Distributed Shortest Link Scheduling Algorithms With Constant Time Complexity in IoT Under Rayleigh Fading

For the shortest link scheduling (SLS), i.e., scheduling a given set of links with minimum time slots, we consider the distributed algorithm design by using the locality of the protocol model with high fidelity under the Rayleigh fading. Different from most previous works, focusing on distributed algorithm design under the deterministic SINR model, which ignores the fading effects in signal propagation, we first show that a successful link of protocol model is also <italic>feasible</italic> under the deterministic SINR model, then it can be scheduled successfully with high probability under the Rayleigh fading, by upper-bounding interference outside interference range of protocol model. Then on the basis of this key conclusion, we design LLS-SLS algorithm to solve SLS within <inline-formula> <tex-math notation="LaTeX">$(2e\Delta ^{T}_{\max }+1)\delta \log _{2} \Delta ^{T}_{\max }$ </tex-math></inline-formula> time slots for a constant <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>. Specifically, <inline-formula> <tex-math notation="LaTeX">$\Delta ^{T}_{\max }$ </tex-math></inline-formula> is the number of a sender’s neighbors within some certain range, and can be upper-bounded. Next, based on the concept of random contention, we design CLLS algorithm to schedule all links after costing <inline-formula> <tex-math notation="LaTeX">$4(\delta +1)\Delta ^{T}_{\max }\ln \left ({\Delta ^{T}_{\max }+1}\right)$ </tex-math></inline-formula> time slots. In addition, extensive simulations evaluate the performance of two proposed algorithms.

Unlike SINR model, although the validity of protocol model is doubtable because that the interference is regarded as existent only between nodes in a local neighborhood, the local nature is suitable for distributed algorithm design since distributed algorithms need commonly a local information rather than a global one.
As a result, most distributed algorithms work in protocol model even if there have been doubts on its validity. Media access under this model, apart from exceptions, is usually done probabilistically. A simple idea is to let nodes transmit randomly, with a probability inversely proportional to the 'competition'. In particular, if every node transmits with some certain probability, we are sure that at least a constant fraction of the transmissions is successful. In addition, protocol model is beneficial to design distributed algorithms due to its locality, but its validity should be carefully examined before putting it to use. To enable distributed algorithm designs, predictable interference control, an open question is whether it is possible to develop a model of measuring interference, which has locality of protocol model and high fidelity, under Rayleigh fading.
In our previous work [3], we increased reliability and latency requirements of IoT under Rayleigh fading by solving link scheduling problem. Specifically, utilizing idea of localized global interference, two distributed algorithms were proposed. In this paper, we now present complementary, novel algorithms for low-latency performance. Following the previous works [3]- [5], we regard a link as being scheduled successfully under Rayleigh fading if its success probability is greater than 1− , where is an acceptable failure probability, since Rayleigh-fading leads to the uncertainty of strength of received signal.
To eliminate effect of stochastic channel fading gain in Rayleigh fading on algorithmic design and analysis, we simplify it to deterministic signal-interference-ratio (SIR) model. The only problem is that we need to show that a successful link of deterministic SIR is also successful with probability 1 − under Rayleigh fading. Moreover, to design distributed algorithms for latency minimization, we use protocol model to localize the global interference of deterministic SIR model, and show that a successful link of protocol model is also successful under deterministic SIR model (called as SIRfeasible simply).
Specially, we first prove that the cumulative interference outside interference range of the protocol model can be upperbounded under deterministic SIR model; then we show that a link can be scheduled successfully with high probability under the Rayleigh fading by ensuring that above interference has no effects on its success probability. Specifically, we bound the Rayleigh success probability of this transmission is at least 1 − by using distance constraint l max as interference range of protocol model, where α is the path-loss exponent, β is the decoding threshold, l max is the maximum link length, c is a constant [6]. Consequently, we present two distributed algorithms for SLS by using the local characteristic of protocol model with high reliability under Rayleigh fading. For settings of α = 5, β = 1, l max = 10m, = 0.1 and c = 0.1, is 1.8559. In summary, the main contributions of this paper are given as follows: • Setting uniform transmission probability for each sender. All senders execute local broadcast algorithm to assign transmitting time to themselves, we design a distributed link scheduling algorithm for SLS (denoted by LLS-SLS). After achieving local broadcast, the number of senders whose transmission times are same is upper bounded and all senders can transmit successfully with high probability after 12δ( 2 + 1) ln T max time slots, where T max is maximum size of neighbors of a sender.
• Based on assumption that the distance between any two nodes is at least 1m, parameter T max can be upper-bounded by 2π 3 √ 3 · [( + 2)l max + 2)] 2 , independent of network size. Specifically, LLS-SLS gives a 1 + 6 e ( 2 + 1)-approximation factor of the optimum. • Finally, utilizing localization characteristic of protocol model, we propose a random contention algorithm for SLS (denoted by CLLS) based on local information. The idea is that each sender accesses the channel in any time slot with probability 1 T max until its first success. In case of a collision, none of the involved senders is successful in this time slot.
The rest of the paper is organized as follows. Section II provides the related work. In Section III, we present network model and the overview of our design. In Section IV, we describe the model transformation from protocol model to SIR model. In Section V, we present two distributed algorithms for SLS problem, and simulations are shown in Section VI. Finally, we summarize our work and conclude the paper in Section VII.

II. RELATED WORK
In the centralized settings, SLS is closely related to capacity maximization (i.e., maximize the number of successful transmissions in a single slot), since we can execute algorithms for capacity maximization repeatedly to solve SLS. However, those works about capacity maximization were centralized and lead to equivalent bounds for SLS with O(log n)approximation (e.g., [4], [6]- [23]).
The interference localization method offers important insights of designing distributed scheduling algorithms (e.g., [24]- [27]). The main idea is to assume the aggregated interference from the senders beyond certain distance can be upper-bounded by a threshold, since the interference power level decreases over distance due to the free-space path-loss. But how to determine this key parameter may be a difficult problem.
Distributed algorithms for greedy maximal scheduling by using interference localization were proposed in [26] and [27]. They introduced the concept of interference neighborhood of a link and constrained the suffered interference inside and outside interference neighborhood being bounded (even the latter can be negligible), respectively. Then, global characteristic of the SINR model is successfully localized, links only need to perform scheduling coordination inside its interference neighborhood. However, the procedure for determining the interference neighborhood is centralized.
By combining the partition and shifting strategies with a pick-and-compare scheme [25], Zhou et al. presented a class of localized scheduling algorithms with provable throughput guarantee subject to physical interference constraints. On observing that distance dominates the interference, they partitioned the links into disjoint local link sets with a certain distance away from each other, and using this implemented localized scheduling.
The analysis of connectivity's scheduling complexity, i.e., the minimal amount of time slots required until a connected structure can be scheduled, in the SINR model that deals with arbitrarily networks has been studied by Moscibroda and Wattenhofer [23]. However, they did not solve the problem of SLS but a similar one. Instead, they proved that scheduling a set of pairs of senders and receivers that all network nodes are strongly connected is possible in O(log 4 n) time steps (independent of the topology) even in arbitrary worst-case networks. However, the lengths of the generated schedules were not compared to the optimal schedule length.
For distributed algorithms of the SLS, Kesselheim and Vöcking introduced a measure called maximum average affectance to analyze random contention-resolution algorithms, in which each packet is transmitted with a fixed probability depending on the maximum average affectance, with O(log 2 n)-approximation [1]. Subsequently, approximation factor was improved to O(log n) by ensuring each link transmitting with some certain probability within logarithmic time slots [2]. Moreover, they concluded that the best possible absolute performance guarantee is logarithmic. Similar results can be obtained in [6], [27].

III. NETWORK MODEL AND DEFINITIONS A. NETWORK MODEL
Considering a set of wireless communications, denoted by L = {l 1 , l 2 , . . . , l n }, where l i represents a transmission from sender s i to receiver r i . The Euclidean distance between node i and node j is denoted by d ij and assume that it is at least 1. Let l max be the maximum link length. The distance between link l i and link l j is the distance from the former sender to the latter receiver. Transmission power is set to P.
Under Rayleigh-fading, the strength of received signal from sender s i to receiver r j is a random variable, denoted by Ph ij /d α ij , and is exponentially distributed with mean P/d α i,j [31], where h ij is the channel fading gain between nodes and α is the path-loss exponent. As the noise power has negligible effect on the results, we then ignore its influence on transmission of a link [11], [11]. Then, the signal-interference-ratio (SIR) of a link l i = (s i , r i ) with respect to a scheduling set S is the ratio of the signal strength received at r i from s i to the signal strength received at r i from other non-intended senders in S. Mathematically, the received SIR of link l i = (s i , r i ) in the presence of S is given by the receiver r i can successfully decode the signal transmitted by its sender s i , if the SIR is above a certain threshold β, this is γ i ≥ β.

B. INTERFERENCE MODEL TRANSFORMATION
Different from the determining SIR model, the strength of received signal under the Rayleigh fading model varies from time slot to time slot, which is described by channel fading gain h, as shown in Eq. (1). In addition, the success probability that a link is scheduled under Rayleigh fading model cannot be up to 1, since inherent nature of fading makes the strength of received signal uncertain. However, the existence of fading gain brings a big challenge on algorithm design and analyses. Therefore, we need to remove the effect of fading gain. Based on the works in [3] and [4], if above success probability exceeds 1 − , a link can be regarded as being scheduling successfully under Rayleigh fading model.
Combing with the SIR model, if we can prove that the success probability of a link determined by SIR model is 1 − at least, the effect of fading gain can be removed. It is reasonable because of the uncertainty of strength of received signal under Rayleigh fading model. Random fading gain and global interference are two challenging problems for designing and analyzing distributed algorithms under Rayleigh fading. To solve those two problems, we first make a transformation from deterministic SIR model to Rayleigh fading model for the first problem, by proving that a SIR-feasible link (i.e., this link can be scheduled successfully in the SIR model) is also scheduled successfully with high probability under Rayleigh fading. Next, for the second problem, we apply the protocol model to localize global interference of deterministic SIR and show that a successful link of protocol model also is SIR-feasible.

IV. TRANSFORMATION FROM PROTOCOL MODEL TO SIR MODEL
In this section, we first utilize the protocol model to localize interference and prove a successful link under protocol model is also SIR-feasible; then, we further show that this transmission is successful with high probability under Rayleigh fading. VOLUME 8, 2020 For protocol model, a link is successful when corresponding intended receiver falls outside the interference ranges of other non-intended senders, which can be simply abstracted to a disc of the radius l max , and settings of will be given in Formula (2). To localize the global interference and design distributed algorithms, we need prove that the interference outside some range can be bounded, and has no effect on the successful transmission of a link. In the following section, we bound the interference outside l max by defining the neighbor of a sender, and show that success probabilities of a link are 1 and 1 − under deterministic SIR and Rayleigh fading, respectively.
Definition 1: Neighbors of a sender v are a set of other senders whose distance satisfies d wv ≤ ( + 1)l max for w, denoted by a set N v .
Define T max as the maximum neighbor size of a sender in the network. In the following analysis, we prove that T max can be upper-bounded by a constant.
Following the existing researches [31]- [33], we employ Aloha as the medium access control (MAC) protocol, in which each sender v simply transmits with probability q and keeps silent with probability 1 − q. Then, Our goal is to show that, although protocol model is intrinsically local, it is available to guarantee efficient medius access using this simple and completely distributed manner.
Definition 2: [34] For a receiver x, the probabilistic interference at x under deterministic SIR, denoted by ψ x , is defined as the expected value of interference experienced by x in a certain time slot, this is where q v is the transmitting probability of sender v. At the end of this section, we present Lemmas 1 and 2 that give sufficient conditions that applying protocol model measures interference under Rayleigh fading from point of success probability of a link, which are the basis of further designing distributed algorithms. The proof of Lemma 1 is based on a technique of bounding the size of disjoint discs within some certain range. The proof of Lemma 2 is based on the characteristic of Rayleigh fading and settings of parameter . The detailed proof is given in the following section. Lemma 1: For a sender v, if the sum of all transmission probabilities of its neighbors starting to transmit can be upperbounded, denoted by q sum , then v can transmit successfully with some certain probability.
In the following section, we will prove q sum is a constant. Lemma 2: A successful link of protocol model is also successful with high probability under Rayleigh fading, when is defined as a function of α, β, l max , c and : Algorithm 1 Local Broadcast within 2l max if v starts to transmit then 5: Update start v = start v + slot end if 10: if v receives rej m and does not broadcast it before then 11: Update start v with max{start i }+slot, i is a neighbor of sender v within 2l max 12: end if 13: end while 14: Senders whose starts are same start to execute Algorithm 2

V. DISTRIBUTED ALGORITHMS FOR SLS A. LOCAL BROADCAST BASED SLS
In this section, we present a distributed algorithm for latency minimization. Each sender starts to broadcast a message with probability q, such that after achieving local broadcast, the number of senders simultaneously transmitting in a certain area of the network is bounded to permit each sender to perform a successful communication within logarithmic time slots with high probability.
We apply the communication model in [38] and [39] and as follows. The pseudo-code of local broadcast is shown in Algorithm 1 with synchronous message passing model, in which each node sends/receives to/from all of its neighbors in each communication round, and the communication rounds are synchronized. If we assume that each round takes one time unit, then the time complexity of the algorithm is the number of time units taken from start to completion. Also, we can set a maximum number of rounds for the algorithm execution according to its time complexity.
Note that, there may exist two colliding cases, one is that a sender receives at least two messages rej m concurrently from different neighbors, which is called as the hidden terminal problem. Another is that at least two senders receive message rej m from each other in a time slot. Next, we show that those two case can be solved with high probability by settings of transmission probability and transmission time to make local broadcast.
In the first case, the probability that a sender receives at least two messages rej m concurrently, denoted by p 1 con , is After costing δ log 2 T max slots for a large enough constant δ, this probability can be reduced to 1 T max δ . Thus, the first case can not happen with high probability.
In second case, we will show that each sender can receive a message from its neighbors with high probability by following Theorem 1.
At the beginning of Algorithm 1 execution, each sender first starts to transmit with probability q and sends rej m to its neighbors inside 2l max . Then, senders receiving rej m change their corresponding start and then transmit again with probability q. The purpose of slot 1 is to ensure the process of message exchange in local broadcast can be achieved with high probability, and above two colliding problems can not exist with high probability. After slot 1 time slots, all senders have known their transmission times by updating their start with the maximum value of start recorded by their neighbors inside 2l max , respectively.
Specifically, if a sender starts to transmit, then it increases its start to start +slot 1 and notifies its neighbors within 2l max its new start. In this way, senders which are neighbors from each other can have different start and execute Algorithm 2 in different slots. The purpose of slot is to ensure those senders whose start is the minimum can execute Algorithm 2, i.e., ensuring they can transmit successfully within slot slots.
Fact 1: [34] Consider two discs D 1 and D 2 of radii R 1 and R 2 , respectively, assume that R 1 > R 2 , we define χ R 1 ,R 2 to be the smallest number of discs D 2 needed to cover the larger disc D 1 . It holds that First, based on the assumption in Section III that the distance between any two nodes is at least 1 and Fact 1, we prove that the time complexity of Algorithm 1 is a constant.
Theorem 1: Local broadcast of Algorithm 1 can be achieved within O (1).
Proof: In each slot, a sender v independently transmits a rej m with probability q and listens with probability q. In the process of message exchange, a transmission from v to one of its neighbors (denoted by x) is made successfully in a certain slot only if exactly v, inside range l max , is the only transmitting node in that slot. Assume that q = 1 2 T max and the probability of a successful transmission by this node in a given slot is v achieves broadcasting successfully at least once with high probability after 2eδ T max ln T max time slots, i.e., 1 − 1 Finally, we show that T max can be regarded as a constant. Recall that the distance between any two nodes is at least 1. For each sender v, by using Fact 1, all possible neighbors in an extended ring of radius ( + 1)l max + l max around sender v are at most and are at least Along with time slots in first case, the total time slots are at most (2e T max + 1)δ log 2 T max and can be regarded as a constant.
Theorem 2: The message complexity of Algorithm 1 is O(n).
Proof: In local broadcast, each sender sends rej m to its neighbors within 2l max . By using Fact 1, the number of neighbors within 2l max is at most Thus, the number of all messages is at most n(3l max + 2) 2 ∈ O(n). Lemma 3: For each active sender v, the sum of transmission probabilities for its active neighbors is bounded, by the execution of the algorithm, this is, 4π . Proof: For active sender v, the distance between any one of active neighbors and node v, by local broadcasting with rej m , is at least 2l max . Then, the distance between the receiver of v and any active neighbors of v is at least l max . Thus, all discs of radius l max centered at those neighbors are disjoint, and the number of neighbors being active in an extended ring of area π ( + 2) 2 l 2 max − l 2 max is at most then, the summed transmission probabilities are at most .
Lemma 3 now yields the claim. Next, we prove that interference measured by a receiver x from active senders which are away the sender of x is at least ( + 1)l max can be bounded.
Lemma 4: Consider a sender v and interference range l max . In a schedule, the sum of transmission probabilities of all active senders outside this range can be bounded by a constant, i.e., if w / ∈N v q ≤ c, then the probabilistic interference VOLUME 8, 2020 experienced by x (v' receiver) under deterministic SIR model, caused by active nodes outside this range can be bounded by (7) Proof: Consider rings Ring k of width ( + 1)l max around v, containing all senders w whose distance satisfies k( + 1)l max ≤ d wv ≤ (k + 1)( + 1)l max , here k ≥ 1 is a constant. In each Ring k , after achieving local broadcast, all discs of radius l max centered at those senders which have same value of start must be located entirely in an extended ring Ring + k of area and any two discs are disjoint since the distance between them are 2l max . By local broadcast, the distance from each active sender w in Ring k (k ≥ 1) to v's receiver, denoted by x, is at least k l max . By applying a standard geometric area argument, we can bound the probabilistic interference ψ Ring k x incurred by active nodes located in ring Ring k (k ≥ 1) as

Summing up the interference over all rings yields
which concludes the proof of the lemma.
Moreover, Lemma 1 now yields the claim according to conclusions of Lemmas 3 and 4.
Corollary 1: A successful transmission of protocol model is SIR-feasible.
Proof: If a link is successful in protocol model, there is no active non-intended senders within l max around the receiver of this link. Based on Equation (7), received SIR at this receiver under deterministic SINR is Therefore, it is successful under deterministic SIR model. Next, we give another method to measure the upper bound of interference received outside interference range of protocol  [31] and [35], the summed interference beyond ( + 1)l max can be expressed as Note that d wx ≥ l max . By regarding d wx as a continuous random variable, we get For maximum link length, its Rayleigh success probability can be expressed as Corollary 2: Setting based on Equation (2), a transmission is successful under Rayleigh fading.
We consider the worst case, i.e., the success probability of a link whose length is maximum. Combining with Formulas (7) and (9) and using in Equation (2), Rayleigh success probability of this link can be ensured with 1 − . Moreover, this link is also SIR-feasible under deterministic SIR from Corollary 4 in [31].
In this way, we can design distributed algorithms that rely strictly on local information by using local characteristic of protocol model under deterministic SIR model, which ensures each selected link has Rayleigh success probability of 1 − . So far, we have shown that neighbors and active neighbors before and after executing Algorithm 1 for each sender v and can be upper-bounded respectively using Formulas (4) and (6), and a transmission is successful if there is no neighbors of its sender transmitting (Corollary 2).
Finally, we propose Algorithm 2 to schedule all active senders with high probability by costing δθ log T max time slots at most. The pseudo-code is given in Algorithm 2.  Theorem 3: All active senders can be scheduled successfully with O(1) whp.
Proof: From Lemmas 1 and 3 and Corollary 2, we can see that a sender v only competes with its neighbors. For receiver x, defining P 1 as the probability that v is the only active sender centered at v within ( + 1)l max . Then The last inequality is by Fact 2. By Inequality (6), we have Thus P 1 ≥ 1− 1 12( 2 +1) . After δθ ln T max time slots, the probability that transmission from v to x is successful at least once is at least Next, we give a worst instance where the last one link can be scheduled after some certain time slots.
In the worst case, all senders within ( + 1)l max are neighbors with each other. Expectantly, only one sender starts to transmit in a time slot. Without loss of generality, assume that the order of transmission starting to transmit is from the left node to the right node, as shown in Fig. 1. When sender v starts to transmit, it notifies its neighbors starting to transmit after slot 1 + slot time slots. Then, Right nodes of sender v start to transmit one by one. Finally, sender w will transmit after slot 1 + T max slot time slots. Theorem 4: In a time slot, Algorithm 2 can give a scheduling set, which consists of a fraction of the optimal one, denoted by 1 3 2 +4 . The approximation ratio of Algorithm 2 is a constant: 1 3 ( 2 + 2). Proof: In a time slot, denote the number of senders can execute transmit() concurrently selected by Algorithm 2 and the optimal schedule by U lls and U opt , respectively. In Algorithm 2, for sender v, if no active neighbors obtained by Algorithm 1 are executing transmit() with probability q, it will transmit successfully. Expectantly, at most one active sender within ( + 1)l max executes transmit() by settings of q = 1 6( 2 +1) , and it is selected into U lls according to Corollary 2. For a optimal schedule, we regard those active neighbors can be scheduled concurrently with sender v in end for 9: end while same slot. Then, according to Inequality (6), Thus, we get From above description, we know the last active senders can be scheduled by costing slot 1 + T max slot time slots. Denote the number of slots until all active senders are scheduled successfully by Algorithm 2 and the optimal solution by U t lls and U t opt , respectively. Thus, approximation factor of SLS problem is given by max + 6( 2 + 1) T max e T max + 6( 2 + 1) < 1 + 6 e ( 2 + 1).

B. RANDOM CONTENTION BASED ALGORITHM FOR SLS
Random contention-resolution algorithms are probably the most intuitive way to share limited resources among several agents in a distributed fashion. The idea is that each node accesses the resource in any time slot with a certain probability q until its first success [36]. The algorithm from [2] is a natural backoff scheme, denoted by Distributed, in the tradition of ALOHA. It is run synchronously, but independently, on each sender of a link. However, their algorithm needs global information for setting transmission probability for each sender and calculating interference at each receiver.
In this section, we present an improved algorithm of random contention-resolution for SLS problem. This algorithm takes advantages of the local decision of the protocol model and the high feasibility of the SIR model. We start by giving a conclusion that if q is chosen small enough, a fraction of the transmissions is successful.
Intuitively, when the probability q v is set right for each sender, a large fraction of the links will transmit successfully, in expectation. This is argued in the following Lemma. What remains is proving that the last one link can be scheduled after some certain time slots. Similar to Formula (11), sender v transmits successfully if there is no active senders within ( + 1)l max , corresponding success probability is at least Due to senders only compete with their neighbors from Corollary 2, there are at most ( T max + 1) senders within ( + 1)l max . The expected number of remaining unsuccessful links in first slot is at most After consuming 4(δ + 1) T max ln T max + 1 time slots, the expected number of remaining unsuccessful transmissions is at most 2 . This is,

VI. SIMULATIONS
In this section, we evaluate how the distributed algorithms performs with uniform power assignment by simulator MAT-LAB. Simulations are carried out on random networks constructed by randomly placing senders on a 1000 × 1000m 2 plane and are done over 200 different networks. Each corresponding receiver is placed by choosing the angle and the distance to the sender uniformly at random from a fixed interval. In this way, a minimal and a maximal distance between sender and its receiver can be specified. The related parameters are given in Table 1. For comparison, we use centralized single slot scheduling algorithm by Goussevskaia, Halldórsson and Wattenhofer (GHW) [6] to solve SLS problem. Their algorithm is a simple greedy algorithm, where the links are processed in a nondecreasing order of length, and each link is included in the set of active senders if the affectance of the link, caused by the current set of active links is less than or equals to a constant c, where c = 1 (2 + max(2, ((2 3 · 9 + 1)β α−1 α−2 ) 1 α ) .
Even though GHW algorithm is an O(1)-approximation algorithm, we realized that for the algorithm to be competitive on our random instances, the constant was too low, resulting in very small sets of active senders.  First, we consider the impacts of the number of links on time complexity and schedule length, as shown in Figs. 2 and 3, respectively. The SIR parameters were set to α = 5, β = 1, P = 6mW, maximum link length is l max = 10m and = 0.1. Over n increasing, the number of time slots needed increases for algorithms LLS-SLS, CLLS and Distributed [2]. This is because that the probability of nodes starting to transmit decreases for greater n, thus they need more time slots to start to transmit. As could be expected, GHW algorithm can not compare with LLS-SLS, CLLS and Distributed. Specifically, GHW computes 1.14, 1.36, and 1.87 times longer schedules than CLLS and Distributed, respectively. Algorithm Distributed computes shorter schedules than LLS-SLS and CLLS since it uses the dynamic transmission probability, namely the schedule length is increased by 32.8% for LLS-SLS and 28.3% CLLS. More important, CLLS only needs local information but without considering the global interference.
Then, setting n = 200 and other settings keep the same as before. The influences of the path-loss exponent on the time complexity and schedule length are shown in Figs. 4 and 5, respectively. As α increases, Smaller path-loss exponent means there are less neighbors for a sender since ( + 1)l max   decreases, i.e., the numbers of time slots needed for LLS-SLS and CLLS decrease. Compared with Distributed, the number of time slots needed for CLLS decreases 38.88%, and its schedule length increases 37.68% on average.
Next, in Figs. 6 and 7, we analyze the influence of the decoding threshold. Greater decoding threshold means there are more neighbors for a sender since ( +1)l max gets greater,   and then the numbers of time slots needed for LLS-SLS and CLLS increase. On average, Distributed needs 1.262 times the number of time slots than CLLS, but the latter computes a 1.745 times schedule length than the former. A sender will compete with more neighbors to try to transmit successfully with a smaller 1 T max , causing more schedules. VOLUME 8, 2020 The influence of the link length is given in Figs. 8 and 9. We can see that Distributed needs more time slots than CLLS for smaller link length, but over l max increasing, a sender will have more neighbors and compete with them with a smaller transmission probability, resulting in more time slots and schedules needed. CLLS computes a 1.853 times schedules than Distributed on average. Moreover, GHW computes 1.311 and 2.43 times schedules than CLLS and Distributed, respectively.
To sum up, the simulations show that although Distributed computes a shorter schedule for different link size, path-loss exponent and decoding threshold, CLLS needs more less time slots and has localized characteristics, which has an advantage in realistic environment.

VII. CONCLUSION
In this paper, by utilizing the protocol model to localize global interference of SIR model, we first show that a successful link under protocol model is also SIR-feasible. Furthermore, we prove that the probability of a SIR-feasible link is at least 1 − under Rayleigh fading. Based on the conclusion, we can design localized distributed link scheduling algorithms of SIR model under Rayleigh fading. Furthermore, combing random contention with localization characteristic of protocol model, SLS can be solved within logarithmic time complexity. Finally, the simulations verifies our theoretical analysis of the designed distributed algorithms.
Only uniform power assignment is considered, hence one future direction is to consider other methods of power control. Another promising research direction is the dynamic network scenarios where nodes move randomly to broadcast or collect data.