Abstract
Data transmission scheme is an effective mode to improve the energyefficiency in packet delivery. In this paper, we investigate the energyefficient data transmission over reliable links and unreliable links in mobilityaware wireless networks. The network topologies in mobilityaware wireless networks are changed from one time slot to another, and they could be described by a sequence of static graphs. We first model these network topologies in a time period as a virtual spacetime graph. On the virtual spacetime graph, energyefficient data transmission problems over reliable links and unreliable links are defined. The aim of the two data transmission problems is to find a spatialtemporal path with the minimum energy cost. Next, we propose an EnergyEfficient Data Transmission algorithm over Reliable Links (EEDTRL) to find the optimal spacetime path. Based on EEDTRL, we also develop a heuristic data transmission protocol over unreliable links (named EEDTUL), in which the path reliability is taken into consideration. Simulation results show that our proposed algorithms perform well in terms of energy cost and transmission count compared with some existing algorithms.
1. Introduction
Data transmission scheme is an effective manner to reduce the energy consumption of packet delivery in wireless networks. However, many intractable problems have been imposed in data transmission because of nodes’ mobility, i.e., the network topologies are changed from one time slot to another, which is an inherent characteristic of wireless networks. Specially, the topology connectivity of the mobile network in each timeslot and even the existence of a routing path between two remote nodes could not be guaranteed. Opportunistic or epidemic transmission schemes are effective forwarding manners for packet transmissions between two remote nodes [1–4], in which data packets are forwarded by utilizing the sporadic contacts between two mobile nodes. However, packet transmissions using opportunistic or epidemic transmission schemes may result in plenty of copy transmissions. Actually, opportunistic or epidemic transmission schemes do not exploit the full potential of the mobility. Although the problem of the absence of routing path could be alleviated through opportunistic or epidemic transmission schemes, how to select the relay nodes, i.e., making forwarding decision over timevarying network topologies is still an intractable problem in mobilityaware wireless networks. Therefore, it is imperative to investigate the data transmission problem in mobilityaware networks.
In the past decades, energyefficient data transmission protocols over reliable and/or unreliable links in static networks have been widely investigated, as shown in [5–8]. However, these network protocols are not suitable for the applications in mobile wireless networks due to the frequent link breakage and rebuilding. In mobile wireless networks, most of the researches for routing design have concentrated on the stable and reliable path selection which may have a long duration [9–12]. In these research works, node mobility is assumed in terms of some classical random mobility models, such as random direction model and random waypoint model. Due to the high randomness, these researches have overmuch emphasis on the path duration in the packet delivery but neglect the importance of another factor—energy consumption.
In the humancentric mobile networks, the temporal characteristics of network topology in many network scenarios could be known a priori. Actually, in [13], the authors show that the potential predictability for human mobility can reach up to 93%. The literature [14] found that about 78%99% of the vehicle location is predicable. Thanks to the development of cloud computing technique, the authors in [15, 16] investigated the mobility prediction in bikesharing systems and in public bus systems, respectively. For perpetual trajectory tracking, the authors in [17] propose an energy and mobilityaware scheduling framework to improve the longterm tracking performance. All of these researches validate that the strong regularities are existed in the daily human and vehicle mobility. In these mobilityaware wireless networks, the literatures [18–20] proposed some mobilityaware energyefficient data transmission schemes. However, these proposed algorithms are based on the connect network topology, and the temporal information of topology change from one time slot to another is not involved explicitly. By exploiting vehicle mobility trajectories, the authors in [21] developed an efficient multicast algorithm in wireless vehicular networks. Using machine learning for mobility prediction, the literature [22] proposed a centralized routing scheme to minimize the overall vehicular service delay. Some other mobilityawarebased network protocols could be found in [23, 24] for lowcost topology control, in [25] for location management, in [26] for clustering, and in [27] for resource allocation.
In this paper, we investigate the energyefficient data transmission problems in mobilityaware wireless networks over reliable links and unreliable links. A series of network topologies, each of which is disconnected with high probability, is modeled as a virtual spacetime graph. In this spacetime topology graph, a spatial link refers as to a wireless link between two nodes at one time slot; while a temporal link means that a node carries its data packet from one time slot to the next time slot. Data transmission problems over reliable links and unreliable links on the virtual spacetime graph are defined, in which the spacetime path with minimum energy cost needs to be found. An EnergyEfficient Data Transmission algorithm over Reliable Links (EEDTRL) is proposed to solve the data transmission problem over reliable links. By extending the EEDTRL, we also develop a heuristic transmission algorithm over unreliable links, i.e., EEDTUL. Some simulations are performed to study the performances of our proposed algorithms. Compared with our conference paper in [28], many new research results are added, such as the optimal data transmission algorithm over reliable links, more elegant optimal objective over unreliable links, and more simulation results.
The remainder of this paper is organized as follows. We review some of the related works in Section 2. Section 3 presents the network model, link model, and energy model. Energyefficient data transmission over reliable links and unreliable links are investigated in Sections 4 and 5, respectively. We provide some simulations to illustrate our algorithm in Section 6 and conclude the paper in Section 7.
2. Related Works
In this section, we summarize some up to date research works related with routing protocol or data transmission in mobility wireless networks.
In delaytolerant mobile networks, the authors in [1] provided a detailed survey of opportunistic transmission algorithms to study the nature of mobility. Specially, it revealed that human mobility is not random at all but have a definite and repetitive pattern. Actually, using opportunistic transmission, a single packet transmission may result in plenty of copies of the packet message. The authors in [29, 30] focused on the controlling twohop forwarding polices, where the problem concerned the decision on whether or not forwarding a given packet to a specific mobile node. By restricting the number of transmission hops, the proposed algorithms in [29, 30] could markedly reduce the copies in packet dissemination. In UAVassisted vehicular delaytolerant networks, the literature [31] developed a routing protocol to improve the reliability of packet transmission by considering both the encounter probability and the persistent connection time. In [32], theoretical upper and lower bounds for the information propagation speed were derived, in which storecarryforward routing model was used. Considering wireless transmissions and sojourns on node buffers, the authors in [33] computed the packet speed and cost according to utilitybased routing rules. However, most of the research results obtained in delaytolerant mobile networks are based on the assumption of random mobility and do not exploit the full potential of the mobility traces.
In mobilityaware networks, a novel graph metric named mobile conductance was conceived to evaluate the information spreading time in [34]. The mobilityconnectivity tradeoff was also quantitatively analyzed in [34] to determine how much mobility may be exploited to compensate for network connectivity deficiency. Bedogni et al. in [35] developed a methodology to infer complete trajectories of individual vehicles and then proposed some temporal connectivity algorithms for packet transmissions. In [36], the authors introduced a concept of energyaware temporal reachability graph (ETRG) and proposed an algorithm to calculate ETRG. According to ETRG, these results revealed the fundamental relations among the system metrics of energy budget, tolerable delay, and data size on the network performance. In [23], a reliable topology design was investigated in delaytolerant networks with unreliable links. According to the known or predictable network topology, several heuristic algorithms were proposed to build reliable and lowcost network topologies. Nevertheless, the energyefficient data transmission protocol does not involve explicitly in these works.
For data transmission or routing design in mobilityaware networks, the authors in [20] developed a relay selection strategy by utilizing partially predictable mobility, in which the directional correlation of destination movement was considered. Simulations showed that the proposed forwarding strategy could achieve a higher delivery utility compared with a forwarding scheme without mobility prediction. In [21], a trajectorybased multicast (TMC) routing was proposed for efficient multicast in vehicular networks. By exploiting vehicle trajectories, TMC routing could achieve a delivery ratio close to that of the floodingbased approach while the cost is reduced by over 80%. Li et al. in [37] concentrated on the communication services of passengers in the train from the base station. According to the regular mobility of the train, the authors in [37] proposed a qualityofservicedistinguished power allocation algorithm to meet each user’s data rate requirement. In [38], a novel socialbased routing approach was proposed, in which a new metric of social energy was introduced by exploiting social behaviors of nodes. Liu et al. in [39] proposed a mobilityaware transmission scheduling scheme, which consists of a relay path planning algorithm and a global time scheduling algorithm. Extensive simulations under realistic human mobility trajectories showed that the transmission scheduling scheme in [39] could achieve high throughput transmission. An aeronautical ad hoc network with rapidly changing topology was modeled as a dynamic graph in [40], and then data transmission problem was formulated as an integer nonlinear programming. A detailed review work is available in the literature [41]. Different from these works, this paper investigates the energyefficient data transmission problem in mobilityaware wireless networks. Specially, a series of dynamic network topologies is modeled as a virtual spacetime graph, which is similar to the literature [23]. The spacetime graph model is also utilized in wireless dutycycle sensor networks for data transmission, as shown in [42, 43]. In our new data transmission problems, both reliable and unreliable links are involved.
3. System Models
In this section, we present the network model, link model, and energy consumption model. A description of the key notations is listed in Table 1.
3.1. Network Model
In mobile wireless networks, the locations of network nodes are changed over time, resulting in topology evolutions. To describe such evolution, a sequence of static graphs is introduced. Let a period of time be divided into discrete and equal time slots, i.e., . Define the network topology at timeslot as an undirected graph , where is the set of nodes, and represents the wireless link between nodes and at timeslot . The dynamic network over a period of time could be modeled as a sequence of static graphs , which describes the topology evolutions due to node mobility. Figure 1(a) provides an example to show the sequence of snapshots of the network topology at each time slot. Note that the topology connectivity at each time slot in mobile wireless networks could not be guaranteed, which incurs the data transmission over them intractable. Here, we assume that the moving track of each node (such as the smart device in public bus) is known in advance, i.e., the topology graph at timeslot is predictable.
(a) A sequence of topology snapshots at each time slot
(b) Virtual spacetime topological graph
We introduce a new graph structure associated with the sequence of network topologies , named virtual spacetime graph . In this virtual graph , the virtual node is the virtualization of node at the end of time slot . As a consequence, each node is replaced with associated virtual nodes. The number of nodes in virtual graph is equal to , i.e., . If , , two directed edges (or links) are generated in , that is, and . This kind of link refers as to a spatial link, meaning that one node can forward a message to node at timeslot . And besides, another kind of link , referred as to a temporal link, is defined in the link set , representing that the node can carry the message in the th time slot. The virtual spacetime graph clearly includes both the spatial information in each time slot and the temporal information due to topology change. Figure 1(b) illustrates the virtual spacetime graph of the sequence of topologies in Figure 1(a). In this example, the red path in Figure 1(b) is a spacetime path from to . That is, the message transmission from node 4 to node 1 needs 3 time slots: node 4 holds its packet at and sends it to node 3 at , and then node 3 sends it to node 1 at . Note that at most one message could be transmitted within one time slot. That is, we assume that the time slot is only long enough for one message transmission.
3.2. Link Model
A wireless link exists at timeslot , i.e., , if and only if the two mobile nodes and are located within the transmission range of each other at timeslot . We assume that a mobile node always fails to transmit its packet to another node which is beyond node ’s transmission range, while packet delivery within each other transmission range succeeds with a probability. We define a reliability probability for each link , which means node can successfully receive a packet sent from node . The values of reliability probability are influenced by the stochastic nature of wireless channel and/or imperfect mobility prediction. According to the value of , we define two link models as follows, i.e., reliable link model and unreliable link model.
In the reliable link model, if a wireless link is , we have ; otherwise, . Under this model, data packets transmitted within the transmission range are always succeeded.
In the unreliable link model, a wireless link connecting two nodes and at timeslot is not necessarily reliable. That is, if a wireless link is , we have ; otherwise, . Under this model, data packets were transmitted from node to node over a wireless link with a successful reception probability .
A wireless link associates with two spatial links, i.e., and , in spacetime graph . Therefore, the reliability probability of wireless link could be converted into and , and we have
For each temporal link in spacetime graph , it may also have a reliability probability for successfully holding its data packet over one time slot. Actually, if buffer overflow or energy depletion incurs at a mobile node, the node may fail to hold its data packet. However, the failures in holding a packet over one time slot are seldom compared with those in data transmission over spatial link. Therefore, it is reasonable to assume that all temporal links are reliable, i.e., for each , . We define the reliability of a spacetime path as the production of all links’ reliability in , i.e., .
3.3. Energy Consumption Model
Both sending a message over a spatial link and holding a message over a temporal link incur energy consumption. The energy cost for holding one bit of data message in one time slot for any node is assumed to be a fixed value , i.e., for any temporal link , the corresponding energy cost for holding a message is , where is the number of bits per message. The energy consumption for sending a massage over a spatial link comes from two parts: transmission and reception. Let be the amount of energy consumption required to receive one bit of data message. The amount of energy consumption for transmitting one bit of data message to meters away is . From [44], we have where is the energy consumed by the sender circuit, is the antenna output energy to reach the receiver unit distance away, and is the path attenuation factor. We define the energy cost for sending a message over a spatial link as where is the distance between nodes and at timeslot , and is the discount rate in time. The energy cost of a spacetime path is the summation of all links’ energy cost in , i.e., .
In next two sections, we firstly define the data transmission problems over reliable links and unreliable links in spacetime graph and then develop two algorithms to solve the proposed transmission problems.
4. EnergyEfficient Data Transmission over Reliable Links
We now define the energyefficient data transmission problem over reliable links on the virtual spacetime graph .
Definition 1. Given a source node , a destination node , the aim of energyefficient data transmission over reliable links is to find a spacetime path with the minimum energy cost from node to node .
Let the spacetime graph over a period of time be connected, such that the spacetime path from source node to destination node could be found over the time period . Here, a spacetime graph is connected over time period if and only if there exists at least one spacetime path for each pair of nodes .
Note that, if the time period is small, the number of selectable spacetime paths is less. For example, in Figure 1(b), if is equal to 2 time slots, the spacetime path between source node and destination node is just one, i.e., . If is equal to 3 time slots, another selectable path between and is added, i.e., . With the increasing of time period , the number of selectable paths is nondecreasing, and then the optimal spacetime path which has the minimum energy cost should be found. Therefore, the energyefficient data transmission problem in a time period can also be viewed as a tradeoff between energy consumption and transmission delay.

In the data transmission problem over reliable links on the virtual spacetime graph , all virtual nodes in the spacetime graph associate with one node but at different moment. Therefore, each spacetime path in that connects node and anyone node constitutes the candidate path set . That is, the source node could transmit its data packet to destination node at timeslot across the spacetime path . The solution of data transmission problem over reliable links is to find the spacetime path in which has the minimum energy cost. Next, EnergyEfficient Data Transmission algorithm over Reliable Links (EEDTRL) is developed to solve the new data transmission problem. Algorithm 1 shows the detailed procedure of EEDTRL.
In EEDTRL, we firstly convert the sequence of static network graphs into a spacetime graph . Then, on the spacetime graph , an extension of Dijkstra’s algorithm is implemented to find the spacetime path with minimum energy cost from node to . In EEDTRL, is comprised of the links of the optimal spacetime path from source node to node with the minimum energy cost . In our assumption, the spacetime graph over a period of time is connected. Thus, the spacetime path from node to node could be found over the time period , which means . Finally, we find the minimum value of and denote it by . Then, the optimal spacetime path from source node to destination node over the time period is with the minimum energy consumption . Since the spacetime graph has virtual nodes, the computational complexity of EEDTRL is in the worst case. Note that the link is a directed link, and the arrow is omitted in the EEDTRL algorithm.
5. EnergyEfficient Data Transmission over Unreliable Links
In this section, energyefficient data transmission problem over unreliable links is defined on the virtual spacetime graph .
Definition 2. Given a source node , a destination node , the aim of energyefficient data transmission over unreliable links is to find a spacetime path with the minimum energy cost and the maximum path reliability from node to node .
The energyefficient data transmission problem over unreliable links has two optimal objectives: the minimum energy cost and the maximum path reliability
Here, is the candidate path set in which each spacetime path connects node and anyone node . To solve the biobjective optimization problem, the second objective is rewrote as
By following a popular approach used to deal with biobjective optimization problems, the two objectives (4) and (6) have been transformed into a single objective, using an importance weight factor [45]. Then, define a composite link weight for each link as where and are the maximum of and the minimum of for , respectively. Note that and are normalization factors used to have the same range for the two objectives.
Based on minimizing on the spacetime graph , an extension version of Dijkstra’s algorithm could be employed, which is similar to Algorithm 1. If the optimal spacetime path with minimum composite weight is found, the corresponding energy cost and path reliability could be further calculated. It should be noted that, because of the link unreliability, if the data packet from source node is failed to be forwarded to the next node by one intermediate node at timeslot , this intermediate node will become the source node at timeslot and find the optimal spacetime path again in the remaining timeslots. We call this heuristic data transmission method over unreliable links as EEDTUL. In the worst case, each transmission has failed, and then the path finding algorithm needs to be executed times. Therefore, the computational complexity of EEDTUL is .
6. Simulations
In this section, some simulations are provided to illustrate the proposed EEDTRL and EEDTUL algorithms. In a network region, some mobile nodes are distributed randomly at the beginning. A sequence of network topologies is generated in the following manner. All nodes move according to the random direction mobility model. That is, each node moves from the current position to the next position with an arbitrary direction and a random velocity selected from the range per time slot. When the mobile node hits the network region boundary, its mobile direction is reversed. By recording each node’s position at each time slot, a sequence of static network graphs could be derived according to the maximum transmission range of each node. The source and destination pairs are selected randomly for each topology instance. Note that the trajectory prediction is not within the scope of this paper. Thus, in our simulation, the random direction mobility model is employed just for generating each node’s mobility trajectory in a time period. Some other important simulation parameters are listed in Table 2 [23, 44].
Figure 2 provides an example of topology evolutions at some consecutive time slots with the number of nodes . It can be seen that the network topology is varied from one time slot to another and is often disconnected in each time slot. Therefore, it is impossible to transmit data packet in a single time slot between two remote nodes. We should consider a sequence of network topologies in multiple time slots for the design of data transmission scheme.
(a)
(b)
(c)
(d)
The proposed EEDTRL and EEDTUL algorithms are evaluated according to the following performance metrics: energy cost, actual transmission count, and path reliability. Actual transmission count between source and destination is equal to the number of spatial links in the spacetime path, while path reliability is the production of all links’ reliability, which is defined in Subsection 3.2. We compare these performance metrics with two common algorithms, which are often employed for data transmission in mobility networks, i.e., epidemicbased transmission and distancebased transmission. These two common algorithms have many versions in different literatures [3, 46]. In our simulation, we extract the main idea of these two algorithms and make appropriate modifications to fit the spacetime graph model. In epidemicbased transmission, the nodes carrying data packet infect the nodes which do not receive the packet utilizing the communication opportunity at each time slot, until the destination node receives this packet. In distancebased transmission, a node, which does not receive the packet, with the minimum distance to the destination is selected as the next forwarding node.
6.1. Simulations on EEDTRL
In this simulation, we first increase the number of nodes from 10 to 30 and keep time period at 10 time slots. Figures 3(a) and 3(b) show the variation trend of energy cost and transmission count versus number of nodes. Before executing the proposed EEDTRL and two comparison algorithms, a sequence of static topology graphs should be converted into a spacetime graph, as shown in Subsection 3.1. These algorithms are implemented on 1000 various spacetime graphs, and in each spacetime graph, one sourcedestination pair is selected randomly. From Figure 3, the proposed EEDTRL has the minimum energy cost and the minimum transmission count compared with the epidemicbased and distancebased algorithms. Actually, EEDTRL can select better communication opportunities for data transmission and hold data packet (do not transmission) for saving energy when the communication opportunities is worse. With the increase of number of nodes, the two metrics of energy cost and transmission count are stable relatively in EEDTRL and distancebased algorithm, while these two metrics are increasing rapidly in epidemicbased algorithm.
(a) Energy cost
(b) Transmission count
The time period is varied from 8 to 12 time slots to study the impact of time period on energy cost and transmission count. The curves of energy cost and transmission count vs. time period are reported in Figures 4(a) and 4(b), in which the number of nodes is 20. Similarly, the proposed EEDTRL generates the most energyefficient spacetime path. From Figure 4(a), the energy cost decreases gradually with the increase of time period in EEDTRL; that is, a longer time period will induce a more energyefficient data transmission.
(a) Energy cost
(b) Transmission count
6.2. Simulations on EEDTUL
For the unreliable link model, the reliability probability of a spatial link is assigned a random value from 0.6 to 1 in this simulation. We first investigate the tradeoff between energy cost and path reliability of the proposed EEDTUL by varying the weight factor from 0.1 to 0.9. The time period and number of nodes are set as time slots and . Figures 5(a) and 5(b) plot the curves of energy cost and path reliability vs. weight factor , respectively. The link weight changes from 0.1 to 0.9, reflecting the everincreasing importance of energy cost in data transmission. From Figure 5, the reduction of energy cost comes at the expense of path reliability. We can see the tradeoff between energy cost and path reliability via the adjustment of weight factor . It should be noted that the energy cost of the spacetime path generated by EEDTUL may not be the actual energy consumption. Because of the unreliable links, the data transmission may fail, and a new spacetime path will be generated. If the path reliability is high, the failure probability of data transmission is low, and then the calculated energy cost is equal to the actual value.
(a) Energy cost
(b) Path reliability
The metrics of energy cost and transmission count in EEDTUL are similar with these in EEDTRL. Thus, we study the path reliability in EEDTUL in contrast to epidemicbased and distancebased data transmission. Figures 6(a) and 6(b) show the simulation results of path reliability vs. number of nodes and time period, respectively. Here, we set time period as 10 time slots in Figure 6(a), number of nodes as 20 in Figure 6(b), and weight factor as 0.5 for both. From Figure 6, the proposed EEDTUL has the maximum path reliability compared with the two common algorithms.
(a) Path reliability vs. number of nodes
(b) Path reliability vs. time period
7. Conclusions
In this paper, the energyefficient data transmissions over reliable links and unreliable links in mobilityaware wireless networks are investigated. A sequence of network topologies deduced by mobility prediction, each of which is usually disconnected, was modeled as a virtual spacetime graph. Data transmission problems over reliable links and unreliable links on spacetime graph were defined, in which a spacetime path with the minimum energy cost would be found. Then, EEDTRL was proposed to find the optimal spacetime path under the reliable link model. Under the unreliable link model, we developed a heuristic data transmission method, named EEDTUL, which could achieve the tradeoff between energy cost and path reliability. The simulation results showed that our proposed algorithms perform well in terms of energy consumption and transmission count compared with some existing algorithms. Specially, we can further improve the energyefficiency of data transmission by increasing the number of nodes and/or prolonging the time period, due to that these ways could result in more communication opportunities for selection.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
This research was supported in part by the National Natural Science Foundation of China (62172457), National Natural Science Foundation of Henan (202300410523), Innovation and Entrepreneurship Training Program for College Students of Henan (S202110478028), Key Scientific Research Project of Henan Educational Committee (21A510003, 22A510018), and Science and Technology Development Project of Zhoukou (2021GG02028).