The forward flow (from Node 1 to Node N) is a CBR session with 1,000-byte packets, while the backward flow (from Node N to Node 1) is a CBR session with 700-byte packets. The data rates are set to 90Kb/s, 80Kb/s, 70Kb/s, and 60Kb/s for the 6-, 8-, 10-, and 12-node chain networks, respectively. These data rates are found via extensive simulations, which achieves the maximum throughput the the corresponding chain network. The mean transmission range is equal to 26.9 m, while the mean carrier sensing range is equal to 59.3 m. When the distance between the transmitter and the receiver is 20 m, the mean interference range is computed to be 35.6 m.

Consider first the case that β = 4 and σ_dB = 0.01 dB. In this case, the received power is almost the same as the mean value, with low randomness exhibited. Since the mean transmission range if 26.9 m, a transmitter hardly reaches other nodes except for its two one-hop neighbors. For example, in the case that Node k + 1 and Node k + 2 concurrently transmit to Nodes k and k + 3, respectively, the success probabilities of the two transmissions are P_DATA1 ≈ 1, P_DATA2 ≈ 1, P_ACK1 ≈ 1, and P_ACK2 ≈ 1. And both transmissions are successful with high probability.

Figure 7 shows the throughput and delay results for the chain network when σ_dB = 0.01 dB. We observe considerable throughput improvements achieved by the proposed protocol over the IEEE 802.11 MAC. We define the throughput improvement ratio as: R thput = ( thoughput _ proposed ) − ( thoughtput _ 820.11 ) ( thoughput _ 802.11 )

The proposed scheme achieves 42.32%, 64.83%, 71.82%, and 47.32% throughput improvement ratios for the 6-, 8-, 10-, and 12-node chain networks, respectively. In addition, the average end-to-end delay is also drastically reduced for all the cases. We define the delay ratio as: R d = ( delay _ proposed ) ( delay _ 802.11 )

From Figure 8(b), we find that the proposed scheme achieves delay ratios of 19.49%, 28.63%, 22.98%, and 25.84% for the 6-, 8-, 10-, and 12-node chain networks, respectively. Most concurrent transmissions are successful in this case.

We next consider the case of β = 4 and σ_dB = 4 dB. Due to the increased σ_dB, the variation of the received power becomes larger. The one-hop distance may be longer in this case, a phenomenon also observed in the work on connectivity in the presence of log-normal shadowing [29, 30]. In the concurrent transmission scenario, the free and scheduled transmitters are regarded as the interfering node for each other's target receiver. Thus, the possibility of feasible concurrent transmission decreases. For example, when d = 20 and r = 40, the success probabilities are P_DATA1 = P_DATA2 = P_ACK1 = P_ACK2 = 0.5376. This is smaller than that when σ_dB = 0.01 dB. Consequently, there are fewer feasible concurrent transmissions when σ_dB gets large.

Figure 8 shows the simulation results when σ_dB = 4 dB, where the achieved throughput improvement ratios are 12.21%, 17.27%, 25.10%, and 21.02%, respectively. From Figure 9(b), we find the ratios of the average end-to-end delays are 89.87%, 88.97%, 81.38%, and 87.91%, respectively. Compared to the case of σ_dB = 0.01 dB, the number of scheduled transmissions are smaller. However, the proposed MAC protocol still achieves considerable improvements in both end-to-end throughput and delay.

We next examine the performance of the proposed scheme using grid topologies as shown in Figure 7(b). In the grid networks, 36, 64, 100, and 144 nodes are deployed in rows and columns, respectively. The distance from neighbor nodes in the right, left, upper and lower directions is 20 m. As shown in 7(b), half of the border nodes are selected as senders, and the corresponding node on the opposite boarder is the target receiver. A half of the sources transmit a CBR traffic of 1,000-byte frames to the border node on the opposite side, while the rest of sources transmit 700-byte frames.

Throughput and delay results for the case of σ_dB = 0.01 dB are plotted in 9. From the figure, we find that the proposed scheme achieves throughput improvement ratios of 20.40%, 10.77%, 12.98%, and 13.88%, for the 36-, 64-, 100-, and 144-node networks, respectively. In addition, we find from 10(b) that the ratios of the average end-to-end delay with the proposed MAC to that of the IEEE 802.11 MAC are 70.33%, 82.07%, 86.82%, and 88.07%, respectively.

The simulation results for the case of σ_dB = 4 dB for the grid networks are presented in 10. We find that normalized throughput improvements of 22.90%, 24.38%, 38.85%, and 61.74% are achieved. We also find from 11(b) that the ratios of the average end-to-end delay with the proposed MAC to that of the IEEE 802.11 MAC are 85.23%, 91.41%, 87.12%, and 84.36%, respectively. The proposed MAC protocol achieves considerable improvements in both throughput and delay for the grid networks with a regular topology structure.

Finally we study the performance of the proposed protocol under four networks with random topology. For the random networks, 64, 81, 100, and 121 nodes are randomly deployed in a square network area, respectively (see the example in 7(c)). Multi-hop CBR traffic flows are randomly generated with randomly chosen source and destination nodes. A half of the transmitters generate CBR session with 1,000-byte frames, and the rest of them generate CBR sessions with 700-byte frames. All the other configurations of nodes are the same as those in the chain networks.

Figure 11 presents the throughput and delay curves when β = 4 and σ_dB = 0.01 dB, and 12 presents the throughput and delay results when β = 4 and σ_dB = 4 dB. From the two figures, we observe the end-to-end throughput improvement ratios of 10.22%, 12.10%, 12.09%, and 14.72% respectively for the four networks when σ_dB = 0.01 dB, and throughput improvement ratios of 11.18%, 12.18%, 32.93%, and 28.70% for the case when σ d B 2 = 4 dB.The proposed scheme also achieves end-to-end delay ratios of 87.96%, 92.56%, 91.25%, and 87.02% when σ_dB = 0.01 dB and 64.30%, 74.34%, 82.32%, and 83.21% when σ_dB = 4dB.

We also examined the impact of shadowing standard deviation σ_dB on the performance of the proposed scheme. For a 64-node network with a random topology (as shown in 7(c)), we keep all the other parameters intact while increase the standard deviation value from 0.01 dB to 12 dB. Since different network environments have different standard deviation values. Such simulations will help answer the question that under what kind of environments the proposed scheme is effective.

The throughput results are shown in 14(a) and the delay results are shown in 14(b). We find that the normalized throughput improvement ratio ranges from 10.2% to 25.2% when the standard deviation is increased from 0.01 dB to 12 dB. The net improvement, however, is more constant and ranges from 1.6 MB to 2.1 MB. We also observe that the standard deviation has very similar impact on both MAC schemes, as the two curves are roughly parallel to each other. When σ_dB gets larger, the throughputs of both schemes decreases. This is because when the channels become more dynamic, more packets will be lost due to transmission errors, leading to goodput degradation. From 14(b), it can be seen that generally about 0.5 s delay reduction is achieved by the proposed scheme over IEEE 802.11 MAC, except for very large standard deviation values. When standard deviation is 10 and 12 dB, both schemes achieve very small delays. We conjecture that because of the large variation of shadowing channels (and thus the transmission ranges), many packets traverse paths with smaller hop-counts, leading to smaller end-to-end delays for both schemes. Overall the proposed scheme outperforms IEEE 802.11 MAC for all the cases considered in this simulation.

As illustrated in Figure 1, the effect of log-normal fading is two-sided: (i) some nodes (e.g., Node B) that are within the transmission range, are not neighbor anymore; and (ii) some nodes (e.g., Node C) that are not a neighbor but now are neighbors (and thus exposed nodes). For the chain network considered in the simulations, Type (i) effect is more likely to happen, resulting in broken links and lost packets. Therefore the performance improvement when σ_dB = 4 dB is smaller than that when σ_dB = 0.01 dB. When the random topology is used, however, a node has more neighboring nodes within the average transmission range and some other nodes outsider but close to the average transmission range. Both Type (i) and Type (ii) events are equal-likely to happen, and we see similar improvements for both shadowing channels and near-deterministic channels.

There are many factors that affect the performance of a MAC protocol in multi-hop wireless networks, such as topologies, traffic patterns, and channel parameters. Overall our simulation results demonstrate that the proposed location-assisted MAC protocol is effective in achieving considerable throughput and delay improvements over the IEEE 802.11 MAC under various circumstances of log-normal shadowing channels. The improvements clearly justify the need and the benefits of opportunistically scheduling concurrent transmissions while considering log-normal shadowing channels.