A Method for Optimizing the Synchronization Efficiency of Multi-hop Wireless Ad Hoc Networks Based on Markov Chains

Abstract

The present invention discloses a method for optimizing the synchronization efficiency of multi-hop wireless ad hoc networks based on Markov chains. Aiming at the node channel contention problem in a multi-hop network environment, this method uses a three-dimensional Markov chain model to model the synchronization message sending competition mechanism, obtains the steady-state probability distributions of the backoff stage and the transmission stage of the master-slave network synchronization process based on the two-way information exchange mechanism according to the non-empty one-step state transition probability, and uses the steady-state probability normalization condition to calculate the probability of a node sending a synchronization beacon frame within a unit time slot. Considering the unfairness of multi-hop network link sending and conflicts, the quantitative relationship between the node conflict probability and the synchronization completion time and the value of the contention window in the network under the given network scale condition is analyzed, and the iterative method is used to determine the mathematical expectation of the completion time of the master-slave network synchronization process. Finally, according to the time expectation expression, the optimal value of the distributed network synchronization backoff parameter under the given network scale condition is determined by the numerical search method. The simulation experiments in the network simulation environment prove the effectiveness of this method, and the comparison results with other algorithms in the modeling and simulation software Matlab prove the advantages of this method in improving the synchronization efficiency.

Description

Technical Field

[0001] The present invention belongs to the field of wireless networks, and particularly relates to a method for optimizing the synchronization efficiency of multi-hop wireless ad hoc networks based on Markov chains. Background Art

[0002] Wireless Ad hoc Networks is a new type of wireless communication network that does not rely on fixed infrastructure. It consists of a group of mobile nodes with wireless transceiver devices, and is characterized by fast and flexible networking and high reliability. Each node in the network is both a terminal and a router, and can forward data packets from other nodes in the network. After the node is powered on, it can quickly form a network in a multi-hop self-organizing manner under the control of a hierarchical network protocol. When some nodes fail or are destroyed and stop working, the operation of the entire network will not be affected, so the network has strong anti-destruction and self-healing capabilities.

[0003] In recent years, wireless ad hoc networks have become an important wireless networking method. UAV Ad hoc Networks is the product of the application of wireless ad hoc network technology in UAV networking. Compared with the traditional network structure, UAV ad hoc networks have the characteristics of strong flexibility and high reliability. They can fly in the air for a long time to perform military tasks such as tracking, reconnaissance, and target indication, and can also be applied to a wide range of civilian fields such as power, communication, and ocean. At present, they have received extensive attention and development. Many key technologies of UAV ad hoc networks, such as data fusion, topology control, target tracking, and node positioning, require the entire network to maintain a unified clock reference to determine the order of event occurrence. Therefore, time synchronization is crucial in the network.

[0004] Appendix Figure 1 The flowchart of the master-slave synchronization process based on two-way information exchange is given, and the specific steps are as follows:

[0005] For the master node, at the beginning of each synchronization cycle, it calculates the optimal contention window W according to the model described below, broadcasts a Request message to inform the slave nodes of the synchronization process, and starts a start timer to wait for the Response message from the slave nodes. Next, it can be divided into the following two cases:

[0006] Case 1: The master node receives the Response message from the slave node and calculates the relative offset and delay size according to the timestamp information. The master node will wait until the timer expires or it receives the messages from all slave nodes.

[0007] Case 2: The master node detects that the channel is busy but does not receive the Response message. In this case, it will reset the timer and broadcast a Syn-col packet to notify the slave nodes that a collision has occurred. After the master node completes the information exchange with all slave nodes, it will calculate all the deviation values and send a Measure message containing the data.

[0008] For the slave nodes, after receiving the Request message from the master node, they enter the contention phase. The steps for the slave nodes to exchange timestamp information with the master node are as follows:

[0009] Step 1: Random backoff. Considering that multiple slave nodes may need to send Response messages to the master node, the slave nodes need to wait for a random time between [0, W - 1] before sending the message. If a slave node detects that the channel becomes busy during the backoff, it needs to suspend the timer and wait.

[0010] Step 2: Send the response message. When the backoff counter reaches 0, the slave node sends a Response message to the master node.

[0011] Step 3: Wait for the Syn-col message. After the slave node sends the Response message, during a collision detection time, if it receives the Syn-col message broadcast by the master node within this time, it can be known that the synchronization message it sent was not successfully received by the master node due to a collision. The slave node needs to reselect a backoff value between [0, W - 1] to send the synchronization frame.

[0012] Step 4: Adjust the clock. After the slave node receives the Measure message, it obtains the calibration information from it and completes the synchronization. Otherwise, the slave node will enter the next-level synchronization process and wait for other master nodes to synchronize with it.

[0013] From the above analysis, it can be seen that the value of the backoff parameter W has an important impact on the synchronization performance. Under the condition that the number of network nodes participating in the synchronization is certain, if the value of W is too large, it will prolong the waiting time of the nodes during the synchronization process, resulting in too long a synchronization completion time; on the contrary, if the value of W is too small, it will exacerbate the collisions of the nodes during the competition to broadcast the synchronization beacon frame, and will also reduce the efficiency of the distributed network synchronization process. In order to obtain the best synchronization efficiency, the present invention uses a discrete Markov chain to model the above synchronization process, obtains an expression for the mathematical expectation of the synchronization process completion time, and obtains the optimal value of the network synchronization backoff parameter through a numerical search method, and improves the synchronization efficiency by optimizing the contention window. Summary of the Invention

[0014] Objective of the Invention: Aiming at the packet collision problem in synchronization that is mostly ignored in existing research, the present invention proposes a method for optimizing the time synchronization efficiency applicable to multi-hop UAV ad-hoc networks. By optimizing the collision probability and the contention window, the overall synchronization completion time of the network is reduced, and the synchronization efficiency is improved.

[0015] Technical Solution: The above invention is achieved through the following technical solutions: A method for optimizing the synchronization efficiency of a multi-hop wireless ad-hoc network based on a Markov chain, including the following steps:

[0016] Step 1: Use a three-dimensional discrete Markov chain to model the synchronization process of the UAV ad-hoc network. The state of the node in the discrete Markov chain is represented by a three-dimensional random variable {s(t), i(t), b(t)}; where, s(t) is used to distinguish the state of the node, 0 represents backoff, 1 represents successful transmission, and 2 represents transmission failure; i(t) represents the remaining number of competing nodes in the current network, and the range is 1 to n; b(t) represents the remaining number of time slots required for the node to maintain the current state during backoff or transmission; According to the transition relationship between node states, the non-empty one-step state transition probability of the discrete Markov chain is obtained.

[0017] Step 2: According to the non-empty one-step state transition probability of the discrete Markov chain, obtain the steady-state probability distributions of the backoff, listening, and transmission phases, and use the normalization condition of the steady-state probability distribution to find the probability that the node broadcasts a synchronization beacon frame within a unit time slot.

[0018] Step 3: Divide the node synchronization completion time into two parts, namely the time experienced by the node before the first reply of the synchronization beacon frame to the master node and the time experienced by the node after the first reply of the beacon frame to the master node; According to the probability that the node sends a synchronization beacon frame within a unit time slot, use the iterative method to determine the mathematical expectations of the above two parts of the time length respectively; Adding the expectations of the two parts of the time length can obtain the expectation of the synchronization process completion time.

[0019] Step 4: According to the expression of the synchronization process completion time expectation, determine the optimal value of the network synchronization backoff parameter under the given network scale condition through the numerical search method.

[0020] The method for optimizing the time synchronization efficiency of the UAV ad-hoc network proposed by the present invention has been implemented in a network simulation environment. The simulation area is 1500×1500m 2 ; Considering different network scales, the total number of network nodes is 26, 41, 58, and 75 respectively, and they are randomly distributed in the simulation area. The physical layer uses the DSSS model, the network layer uses static routing, and the transport layer uses the UDP protocol; the node crystal oscillator deviation is 10ppm, the network synchronization period is 1s, the simulation time is 300s, and the remaining simulation parameters are shown in Table 1.

[0021] Table 1 Simulation Parameters

[0022]

[0023] Appendix Figure 4 The comparison between the simulation values of the completion time of the multi-hop wireless ad-hoc network synchronization process obtained by changing the synchronization backoff parameter W under different network scale conditions and the calculated values obtained by the present invention is given. The consistency between the simulation values and the calculated values demonstrates the effectiveness of the method of the present invention for determining the completion time of the multi-hop wireless ad-hoc network synchronization process under different values of the backoff parameter W. Under the condition of the change of the synchronization backoff parameter W, the calculated value of the completion time of the synchronization process obtained by the present invention is a concave function, which demonstrates the effectiveness of the present invention in using the numerical search method to determine the optimal value of the distributed network synchronization backoff parameter.

[0024] Appendix Figure 5 and 6 The comparison results between the algorithm proposed by the present invention and other algorithms in terms of the synchronization completion time and the synchronization success rate with the change of the network hop count are given. It can be concluded that the present invention has significant advantages in reducing the synchronization completion time and improving the synchronization success rate. Brief Description of the Drawings

[0025] Figure 1 is the flowchart of the multi-hop network synchronization process based on the two-way information exchange mechanism provided by the embodiment of the present invention

[0026] Figure 2 is the schematic diagram of the conflict area provided by the embodiment of the present invention

[0027] Figure 3 is the state transition diagram of the three-dimensional Markov chain model provided by the embodiment of the present invention

[0028] Figure 4 is the comparison diagram of the simulation and numerical calculation results of the WOTS algorithm provided by the embodiment of the present invention

[0029] Figure 5 is the comparison diagram of the simulation results of the WOTS algorithm provided by the embodiment of the present invention with the TPSN algorithm and the AHTS algorithm in terms of the change of the synchronization completion time with the network hop count;

[0030] Figure 6 is the comparison diagram of the simulation results of the WOTS algorithm provided by the embodiment of the present invention with the TPSN algorithm and the AHTS algorithm in terms of the change of the synchronization success rate with the network hop count;

[0031] Figure 7 is the comparison diagram of the simulation results of the WOTS algorithm provided by the embodiment of the present invention with the TPSN algorithm and the AHTS algorithm in terms of the change of the synchronization completion time with the network scale;

[0032] Figure 8It is a comparison diagram of simulation results of the WOTS algorithm provided by the embodiments of the present invention with respect to the variation of the collision probability with the network scale; Detailed implementation manners

[0033] The core idea of the present invention is as follows: aiming at the randomness and collision problems of message transmission in a multi-hop network, the synchronization efficiency of the multi-hop network is modeled and analyzed, a three-dimensional Markov chain synchronization competition stage model is proposed, and the appropriate competition window size to achieve the optimal efficiency is deduced to achieve the optimal synchronization efficiency. The method for optimizing the time synchronization efficiency of the unmanned aerial vehicle ad hoc network proposed by the present invention has been implemented in a wireless network simulation environment.

[0034] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0035] Step 1: Model the synchronization process of the unmanned aerial vehicle ad hoc network by using a three-dimensional discrete Markov chain.

[0036] For the time synchronization algorithm based on the two-way information exchange mechanism, construct a three-dimensional Markov chain model as shown in Figure 3 . The state of a node in the discrete Markov chain is represented by a three-dimensional random variable {s(t), i(t), b(t)}, where t and t + 1 respectively correspond to the start of two consecutive synchronization time periods. s(t) is used to distinguish the state of the node, 0 represents the backoff state, 1 represents the transmission success state, and 2 represents the transmission failure state; i(t) represents the remaining number of competing nodes in the current network, with a range of 1 to n; b(t) represents the remaining number of time slots required for the node to maintain the current state during the backoff or transmission process. The meanings of the other variables in the appendix are as follows: Figure 2 The meanings of the other variables in the appendix are as follows:

[0037] n: The total number of slave nodes within the transmission range of the current master node;

[0038] W: The network synchronization backoff parameter;

[0039] p(f i ): The probability that the node detects that the channel is busy and suspends waiting during the backoff process before broadcasting the synchronization beacon frame;

[0040] p(s i ): The probability that there is another slave node within the transmission range of the master node that successfully sends a synchronization frame during the backoff process before the node broadcasts the synchronization beacon frame;

[0041] p suc (i): The probability that the node successfully sends a synchronization frame during the transmission process.

[0042] Use p(b|a) to represent the one-step state transition probability from state a to b, then Figure 2The non-empty one-step state transition probabilities of the shown discrete Markov chain during the backoff process and the transmission process can be respectively expressed as follows:

[0043]

[0044] Equation (1) represents the state transition of the node during the backoff process. Among them, the first equation represents the probability of subtracting 1 from the remaining value of the current backoff counter when the duration of the node listening to the channel continuously being idle reaches the length of one backoff slot; the second equation represents the probability of subtracting 1 from both the current number of competing child nodes and the backoff counter value when there is a participating child node in the network that successfully sends a synchronization message; the third equation represents the probability of subtracting 1 from the current node's backoff counter value when the number of competing slave nodes is 1; the fourth to seventh equations represent the probabilities that after the value of the node's backoff counter decreases to 0 and enters the transmission state, the probability of successful transmission and entering the next synchronization period, and the probability of transmission failure and needing to reselect the backoff window to wait for retransmission.

[0045]

[0046] Equation (2) represents the state transition of the slave node during the transmission process of sending a synchronization frame. It is assumed that the transmission process lasts for a total of D slot lengths, and there are i nodes in the network competing to send synchronization request frames. When the remaining value of the backoff counter decreases to zero, the node enters the transmission process. In the first slot of the transmission, the probability of an instantaneous collision occurring is p ic (i); in any subsequent transmission slot, the probability of a continuous collision occurring is p c (i), and the node may enter the transmission failure state at any time. Only when there is no collision in the first slot and no collision occurs in all subsequent transmission slots, can the time synchronization between the master and slave nodes be successfully completed; otherwise, it is a transmission failure and the node re-enters the backoff stage.

[0047] Step 2: Based on the non-empty one-step state transition probabilities of the discrete Markov chain, obtain the steady-state probability distributions of the backoff, listening, and transmission phases, and use the normalization condition of the steady-state probability distribution to find the probability that the node broadcasts a synchronization beacon frame within a unit slot.

[0048] Let P(j, i, k) represent the steady-state probability of the above Markov chain, P(j, i, k) = lim t-＞∞ P(s(t) = j, i(t) = i, b(t) = k), where j ∈ [0, 2], i ∈ [1, n], k ∈ [0, W - 1]. From this, it can be obtained that the steady-state probabilities of each state of the node during the backoff process can be expressed by Equation (3).

[0049]

[0050] Among them, when the backoff counter reading k is W - 1 and there are i competing nodes in the network, the steady-state probability of the node can be obtained as

[0051]

[0052] The node will initiate the transmission of the synchronization frame when the backoff counter value decreases to 0. Therefore, when there are i competing nodes in the network, the synchronization frame transmission probability τ(i) can represent the relative distribution of the state probability when the backoff counter decreases to 0

[0053] τ(i) = p(0, i, 0) (5)

[0054] When there is link communication within the carrier sensing range of the node, the node will detect that the channel is busy and suspend waiting. Therefore, the suspension probability p(f i ) can be expressed as

[0055]

[0056] Among them, RCS represents the carrier sensing range of the node. In the case where the carrier senses that the channel is busy, the probability p(s i ) of a node participating in the synchronization competition within the transmission range of the master node successfully transmitting can be expressed as

[0057]

[0058] According to the state transition equation of Equation (2), the steady-state probability of the node being in the transmission success state in any one time slot can be expressed as

[0059] P(1, i, l) = P(S i )[1 - p ic (n)][1 - p c (n)] D-l-1 , 1 ≤ i ≤ n, 0 ≤ l ≤ D - 1 (8)

[0060] Among them, [1 - p ic (i)] means that the node does not generate an instantaneous collision in the first time slot of the transmission process, and [1 - p c (i)] means that there is no continuous collision in the remaining time slots except the first time slot. Therefore, the probability of the node being in the transmission success process during the entire transmission process is

[0061]

[0062] If a collision occurs in a certain time slot during the transmission process, the node transmission fails, and the steady-state probabilities of its various states are

[0063] P(2, i, l) = P(S i ){1 - [1 - pic (i)][1 - p c (i)] D-1-l},1 ≤ i ≤ n,0 ≤ l ≤ D - 1 (10)

[0064] The probability that a node fails to transmit during the entire transmission process is

[0065]

[0066] In the case where there are i competing nodes in the network, the probability τ'(i) that a node is in the process of transmission is equal to

[0067]

[0068] Whether a node can successfully transmit a synchronization request frame depends on whether an instantaneous collision and a persistent collision occur at the receiving node. Let I denote the instantaneous collision region and P denote the persistent collision region. τ(i) and τ'(i) respectively represent the transmission probability and the probability of being in the process of transmission of a node in the case where there are i competing nodes in the network. In the first time slot of the transmission process of the sending node, if a node in region I initiates data transmission simultaneously, or a node in region P is in the process of transmission, an instantaneous collision will occur

[0069]

[0070] In any time slot other than the first one, if nodes in region P start to send data simultaneously, a persistent collision will occur

[0071]

[0072] Assume that the entire transmission process takes D time slots. A data packet can be correctly received if and only if no collision occurs during the entire transmission process of the node, that is, no instantaneous collision occurs in the first time slot and no persistent collision occurs in the remaining (D - 1) time slots. Therefore, the probability p suc (i) is

[0073] p suc (i) = [1 - p ic (i)][1 - p c (i)] D-1 (15)

[0074] Summing up the probabilities of all states of the node during the backoff process and the transmission process, we can obtain from the normalization condition

[0075]

[0076] According to the normalization condition of the entire Markov chain and the above state transition equation, the probability of the node in each state and the conflict probability can be calculated, thereby calculating the synchronization completion time of the entire network.

[0077] Step 3: Use an iterative method to determine the mathematical expectation E[T] of the completion time of the above time synchronization process.

[0078] definition The time required for the child node to successfully complete the transmission of the synchronization request message from the P(0, i, k) state is calculated using the three-dimensional discrete Markov chain model described above. For the convenience of expression, the following symbolic variables are set: δ represents the length of the idle time slot; ε represents the transmission time of the slave node sending the Response message; ζ represents the transmission time of the master node sending the syn-col message when a conflict occurs. Let E[σ i ] represents the expected value of the length of the child node backoff slot when there are i competing child nodes in the network. Since the probability of a node listening to an idle channel is 1-p(f i ), so we can take E[σ i ] is expressed as:

[0079]

[0080] When i=1, the expected time for the child node to complete the transmission It can be expressed as:

[0081]

[0082] When i∈[2,n], The expression can be expressed as:

[0083]

[0084] In each synchronization cycle, the n slave nodes in the network need to exchange information with the master node. Therefore, for any slave node participating in the competition, when it starts the synchronization process and selects the initial backoff counter value as k, the above obtained That is, the time required for the slave node to successfully complete the transmission of the synchronization message. Therefore, the expression of the average synchronization time E[T] of the competing nodes in single-level synchronization can be obtained as follows:

[0085]

[0086] Combining the above equations (17), (18), and (19), all the above can be calculated by iteration. The value of .

[0087] Step 4: Determine the optimal value of the network synchronization backoff parameter under the given network scale condition through a numerical search method according to the expression of the expected time to complete the synchronization process.

[0088] According to the expected time to complete the distributed network synchronization process E[T] determined in Step 3, the present invention determines the optimal value of the distributed network synchronization backoff parameter under the given network scale condition through a numerical search method. The specific method is as follows: Starting from the minimum value of 1 for the value of the distributed network synchronization backoff parameter W, calculate the corresponding values of the expected time to complete the distributed network synchronization process E[T] in turn, where W = 1, 2, 3,.... If the value of E[T] satisfies

[0089] E[T] (W+1) -E[T] (W) >0 (21)

[0090] then W is the optimal value of the network synchronization backoff parameter under the current network scale condition.

[0091] The content not described in detail in this application for the present invention belongs to the prior art well-known to those skilled in the art.

Claims

1. A method for optimizing the synchronization efficiency of a multi-hop wireless ad hoc network based on a Markov chain, comprising the following steps: Step 1: Model the synchronization process of the UAV ad hoc network using a three-dimensional discrete Markov chain; Step 2: Obtain the non-empty one-step state transition probability of the discrete Markov chain according to the transition relationship between node states; Step 3: According to the non-empty one-step state transition probability of the discrete Markov chain, obtain the steady-state probability distributions of the backoff, listening, and transmission phases, and use the normalization condition of the steady-state probability distribution to find the probability that a node sends a synchronization beacon frame within a unit time slot; Step 4: According to the probability that a node sends a synchronization beacon frame within a unit time slot, use the iterative method to determine the expectation of the synchronization process completion time; Step 5: According to the expression of the synchronization process completion time expectation, determine the optimal value of the network synchronization backoff parameter under the given network scale condition through the numerical search method; Furthermore, in the said Step 2, the non-empty one-step state transition probability of the discrete Markov chain includes the following specific steps: Step 2-1: The state of a node in the discrete Markov chain is represented by a three-dimensional random variable {s(t), i(t), b(t)}; where s(t) is used to distinguish the state of the node, 0 represents backoff, 1 represents successful transmission, and 2 represents transmission failure; i(t) represents the remaining number of competing nodes in the current network, ranging from 1 to n; b(t) represents the remaining number of time slots required for the node to maintain its current state during backoff or transmission; N is the total number of slave nodes within the transmission range of the current master node; W represents the size of the network synchronization backoff window; p(f i ) represents the probability that the node detects a busy channel and suspends waiting during backoff; p(s i ) represents the probability that another slave node within the transmission range of the master node successfully sends a synchronization frame during the backoff of the node; p suc (i) represents the probability that the node successfully sends a synchronization frame during transmission; p(b|a) represents the one-step state transition probability from state a to b; Step 2-2: The non-empty one-step state transition probability of the discrete Markov chain during the backoff process is expressed as: Step 2-3: The non-empty one-step state transition probability of the discrete Markov chain during the transmission process is expressed as: Furthermore, in the said Step 3, the calculation of the probability that a node sends a synchronization beacon frame within a unit time slot includes the following specific steps: Step 3-1: Denote the steady-state probability of the above Markov chain by P(j, i, k), where P(j, i, k) = lim t-＞∞ P(s(t) = j, i(t) = i, b(t) = k), j ∈ [0, 2], i ∈ [1, n], k ∈ [0, W - 1]; thus, the steady-state probabilities of each state during the backoff process of the node are expressed as: Among them, when the backoff counter reading k is W-1 and there are i competing nodes in the network, the steady-state probability of the node can be obtained as: Step 3-2: The synchronization frame transmission probability τ(i) is expressed as the relative distribution of the state probabilities when the backoff counter is reduced to 0: τ(i) = p(0,i,0) Step 3-3: RCS represents the carrier sensing range of the node. The probability p(f i ) that the node carrier senses the channel is busy and suspends waiting is as follows: When the carrier senses that the channel is busy, the probability p(s i ) of a node within the transmission range of the master node successfully transmitting is expressed as: Step 3-4: The steady-state probability that a node is in the transmission success state within any one time slot is expressed as: P(1,i,l) = P(S i )[1 - p ic (n)][1 - p c (n)] D-l-1 , 1 ≤ i ≤ n, 0 ≤ l ≤ D - 1 Among them, [1 - p ic (i)] means that the node does not generate an instantaneous collision in the first time slot during the transmission process, and [1 - p c (i)] means that there is no continuous collision node in the time slots other than the first time slot; the steady-state probability of transmission failure is: P(2,i, l) = P(S i ){1 - [1 - p ic (i)][1 - p c (i)] D-1-l},1 ≤ i ≤ n,0 ≤ l ≤ D - 1 Step 3-5: In the case where there are i competing nodes in the network, the probability τ’(i) that a node is in the transmission process is equal to: Step 3-6: Let I represent the instantaneous collision area and P represent the persistent collision area. τ(i) and τ’(i) respectively represent the transmission probability of a node and the probability of being in the transmission process in the case where there are i competing nodes in the network; the instantaneous collision probability of a node sending a message is: In any time slot except the first one, if a node in area P starts to send data, persistent collision will occur: Step 3-7: Assume that the entire sending process requires D time slots. A data packet can be correctly received if and only if there is no collision during the entire transmission process of the node. Therefore, the probability p suc (i) is as follows: p suc (i) = [1 - p ic (i)][1 - p c (i)] D-1 Step 3-8: Sum the probabilities of all states of the node during the backoff process and the transmission process, and obtain from the normalization condition: Furthermore, in the said Step 4, the calculation of the mathematical expectation of the synchronization completion time includes the following specific steps: Step 4-1: Divide the node synchronization process completion time into two parts, namely the time experienced by the node before it first replies to the synchronization beacon frame to the master node and the time experienced by the node after it first replies to the beacon frame to the master node; Step 4-2: Use to represent the time required for the child node to successfully complete the transmission of the synchronization request message from the state of P(0, i, k); use δ and ε to represent the idle slot length and the transmission time of the slave node to send the Response message respectively; ζ represents the transmission time of the master node to send the syn-col message when a collision occurs; let E[σ i represent the expected value of the backoff slot length of the child node when there are i competing child nodes in the network: Step 4-3: When i = 1, the expected value of the time for the child node to complete sending is expressed as: When \(i\in[2,n]\), is expressed as: Step 4-4: For any slave node participating in the competition, when it starts the synchronization process and selects the initial backoff counter value as k, the is the time required for the slave node to successfully complete the transmission of the synchronization message. Therefore, the expected synchronization time E[T] can be obtained as follows: Step 4-5: Combine the above equations and calculate all the values in Step 4-3 through iteration.

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