https://orcid.org/0009-0002-1916-7938
Figures
Abstract
Information floating (IF) is a method of delivering information to mobile nodes in a desired area while avoiding unnecessary communication and information dissemination by restricting direct wireless transmission to a transmittable area (TA). This restriction, however, also leads to the termination of IF, which is a longstanding problem that must be overcome. As a solution, methods have been developed to predetermine the optimal TA size based on environmental parameters such as node density. If the density changes over time, then the estimation of the density and the optimization of the TA must be repeated. Therefore, we previously proposed a method that guarantees that the IF never ends in principle, even if the node density changes over time, by dynamically controlling the TA size. However, this method is only applicable in a one-dimensional network. Here, we propose a method that guarantees, even in two-dimensional networks, that the IF never ends. To accomplish this, we introduce two key functions. The first autonomously restarts the IF even if it has temporarily terminated. The second function dynamically controls the TA size. We also highlight the necessity of introducing a lifetime for the TA generated by the dynamic control method if the density changes over time, and we improve the proposed method accordingly. We show the effectiveness of the proposed methods in terms of continuity and tracking performance through theoretical and simulation evaluations.
Citation: Miyakita K, Meguro D, Tamura H, Nakano K (2026) Autonomous restart of information floating and dynamic control of transmittable area. PLoS One 21(1): e0341468. https://doi.org/10.1371/journal.pone.0341468
Editor: Abu Sayed Chowdhury, National Marrow Donor Program, UNITED STATES OF AMERICA
Received: June 24, 2025; Accepted: January 7, 2026; Published: January 29, 2026
Copyright: © 2026 Miyakita et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
If mobile nodes transmit information to other mobile nodes using direct wireless communication as they move, the information becomes spatially diffused. This type of information diffusion enables a system to deliver information to distant mobile nodes [1–6]. In this paper, we call this “epidemic communication.” Epidemic communication is an effective method in delay-tolerant networks (DTNs). However, this method has problems such as the spread of information to unrelated areas and the increase in unnecessary transmissions. One of the methods used to solve these problems is information floating (IF), which restricts the location of direct wireless transmissions to within the transmittable area (TA) [7–21]. In order to execute IF, each node must know its own location using a positioning technology such as GPS.
Fig 1 shows an example of IF. Suppose that IF delivers a message M.
where data is the data to be delivered and TA is the information on the position and size of the TA. In Fig 1, black and white circles indicate mobile nodes with and without M, respectively. The direct wireless communication range is r. The center of TA is the coordinate O. A is the first mobile node having message M. At time , the distance between A and B becomes r and A is in TA, so A sends M to B. In the same manner, at time
, A and B send M to D and C, respectively. On the other hand, C does not send M to E, even though they are within each other’s communication range, because C is outside the TA. At time
, all nodes having M leave the TA. If no node having M enters the TA after this time, IF will never restart. This is the end of IF.
(a) , (b)
, (c)
.
As described above, IF has a critical problem in that it ends if all nodes having information leave the TA while none enters the TA from outside. To overcome this problem, almost all papers on IFs have studied how to determine an appropriate TA size so that the IF continues for a sufficiently long time depending on the environmental parameters, in particular the node density. Note that this approach can only be applied if the node density is obtained by some method. Moreover, if the density changes over time, then the above estimation and optimization steps must be repeated.
One solution to achieve the proper adaptation is to dynamically control the TA. Consequently, various methods to accomplish this have been proposed. In one such method [18,20], each node having information sets the size of the TA based on its past experience of contacts with other nodes, and thus the node is expected to contact other nodes while in the TA. However, this is merely an expectation, and this method still has the same critical problem as using a fixed TA.
As an alternative, in a previous study [17] we proposed another dynamic control method for TA, where the IF never ends in principle. In this method, a node that has not sent information to other nodes within the TA expands the TA to its own position until it sends information to another node. Conversely, a node that has received information in the expanded TA shrinks the TA to its own position while moving toward the center of the TA, i.e., reference point O. By dynamically controlling the TA in this way, there is always at least one node having information within the TA, so it is guaranteed that the IF never ends. Although this method works well for a one-dimensional network, extending it to two-dimensional networks is not trivial, and there remain issues that must be overcome.
In addition, these dynamic controls have not been evaluated in situations where the density changes over time, although they are needed for these situations.
In this paper, we consider how to develop an IF method that never ends and can dynamically control a TA even in two-dimensional networks. As an extension to the two-dimensional case, we consider road networks spreading over a two-dimensional plane. In particular, we give attention to lattice networks. In this paper, we introduce the concept of autonomous restart (AR), which restarts the IF even if it has temporarily terminated. This restart process is carried out autonomously. In addition, the size of the TA is dynamically controlled using the information on the position where the AR function is executed.
The novel technical advances and contributions of this paper are as follows.
- To achieve an IF method that never ends completely, even in two-dimensional networks, we introduce AR.
- Simultaneously, we also propose a dynamic control method for TA. This allows the TA to be adjusted autonomously even if the density changes over time, and information can be delivered to many nodes while avoiding unnecessary transmissions.
- We identify the following serious problem in the case of time-varying density. In general, dynamic TA control expands the TA as the density decreases. On the other hand, it is also designed to shrink the TA as the density increases. In actual situations, however, it is not easy to shrink the TA in the latter situation. Here, we present simulation results that illustrate this problem and propose a way to overcome this difficulty.
- We present evaluations of the proposed methods as follows. First, we clarify how AR can restart a temporarily terminated IF through a theoretical analysis. Second, we demonstrate the effectiveness of the proposed methods through simulation. In particular, we show that they work well even in cases where the density changes drastically over time and that they significantly improve performance compared to traditional methods using a fixed TA.
To demonstrate the practical relevance of IF, we outline potential deployment scenarios and the practical challenges of its implementation. IF can be used to deliver local information to unspecified mobile nodes passing through a specific area, without the need for communication infrastructure. The following applications have been considered in the literature: distribution of local information or advertisements [7,12,14–16], distribution of traffic or accident information in vehicular ad hoc networks (VANETs) [7,11,14–16,19], distribution of emergency information during disasters [7,14,16,19,21], and applications to sensor networks [12]. Furthermore, the use of IF to provide available route information by virtually accumulating each vehicle’s travel history has been proposed [21]. In such applications, node density is likely to change over time; therefore, preventing IF from terminating completely in these situations is a significant practical challenge in its implementation.
The rest of this paper is organized as follows. In Sect 2, we first explain the proposed methods in a one-dimensional network to demonstrate our basic idea. In Sect 3, we extend the proposed methods to a two-dimensional road network that includes intersections. We also explain an additional operation for shrinking a TA in the proposed method, and we thus propose an improved method. In Sect 4, we evaluate the proposed methods using theoretical analysis and simulations. Sect 5 concludes this paper. For the convenience of readers, the acronyms used in this paper and their meanings are listed in Appendix A, and the theoretical computation of a key equation in Sect 4.2 is shown in Appendix B.
2 Proposed method in one-dimensional road model
In this section, we first consider a one-dimensional model to explain the basic idea of the proposed method. As in the previous work [17], let us suppose that reference point O is at the center of the TA.
2.1 Autonomous restart (AR-1d)
First, we explain the function of AR, which restarts an IF that has temporarily terminated. We refer to the AR used in a one-dimensional model as AR-1d. Note that AR-1d is different from the previous method [17] and is adopted with a view toward its extension to two dimensions.
In AR-1d, we add a flag to message M as follows.
flag takes the value 1 or 0, and it is used to determine whether the node has the role of restarting IF. A node having M with flag = 1 has to restart IF. To do this, this node continues searching for approaching nodes and sends M with flag = 1 to the first contacted node after passing O, even if this node is outside the TA. On the other hand, a node having M with flag = 0 executes normal IF.
We explain the operation of AR-1d using the example in Fig 2. A is the node that starts the IF. Therefore, A is the only node having M with flag = 1 at the initial moment . Between
and
, A does not contact any node; therefore, at time
, A has left TA and the IF has temporarily terminated. However, because A has M with flag = 1, A sends M with flag = 1 to B, which is approaching O, even though A is outside the TA, and then sets its flag to 0. At time
, B enters TA and restarts IF, and M is sent from B to D. Here, since B has not yet passed O, B does not send M with flag = 1 to C but instead sends M with flag = 0. Then, at time
, B sends M with flag = 1 to E, since E is the first contacted node after B has passed O.
(a) , (b)
, (c)
, (d)
, (e)
(with TA expansion).
As seen from these procedures, a node having M with flag = 1 always exists in the system, and this node is always moving toward O. Therefore, even if the IF has temporarily terminated, it will always restart, and thus the IF never terminates completely. Also, in AR-1d, the number of nodes having M with flag = 1 does not increase, in order to avoid increasing useless transmissions. As a result, there is always only one node having M with flag = 1 in the system.
Note that to execute AR-1d, each node must know the moving directions of itself and its neighbors, as well as its own position. Therefore, in AR-1d, it is assumed that each node measures its own moving direction from the time variation of its location information and exchanges this information with its neighbors.
2.2 Dynamic control of TA (DC-1d)
In the example shown in Fig 2, AR-1d can prevent the complete termination of IF, but there are nodes, such as node C, that cannot receive information by passing through the TA during the temporal termination of IF. To reduce the occurrence of such nodes, we consider a method of dynamically controlling the TA based on the location where M with flag = 1 is transmitted. For example, in Fig 2, if the TA is expanded to the location of A at , C will be able to receive M from B, as shown in Fig 2e. We call this method DC-1d.
As mentioned above, in DC-1d, when M with flag = 1 is sent in the operation of AR-1d, TA is updated to include the location where M with flag = 1 was sent. Here, if the TA is updated to include only the latest location where M with flag = 1 was sent, the TA will be updated significantly and frequently due to the randomness of the node locations, and thus the system will become unstable. To prevent this, each node stores a history of the locations where M with flag = 1 was sent in the past n times and then updates the TA to include all of these n locations, where n is a predetermined positive integer. Specifically, we determine the new TA as the region to the furthest point in the history of n positions, while making it symmetrical with O as the center (Fig 3). Such a method of determining TA facilitates an easy extension to two dimensions.
(i) indicates the ith most recent position where M with flag = 1 was transmitted.
To perform the above algorithm, in DC-1d we add new parameters to message M as follows.
where tTA is the time when the TA information was updated and is the history of the most recent n positions where M with flag = 1 was transmitted. When two nodes having different TA values contact each other, they compare the tTA values and update the TA and tTA values to those of the node having the more recent tTA value.
3 Extension to two-dimensional road model with intersections
In this section, we extend AR-1d and DC-1d to two dimensions. As with the one-dimensional model, we consider a reference point O in the two-dimensional model, and we assume that this reference point is at the center of the TA. Unlike the case of the one-dimensional model, the two-dimensional model has some nodes that do not pass through O. Therefore, the purpose of AR in the two-dimensional model is to deliver information to the nodes that pass through O.
In the one-dimensional model, the node receiving M with flag = 1 always goes to O immediately. On the other hand, in the two-dimensional model, such a node does not always go to O but may move in a different direction from O, since it may turn before reaching O. In this paper, we assume that nodes move along roads without detours. Therefore, once a node has moved away from O, it never returns to O, and another node needs to return the M with flag = 1 to O.
For example, consider a lattice road network as shown in Fig 4, where s1, s2, s3, are road segments. O is assumed to be at the center of a road segment. If node A having M with flag = 1 cannot send M to an approaching node in s1, then A enters s2, s3, or s4 while keeping M with flag = 1. Suppose that A enters s2 and sends M with flag = 1 to another node B. If B enters s1, M with flag = 1 arrives at O within a short time. Otherwise, B enters s3 or s4, and M with flag = 1 moves away from O. Such behavior in lattice networks is different from that in AR-1d. Hence, we need to consider the effects of this kind of behavior toward achieving AR in two-dimensional networks.
3.1 Autonomous restart (AR-2d)
To overcome the above difficulty in the two-dimensional case, we extend AR-1d and DC-1d to AR-2d and DC-2d, respectively.
AR-2d is basically the same as AR-1d, and a node having M with flag = 1 sends it to the first contacted node after passing O. However, in AR-2d, if a node having M with flag = 1 turns in a direction away from O at an intersection before reaching O, it sends M with flag = 1 to the first contacted node after making this turn.
As shown in the above explanation, we must evaluate how long it is necessary to wait for an M with flag = 1 to return to O after the previous M with flag = 1 has left O. Therefore, we discuss this issue using the theoretical analysis in Sect 4.2.
3.2 Dynamic control of TA (DC-2d)
DC-2d is basically the same as DC-1d, but the way of determining the TA is different from that for DC-1d, since it corresponds to a road network with intersections. In DC-2d, TA is determined as the region from O to the furthest point in the n most recent positions where M with flag = 1 was transmitted, while making it symmetrical with O as the center. Specifically, if di is the distance along the road network from O to the ith latest position where M with flag = 1 was transmitted, then the TA is every region within distance along the road network from O.
3.3 Additional operation for shrinking TA and improving DC-2d: Proposal of DC-2d-i
In this subsection, we explain the additional operation done for the shrinkage of TA in the proposed method and further propose an improved version of DC-2d.
Suppose that at time , M with flag = 1 is transmitted and the TA is set to TA1 by the function of DC-2d. After that, at time
(
), M with flag = 1 is transmitted, and the TA size is set to TA2. Here, we assume that the node density increases between
and
, and as a result, TA2 becomes smaller than TA1. An example is shown in Fig 5. This TA2 is the new TA, but since M, which includes information about TA2, is basically only transmitted within TA2, the information that the TA is updated to TA2 is not delivered outside TA2. In other words, in region TA1 − TA2, TA1 continues to be used. However, this is undesirable because it increases unnecessary transmissions.
To avoid the above problem, we improve DC-2d so that the TA information is reset after a certain lifetime since the TA was made. We call this improved method DC-2d-i. Here, resetting the TA information means setting the TA size to 0, i.e., making the TA only O. In this paper, the lifetime is determined dynamically in the same way as the TA size. If the lifetime is too short, all TA information in the service area is reset before the next TA is made, resulting in many nodes not receiving M. With this in mind, we set the lifetime to approximately the time until the next TA is made, as follows.
Assume that at time , M with flag = 1 is transmitted and the latest TA information TA0 is made. Let d0 be the distance from O to the edge of TA0 (see explanation in Sect 3.2, from which we derive
). The node receiving M with flag = 1 at
will make the next TA, and it is expected that this will be done, at the latest, before this node leaves TA0. This time is approximately
because the distance from one edge of TA0 to the other is 2d0. Here, v is the velocity of each node, and this is assumed to be known or estimable. As a result, in DC-2d-i, the TA information TA0 made at
is reset at
. Then, if the node receives TA information made after
or makes a new TA by receiving M with flag = 1, it updates the TA information.
In Sect 4.3, which presents simulation results, we discuss this phenomenon and the effectiveness of the improved method.
4 Performance evaluation
4.1 Assumptions
In this section, we evaluate the proposed method. As a model of a two-dimensional road network, we use the lattice model shown in Fig 6. This is a 13 × 12 lattice structure, where each side of the lattice has a length of 2a = 500 m. Mobile nodes enter the lattice network from each of the roads at the edge of the lattice network according to a Poisson process with density λ. Each mobile node moves without detour as follows. There are four groups of nodes. For Groups 1, 2, 3, and 4, a node moves to the right or up, left or up, right or down, and left or down, respectively. The node densities of these four groups are the same. For Group i and intersection j, a node turns with probability pturn,i,j and goes straight with probability 1 − pturn,i,j after passing the intersection, where pturn,i,j is a predetermined value. At the initial time of the simulation, the first node having M with flag = 1 leaves O and then moves to the left.
As described in the preceding section, the proposed method attempts to overcome the problem of M not returning to the TA. The achievement of this goal should be considered in the evaluation of the proposed method. In the above mobility model, mobile nodes move along the road network without detours and never turn back; therefore, this model is suitable for our objective because mobile nodes never return to the TA once they leave it. Note that if we use the random waypoint (RWP) mobility model, which is often used to evaluate the performance of mobile networks, mobile nodes often turn back to the previous area. Therefore, the RWP model is not suitable for our objective.
Regarding communication and node movement, to initially evaluate the fundamental performance of the proposed method, we consider the following simplified assumptions (Basic Factors BF1 to BF5).
- BF1: Communication is possible if and only if the nodes are within a constant communication range r = 100 m.
- BF2: There is no shadowing effect, and communication always succeeds when the straight-line distance between the nodes is within r.
- BF3: Each node moves at a constant velocity
m/s = 36 km/h.
- BF4: For all node groups i and all intersections j, pturn,i,j is constant and is 2/3. As can be easily seen, this causes the nodes on the road to obey a Poisson process with an even density of λ in each direction on each road segment when looking at a snapshot.
- BF5: There is no GPS error, and all nodes can always accurately determine their own position.
The theoretical analysis and the simulation results with these simplified assumptions are shown in Sects 4.2 and 4.3, respectively.
In addition, to investigate the effects of variation and randomness in real situations, we also consider the following additional assumptions (Additional Factors AF1 to AF5).
- AF1: A probabilistic link error model is introduced [22]. This model assumes that the communication success rate is
, where x is the distance between two nodes. η is the path loss exponent, and we set it to
. Each node having information attempts transmission every
s, and for each attempt the success or failure of communication is determined by the above probability.
- AF2: A shadowing effect is introduced. We assume that all areas outside the roads in the lattice road network shown in Fig 6 are obstructed by buildings. Consequently, communication between diagonal directions is impossible, and only nodes moving along the same road can communicate with each other.
- AF3: Variation in node speed is considered. There are two types of nodes, moving at speeds
m/s and
m/s, coexisting in equal traffic volumes. This scenario assumes all road segments are dual carriageways, where overtaking the vehicle in front is possible. However, to simplify the simulation, lane widths are not considered. Note that the mean time required for each node to travel a unit distance x [m] is
s, which is identical to the mean time for BF3, namely
s.
- AF4: An uneven node density model is considered. To create this, we set pturn,i,j to a random value (a uniform random number between 1/4 and 3/4) for all i and j. As a result, the density distribution becomes uneven, as shown in Fig 7.
- AF5: A GPS error model is introduced [23]. This model assigns errors following independent normal distributions (mean 0, standard deviation σ) in both x-axis and y-axis directions. Here, we set
m. Each node having information determines whether it is within the TA and its direction of movement every
s, with the GPS error determined by the above distribution at each time.
The simulation results with these additional assumptions are given in Sect 4.4.
4.2 Evaluation of AR-2d’s ability to restart IF by theoretical analysis
Before showing the simulation results of the proposed method, we evaluate AR-2d’s ability to restart IF using a theoretical analysis. To do this, we theoretically compute how far away from O the M with flag = 1 moves before returning to O. In addition, by using the derived formula, we show that the temporarily terminated IF is always restarted, except in the case of extremely low density. In this subsection, we assume only BF1 to BF5. The numerical results of the formula are also used in the discussion on the simulation results in Sect 4.3.
Let A0, A1, A2, and A3 be the events where M with flag = 1 that departed from the left side of O returns to O without leaving the orange, red, blue, and green road segments in Fig 4, respectively. In general, for , Ai is the event where M with flag = 1 that departed from the left side of O returns to O without going further than i road segments from intersection C. We compute and evaluate
, where
is the probability that event ⋅ occurs. Of course, since this is a symmetric model, we can also evaluate M with flag = 1 that departed from the right side of O in the same way using
. Theoretical computation of
is given in Appendix B.
The numerical results for are shown in Fig 8. The horizontal axis is λ. From this figure, we can see that
approaches 1 only when the density is significantly large (roughly greater than 0.01 m−1). Consequently, at such a density, M with flag = 1 that departed from the left side of O returns to O without leaving the orange road segment. Furthermore, it returns to O without leaving the red, blue, and green road segments when λ is greater than 0.004 m−1, 0.0025 m−1, and 0.002 m−1, respectively. From these results, in the operation of DC-2d and DC-2d-i, when the values of λ are approximately 0.01, 0.004, 0.0025, and 0.002 m−1, the TA is expected to be the regions within the orange, red, blue, and green road segments, respectively.
As shown in Appendix B, we can also compute the value of for
, and this value is 1 if
m−1. This theoretically guarantees that if the service area is sufficiently large, M with flag = 1 always returns to O, except in the case where the node density is extremely small.
4.3 How successfully the proposed method restarts IF and dynamically controls TA: Simulation results
Here, we show how successfully the proposed method restarts IF and dynamically controls TA by using computer simulations. This subsection also assumes only BF1 to BF5. We consider the situations where the node density changes over time with the following two patterns of density change.
We evaluate the following seven metrics. The first is the mean of the time until the IF is completely finished, namely the time until all nodes having information leave the service area, denoted by E(Tf). The second is the mean of the size of TA (i.e., the distance from O to the edge of TA along the road network) for all nodes having M at each time t. This verifies that the TA information is updated appropriately. Note that when calculating this metric for DC-2d-i, we exclude the nodes whose TA size is 0 due to resetting the TA information. The third is the ratio that nodes passing through O receive M before leaving the service area, denoted by prec. Since the purpose of IF in the two-dimensional model is to deliver information to nodes passing O as mentioned above, we use this metric as the information reception rate. The fourth is Ruseless:
Ruseless indicates how many useless transmissions are made to complete one necessary transmission, and it is thus used to evaluate the number of useless transmissions. The fifth, sixth, and seventh metrics are the numbers of transmissions in M of data, TA information, and flag = 1 information, which are denoted by Ndata, NTA, and Nflag=1, respectively. These are used to evaluate the communication overhead associated with flag information and TA updates by comparing NTA and Nflag=1 with Ndata. Note that, although these three types of information can be transmitted simultaneously, we count them separately. The following figures show the mean values of each evaluation metric for 10 simulations with a simulation time of 40,000 seconds.
In DC-2d and DC-2d-i, we used n = 10. We tried several other values of n, but the results did not change significantly.
In the following, we evaluate the continuity and tracking performance of the proposed method using the above metrics and discuss this method’s computational and communication overheads.
4.3.1 Continuity performance.
To evaluate the continuity performance of (AR-2d)+(DC-2d-i) in situations where the density changes, we show the simulation results of E(Tf) in Fig 9. (AR-2d)+(DC-2d-i) is realized by adding the two functions AR and DC to the conventional IF using a fixed TA. Therefore, to investigate the effects of AR and DC, we also show the results for the conventional IF using a fixed TA in Fig 9 for comparison. Here, we do not compare the proposed method with other IF methods because no IF method has been adapted to the case of time-varying density and the mobility model where nodes never return to the same area. For example, the IF methods proposed in two previous works [18,20] are based on the RWP model, which cannot be used for evaluation of our proposed method as mentioned. Similarly, we do not compare the proposed method with representative protocols for DTNs such as PLOPHET [4] and Spray and Wait [5], since these protocols are designed to deliver information to a specified destination node, and thus their purpose differs from that of IF.
(a) Pattern 1, (b) Pattern 2.
In Fig 9, the horizontal axis is the size of the TA in the conventional IF (distance from O to the edge of the TA). From this figure, in both Patterns 1 and 2, (AR-2d)+(DC-2d-i) achieves a high E(Tf). In fact, we have E(Tf) = 40,000 s, which means that the IF never finished completely during the simulation. Note that, in Pattern 2, even though there are time periods when the density is less than the density threshold of m−1 derived in Sect 4.2, (AR-2d)+(DC-2d-i) successfully prevents the complete termination of IF. This is because the approximation of
used in the theoretical computation is a safe approximation.
On the other hand, in the conventional IF, we can increase E(Tf) by increasing the size of TA; however, because the size of TA required to achieve E(Tf) = 40,000 s differs between Patterns 1 and 2, it would be difficult to find the optimal size of TA without prior knowledge of the change in density. The results for Ruseless are also shown in Fig 9. Even when using the optimal TA sizes for the conventional IF (800 m for Pattern 1 and 1100 m for Pattern 2), the value of Ruseless is significantly larger than for (AR-2d)+(DC-2d-i). From these results, (AR-2d)+(DC-2d-i) successfully prevents the complete termination of IF, and it also significantly reduces unnecessary transmissions compared to the method where the TA is fixed.
4.3.2 Tracking performance.
Next, we show the time-tracking performance of TA size in Fig 10. To confirm the effect of the improvements to the proposed method described in Sect 3.3, we show the results for both (AR-2d)+(DC-2d) and (AR-2d)+(DC-2d-i). From this figure, (AR-2d)+(DC-2d-i) succeeds in increasing and decreasing the TA size in response to the decrease and increase in density, respectively. On the other hand, in (AR-2d)+(DC-2d), the TA size increases in response to the decrease in density but does not sufficiently decrease even when the density increases. This indicates the necessity of introducing a lifetime of the generated TA, as explained in Sect 3.3. In addition, the relationship between the values of λ and TA size in (AR-2d)+(DC-2d-i) is roughly within the range estimated in the theoretical evaluation in Sect 4.2.
(a) Pattern 1, (b) Pattern 2.
Next, we show the time variation of prec in Fig 11. In this figure, we divide the horizontal axis into 500-s intervals and plot the prec value as the height of the step for nodes that entered the service area during each time interval. Although the simulation time is 40,000 s, Fig 11 shows the results only up to 30,000 s for readability. From this figure, we can see that in both Patterns 1 and 2, although the density decreases in the middle of the simulation, a high prec is always maintained. In Pattern 2, prec slightly decreases around 10,000 s. This is because the decrease in density in Pattern 2 is too sudden and significant, causing a slight time lag in the dynamic control of TA for DC-2d and DC-2d-i. Nevertheless, DC-2d and DC-2d-i can track such a sudden change in density, and after about 11,500 s, prec returns to nearly 1.
(a) Pattern 1, (b) Pattern 2.
Note that if we apply the conventional IF to Patterns 1 and 2, the performance obviously varies depending on the fixed TA size, but, for example, if we optimize the TA size for the initial density, the IF ends completely just after s due to the sudden reduction in density, and prec becomes 0 after this time.
Finally, to observe how the above factors affect useless transmissions, we show the time variation of Ruseless in Fig 12. From this figure, Ruseless becomes large only between 10,000 s and 20,000 s in (AR-2d)+(DC-2d-i), while it remains large after 20,000 s in (AR-2d)+(DC-2d). This indicates that (AR-2d)+(DC-2d-i) suppresses useless transmissions while continuing IF even when the density changes.
(a) Pattern 1, (b) Pattern 2.
Table 1 shows prec and Ruseless for (AR-2d)+(DC-2d) and (AR-2d)+(DC-2d-i) over the entire simulation, and we can see that (AR-2d)+(DC-2d-i) keeps prec close to 1 while greatly reducing Ruseless.
4.3.3 Discussion on computational and communication overheads in proposed method.
We also consider the computational load required to execute the proposed method and the communication overhead associated with flag information and TA updates. First, we discuss the computational load. In (AR-2d)+(DC-2d-i), the following processing is required in addition to the conventional IF:
- When a node that already has information receives newer TA information, it updates its own TA information.
- The TA information is reset when the TA’s lifetime expires.
Additionally, the following processing is required only for a node having M with flag = 1:
- When the node moves away from O, it sends the M with flag = 1 to a node approaching O.
- At that time, it adds its own position to the transmitted position history information and calculates a new TA.
These processes are not complicated, and the total size of the flag and transmitted position history information is extremely small, amounting to at most a few dozen bytes. Therefore, regardless of the network scale, it is considered unlikely that computational load would make practical implementation difficult.
Next, to discuss communication overhead, we show Ndata, NTA, and Nflag=1 for (AR-2d)+(DC-2d-i) in Fig 13. From Fig 13, we can see that Nflag=1 is significantly smaller than Ndata. This can be understood by considering the meaning of flag = 1: It is basically sent to a node that is moving to O from outside the TA. Such a node does not yet have the data. This indicates that flag = 1 information is transmitted simultaneously with the data in almost all cases. Therefore, regardless of the network scale, Nflag=1 is much smaller than Ndata.
(a) Pattern 1, (b) Pattern 2.
We can also see from Fig 13 that NTA is approximately twice Ndata in many cases. This means that the number of TA update transmissions alone is nearly equal to Ndata, since a node that receives data also receives TA information simultaneously. The reason for this result is considered as follows. As explained in Sect 3.3, the new TA information is generated at intervals roughly equivalent to how long a node remains within the TA after receiving information. Therefore, a node that receives information will receive updated TA information approximately once before exiting the TA. In some cases, including those involving significantly high density or the moment just after density increases rapidly, NTA can exceed twice Ndata. However, even in these cases, NTA is at most on the order of a few times Ndata.
From the above discussion, we believe that implementing the proposed method will not impose an unworkable load in terms of either computational or communication overhead, regardless of the network’s scale.
4.4 Results with additional factors (AF1 to AF5)
In this subsection, we present simulation results for the proposed method using AF1 to AF5, defined above, to verify its effectiveness even when ideal assumptions are not satisfied. The simulation results are shown in Figs 14 to 18. The evaluation metrics and presentation are the same as in Figs 9 to 13. From Figs 14 to 18, the fundamental trends are largely the same as in the cases for BF1 to BF5 (Figs 9 to 13), although there are some variations in the values. This confirms that (AR-2d)+(DC-2d-i) remains effective even with the additional factors AF1 to AF5.
(a) Pattern 1, (b) Pattern 2.
(a) Pattern 1, (b) Pattern 2.
(a) Pattern 1, (b) Pattern 2.
(a) Pattern 1, (b) Pattern 2.
(a) Pattern 1, (b) Pattern 2.
Here, to confirm how each factor of AF1 to AF5 influences the results, we show the simulation results for (AR-2d)+(DC-2d-i) when applying each factor individually in Fig 19. In this figure, we specifically show only the time variation of the mean size of TA, since this is a particularly characteristic result. Fig 19 shows that the size of the TA is clearly larger when AF5 is applied than when AF1 to AF4 are applied, particularly during the high-density time intervals of 0 to 10,000 s and 20,000 to 40,000 s. This is assumed to occur because GPS errors cause misjudgments in determining the direction of movement, sometimes resulting in a node moving away from O failing to send M with flag = 1 to a node moving toward O. Despite this influence, however, the proposed method successfully prevents the complete termination of IF and achieves superior performance to the fixed TA method, as shown in Figs 14 to 18.
(a) Pattern 1, (b) Pattern 2.
As shown above, the proposed method remains effective even when incorporating factors of variation and randomness within the range of AF1 to AF5. Evaluating and improving the proposed method to account for other factors, such as radio wave interference and irregular road networks, remains a future problem.
5 Conclusions
In this paper, we proposed a method to autonomously restart information floating even if it has temporarily terminated in two-dimensional lattice networks. This work was carried out to solve the critical problem of information floating where it is completely terminated. In addition to the autonomous restart method, we also proposed a method to dynamically control the transmittable area for information floating. For the dynamic control method, we emphasized the necessity of setting a lifetime for the generated transmittable area, which improved the proposed method.
The results of theoretical computations and simulations show that the proposed methods can prevent information floating from completely terminating even in situations where the density of mobile nodes decreases suddenly and significantly. In addition, we showed that the dynamic control method can deliver information to almost all of the mobile nodes that need the information, while suppressing unnecessary transmissions.
As mentioned, future work includes conducting evaluations that account for a wider variety of factors, such as radio wave interference, as well as extending the proposed method to road networks of various shapes.
Appendix
B. Computation of 
Since precisely computing is difficult, we introduce a safe approximation based on two assumptions. First, we assume that information is only transmitted when two nodes pass each other, which means that the effect of the communication range r is ignored. Second, we assume that M with flag = 1 always returns to O by retracing the intersections it took to move away from O. Under these assumptions, we can easily compute
.
First, can be computed as the probability that a node departed from the left side of O passes another node before reaching intersection C. Therefore, we clearly have
Next, we compute for
. This can be expressed as
where is the probability that Ai occurs given that A0 does not occur (i.e., given that M with flag = 1 that enters s1 from O leaves s1 without returning to O).
Then, to compute , we define Si as the set of outer road segments of si as viewed from O. For example,
,
, and
. We also define ppass and ps as follows.
- ppass: Probability that a node entering S1 after leaving C will pass another node before leaving S1. We clearly have
.
- ps: Probability that the above node, moving on S1 and toward C, will go toward O after reaching C. We clearly have ps = 1/3.
First, we compute . Consider the situation where a node having M with flag = 1 (denoted by n0) enters S1 after leaving s1. If n0 passes another node (denoted by n1) on a road segment in S1 (i.e., s2, s3, or s4) and n1 moves toward s1 immediately after reaching C, then M with flag = 1 will return to O in one step. This probability is computed by
. If n1 moves to a road segment of S1 other than s1 after reaching C, n1 passes another node (denoted by n2) on S1 and n2 moves toward s1 after reaching C, then M with flag = 1 will return to O in two steps. This probability is computed by
−
. In general, the probability that M with flag = 1 returns to O in m steps is computed by
−
. Therefore,
can be computed by adding up these probabilities for
as follows.
Next, we compute . The basic idea is the same as the computation of
, but the probability that an M with flag = 1 entering S1 after leaving C returns to C without leaving S1 to S4 is computed as
, not ppass. Therefore,
is computed by the following equation.
In the same manner, we generally obtain the following recurrence equation for for
:
For clarity, the equations of are summarized in Table 3.
By solving this recurrence equation and substituting the solution of into Eq (8), we have the general solution of
as follows.
Furthermore, we can compute , which means the probability that M with flag = 1 eventually returns to O despite how far it has travelled, as follows.
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