I Introduction
With the development of wireless communication technologies, wearable and pocket communication devices such as Eglasses, Ewatches, smart belts and etc., have penetrated into our daily lives. All these communication devices require to exchange vital information with central servers, such as the users’ physical condition, the location information and etc.
Delay tolerant networks (DTNs) [1717] have been proposed for realising communication in intermittent networks. In DTNs, a direct communication link between any pair of nodes is not always available, due to extreme terrestrial environments or movements of the communication nodes. In order to improve the performance of the information delivery, Akhtar et al. [1111] has proposed to exploit the users’ social properties for improving the information delivery performance of the DTNs, which yields a new concept of mobile social network (MSN). Nikolaos et.al. [2828] proposed the architecture and social properties of MSN, while also threw some key research challenges. There are two communication types in MSNs, namely device to device (D2D) communication [3131] and device to access point (D2A) communication, as potrayed in Fig.1. In the D2D communication, a pair of different users may establish a direct communication link between their devices without the aid of any centralised infrastructure, if they enter each other’s transmission range. For example, Jie et al. [1818] has proposed a content dissemination scheme by exploiting the D2D communications among the users. By contrast, in the D2A communication, a user may communicate with an centralised AP, if it enters the AP’s transmission range. For instance, Zhao et al. [1212] has studied a joint AP deployment and routing scheme for efficiently delivering information to the users in MSNs. Later on, Ying et al. [1313] have proposed another AP deployment scheme by exploiting the social popularity of specific locations. Furthermore, Fan et al. [1414] [1515] have developed a joint scheme for both the AP’s deployment and the storage allocation in order to efficiently deliver popular content towards the users. In contrast to the conventional highpower cellular communication, the APs in the MSN often operates at a low transmit power. As a result, the transmission range of the APs is limited. Hence, users have to enter the transmission range of the APs so as to download important information to their wearable devices. Therefore, users’ mobility largely determines the communication performance of the MSN.
Furthermore, due to their limited size, wearable devices are normally equipped with batteries having limited capacity. The batteries may quickly drain for the sake of powering complex signal processing and sensing tasks. Yang et.al. [2929] proposed an approach to offload services for resourceconstrained mobile devices, while the resource management can also be operated on the cloud to reduce the energy consumption [3232]. However, this is not an ideal solution as some tasks are simply not suitable to run on servers and task offloading also incurs extra energy consumption. Therefore, it is essential to incorporate the function of wireless energy transfer (WET) to the conventional communication oriented APs, which yields the hybrid access points (HAPs). Many radio frequency (RF) signals based WET techniques have been studied in different OSI layers [3030], such as the energy beamforming [555] and channel coding [222] in the physical layer, the network architecture [333] and the MAC protocol design [444][777] as well as the resource allocation [666] in the network and MAC layer. Moreover, coordinating both the WIT and WET in the same RF spectral band has attracted much attention from both the academia and industry, which triggers substantial works in simultaneously wireless information and power transfer (SWIPT) and wireless powered communication (WPC). All these related works [888, 999, 1010] aim for striking a balance between the WET and the WIT performance by exploiting the hot communication techniques, such as the massive MIMO technique [1010], full duplex technique [2424]
and channel estimation technique
[888].However, many of the energy concerns focus in the future cellular networks [3333], while few existing works study the WET in the scenario of MSNs, where both the WIT and WET performance is dominated by the users’ mobility patterns. Fan et al. [1616] have firstly proposed to exploit the socalled enhanced throwboxes for realising both the WIT and WET in the MSN. However, their model is extremely simplified without considering a realistic mobility model, the wireless channel attenuation and the distinctive features of the WIT and the WET. In order to fill this gap, our novel contribution can be summarized as follows

A user mobility model in the city streets is exploited for characterising the users’ mobility patterns, in which users move randomly along the city streets and they may randomly opt to turn or to go straight, when they reach a crossroad.

A twodimensional Markov chain is exploited for analysing the seminal metrics of the users’ mobility patterns, including the users’ total sojourn duration within the WIT range of the HAP and that within the WET range.

Both the WIT and WET efficiencies of the MSN studied are derived in closeform. Then, three different HAP deployment schemes are proposed, namely the Ideployment scheme for maximising the WIT efficiency, the Edeployment for maximising the WET efficiency and the Bdeployment for striking a balance between the WIT and WET efficiencies. The algorithms for obtaining both the optimal and suboptimal solutions are developed.
The rest of the paper is organized as follows. In section II, both the network model and the users’ mobility model are introduced. Then, the users’ mobility patterns are analysed by exploiting a twodimensional Markov chain in Section III. In section IV, three HAP deployment schemes having different goals are proposed, whose solution is obtained in Section V. After the simulation results presented in Section VI, our paper is concluded in Section VII.
Ii System Model
Iia Network Model
We focus our attention on the typical D2A communication in a mobile social network (MSN), where users are equipped with wearable devices and moving around street blocks. Users may download important information and charge their batteries via RF signals emitted by the HAPs. Due to the limited transmission range of the HAPs, users’ mobility and the popularity of the hotspots (e.g. shopping mall, parks and sport centres) may jointly determine the performance of both the WIT and WET. As a result, an optimal deployment scheme of the HAP should consider both the users’ mobility and the popularity of the hotspots. We assume users in the MSN studied, which are denoted by a set of . We also assume that sites in the MSN studied as the candidates for the deployment of the HAPs, which are denoted by a set of . Furthermore, we have
HAPs in total pending to be deployed. A vector
is exploited for characterising a specific deployment scheme of the HAPs, where indicates that a HAP is deployed at the site , while indicates that no HAP is deployed at the site . Obviously, if we have more HAPs than the number of sites, say , each site can have a single HAP to be deployed. By considering the deploying cost, we normally have fewer HAPs than the number of sites, say . In order to maximise the benefit from these HAPs, an optimal deployment scheme is required.The HAPs are equipped with a pair of radio fronts operating in different spectral bands for facilitating the WIT and the WET, respectively. Specifically, the WET operates in a low frequency band, which may substantially reduce the path loss induced channel attenuation [2525]. By contrast, the WIT operates in a high frequency band in order to obtain a high bandwidth for increasing the throughput. The energy harvesting circuit and the information reception circuit at a wearable device have diverse requirement on the minimum power of the received RF signal. For example, the minimum power of the received RF signal at a wearable device can be 80 dBm for successful information recovery. By contrast, the minimum power requirement on the received RF signal is at least 40 dBm for activating the energy harvesting circuit [2727]. As a result, given the same transmit power, the HAP has distinct WIT range and the WET range. If a HAP is deployed at the site , its WIT range is denoted as , while its WET range is denoted as . Since the energy harvesting requires higher received signal power than the information decoding, we normally have . Furthermore, homogeneous HAPs are conceived in the MSN studied. As a result, all the HAPs pending to be deployed have an identical WIT range , denoted as , and they also have an identical WET range,^{1}^{1}1
The wireless channel power gain is mainly affected by the pathloss as well as the multipath fading. We assume that the multipath fading has the same statistical properties at each crossroad. Hence, given the information outage probability of the received WIT SNR lower than an identical threshold, the WIT ranges of the HAPs are identical. So are the WET ranges of the HAPs. Therefore, we assume that the WIT and the WET ranges of all the HAPs are identical.
denoted as . Moreover, in the MSN studied, when a user enters the WIT range of the HAP, it can only download its requested information from the HAP. If the user enters the WET range of the HAP, it can simultaneously download the information and harvest energy from the HAP, since we have . In this MSN, we aim for deploying the HAPs at the optimal sites in order to maximise the performance of both the WIT and WET.IiB Mobility Model
We assume that all the users move around a grid based city street blocks, by obeying a mobility model more generic than the conventional Manhattan mobility model [2222]. In our mobility model, the users move along the streets. Once they arrive at a crossroad, they may choose any available direction to move forward. Since users from four different directions may gather at a crossroad, deploying a HAP at a crossroad is capable of efficiently serving more users. As a result, all the crossroads are regarded as the potential sites for the deployment of the HAPs.
Our city street blocks are illustrated in Fig.2, where the green lines indicate the streets in the city, while the red and blue circles represent the WIT ranges and the WET ranges of the HAPs, respectively. The entire area is equally divided into several square regions having the cross roads as their centres, as portrayed in Fig.2. We assume that there are crossroads in the city street blocks, where represents the number of crossroads in the vertical dimension, while represents the number of crossroads in the horizontal dimension. All the crossroads can be denoted by the set , which are also potential sites for the deployment of the HAPs, and users’ movements are bounded in the area studied. Generally, a crossroad may have arbitrary number of streets connected to it. However, in our model, we consider that a crossroad is only connected to its nearest peers in our grid based street blocks. Moreover, a crossroad also has a socalled crowded range for characterising its popularity. Moving within the crowded range may substantially reduce the speed. A higher crowded range represents that more people gather at this crossroad. The crowded range of the crossroad is represented by an orange circle having the radius of . The crowded range is an average value according to the statistics of the crossroad . When a user moves outside of the crowded ranges of the crossroads, its speed is assumed to be . If enters the crowded range of the crossroad , it may reduce its speed to , which is determined by the user itself and by the popularity of the crossroad . Without loss of generality, every street connecting a pair of neighbouring crossroads has an identical length of . Since the shortrange communication technique, such as WiFi, bluetooth and etc., is invoked for enabling the D2A communication in the MSN, the WIT range is relatively low. As a result, we may reasonably assume that the street length satisfy the condition of .
Iii User Mobility Analysis
In this section, we model the users’ mobility pattern by a twodimensional discrete Markov chain, which may help us in analysing a user’s sojourn duration within the WET range of a HAP and that within its WIT range, respectively.
Iiia TwoDimensional Markov Chain Modelling
As portrayed in Fig.2, the entire area is equally divided into a set of square regions having cross roads as their centres. As a result, a single street is halved into two segments belonging to a pair of neighbouring square regions. The side length of each square region is , which is equal to the length of a street. The square region having the crossroad as its centre is denoted by . All the direction choosing probabilities of the user can be denoted by a set of . Specifically, represents the probability of the user moving from the crossroad to its neighbour . By defining the user ’s stay within the square region as one of its mobility states, we may model the user ’s mobility pattern by a twodimensional Markov chain, which is illustrated in Fig.3. Consequently, also represents the probability of transiting from the state to the state .
The stationary probabilities of all the states in the Markov chain of can be denoted by a vector , whose element represents the corresponding stationary probability of the state . Similarly, we define a matrix of the state transition probabilities by rearranging the entries in the original set of the state transition probabilities. The entry of is then mapped on the entry of . Given that our Markov chain is ergodic and irreducible, the stationary distribution can be readily calculated by
(1) 
In the rest of the paper, we denote the stationary probability of the state by , which is equal to the element of in the vector .
IiiB Total Sojourn Duration of the Square Regions
Since the users only change their moving directions when they arrive at any of the crossroads, the travel distance of the user within any of the square regions is . For a specific square region , since we have , the crowded range of the crossroad is also included in this square region. As a result, when the user visits the square region , its sojourn duration within this region is derived as
(2) 
The sojourn distribution of the user sojourning within a specific square region can be defined as the set . Note that the probability is different from the stationary probability of the state in the Markov chain of Fig.3, since the derivation of only considers the state transition probabilities but the derivation of further considers the time duration of staying in a specific state. Based on the steady probability of the user staying within the square regions and its corresponding sojourn durations , the probability of the user sojourning within the square region can be derived as
(3) 
Given a specific observation duration , the total sojourn duration of the user in the square region can be expressed as .
IiiC Total Sojourn Duration within the WIT and WET ranges
We firstly aim for obtaining the sojourn duration of the user within the WIT range of the HAP, if it is deployed at the crossroad . The travel distance of the user within the WIT range should always be . If the WIT range is shorter than the crowded range , the user always move within the WIT range at a speed of . If the WIT range is longer than the crowded range , the user may move for a distance of at a speed of and it may move for another distance of at a speed of . As a result, the sojourn duration of the user within the WIT range can be expressed as:
(4) 
By sticking to the same methodology, the sojourn duration of the user within the WET range is formulated as:
(5) 
Furthermore, the total sojourn duration of the user within the WIT range of the HAP deployed at the crossroad during the entire observation duration is derived as:
(6) 
Similarly, the total sojourn duration of the user within the WET range of the HAP deployed at the crossroad during the entire observation duration can be derived by substituting for into (6). According to the above analysis, the following remark is obtained:
Remark 1
For a specific crossroad , when the crowded range of other crossroads are fixed, the total sojourn duration of the user in the WIT range of is a convex function with respect to . The total sojourn duration can be maximised when the crowded range is equal to the WIT range . By contrast, the total sojourn duration of the user within the WIT range of any other crossroad is a monotonously decreasing function with respect to the crowded range of the crossroad . Same conclusion can be made on the total sojourn duration of the user within the WET range of the HAP.
Proof:
Please refer to appendix A for the detailed proof.
Iv Problem Formulation
By considering the users’ mobility pattern and the popularity of the crossroads, we aim for obtaining an optimal HAP deployment scheme for maximising either the overall WIT or the overall WET performance of the MSN.
Iva WIT Efficiency
The successful WIT efficiency is defined as the ratio between the actually delivered information amount and the maximum possible delivered information amount of all the users in the MSN studied. We assume that a user may download the information from the HAP with a constant rate , which is readily realised by invoking power control, the WIT efficiency of all the users during the observation duration is formulated as
(7) 
where denotes the deployment state of the HAP at the crossroad . Furthermore, indicates that an HAP is deployed at , while indicates that no HAP is deployed at . All the form an HAP deployment matrix , having a size of . The numerator of (7) indicates the actually delivered information amount of all the users, given a specific deployment scheme of the HAPs, while the denominator of (7) indicates the theoretically achievable information amount of all the users, if at least an HAP is deployed at each crossroad. According to the above analysis, we have the following remark:
Remark 2
If no HAP is deployed at the crossroad , the WIT efficiency is a monotonously decreasing function with respect to . By contrast, if an HAP is deployed at the corssroad , the WIT efficiency is a convex function with respect to . Furthermore, the WIT efficiency can be maximised, when we have .
Proof:
Please refer to appendix B for the detailed proof.
As a result, in order to improve the WIT efficiency, we should make the crossroads less crowded if no HAP is deployed, while we should let , if an HAP is deployed at the crossroad .
IvB WET Efficiency
We define the WET efficiency as the total harvested energy amount by all the users in the network. Due to the path loss incurred channel attenuation, the power of the received RF signals at the users is determined by the signals’ propagation distance. For example, when the user enters the WET range of the HAP deployed at the crossroad , the distance from to the HAP firstly reduce. Since the user keeps moving, the distance between and the HAP at will first decrease from to 0 and increase to again. According to [2121], the pathloss between the HAP and , when they are metres away from each other, can be expressed as
(8) 
where and are the transmit and receive antenna gains, respectively, while is the speed of the light, is the carrier frequency, and is the path loss exponent. Furthermore, and are the reference distance and the distance between and the HAP, respectively. Denoting the path loss at the distance of as , the pathloss model of (8) can be reformulated as
(9) 
In the nearfield of the transmitters, say , we assume that the pathloss is . For the shortrange communication, is normally a single metre [2121], which is even lower than the minimum among the WIT range , the WET range and the crowded range . As a result, the energy harvested by the user , when it enters the transmission range of the HAP deployed at the crossroad , can be formulated as
(10) 
where indicates the transmitting power of the HAP, while indicates the efficiency of rectifying the received RF signals to the direct current (DC) and we make without loss of generality. Finally, the closedform equation of the energy harvested by the user can be derived from (10) as
(11) 
As shown in (11), the amount of energy harvested by the user at the crossroad increases, as the crowded range of increases. If we keep increasing beyond , the energy converges to a constant value.
Given the total observation duration , the sojourn duration of the user within the WET range of the HAP deployed at the crossroad is derived as . As a result, the number of times that moves within the WET range of the HAP deployed at the crossroad during the entire observation duration can be calculated as . Therefore, the total energy harvested by the user during the total observation duration can be formulated as
(12) 
Since the energy harvested by the users have to be stored in their batteries, energy harvesting may not operate, once their batteries are fully charged. Without loss of generality, we assume that all the users’ battery capacity are . Hence, the actual energy harvested by the user can be expressed as . Furthermore, the WET efficiency can be obtained as
(13) 
According to the above analysis, we have the remark as follows:
Remark 3
When the crowded range of the crossroad satisfies the condition of , the WET efficiency does not increase as the crowded range increases, regardless of whether an HAP is deployed at the crossroad or not.
Proof:
Please refer to appendix C for the detailed proof.
Remark 3 tells us that in order to improve the WET efficiency, we should make the crowded range of the crossroad smaller, when it satisfies the condition .
IvC HAP Deployment Schemes
We will study three deployment schemes of the HAPs for serving different purposes, namely the balanced deployment scheme (Bdeployment), the energy harvesting oriented deployment scheme (Edeployment) and the information downloading oriented deployment scheme (Ideployment).
In order to maximise the WIT efficiency by ensuring that the total energy harvested by the users is higher than a threshold, we propose the Bdeployment scheme, which can be formulated as the following optimisation problem:
Objective:  (14)  
(14a)  
(14b)  
(14c) 
where the parameter is introduced for adjusting the designer’s preference on the WIT performance and the WET performance. A higher value of indicates that the designer is inclined to obtain a higher WET performance, while a lower value of indicates that the design is inclined to the WIT performance. The parameter in (14c) is the maximum energy that can be harvested by the users, which can be obtained by solving the following optimisation problem:
Objective:  (15)  
Subject to:  (15a)  
(15b) 
The optimisation problem (15) is regarded as the Edeployment scheme. Note that if we have in (14c), the optimisation problem (14) is degenerated to (15). If we have in (14c), the Bdeployment scheme is degenerated to the Ideployment scheme, which can be formulated as
Objective:  (16)  
Subject to:  (16a)  
(16b) 
V Solutions of the HAP Deployment
Va Exhaustive Searching Method
Since the deployment solution is an integer matrix, the above three problems cannot be solved in a polynomial time. The exhaustive search is capable of traversing all the possible deployment solutions and it thus choose the optimal one. This is the most inefficient method since its complexity is as high as . However, when the solution space is small, exhaustive search can be adopted for finding the optimal solution with the tolerant complexity. Let us define as the generic objective function with respect to the HAP’s deployment indicator matrix . Hence, we have for both the Ideployment problem and the BDeployment scheme. We also have for the Edeployment problem. The pseudo code of the exhaustive search is provided in Algorithm 1.
Since the exhaustive search is an aimless searching method, we have to traverse all the possible deployment solution before finding the optimal one. Since the objective function of the Ideployment problem (16) is linear, a directional searching method can be adopted for reducing the complexity. By letting , the WIT efficiency expressed in (7) can be reformulated as
(17) 
In order to maximize the WIT efficiency , we should deploy an HAP at the crossroad associated with a higher coefficient . The pseudo code of the directional searching is provided in Algorithm 2.
By adopting the directional searching, the complexity can be reduced to . By contrast, since the objective functions in the Edeployment problem is nonlinear, the directional searching cannot be adopted for finding the optimal deployment solution. Furthermore, since there is a nonlinear constraint on the energy harvesting performance, the directional searching cannot be adopted for finding the optimal solution of the Bdeployment problem.
VB HAP Deploying Probability
Observe from (14)(16) that the optimisation problems of the Bdeployment scheme, the Edeployment scheme and the Ideployment scheme all belong to the integer programming, since the deployment indicator at the crossroad is either zero or one, which cannot be solved in polynomial time. As a result, we transform these three integer programming problem into the real number based optimisation problem by relaxing the original integer constraint on to the constraint that is a real number within the region . This relaxation is reasonable since a real valued can be regarded as the probability of an HAP being deployed at the crossroad .
We will elaborate on how the constraint relaxation operates in solving the optimisation problem of the Bdeployment as an example, since both the Edeployment scheme and the Ideployment scheme are special cases of the Bdeployment scheme. We introduce a new variable for representing the actual energy harvested by the user . The constraint (14c) can be then rewritten as
Accordingly, the original optimisation problem of (14) can be reformulated as
Objective:  
Subject to:  
(18) 
where is defined in Section VA. Since both the objective function and the constraints are linear with respect to and , the optimisation problem (VB) can be solved by adopting the simplex method [2323].
Similarly, the reformulated optimisation problem of the Edeployment scheme can be expressed as
Objective:  
Subject to:  
(19) 
and the reformulated optimisation problem of the Ideployment scheme can be expressed as
Objective:  
Subject to:  
(20) 
Both these two optimisation problems can also be solved by adopting the simplex method.
VC Branchandbound approach
Since the exhaustive searching cannot solve the optimisation problem of the HAP’s deployment in a polynomial time and the directional searching is only capable of solving the optimisation problem of the Ideployment, we need a generic lowcomplexity algorithm for finding the optimal solution of all the three deployment scheme. As a result, the branchandbound approach is adopted for solving these optimisation problems by sacrificing the optimality to some extent. The key steps of the branchandbound approach are summarised as follows, where the Bdeployment optimisation problem is chosen as an example:
STEP 1: Denote the upper bound and the lower bound of the HAP’s deployment indicator at the crossroad as and , respectively. Let us initialise and for each . The set of all is denoted as , while the set of all is denoted as .
STEP 2: We then transform original integer programming problem into the following real valued optimisation problem as
Objective:  
Subject to:  
(21) 
where . We can use the same method for solving the real programming problem in Section VB, while the optimal solution of all the of (VC) is denoted as the set .
STEP 3: If the resultant in is an integer matrix, is the final solution. Otherwise, let and . Then, arbitrarily choose a noninteger element in . Let and , where and indicates the largest integer smaller than and the smallest integer larger than , respectively.
STEP 4: Divide the optimisation problem (VC) into the following two subproblems
Objective:  
Subject to:  
(22) 
and
Objective:  
Subject to:  
(23) 
Both of these two subproblems can be solved by exploiting the simplex method of Section VB. The optimal solutions of these two subproblems are denoted as and , respectively. If we have , we have , , and . Otherwise, we have , and .
The pseudo code of our branchandbound approach is provided in Algorithm 3. The complexity of the branchandbound approach is , which is much lower than that of the exhaustive searching method.
Vi Performance Evaluation
Symbols  The meaning of these symbols 

number of users  
number of candidate sites to deploy HAPs  
number of HAPs to be deployed  
the th user,  
the th candidate site to deploy HAP  
,  number of crossroads in vertical and horizontal dimension 
the crossroad with the coordinate of  
the square region of the crossroad with the coordinate of  
WIT range of the HAP  
WET range of the HAP  
crowded range of the crossroad with the coordinate of  
HAP deployment vector  
stationary probability of staying in  
stationary probability of staying in in any time  
sojourn duration of visiting  
sojourn duration of visiting the WIT range of  
sojourn duration of visiting the WET range of  
observation duration  
total sojourn duration of staying in  
total sojourn duration of staying in the WIT range of  
total sojourn duration of staying in the WET range of  
WIT efficiency  
energy harvesting amount of visiting at each time  
WET efficiency 
In this section, numerical results are provided for evaluating the performance of the three deployment schemes of the HAP. The users in the MSN studied move within the city street blocks having crossroads in our simulation and each crossroad has a different crowded range. The moving speed of the users in different spaces are arbitrary. However, the speed of a user moving within a crowded range is lower than the speed moving outside of a crowded range. Furthermore, the probability of a user changing its moving direction at a crossroad is also arbitrarily chosen. The parameter settings are summarised in TABLE.II, if there is no specific statement.
Number of users  100 

Total simulation time  10 hours 
Number of HAPs  8 
Street length  200m 
WIT range  50m 
WET range  10m 
Crowd range  5m60m 
Reference distance  1m 
Path loss at  0.003 
Battery capacity  1J 
Importance parameter  0.97 
Energy transmitting power  1W 
Via Accuracy of Markov analysis
We evaluate a single user’s stationary probability of staying at different crossroads by both the Markov chain based analysis and by the MonteCarlo simulation. Observe from Fig.4 that our analytical results perfectly match the simulation results, which demonstrates the accuracy of our Markov chain aided analysis.
ViB Exhaustive Searching v.s. BranchandBound
We firstly compare the exhaustive searching method to the branchandbound counterpart in terms of the WIT and the WET efficiency. Specifically, the exhaustive searching is capable of finding the optimal deployment solution, while the branchandbound based method may only find the suboptimal deployment solution. Observe from Fig.5 that the solutions obtained by relying on the branchandbound approach are always worse than those of exhaustive searching method, since the branchandbound approach sacrifice the optimality for a lower complexity. Note that in the Ideployment scheme, the solutions of these two methods perform the same. This is because the first iteration of the branchandbound approach is capable of producing an integer deployment solution . Moreover, observe from Fig.5 that the performance gaps between the solutions of the branchandbound method and those of the exhaustive searching method are negligible. Hence, when the solution space is very large, the branchandbound approach can be adopted for reducing the computational complexity without remarkable performance degradation.
ViC Popularity of Crossroads
We randomly choose a crossroad and change its crowded range for evaluating its impact on the users’ sojourn duration and the probability of an HAP being deployed at this crossroad, while the crowded ranges of other crossroads are fixed. Observe from Fig.6 that, users’ sojourn duration within the WIT range and the WET range of the HAP deployed at the chosen crossroad both firstly increase and then reduce as its crowded range increases. Furthermore, when the crowded range is the same as the WIT range (or the same as the WET range ), the sojourn duration within the WIT range (or the WET range) is maximised. By contrast, the user’s sojourn duration within the WIT range and that within the WET range of the HAP deployed at another crossroad both reduces as we increase the crowded range of the chosen crossroad. These observations are in line with Remark 1.
Furthermore, we plots the probabilities of an HAP being deployed at the chosen crossroad against its crowded range in Fig.7. Observe from Fig.7 that the probability of the HAP being deployed is a nondecreasing function with respect to the crowded range of the chosen crossroad, regardless of any specific deployment scheme. Moreover, the HAP’s deploying probability in the Ideployment scheme is either 0 or 1. Furthermore, the deploying probability of the HAP in the Bdeployment scheme is a compromise between the Ideployment scheme and the Edeployment scheme, since it aims for balancing both the information downloading requirement and the energy harvesting requirement of the users.
ViD Information Downloading and Energy Harvesting
In Fig.8 and Fig.9, we plot both the WIT and the WET efficiencies against the number of the HAPs pending to be deployed. We choose the Sdeployment scheme originally proposed by Ying et al [2626]. In this scheme, the HAPs are deployed by only considering the visiting frequency of users at each crossroad. We can readily obtain that our proposed HAP deployment schemes are better than the Sdeployment in terms of both the WET and WIT efficiencies. Observe from Fig.8 that the Ideployment scheme outperforms its counterparts in terms of the WIT efficiency, while the Edeployment scheme performs best in terms of the WET efficiency, as illustrated in Fig.9. Furthermore, as shown in both Fig.8 and Fig.9, the Bdeployment scheme achieves a compromise performance between the Ideployment scheme and the Edeployment one. Moreover, both the WIT and WET efficiencies increase, as we increase the number of the HAPs.
In Fig.10 and Fig.11, we investigate the impact of the preference parameter on the performance of the Bdeployment scheme. Observe from Fig.10 that the Ideployment performs best in terms of the information downloading performance. When is lower than , which indicates a relatively relaxed energy constraint, the Bdeployment scheme has the same solution as the Ideployment scheme. By contrast, as continuously grows, the information downloading performance of the Bdeployment reduces until reaches that of the Edeployment. In Fig.11, the opposite trend is observed. As the preference parameter continues to increase beyond , the energy harvesting performance of the Bdeployment scheme increases until it reaches that of the Edeployment.
Vii Conclusion
The HAPs are deployed for the sake of simultaneously satisfying both the information downloading and the energy harvesting requirements of the users. These users move within the grid based city street blocks, whose mobility patterns are analysed by exploiting a twodimensional Markov chain. According to the users’ mobility patterns and the popularity (represented by the crowded range) of the crossroads, the Bdeployment scheme of the HAPs has been proposed, for striking a balance between the WIT and WET efficiencies. Both the theoretical and simulation results demonstrate that deploying the HAPs at the crowded crossroads may substantially improve both the WIT and the WET efficiencies. Additionally, changing the parameter is capable of flexibly adjusting our preference on either the WIT or the WET efficiencies in the Bdeployment scheme.
Appendix A
We will choose the analysis of information range for example, while that of energy range will be the same.
Let as our target crossroad, while the crowded range of other crossroads are fixed. When is increasing, there are two possible cases, namely and . Hence, we will analyse the total sojourn duration of the each user in WIT range of a crossroad for these two cases, respectively.
Cases 1: . With the aid of (2),(4) and (6), we may readily obtain the total sojourn duration of the user within the WIT range of the HAP deployed at the crossroad as
(24) 
Observe from (24) that, is only correlated with at the denominator of the second term in (24). Since is an monotonously increasing function with respect to according to (2), is also a monotonously increasing function with respect to .
By considering another crossroad , the total sojourn duration of the user within the WIT range of the HAP deployed at this crossroad can be expressed as
(25) 
Since the crowded range of the crossroad does not change, is also unchanged. As a result, reduces as we increase , since is a monotonously increasing function with respect to .
Cases 2: . The total sojourn duration of the user within the WIT range of the HAP deployed at the crossroad can be expressed as