We propose a fully distributed medium access scheme such that the node with high score is likely to access the channel earlier than those with lower score. To facilitate understanding we begin with the limiting case such that the length of minislots vanish however the contention window remains fixed. Namely we let δ = t_c/m and m → ∞. Thus choosing a minislot to transmit data corresponds to selecting a real number from the continuous interval [0, t_c] which is the time instant to start transmission. In this case it is easy to find a fully distributed scheme without collision: for example Node i sets a timer for channel access at the beginning of each frame where the timeout is set to (1 – y_i)t_c. Then with probability 1 Node i* = argmax_i∈I [y_i] will win the channel access. Thus if Node i* is k hops away from the sink, the maximum-score data reaches the sink in exactly k frames with probability 1. If we consider a balanced tree for the data gathering tree, the maximum score data is retrieved in k ∼ O(log(n)) frames.

In practice however, the minislot δ cannot be indefinitely small. δ should account for propagation delay and should be sufficiently long such that the transmission from a station can be detected by its neighboring nodes [28]. For example in 802.11a, δ is given by 9 μs. We propose a fully distributed priority based scheme when δ is strictly positive and m is finite. The idea in the above case of continuous contention window with vanishing δ was to defer transmission based on the score. For δ > 0 with finite m, however, one should not defer transmission in a deterministic manner, otherwise once multiple nodes collide at a frame they will repeat colliding in all subsequent frames, resulting in a deadlock. In the proposed scheme a node defers the participation in channel contention, by remaining idle during the first several minislots, say k < m minislots. Note k is determined based on the score. Since m is fixed, this has an effect of shrinking the individual contention window to m – k for the sensor. We refer to such shrunk contention window size as the effective contention window (ECW) for the node. The sensor selects a minislot for transmission randomly and uniformly over the remaining m – k minislots.

Specifically the proposed scheme operates as follows. We introduce global system parameters denoted by β which is a strictly positive integer, and γ which is a strictly positive real number. Details on these parameters will be explained later. For Sensor i let us denote the data to be transmitted/forwarded by the score y_i ∈ [0, 1]. The counter c ∈ ℤ₊ is incremented each time collision occurs, and is reset to 0 once a transmission is successful.

Set the window length w as (2) w   =   ( 2 c   ⋅   ⌈ m   ⋅   ( y i ) γ ⌉   +   β ) ∧ mwhere a ∧ b denotes min(a, b) and ⌈·⌉ denotes the ceiling function.

The details are provided as follows. The idea is that a node delays channel contention by the number of minislots that increases with decreasing y_i. Thus the sensors with higher y_i gets higher chance of transmission, since such sensors are likely to participate in contention earlier during the contention phase. The choice of the effective window size w given by Equation (2) renders the size be roughly proportional to y i γ, if we ignore β. The term 2^c accounts for binary exponential back-off (BEB), i.e., ECW size increases exponentially with the number of successive collisions.

An example is illustrated in Figure 2 assuming m = 10, γ = 1, c = 0 and ß = 1. There are three contending sensors with ID 1, 2 and 3. The scores for the sensors are given by y₁ = 0.9, y₂ = 0.5 and y₃ = 0.1. From Equation (2), ECW size for Sensor 1 is m, thus it fully utilizes the contention phase period. Sensor 2 has the score 0.5, thus from Equation (2), its ECW size is given by 6. Sensor 2 picks a minislot randomly from minislot index 5, 6, . . ., 10. Thus the probability of selecting a minislot within ECW is 1/6. Sensor 3 has a low score of y₃ = 0.1, thus gets ECW size of 2. The purpose of the parameter ß is to set a lower bound on the size of ECW. Since ß ≥ 1, the lower bound on w is at least 1.

The second technique is called selective forwarding. The idea is that one can suppress unnecessary data transmission by exploiting the property of max function. Suppose at jth frame a sensor received a packet which needs to be forwarded to the parent of the sensor. Assume that the sensor has kept history of the maximum score among all the received data up to (j – 1)th frame. Suppose the score contained in the packet received at jth frame is less than the maximum score over all the packets which have been received and forwarded by the sensor prior to the jth frame. Since the sink needs to retrieve only the maximum score data, this packet will be eventually discarded at the sink. Rather than wasting resource to forward data to the sink, the sensor can silently drop the packet. This is called the selective forwarding. The max operation make such technique particularly simple, since it is easy to store the history of the maximum data.

Let us denote the maximum among the scores which Node i has been successfully received for forwarding up to frame t ∈ ℤ₊ by z_i(t). We define z_i(0) = F_Xi (x_i). Suppose at frame s ∈ ℤ₊, the node has received the data z requested for forwarding. In case where z_i(t) > z, Node i will discard the data, and if z_i(t) already has been transmitted, the sensor will go idle: otherwise the sensor will proceed to transmit z_i(t). In case where z_i(t) ≤ z, Node i updates z_i(s) to z, and this data will be queued for transmission. A flowchart in Figure 4 illustrates the above procedure of selective forwarding. Note z in the flow chart denotes the received data (score). Also note that a sensor may receive data while attempting to transmit, and the procedure shown in Figure 4 captures such aspect combined with the selective forwarding.

Figure 5 illustrates an example of selective forwarding. There are four sensors with ID 1, 2, 3 and 4. The packet contains data with a format (i, y_i) where i is sensor ID and y_i is the score of the sensor. At frame k, Sensor 3 transmits the packet with data (3, 0.9) to Sensor 1. At frame k + 1, Sensor 1 forwards the data (3, 0.9) to its parent, Sensor 4. At this frame Sensor 1 updates z₁(k) to the forwarded data 0.9, assuming z₁(k – 1) < 0.9. At frame k + 2, Sensor 2 transmits data (2,0.7) to Sensor 1. However the score 0.7 is less than z₁(k) = 0.9, Sensor 1 silently drops the packet and remains idle at frame k + 3. In this specific example only 3 packets have been transmitted using selective forwarding instead of 4 packets. Suppressing unnecessary packet transmissions will help reducing the contentions in upstream traffic to the sink, which will likely to reduce latency. This is demonstrated later in Section 5.

Since the wireless channel is a shared medium, every transmission is in fact a broadcast. For a node with a close by transmitter, there is a chance of overhearing the transmitted packet even though the node is not the intended receiver. Overhearing is useful since such opportunity is obtained without contention, however a sensor can suppress transmission of unnecessary data. The scheme operates as follows. Similar to selective forwarding every sensor stores the maximum score up to current frame. Let us denote the maximum among all the score received/forwarded at Sensor i up to time t ∈ ℤ₊ again by z_i(t). If Sensor i overhears data z that is greater than z_i(t) at frame s, the sensor can update z_i(s) to z. Note such update increases z_i(·), which increases the chance of discard unnecessary data using selective forwarding. Thus exploiting overhearing is a complementary scheme to selective forwarding.

Overhearing may not be always possible even though there is ongoing transmission by a neighboring sensor. The overhearing opportunity is affected by concurrent transmissions of other neighboring sensors, interference, etc. Given a set of transmitting sensors in the network, the set of nodes which can successfully overhear nearby transmission varies with the interference model used. The following is the condition under which overhearing is allowed in our node-exclusive interference model. Consider a transmitting sensor i and the set of neighboring sensors denoted by N(i). Any v ∈ N(i) can overhear i’s transmission if v is neither transmitting nor receiving at the same frame. The condition is from the limitation that in node-exclusive interference model, nodes cannot transmit and receive simultaneously. Note in other interference models such as two-hop interference model which is suitable for modeling IEEE 802.11 DCF (Distributed Coordination Function), one can find similar conditions for overhearing.

Figure 6 illustrates an example of overhearing. There are four sensors with ID 1,2,3 and 4. The packet contains data with a format (i, y_i) where i is sensor ID and y_i is the score. At frame k, Sensors 2 and 3 contended for transmission to Sensor 1, and Sensor 3 won frame k. Thus Sensor 3 transmitted the packet with data (3, 0.9) to Sensor 1, while Sensor 2 waits for the next frame. At frame k +1, Sensor 1 forwards the data (3, 0.9) to its parent, Sensor 4. Sensors 2 and 3 are assumed to be idle, i.e., neither transmitting or receiving at frame k + 1, which means they are eligible for overhearing the packet containing (3, 0.9) transmitted by Sensor 1. Having overheard data that is greater than the current data to transmit, Sensor 2 updates z₂(k + 1) to 0.9. Since there is no need to forward (2,0.7) to Sensor 1, Sensor 2 remains idle at frame k + 2. Now suppose Sensor 2 had a child node. If Sensor 2 had been receiving data from its child sensor at frame k + 1, it would not have been able to overhear Sensor 1’s transmission, e.g., see the rightmost of Figure 6. In that case Sensor 2 might have transmitted at frame k + 2.

In general the graph describing the data gathering tree may not be the same as the node adjacency graph. Namely there can be a node adjacency graph 𝒢̂ such that the data gathering tree 𝒢 is a subgraph of 𝒢̂. This implies that there may be further possible overhearing opportunity in the network other than that resulting from data gathering tree. For example, in Figure 6, Sensors 2 and 3 may be adjacent nodes in 𝒢̂, although such relation is not specified in the forwarding graph 𝒢. When the node adjacency graph is dense the network is a rich environment for overhearing. Note for dense graphs the network throughput may be severely limited by contention and interference. However overhearing compensates such limitation particularly for the applications retrieving the extreme data value, since unnecessary transmissions can be further suppressed by overhearing in denser networks, which in turn reduces subsequent contention for channel access.

The whole procedure which combines priority-based channel access, selective forwarding and overhearing is given in Algorithm 1. The SelectSlot() subroutine used in Algorithm 1 is provided in Algorithm 2.

Recently distributed scheduling algorithms for wireless networks with contention based medium access has gained much interest. A series of work [14–21] has been proposed on assigning channel access priorities to nodes based on the individual loads, e.g., queue lengths. Note however, these studies have been carried out under different context: the emphasis has been on achieving a fraction of the capacity region of the network using only local information, e.g., queue lengths of contending neighbors. The capacity region is defined to be the union of the set of all input rates which can be supported by the network, i.e., input loads under which the system can be stabilized.

Among these studies, [17] and [16] use timers with exponentially distributed timeouts as a means to assign priorities in channel access. If a node has data to transmit, it first sets a timer whose timeout is exponentially distributed independently of other nodes. The node waits until the timer expires, and begins transmission unless the channel is sensed to be busy. The rates of such exponential distribution are set according to certain function of the backlog of the nodes. Thus multiple neighboring nodes independently compete with exponentially distributed timers, and the one whose timer expires first wins the channel. Thus by controlling the rate of timer expiry, the priorities of channel access are implicitly assigned to nodes based on their individual loads.

A similar idea has been used for policy V [18] and Q-SCHED algorithm [20]. They also consider a two-phase contention model in which a node picks a minislot to transmit during contention phase. Nodes wait for a random number of minislots before transmission. The distribution of waiting time is a truncated version of geometric distribution which can be regarded as a discrete analogue of (truncated) exponential distribution. However in these schemes there is nonzero probability that a node does not attempt transmission [18,20]. In such schemes, since the nodes do not always participate in channel contention, the probability of collision will be relatively low. However one should be more aggressive in attempting transmissions in our problem, particularly for nodes with high scores. The reason is that, in the proposed scheme, selective forwarding and overhearing are used in addition to priority based channel access. This in turn increases the probability that high score nodes will be scheduled in the subsequent frames, i.e., the effect is disseminated throughout the network. Once a transmission of high score packet is successful, it can suppress unnecessary traffic generated from the neighboring nodes. Also the system parameters γ and ß should be set in an aggressive way. Indeed we have heuristically found that setting γ = 3 and ß = 1 leads to better performance. Note when γ > 1 and as ß gets smaller, the sensors are more aggressive in transmission attempts. In summary, the synergistic effect among the components of the proposed scheme enables us to be more aggressive towards minimizing the latency.

Furthermore these works focus on achieving a long-term throughput that is sustainable by the network. However our problem pertains to retrieving information in a short period of time. The traffic is bursty in time and space, thus the regime of interest is quite different from such setup. Indeed we will demonstrate using simulation that those scheme may not be effective in reducing the latency of retrieving extreme statistics.

In this subsection we discuss several extensions of the proposed framework. We first consider k-maximum data retrieval problem as follows. Let us denote k-th largest data from the scores y₁, . . ., y_n by y_[k]. The k-maximum data retrieval problem is for the sink to collect y_[1], y_[2], . . ., y_[k]. Similar overhearing and selective forwarding can be applied with a small change. The selective forwarding can be modified such that a node forwards only the data larger than the k-th largest data that it has received/forwarded. Note every sensor needs to maintain only k sets of variables to keep track of the history of k largest data since max is a divisible function. Similar modification can be made for the overhearing technique.

Another extension is a scenario where the sensors generate data samples periodically. Assume the sensors generate data samples at every unit time. When the signal for data retrieval is broadcast to the network, the sensors report the most recent data samples, say the last b samples, to the sink. The time instants of data sample generations are called timestamps. Thus the data samples are periodically generated and stored over a sliding time window of the length of b timestamps where the timestamps are one unit time apart. The goal of sink is to retrieve b maximum scores, one for each timestamps. Specifically suppose the timestamps for the maximum data retrieval are denoted by t₁, t₂, . . ., t_b. Denote the score at sensor i for timestamp t_j by y_i(t_j). The objective for the sink is to collect y*(j) = max_i∈I [y_i(t_j)] for j = 1, 2, . . ., b. The latency is measured as the time when all of b maximum scores y*(j), j = 1, 2, . . ., b are retrieved at the sink. We assume a packet can contain only one data sample, and only one packet can be transmitted per frame. The timestamp is assumed to be included in a packet. In this application the sensors can apply rules for selective forwarding and overhearing separately to packets with different timestamps. A set of packets with the same timestamp can be treated as an individual flow, thus the proposed scheme can be applied separately to each flow. However the latency is affected by the parameter b. Since initially b packets need to be transmitted by every node, when b is large the network may undergo congestion for a long period of time. We later study the impact of b on the performance using simulations.

By simulation we demonstrate the efficacy of the proposed scheme. We assume a complete binary tree as the data gathering tree. The number of nodes considered have been n = 2⁷ – 1 = 127 and n = 2⁹ – 1 = 511. The number of minislots were varied from either m = 10 or m = 30. We compare the performance of the proposed scheme with a baseline scheme defined as follows. The baseline scheme uses UNIFORM scheme for the medium access, and uses a simple in-network aggregation of combining available data without selective forwarding or overhearing. The latency has been measured in terms of the number of frames it takes for the extreme data arrives at the sink. Recall the latency is denoted by a RV T, and for a given deadline d we would like to maximize ℙ(T = d). Thus we examine the CDF of the latency for comparison.

Figure 7 shows the CDF of the latency when b = 1 for n = 127 and 511, each for m = 10 (left) and 30 (right). We see that the proposed scheme clearly outperforms the baseline scheme. For example when the probability of retrieval of the maximum data is given by 0.8, the proposed scheme achieves the latency that is roughly 19%–20% (resp. 15%–17%) lower than the baseline scheme when m = 10 (resp. m = 30). The overall latency is reduced for larger m since the contention window has become wider. The relative performance of the proposed scheme is slightly better for smaller m, which suggests that as the level of contention become higher, the proposed scheme becomes more effective.

Figure 8 shows the CDF of the latency when b = 3 for n = 127 and 511, each for m = 10 (left) and 30 (right). The reduction in latency at the probability of retrieval of 0.8 is 32–33% (resp. 30–32%) when m = 10 (resp. m = 30). Clearly the gains have increased compared to the case where b = 1. With larger b the network undergoes higher level of contention and temporary congestion, thus the gain in latency reduction due to the proposed scheme increases under high load on the network. The mean latency has been compared in Table 1. The relative reduction in mean latency by the proposed scheme ranges 15.1–18.0% when b = 1, and 31.1–34.3% when b = 3.

Note that there are many ways to assign priorities of channel access to nodes. In this subsection we compare several schemes for priority assignment based on previous work on multi-hop networks [18,20]. These works also adopt a two-phase time slotted model for channel access. The contention window is similarly defined as consisting of a fixed number of minislots. The key difference is the algorithms for choosing a minislot for transmission. We apply these algorithms to our problem and compare the performance with the proposed scheme. We first need the following definition. Let us denote the set of the child nodes of Sensor i by 𝒞(i) and the parent node by 𝒫(i) in the aggregation tree. Note the nodes in 𝒞(i) and 𝒫(i) are the interferers to Sensor i. Let us define (10) q i   =   y i y i   +   max ( y 𝒫 ( i ) ,   ∑ k ∈ 𝒞 ( i ) y k )

Roughly speaking q_i can be viewed as a ‘normalized’ score of Sensor i with respect to its interferers.

The first algorithm we consider is policy V proposed in [18]. In policy V a node attempt a transmission at every minislot with the following probability: (11) ( m   −   1 2 m )   ⋅   q i

The second algorithm is Q-SCHED [20]. In the algorithm a discrete RV M is defined which takes a value from {1, 2, . . ., m + 1} and has the following distribution: for k = 1 . . ., m, (12) ℙ ( M = k )   =   exp ( − g ( m ) q i m ( k   −   1 ) )   −   exp   ( − g ( m ) q i m k )where g(m) = log(2m)/2. ℙ(M = k) is the probability that the minislot k is selected. Also ℙ ( M   = m + 1 )   =   1   −   ∑ k = 1 m ℙ ( M   =   k ) is defined to be the probability that the node will not attempt transmission in the frame. Note the algorithms above are modified to fit our model: in the original definition of q_i, instead of scores, a ratio of queue lengths normalized by link capacity is used. The analysis of [18,20] shows that with the above algorithm, the probability of Sensor i becoming the winner is guaranteed up to an amount that is proportional to q_i. Specifically the probability is at least q_i(0.5 – f (m)) where f (m) → 0 as m → ∞. Roughly speaking, the priority of channel access is assigned to Sensor i such that the channel access is guaranteed with a probability close to 0.5q_i.

We compare the performances of different algorithms for channel access. Figure 9 shows the mean latency associated with the channel access schemes. The selective forwarding and overhearing are used with varying medium access schemes. ‘Mod. V’ (resp ‘Mod. Q-SCHED’) represents the modified policy V (resp. Q-SCHED algorithm). We observe that the proposed scheme incurs the lowest mean latency both for m = 10 and 30. Somewhat surprisingly, UNIFORM scheme performs better than modified policy V and Q-SCHED. The result can be explained as follows. Figure 10 shows the statistics on the number of collision and the amount of traffic generated at the first frame. The first frame is defined to be the frame at which the data gathering is initiated. Note the generated traffic is potentially the heaviest at the first frame. On the left of Figure 10 is the mean number of collisions in the first frame. We see that the number of collision of the proposed scheme is higher than UNIFORM scheme, while Q-SCHED and policy V have considerably lower number of collisions. However seemingly high number of collisions for the proposed scheme is in fact relatively small compared to the number of successful transmissions in the first frame which is shown on the right of Figure 10. We observe that the number of successful transmissions in the first frame for policy V and Q-SCHED are lower than the proposed and UNIFORM scheme. The reason is that, under policy V and Q-SCHED algorithm, there is nonzero probability that a node is ‘idle’, i.e., it does not participate in channel contention. While in the proposed and UNIFORM scheme, every node attempts transmission at every frame. This also accounts for the lower number of collisions for policy V and Q-SCHED algorithm. Figure 11 shows the mean score of the successful transmitted packet in the first frame. Since the scores of sensors in the first frame are n i.i.d. uniform RVs on [0,1] by definition, we see that the mean score for UNIFORM scheme is approximately 0.5. We see that the modified policy V and Q-SCHED has a mean score that is only slightly higher than the proposed scheme, which is possible however, by reducing the number of successful transmissions in the first frame. Figure 11 shows that the proposed scheme finds compromise on these aspects: it achieves higher mean score than UNIFORM scheme without excessively reducing the number of successful transmissions in the first frame. Although not shown here, we observed that the number of collisions subsides faster in subsequent frames for the proposed scheme, compared to other schemes. This is a synergistic effect with selective forwarding and overhearing, since successful transmissions of high-score data contribute to suppressing traffic and reducing contention by these techniques, which eventually leads to decreased latency.

Here we make an important observation. When one intends to further prioritize high-score sensors, it is likely to lower the number of successful transmissions. Reducing the number of successful transmissions, especially by increasing the number of ‘idling’ nodes, seems to be more harmful to latency reduction as we see that UNIFORM outperforms policy V and Q-SCHED in Figure 9. Thus it is crucial that one should aggressively prioritize high score nodes, however should not excessively lower the number of successful transmissions. The proposed scheme is observed to exhibit such property.

In this subsection we study the effect of a finite m on the analytical estimate Equation (3). We consider a set of nodes with n = 6 having scores (0.3,0.4,0.5,0.6,0.7,0.8). We assume every node interferes with every other node. The probability of winning a frame under the proposed medium access scheme is numerically computed. On the left of Figure 12 is a plot of the probability of winning a frame as a function of the score with varying m. For small m, there is a gap between the probability of winning and the analytical estimate. This is due to the fact that the analysis assumes m → ∞ and does not account for collisions. However the gap appears to be small for a substantially small value of m, e.g., m = 10, considering the relatively large number of contending users, i.e., n = 6. For larger value of m the estimate becomes fairly accurate.

Let us consider the probability of winning a frame conditional on that no collision has occurred. Heuristically one could expect that Equation (3) becomes a better estimate, since by definition the collisions are already accounted for when the conditional probability is computed. On the right of Figure 12 is a plot of the probability of successful transmissions conditional on that no collision has occurred. Indeed the figure seems to support the argument: Equation (3) is virtually identical to the actual conditional distributions for all the simulated values of m. Let us denote the probability of Sensor i winning a frame for finite m by p_i. We conclude that Equation (3) can be used as an approximation to p_i/Σp_i. In summary the analytical estimate Equation (3) is a reasonably good approximation even for small m, and is an excellent approximation for the conditional distribution, i.e., the relative probability of a successful transmission among sensors.