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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI10.1109/ACCESS.2020.2998575, IEEE Access

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Research on the Impact of InteractionTimes in Distributed Wireless Ad HocNetworksXIN LI1, XU LI11School of Electronic and Information Engineering Beijing Jiaotong University, Beijing, 100044 China

Corresponding author: Xu Li (e-mail: [email protected]).

This work was supported in part by the National Key R&D Program of China under Grant 2018YFC0826303, and in part by theFundamental Research Funds for the Central Universities under Grant 2018JBZ102.

ABSTRACT The distributed wireless ad hoc network caters to users’ demands for high reliability, lowlatency, and low cost. It has the characteristics of low operation and maintenance costs, strong scalability,no center, and self-organization, and has wide application prospects in 5G and industrial Internet of Things.The access process is a key process for the operation of distributed wireless ad hoc networks, which has ahuge impact on its performance. The distributed election mechanism is an effective solution to avoid accesscollision and achieve efficient access. The number of interactions in the election mechanism will affect theamount of information obtained by the nodes, which will affect the performance of the network’s electionsuccess probability, delay, and efficiency. When the number of interactions is small, the node obtains lessinformation, and the collision and waste of time slots are serious, resulting in deterioration of networkperformance. Increasing the number of interactions can increase the amount of information acquired bythe nodes, but it will also increase network delay and network overhead. However, the current researchon the number of interactions is insufficient. Therefore, this paper studies the impact of the number ofinteractions on the legal election set and establishes the analytical models of election success probability,delay, and efficiency. The range of interaction times is determined by maximizing the efficiency, electionsuccess probability and minimizing delay. Finally, simulation results show that the number of interactionsshould be adjusted according to the network scale and efficiency.

INDEX TERMS Distributed wireless ad hoc networks; Interaction times; Election success probability;Delay; Resource efficiency

I. INTRODUCTION

A. MOTIVATION

D ISTRIBUTED wireless ad hoc network is a mobilecommunication network that does not depend on basic

network facilities. It can quickly build a wireless communi-cation network at any time and any place [1]. Compared withcentralized networks, distributed wireless ad hoc networksdo not have strict control centers, and all communicationnodes are equal in status. A single node failure does not havea great impact on the network, and it has good resistanceto destruction and scalability. Therefore, it has been widelyused in the Internet of Things, military communications,rescue and disaster relief, and emergency communications[2], so it is necessary to conduct research on distributedwireless ad hoc networks. The main challenge of wireless

communication is how to find a cost-effective way to sharechannel resources [3]. The election mechanism is an effectivesolution to avoid access collision and improve access effi-ciency. Compared with the distributed resource schedulingmechanism of IEEE802.11 protocol based on the competition[4] and the IEEE802.15.4 protocol based on the competitionand reservation [5], the IEEE802.16 mechanism [6] based onelections and reservations can improve the resource utiliza-tion efficiency of the channel while reducing conflicts [7].During the access process, due to direct channel occupancy,the competition mechanism has a severe collision and alow probability of success. The reservation mechanism usesmultiple signaling interactions to reserve channels to reduceaccess collisions, but the signaling consumption is high. Theelection mechanism uses an election algorithm to determine

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Xin Li, et al.: Research on the Impact of Interaction Times in Distributed Wireless Ad Hoc Networks

the election set for election according to the next time thenode sends control signaling. This allows only a single nodeto send data per time slot [8], which reduces the signalingconsumption while reducing the collision loss during theaccess process.

In the election mechanism, the number of interactionsaffects the amount of information obtained by the node, andthus affects the node’s judgment on whether its neighborscan join the election set. When the number of interactionsis less, the amount of information obtained by the nodes isless, the collision between the nodes and the waste of timeslots are serious, the probability of node election success andthe efficiency is low. Increasing the number of interactionscan increase the amount of known information of the node,eliminate the nodes whose next transmission time is not inthe current election time slot. At the same time, it reducescollisions and waste caused by forwarding failures and biterrors, and improves the probability of node election success.However, the increase of interactions will cause an increasein resource consumption, reduce the resource efficiency ofthe network, and increase the network delay.

B. RELATED WORKAt present, many scholars are studying the election mech-anism of distributed wireless ad hoc networks. Reference[9] considered the delay performance of the network andstudied the three-way handshake time of the distributedwireless ad hoc network MAC layer. However, it did notconsider the impact of the election mechanism on the accessdelay, and set the access time of the scheduling messageto a constant. Reference [10] proposed a scheme to adjustthe priority based on the channel state and the buffer state,which improved the network throughput to some extent andreduced the packet loss rate. However, it did not consider theimpact of the number of interactions in resource allocationon the allocation result, nor did it consider the optimizationof network delay performance. Reference [11] proposed animproved scheduling algorithm based on a variable back-off algorithm, which improved the latency and throughputof the network, but it did not consider the impact of thenumber of interactions on the probability of node accesssuccess. Reference [12] proposed an adaptive hold-off indexalgorithm, which adjusted the hold-off index of the networkbased on the location of the node along with the networktraffic. This algorithm reduced the maintenance delay of thenode, but it made the node less consistent due to less inter-action, which affected the network performance. Reference[13] proposed a cross-layer scheduling scheme, and adjustedthe hold-off index of the nodes according to the channel-quality information (CQI) of the physical layer. While it didnot consider the impact of the number of interactions onthe legal election set. Therefore, from the existing literature,for the study of distributed election mechanism, there is noliterature on the important parameter of the number of inter-actions. However, the impact of interaction times in practicalapplications cannot be ignored. Especially in the battlefield

scene where the electromagnetic interference is serious, thesignal is unstable, and the topology changes rapidly. If theinfluence of network interaction is not considered, the rapidchange of network topology makes it unable to let nodesobtain enough information for a single interaction to maintainelection sets. The consistency of the election set is poor. Atthis time, a high probability of data transmission will causenetwork collision or waste of time slots.

C. CONTRIBUTIONBased on the above, the novel contribution of this paper arefollowing.

1) We first establish a multiple interaction model of thedistributed election mechanism and tell the relationshipbetween the number of interactions and the election set.

2) In order to analyze the impact of the number of inter-actions on network performance, we establish a modelof election success probability, delay, and resourceefficiency.

3) We establish a comprehensive optimization model andselect the optimal number of interactions by maxi-mizing resource efficiency, election success probabilityand minimizing delay.

4) We analyze the theoretical and simulation results andfind that the number of interactions has a great influ-ence on the performance of the network, and there isan optimal number of interactions to make the networkperformance optimal.

The organizational structure of this article is as follows.The second part introduces the system model, the third partestablishes the election success probability, scheduling delay,efficiency and comprehensive optimization model based onthe number of interactions, the fourth part carries on thenumerical simulation analysis to the model, and the last partsummarizes the completed work.

II. SYSTEM MODELA. NETWORK MODELThis paper uses a distributed wireless mesh network topol-ogy, and all nodes are distributed independently. Becausenetwork shape has no effect on network performance, forconvenience, this paper uses the network topology modelshown in Fig.1. We assume that the number of nodes in thenetwork is N = π(hr)2ρ, where r denotes the effectivecommunication radius of the node, h denotes the maintainedneighbor range of the node, and ρ denotes the density of thenode.

B. DISTRIBUTED ELECTION MECHANISMThe distributed election mechanism can reduce the collisionloss of scheduling message access and ensure reasonablescheduling and collision-free transmission of control mes-sages. Each time the node sends control signaling, it electsm transmission opportunities after it. And the election objectis all neighbors that meet the election conditions within thetwo-hop range of the node. The election process is as follows.

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FIGURE 1. An example of the Distributed Ad Hoc Network model

1) The nodes randomly back off and wait for the MSH-NCFG transmission time.

2) When the node receives the NCFG messages fromneighbor nodes, it will update the neighbor informationaccording to the content of the neighbor carried inthe NCFG message, and record the next transmissiontimeNextXmtT ime(NXT ) and the earliest next nextsending timeEarliestSubsequentXmtT ime(ESXT )of all neighbor nodes. Based on this, the effective set ofcompeting nodes that have a competitive relationshipwith this node is determined.

3) The node chooses the first idle slot after back-off,and competes with the eligible two-hop neighborsin turn. The competition algorithm uses the Mesh-Election algorithm. If the competition is successful, thenode continues to elect until the specified number ofinteractions is reached and back off. If the competitionfails, it continues to compete for the next time slot untilit succeeds.

The important thing in election process is to determine thenodes that meet the requirements in the two-hop neighbors.The node uses two parameters NextXmtMx(NXM) andXmtHoldoffExponet(Exp) to calculate the next and theearliest next transmission time of the neighbor.

The range of the next transmission time NXT follows

2Exp ×NXM < NXT ≤ 2Exp × (NXM + 1) (1)

The range of the earliest next transmission time ESXT isexpressed as

ESXT = NXT + E(H) (2)

Among them, E(H) denotes the expectation of the lengthof the maintenance interval H . Since all nodes run the sameprotocol in a distributed network, all parameter settings arethe same. Then E(H) can be expressed as

E(H) = mNnbr (3)

FIGURE 2. The relationship between the number of interactions and theelection mechanism

Among them, m represents the average number of interac-tions of a single node, that is, after a node successfully sendsa DSCH message, each node is required to interact at leastm times to obtain a sufficient amount of information beforestarting a new election to obtain a time slot.

If theNXT or theESXT of the neighbor nodes contain thecurrent election time slot, we add them to the election set.

III. MODEL ANALYSISFrom the perspective of the protocol, this section analyzes theimpact of the number of interactions on network performanceby establishing models such as the probability of electionsuccess, delay, and resource efficiency to guide the value ofparameters in engineering applications.

A. ELECTION SUCCESS PROBABILITY

Fig.2 is an example of the relationship between the number ofinteractions and the election mechanism. As can be seen fromthe figure, nodes interact with neighbors during the backoffprocess to collect neighbor maintenance messages. The nodestarts the election in the first time slot after the interaction iscompleted. If the election is successful, the election is contin-ued until the specified number of interactions is completed.If the election is not successful, the node continues to electthe next time slot until it succeeds.

In the network with node density of ρ, if the node main-tains neighbors in the h-hop area, the number of neighborsin the node maintenance area is Nnbr = π(hr)2ρ − 1.The number of interactions is m, and the effective electioninterval V = 2Exp. Let Q denote the election length, whichrepresents the number of time slots needed for one nodefrom the end of the back off process to the next successfulcompetition, and we have 0 ≤ Q ≤ V in consideration of thelimited election interval. This paper sets the same parametersfor all nodes, then the nodes in the network occupy time slotresources equally, and the probability of successful access inq time slot can be expressed as

pq =1

Ncpt(q)(4)

In which Ncpt(q) denotes the number of nodes participat-ing in the election in slot q. The probability of the node firstsuccessful election in q slot can be expressed as

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P (Q = q) =

q−1∏i=1

(1− pi)pq, q ∈ (0, V ] (5)

Since the parameters of all nodes are the same, the proba-bility of election success in each time slot is the same understeady-state conditions, that is, pi = pq = p. The probabilitymass function of Q is expressed as

P (Q = q) = (1− p)q−1p, q ∈ (0, V ] (6)

The number of interactions before the election affects thenumber of nodes participating in the effective election, whichis denoted as Ncpt(q). According to the above system model,the effective competition nodes mainly include two parts:• The nodes with known scheduling parameters;• The nodes with unknown scheduling parameters;In the actual network, due to problems such as untimely

message forwarding and signal error codes, the nodes withknown scheduling parameters in the network mainly includeone-hop neighbors whose messages can be received correctlyand two-hop neighbors whose messages can be successfullyreceived and forwarded by one-hop neighbors. Therefore, thenumber of nodes with known scheduling parameters can beexpressed as

Nk = N1(1− pf ) +N2(1− pn)(1− pf )(1− p1− p2) (7)

Where N1 is the number of one-hop neighbors of thisnode,N2 is the number of two-hop neighbors of this node, p1denotes the probability of inconsistent messages due to un-timely forwarding, p2 denotes the probability of inconsistentmessages due to network errors, pf denotes the probabilityof sending failure due to channel instability and topologychanges, and pn denotes the probability of a node newlyjoining the network.

In the distributed wireless ad hoc network, without loss ofgenerality, we choose a node in the network as a typical nodefor analysis. For nodes with known scheduling parameters,only when a typical node and its neighbors form the relation-ship shown in Fig.3, the typical node adds its neighbors to theelection set. The competitive relationship between a typicalnode and a neighbor node includes the following two cases,where node k represents a typical node and node 1 representsa neighbor node.

The e in Fig.3 represents the time from the current timeto the next transmission opportunity. The size of e greatlyaffects whether the NXT and the ESXT of the neighborinclude q slot.

As shown in Fig.4, Z(t) denotes the number of NCFGtransmissions of node k at slot t, then Z(t) is a countingprocess with time τ , where τ is the election period. Thecounting process of each node eventually stabilizes, obeyingi.i.d. in each cycle, and finally forms a smooth traversalupdate process.

FIGURE 3. Description of nodes participating in the election

FIGURE 4. Description of the renewal process

In the election process, Z(t) is an update process, t is themoment that is selected arbitrarily during this election cycleτ , and e represents the remaining time from time t to thenext election. The limit distribution of remaining time [14]satisfies

P (e ≤ y) = 1

µ

y∑i=1

P (τ ≥ i) (8)

In which µ = E(τ) = H + E(Q). Since the distributionof the election cycle τ satisfies

P (τ ≥ i) ={

1 i ≤ H∑Vs=i−H(1− p)s−1p i > H

(9)

Thereby

P (e = y) = P (e ≤ y)−P (e ≤ y−1) =1

µP (τ ≥ y) (10)

According to Fig.3, it can be concluded that the propor-tion a of nodes with known parameters participating in theelection satisfies

α =P (e < q|Q = q) + P (q +H − V < e ≤ q +H|Q = q)

=V

µ+

1

µ

q+H∑i=H+1

V∑q=i−H

(1− p)qp

=V

µ+

1− (1− p)q

µp− q

µ(1− p)V , 0 ≤ q ≤ V

(11)In the distributed wireless ad hoc network, a typical n-

ode obtains maintenance information of a two-hop neighbor

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FIGURE 5. Schematic diagram of the relationship between two-hop nodesand typical nodes

FIGURE 6. Schematic diagram of the relationship between two-hop nodesand typical nodes

through its one-hop neighbor. Only when the typical nodeforms a relationship with its one-hop neighbor and two-hopneighbor as shown in Fig.5, the two-hop neighbor informa-tion obtained by the typical node is timely. That is, after thetwo-hop neighbors of a typical node send information, thereis a one-hop neighbor of the node sending information inorder to forward the information of the two-hop neighbors.

Therefore, the probability that the two-hop node informa-tion is not timely due to forwarding is expressed as

pout = 1−P (q +H − V <e1 ≤ q +H, e2 ≤ e1 − V |Q = q)

= 1− q +H − Vµ

− 1− (1− p)q

µp+q

µ(1− p)V

(12)As shown in Fig.6, there are the one-hop common neigh-

bors l in the intersection area of the two-hop neighbor mand the typical node k, the common neighbor forwards themessage of the two-hop neighbor to the typical node. Thearea of intersection is as follows.

A(d) = 2r2 arccosd

2r− r

2

√4d2 − r2 (13)

Where d is the distance between two-hop neighbors. Thenthe number of common one-hop neighbors in the intersecting

area is as follows.

Nc = ρA(d) (14)

The probability that all common one-hop neighbors are notforwarded in time is expressed as

p1 = pNcout (15)

Assume that all common one-hop neighbors cannot for-ward messages normally due to bit errors. Then the transmis-sion failure probability due to the channel is expressed as

p2 = pNc

f (16)

Nodes with unknown scheduling parameters mainly in-clude nodes whose one-hop neighbors failed to send dueto bit errors, unknown two-hop newly-entered nodes, andnodes among two-hop neighbors that were not successfullyreceived due to forwarding and bit errors. The nodes withunknown scheduling parameters can be expressed as

Nuk = N1pf +N2pn +N2(1− pn)pf+N2(1− pn)(1− pf )(p1 + p2)

(17)

Then the number of nodes participating in the election isexpressed as

Ncpt =1

pele= αNk +Nuk + 1 (18)

Therefore, the probability of successful election of nodesin the network is expressed as

pele =1

αNk +Nuk + 1(19)

It can be seen from formula (19) that the probability ofelection success pele is related to the number of elected nodesets. Reducing the number of elected node sets can increasethe probability of node election success. In addition, since µis a parameter related to m, α and µ are generally inverselyproportional. As the number of interactions increases, αdecreases and the set of nodes participating in the electiondecreases. In other words, the increase in the number ofinteractions will make the probability of nodes with knownscheduling parameters participating in the election smaller,thereby reducing the number of nodes participating in theelection in a single time slot and increasing the probabilityof election success.

B. DELAY PERFORMANCE ANALYSISThe distributed network election mechanism can ensure thatonly one node sends control messages for each time slotwithin a given neighbor maintenance range. On one hand,the number of interactions affects the size of the maintenancedelay. On the other hand, the number of interactions affectsthe election success probability of the node, thus affecting thetransmission probability of the control slot, and affecting the

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FIGURE 7. Schematic diagram of election delay

election delay. This section considers the impact of interac-tion times, determines the probability distribution of electionmessages, and analyzes the impact of interaction times onnetwork latency performance.

The network delay is mainly divided into two parts, main-tenance delay and election delay. Maintenance delay refersto the delay in which a node receives neighbor messages toupdate election information. The election delay is determinedby three factors, the first is the arrival time of the maintenancemessage, the second is the election length of the electionmechanism, and the third is the number of node interactionsin the network. The election delay is shown in Fig.7.

Where T1 denotes the election delay of the election mech-anism, and its probability distribution is expressed as

P (T1 = k)

=

1

µ, 0≤k≤H

1

µ((1− pele)k−H−1−(1− pele)V ), H<k≤H + V

(20)According to the above analysis, the expectation of T1 can

be obtained as

E(T1) =H+V∑k=1

kP (T1 = k)

=H(H+1)

2µ+(1−pele)V−1

µpele− (2H+1+V )V

2µ(1−pele)V

(21)Therefore, the network delay can be expressed as

T = H +mE(T1) (22)

C. RESOURCE EFFICIENCY ANALYSISIn a distributed wireless ad hoc network, when the numberof network interactions is small, the amount of informationacquired by the nodes is low, the network is wasted or col-lided seriously, and the network efficiency is low. Increasingthe number of interactions can improve network efficiency,but blindly increasing the number of interactions does notincrease the amount of effective information obtained by thenodes, and also leads to a reduction in network efficiency.

This section analyzes the impact of interaction times onnetwork resource efficiency.

We define resource efficiency as the ratio of the amount ofeffective information to the total cost. Network consistency,channel quality and other impacts are specifically consideredin collision overhead and waste overhead.

η =I

C1 + C2 + C3(23)

Among them, I denotes the amount of effective infor-mation, C1 denotes the network maintenance overhead, C2

denotes the network collision overhead, and C3 denotes thenetwork wasted overhead.

In the election mechanism, the effective amount of infor-mation refers to the election information that can determinethe election set, that is, the neighbor node can successfullysend the election parameters to the current node after multipleinteractions. The amount of effective information is relatedto the number of network nodes and the probability of suc-cessful transmission of the nodes. The amount of effectiveinformation I can be expressed as:

I = π(hr)2ρI0(1− pmf ) (24)

where I0 denotes the amount of information maintained bya single node.

The total network cost mainly includes the normal main-tenance cost of the network, collision cost and waste costdue to consistency and bit errors. We assume that each timethe maintenance cost of a single node is K, so the normalmaintenance cost of the network is expressed as

C1 = π(hr)2ρmK (25)

The only reason for the collision in the network is whena new node is added to the network or the node that has justbeen maintained this time. At this time, the election set ofthe two-hop neighbors is inconsistent due to the effect ofmessage forwarding. When there is a new node in the two-hop neighbor of a typical node and the typical node does notknow the existence of the new node, if the typical node electsitself and the one-hop neighbor of the known new node electsa new node, this will cause a collision . The typical node doesnot know that the existence of the new node is mainly dueto two situations, one is that the message is not forwardedin time, and the other is due to the high signal error rate.At this time, for a typical node, the acquired informationis consistent with the original. The number of nodes withknown scheduling parameters and the number of nodes withunknown scheduling parameters does not change.

For one-hop neighbors who know the existence of newnode, the number of nodes with known parameters is asfollows.

N ′k = N1(1−pf )+N2(1−pn)(1−pf )(1−p1−p2) (26)

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The number of nodes with unknown parameters is asfollows.

N ′uk = Na +N1pf +N2(1− pn)(1− pf )(p1 + p2)

+N2pn(1− pf )(1− p1 − p2) +N2(1− pn)pf(27)

Among them, Na = N2pn(1 − pf )(p1 + p2) denotesthe number of newly added nodes that successfully sentmessages to a one-hop neighbor but were not successfullyforwarded by the one-hop neighbor.

Therefore, in a one-hop neighbor of a typical node, thenumber of nodes participating in the election is as follows.

Ncpt−1 = α′N ′k +N ′uk + 1 (28)

Where α′ denotes the proportion of nodes with knownparameters participating in the election under the currentprobability of successful election. The collision caused bymessage forwarding is because a typical node elects itselfin a certain time slot and a one-hop neighbor of the typicalnode elects a two-hop neighbor that the typical node does notknow. The collision probability can be expressed as:

pc =Na

α′Nk +N ′uk + 1(29)

Then the expectation of collision cost can be expressed as:

C2 = pcπ(hr)2ρmK (30)

The reason for the waste in the network is that when allthe nodes are maintained, the two-hop neighbor of a typicalnode leaves the network but is not forwarded to the nodeby its one-hop neighbor. At this time, the election set ofthe two-hop neighbors is inconsistent. When a typical nodeelects a two-hop neighbor node that has left in a certaintime slot, no node occupies the channel and causes waste.At this time, for a typical node, the acquired informationis consistent with the original. The number of nodes withknown scheduling parameters and the number of nodes withunknown scheduling parameters does not change.

The number of neighbor nodes of a typical node partici-pating in the election is as follows.

Ncpt−t = αNk +Nuk + 1 (31)

For one-hop neighbors who already know that a nodehas left, the number of nodes with known parameters is asfollows.

N ′k = N1(1− pf ) +N2(1− pn)(1− pf )(1− p1 − p2)+N2(1− pl)(1− pf )(1− p1 − p2)−Nk

(32)Among them, Nl = N2pl(1 − pf )(p1 + p2) denotes

the number of newly leaving nodes that successfully knownby one-hop neighbors but were not successfully forwarded,and pl denotes the probability of a node newly leaving thenetwork. Therefore, the waste caused by message forwarding

is because the typical node elects a two-hop neighbor whohas left. The probability of waste is as follows.

pw =Nl

αNk +Nuk + 1(33)

Then the expectation of wasted cost can be expressed as:

C3 = pwπ(hr)2ρmK (34)

D. COMPREHENSIVE OPTIMIZATIONIncreasing the number of interactions will increase the net-work delay, but at the same time will also increase theaccuracy of maintenance, which will reduce the waste of timeslots, improve resource efficiency, and reduce delay. Networkdelay includes maintenance delay and election delay. Onone hand, the number of interactions affects the maintenancedelay. On the other hand, the number of interactions affectsthe accuracy of message maintenance and the resource effi-ciency of the network, thereby affecting the election delay.This section takes resource efficiency as a constraint andanalyzes the number of interactions m when the networkdelay is minimum. The objective function is T , the constraintis η ≤ σ, σ is the maximum value of resource efficiency, andthe node density of the network should satisfy ρ ≥ 10−5. Theoptimization problem can be expressed as

min T

s.t.η ≤ σρ ≥ 10−5

(35)

Then the K-T condition of this optimization problem is asfollows.

∂T

∂m+ w1

∂(η − σ)∂m

+ w2∂(100000ρ− 1)

∂m= 0

w1(I

C1 + C2 + C3− σ) = 0

w2(100000ρ− 1) = 0

w1, w2 ≥ 0

(36)

According to ∂2T∂m2 ≥ 0 and ∂2η

∂m2 ≥ 0, it can be knownthat both delay and resource efficiency in the above formulaare convex functions related to the number of interactionsm, then the optimization problem has an optimal solution.This is because the more interactions, the greater the delay,but the more accurate the node’s election set, the higher theprobability of election success, and the higher the utilizationof time slot resources. The resource utilization can reducenetwork waste and collision, thereby reducing network delay.Therefore, there is a suitable m to minimize the networkdelay under the condition of maximum resource efficiency.

IV. NUMERICAL ANALYSISIn this section, we use MATLAB to simulate the abovemodels. The purpose of simulation analysis is to analyze theperformance changes of the above models under different

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TABLE 1. Parameter Settings

Notation Meaning Valueρ Node density (nodes/km2) 0-100r Effective communication radius (m) 250h Number of hops maintained by neighbors 2Exp Backoff index 0/1/2V Election interval 2Exp

m Number of interactions 1-3

FIGURE 8. The variation of the election success probability with network nodedensity given m=1,2,3.

parameters. Table 1 summarizes the important parametersused in the simulation process.

Fig.8 depicts the variation of election success probabilitywith the number of interactions under different node den-sities. As can be seen from the figure, as the node densityincreases, the election success probability decreases. The rea-son is that when the density of the network nodes increases,the number of effective competing nodes increases and thisnode needs to compete with more nodes for the transmissiontime slot. So the election success probability will decrease.When the density of network nodes is given, as the number ofinteractions increases, the election success probability of thenode increases. Because the more the number of interactionsis, the more information is obtained, which reduces the setof legal election nodes participating in the election, therebyeffectively alleviating the fierce competition.

Fig.9 depicts the change of delay with node density un-der different interaction times under the condition of lowprobability of sending failure. As can be seen from thefigure, no matter how large the number of interactions is, thedelay increases as the density of nodes increases. Under thesame network density, the greater the number of interactions,the greater the delay. This is because the increase in thedensity of nodes expands the set of legally elected nodes.Nodes need to be elected together with more nodes to obtainthe transmission right of the time slot. The chance of thenode obtaining the time slot is reduced, which increasesthe delay. Since the transmission failure probability is low,

FIGURE 9. The variation of the delay with network node density givenm=1,2,3 under pf = 0.1

the probability of neighbor transmission failure is low. Lessinteraction has been able to meet the maintenance needsof the network. At this time, the increased interaction canincrease the probability of successful elections and reduce thedelay. However, it can no longer offset the maintenance delaycaused by the increased number of interactions. Therefore,under the same node density, the delay increases as thenumber of node interactions increases.

Fig.10 depicts the change of delay with node densityunder different interaction times under the condition of lowprobability of sending failure. As can be seen from the figure,no matter how large the number of interactions is, the delayincreases as the density of nodes increases. The reason isthat as the density of nodes increases, the set of legallyelected nodes increases, and the node needs to be elected withmore nodes to obtain the transmission right of the time slot,and the chance of the node obtaining a time slot decreases,which increases the delay. When the density of nodes issmall, less interaction can already satisfy the nodes to obtainenough information to determine the election set. However,with the increase of node density, fewer interactions canno longer meet the needs of nodes to obtain information,resulting in fierce competition among nodes and increaseddelay. Therefore, the range of interaction times needs to becarefully selected according to different network sizes toobtain the best performance.

Fig.11 depicts the relationship between different inter-action times and network delay under resource efficiencyconstraints. The network node density in the figure is ρ =50nodes/km2. As can be seen in the figure, as the resourceefficiency increases, the network delay decreases according-ly. In the case of high resource utilization, the network delaywith fewer interactions is smaller, and under the conditionof low resource utilization, the network delay with higherinteractions is smaller. This is because with a certain networkoverhead, with the increase of resource efficiency, the amount

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FIGURE 10. The variation of the delay with network node density givenm=1,2,3 under pf = 0.6

FIGURE 11. The variation of the delay with resource efficiency given m=1,2,3.

of effective information obtained by the network increases,and the waste or collision of time slots is small. At this time,fewer interactions can meet the maintenance needs of thenode. In the case of high resource efficiency, the network hasa small waste of time slot resources, low collisions betweennodes, and a sufficient amount of effective information thatcan be obtained with fewer interactions. Increased interactiondirectly causes an increase in network delay. In the case oflow resource efficiency, multiple interactions can improve theaccuracy of information learned by nodes, reduce networkwaste and collision, and reduce network delay.

V. CONCLUSIONIn a distributed wireless ad hoc network, the number ofinteractions affects the amount of information acquired bythe nodes and has a great impact on network performance.This paper models and analyzes the influence of the numberof interactions in the distributed election mechanism, and

deeply considers the problem of inconsistent election setdue to network forwarding and error codes in the rapidlychanging topology. The results show that the distributedelection success probability, delay and efficiency modelsestablished in this paper can effectively reflect the impact ofinteraction times and node density on network performance.As the density of nodes increases, the probability of nodeelection success and the resource efficiency of the networkincrease with the number of interactions. However, we cannotincrease the number of interactions without limit, otherwiseit will cause problems such as worse delay performance andlower resource efficiency. This paper provides a reference forselecting the appropriate number of interactions according tothe network node density in engineering practice.

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[5] A. Karalis, "ATP: A Fast Joining Technique for IEEE802.15. 4-TSCHNetworks," 2018 IEEE 19th International Symposium on "A World ofWireless, Mobile and Multimedia Networks" (WoWMoM), Chania, 2018,pp. 588-599.

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ternational Conference on Control, Automation and Systems Engineering(CASE), Singapore, 2011, pp. 1-5.

[14] M. Cao, W. Ma, Q. Zhang and X. Wang, "Analysis of IEEE 802.16Mesh Mode Scheduler Performance," in IEEE Transactions on WirelessCommunications, vol. 6, no. 4, pp. 1455-1464, Apr.2007.

XIN LI received the B.Sc. degree in communica-tion engineering from Beijing Jiaotong University,China, in 2018, where she is currently pursuingthe M.Sc. degree in communication and informa-tion systems. Her research interests are mainlyperformance analysis of wireless self-organizingnetworks.

XU LI received the B.Sc. degree from Jilin U-niversity, China, in 1991 and the Ph.D. degreefrom Northeastern University, China, in 1997. S-ince 2003, she has been a Professor and the Headof the Wireless Self-Organizing Communication-s Group, School of Electronic and InformationEngineering, Beijing Jiaotong University, China.She holds over 40 patents. Her research interestsinclude wireless self-organizing networks, sensornetworks, and unmanned aerial vehicle systems.

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