Wireless mesh networks (WMNs) are becoming increasingly popular in modern cities, since many of them have emerged to meet the growing demand for broadband wireless services. WMNs provide a promising solution to address the “last-mile” issue of access networks [1]. However, the throughput of WMNs is limited by the disadvantages of radio frequency (RF) technology, including high cochannel interference and bit error rates, cluttered RF spectrum and bandwidth scarcity. One promising solution is to design advanced WMNs using multiple wireless technologies without the above-mentioned disadvantages of RF technology.

Communicating over the optical domain, the so-called free space optical (FSO) channel, with nearly boundless frequency spectrum and high data rates, has been proposed as a viable alternative for high-speed “last-mile” connectivity [2]. However, the benefits of FSO do not come without a price, its link performance is decreased significantly by the random temporal and spatial irradiance fluctuation from atmospheric turbulence. Besides, Line of sight (LOS), which is not a requirement for RF transmissions, is a restriction for FSO transmissions [3].

The hybrid approach of RF and FSO can avoid the weakness of each individual channel type, therefore, the hybrid RF/FSO wireless mesh networks are promising technologies to increase network throughput. There are various WMN architectures [4], we focus on infrastructure WMNs which are illustrated in Fig. 1. As shown in Fig. 1, the infrastructure WMNs are composed of mesh gateway (mesh routers with gateway/bridge function), mesh routers (access points) and mesh clients (mobile or others). The mesh routers which form a backbone for the mesh clients in this infrastructure WMN are stationary. Because FSO is weakly used in mobile environments, but is well adapted to stationary conditions, it can be used to augment the backbone of the infrastructure WMNs and to improve network throughput.

In this paper, assuming that flows, interference model and physical topology of the network are known in the control center, and the optimal configurations then passed along each mesh router can be computed here. We claim that the centralized solution is only deemed more appropriate but not necessarily the best. First we use an additive interference model to find a set of links that can be scheduled at the same time, then a mixed integer linear program (MILP) is introduced to obtain the optimal placement of FSO links. The major contributions of this work are as follows:

a. Transmitted power control of mesh nodes is proposed in our resource assignment optimization scheme. We select the most appropriate transmission power according to the additive interference model, which can maintain network connectivity, minimize interference and lay foundations for maximizing total network throughput. We provide various numerical results to demonstrate that our selection of the transmitted power is the most appropriate to maintain network connectivity, minimize interference and imply more scheduling flexibility.

b. We introduce the fading models for FSO and RF links into the formulation of hybrid RF/FSO wireless mesh networks, and the proposed optimization model includes the impact of the reliability versus link distance and transmitted power. The outage probability of FSO links and the link availability of RF links which is related to the distances of RF and FSO links and the transmitted power of mesh nodes are introduced to model the realistic scenarios. Note that the outage probability and the link availability of diverse links are different from each other. To the best of our knowledge, this is the first work that introduces the ergodic link availability of FSO and RF links based on gamma-gamma atmospheric turbulence channel and the Rayleigh fading model, respectively, into this kind of hybrid RF/FSO optimization. Many scenarios are considered which demonstrate that the method we use can obtain higher optimal throughput.

c. We use MILP solvers (CPLEX and Gurobi) to examine theoretical results and compare their performance of finding optimal solutions and computation times. To the best of our knowledge, this is the first work that computes this realistic model of hybrid FSO/RF optimization using CPLEX and Gurobi solvers, and compares their performance and computation times in this field of study. We show by simulation that CPLEX and Gurobi can optimally solve all small and medium instances and even some large-scale instances in a relatively short period of time, and we also show how to choose the appropriate solvers.

The rest of this paper is organized as follows. In Section II, we discuss related work. Section III provides the design parameters of FSO links. Section IV provides the network models and assumptions, and the problem is presented and formulated as MILP. In Section V, we provide simulations and discussions to show the performance of the proposed formulation and comparison of the optimization solvers. Finally, Section VI presents conclusions and directions for future work.

Adding FSO links to RF WMNs to enhance the performance of the networks is studied in [4-12]. An MILP is introduced in [4] to address link allocation, routing and scheduling problems for hybrid RF/FSO wireless mesh networks. However, it may not be feasible to apply these results to practical WMNs for three reasons: the authors assume network nodes transmit with the biggest power, which may increase interference and minimize the total throughput of the overall network; the link availability of the different links is assumed to be the same and as a fixed value, however, it may be impractical as the link availability is determined by link distance and transmitted power and different from each other; there is something wrong with the simulation results of the independent sets, which is the basis of the overall simulation.

Mohammed N. Smadi and Sasthi C. Ghosh [5] proposed a hybrid RF/FSO mesh networks system which formulated a joint clustering and gateway placement problem to meet the specified capacity. Their approach does not involve power control. In [6], in order to meet target Quality of Service (QoS), end-to-end delay and throughput of the hybrid RF/FSO mesh networks, a scheme based on adaptive adjustments to both transmission power and the optical beam-width at individual nodes is proposed. The overall system is formulated as an integer linear program (ILP). The problem of RF links adaptive scheduling and optimal placement of FSO links is formulated in [7] as an instance of ILP. However, the fading nature of the communication links is not taken into consideration in the above mentioned two papers. In [8], FSO links are added to WMNs as backup links when RF links are deteriorated, the objective is meeting the cumulative RF interference constraints using the smallest number of FSO links. Nevertheless, it does not take account of the throughput optimization problem. In [9], a reconfigurable hybrid RF/FSO system is proposed to enhance the capacity and availability, however, the upper bound derived by the hybrid routing scheme is impractical for practical WMNs. Abhishek Kashyap and Mark Shayman [10] proposed a routing scheme considering the security and higher bandwidth of FSO links and reliability of RF links to enhance the performance of the hybrid RF/FSO networks. In [11], the same authors utilized RF links as the instantaneous backup of the FSO links, and provided topology control algorithm for the backup links in order to maximize the backup performance. Perhaps the most serious disadvantage of these above systems is that each node is equipped with RF and FSO transceivers which will increase the overall cost. To address the power allocation problem of hybrid RF/FSO network, a system is provided in [12] with the goal of maximizing overall throughput. The main shortcoming with this system is that the result is impossible to utilize in realistic WMNs because of the overlook of the fading nature of communication links.

On the algorithmic side, CPLEX has been applied to solve optimization in some papers in the field of this paper, such as [5, 7]. However, these studies neither utilize Gurobi [13, 14] to obtain the optimal solutions nor compare the performance and computing times in the field of this paper. As described later in Section V, the simulation results obtained by CPLEX and Gurobi are presented to show that they can scale well with the problem size. And the performance and computation times of CPLEX and Gurobi are compared to show how to choose the appropriate optimization solvers.

Laser beams are transmitted through atmosphere in FSO systems, FSO links suffer from phase fluctuation and intensity scintillation caused by atmospheric turbulence and misalignment errors, which can severely deteriorate link availability and fade the signal at the receiver. An automatic tracking mechanism is assumed to correct misalignment errors in this work, and based on that, metrics such as link distance, link availability and emissive power can be considered to evaluate the performance of FSO links with atmospheric turbulence.

In FSO systems, transmit electronics supply an electrical current to modulate the instantaneous intensity of a laser diode, the optical beam propagates through an atmospheric turbulence channel. At the receiver, a photodiode detects the optical intensity which is converted into a proportional electrical current. We assume additive white Gaussian noise (AWGN) for the shot noise and the channel is stationary, memoryless and ergodic, with identically and independently distributed intensity fading statistics. We also consider that the channel state information (CSI) is available at both the transmitter and the receiver. The statistical channel model is given by [15, 16]

where y is the electrical signal at the receiver, s^FSO = η^FSO I^FSO is the instantaneous intensity gain, η^FSO is the effective photocurrent conversion ratio of the receiver, I^FSO is the normalized irradiance, x is the modulated signal and n is zero-mean AWGN with variance . Based on above, the atmospheric channel model is defined by [17]

where P_r(t) is the average power of the receiver, P_s(t) is the emissive power, X(t) is atmospheric channel parameters and n(t ) is the average power of the noise.

The atmospheric turbulence-induced fading channels for both the weak-to-moderate and moderate-to-strong turbulence FSO channels are suitable to model by the gamma-gamma distribution [18, 19]. Therefore, in this work, we consider an FSO communication system with circular polarization shift keying (CPoLSK) modulation based on gamma-gamma atmospheric turbulence channel. The probability density function (PDF) for the effect of atmospheric turbulence is [18]

where K_v(⋅) is the modified Bessel function of the second kind of order v, and Γ(⋅) is the gamma function. Positive parameters α and β can be directly related to the large scale and small scale intensity scintillations of an optical wave through the expressions below [17]

where is the unitless Rytov variance used to classify intensity scintillation strength, is the refractive-index structure parameter. k^FSO = 2π / λ^FSO is optical wave number, λ^FSO is the optical wavelength and d^FSO is the link distance. In general, varies from 10^-17m^-2/3 to 10^-13m^-2/3 for weak up to strong turbulence, respectively, with a defined typical average value of 10^-15m^-2/3. By integrating Eq. (3), the cumulative distribution function (CDF) of gamma-gamma distribution can be derived as [20]

where is Meijer G-function [18].

Perturbation approximation theory [21] is a widely accepted theory of phase fluctuation based on Rytov theory derived by Tatarskii. The phase fluctuation distribution is described as a Gaussian distribution according to this theory [22], and the phase noise caused by turbulent atmosphere is random. The distribution of phase noise is given by [17]

where δ² is the variance of the phase noise.

In this work, the signal-to-noise ratio (SNR) γ for an analog channel is γ = P_r(t) / n(t) and we assume P_s(t) and X (t) are irrelevant, the analytic expression is thus given as γ = X(t)(P_s(t)/n(t)) + 1, where X(t) can be derived from Eqs. (1) and (6). Our expression for γ can thus be given as , where R is the responsivity of the photodetector, B^FSO is the bandwidth of the photodetector, is the side power spectral density of n_th(t), and n_th(t) is the detector noise that satisfies a Gaussian process.

We consider the outage probability as an influential metric for the system design, as it represents the probability that the system instantaneous SNR falls below a critical threshold of a predetermined target SNR. The SNR threshold μ will have an impact on the outage probability, thus, the link outage probability of FSO links for weak-to-moderate turbulence strength is given by [17]

We define the normalized threshold ν as and take advantage of the properties of the Meijer G-function , we can write the outage probability in gamma-gamma atmospheric turbulence as [17]

We define the link availability of FSO links as a probability that an FSO link transmits successfully at a SNR which is greater than the predetermined target SNR, and it is expressed as . Therefore, the link availability can be easily derived as

that is,

Adjusting the transmission power of nodes, the total number and placement of FSO links has both benefits and disadvantages for hybrid RF/FSO wireless mesh networks. A node with higher transmission power has more nodes in the transmission range (i.e. higher node degree), which can reduce the average number of hops and thus maximize the throughput of the network. However, higher node degree can lead to more channel contentions and an increase of interference, thus minimizing the network throughput.

Adding FSO links to the backbone of WMNs can improve network throughput, however, the benefits do not come without a high price of the FSO links [23, 24]. Meanwhile, the placement of FSO links has an impact on the network throughput. According to the analyses above, there is a trade-off between the transmission power, the number and the placements of FSO links and throughput improvement. Therefore, our objective is to maximize overall network throughput by selecting the appropriate transmission power of mesh nodes and the total number of FSO links, and to determine the optimal placements of FSO links.

In our system model, the hybrid RF/FSO wireless mesh networks are modeled as a directed connectivity graph G = (N, L) , where N is the mesh router nodes set and L is the feasible links set in the network. We maintain the location of each node, it means that the physical topology of the network is given. Let l_i,j ∈ L (i, j ∈ N) represent the directed link from node i to node j. We consider L^RF and L^FSO as the feasible link sets of RF sub-network and FSO sub-network, so L = L^RF U L^FSO. We adopt the protocol and physical model for successful reception of a transmission over one hop, i.e., the link is feasible if node i can reach node j within one hop.

In the following subsections, we present the protocol and physical model for the network and we restrict ourselves to conflict-free scheduling in all cases at the same time. The joint transmission slot assignment, FSO links allocation, power control and properly number and allocation of FSO links choosing problems are formulated as an instance of MILP, whose solution can be used to optimize resource assignment.

The protocol model and physical model for an RF sub-network is discussed below. The origin node and the destination node of the RF link is denoted as o(l ) and d(l ), sometimes let i = o(l ) and j = d(l ). These models are similar to those introduced in [25].

a. Protocol Model: In the protocol model, each RF node i has a communication range r_i and an interference range , generally . A transmission is successful if the following conditions are satisfied:

dij ≤ ri. and for all nodes h that transmit at the same time.

where d_ij is the distance between node i and node j. Note that the condition implies that an RF node may not receive and transmit data at the same time, and it requires that both the receiver and sender are free of interference. The capacity of the link, r_i and are determined by the power P_i of transmit node i , hence, the transmitter i, the receiver j and the power P_i of transmit node i determine each RF link . Based on this, we can note that the links with the same sender i and receiver j but different transmitted power P_i are regarded as the diverse links, that is, the RF link is identified as a logical link rather than a physical link.

b. Physical Model: Each RF link is identified by:

o(l), d(l) the origin node and the destination node, which is denoted by i , j. the transmitted power used by , it is determined by an independent set (ISet) or reliable set (RSet) which is discussed below. the link rate of the RF link in bits per second. We assume that a link characterized by (o(l), d(l), , ) is transmitted successfully if its SNR meets the following condition [4]:

where G_ij is the channel gain of RF link , N₀ is the receiver thermal noise power spectral density in the operating frequency band, assumed to be fixed and the same for all nodes. is the threshold of SNR corresponding to the transmission rate . The channel gain between the node i and node j separated by the distance d_ij is assumed to be given by G_ij = ( d_ij / d₀)^-ηRF where d₀ is the close-in reference distance and η^RF is the path loss exponent. Note that there are potentially multiple links between two nodes and that differ from each other by power or link rate , so it is more a logical entity rather than a physical link. However, this is only a realistic assumption in urban or suburban areas with rooftop antennas [1], because there is an implicit assumption that the channel gain is quasi-static.

The additive interference model used for the protocol and physical model is described by the concept of an ISet [1, 25], which is a set of links which can all transmit simultaneously without interference.

To describe the ISet of protocol model, a conflict graph is defined [25], E, whose nodes correspond to the links in the connectivity graph G. If the links and cannot operate at the same time, that is, if they interfere with each other, there is an edge between the nodes and in E. An ISet is a set of nodes in E in which no edge exists between any two nodes. All ISets include all feasible FSO links, because FSO links do not interfere with RF links. Based on the protocol model described above, if the link and the link are in the same ISet, the following conditions are satisfied: , , , and .

We assume that the interference on a given link is the cumulative interference from all the links that are active at the same time. Hence, the links in the same ISet for the non-fading physical model must satisfy the following conditions:

A set is an ISet only if no two links in this set share a node, that is, if the link and the link are in the same ISet, i ≠ p ∧i ≠ q ∧ j ≠ p ∧ j ≠ q.

where is the signal-to-interference-plus-noise ratio (SINR) of . For the fading physical model, SINR is defined as

where is the time-variable channel fade between nodes i and j, which is a random variable. According to [4], we assume that RF link availability is , which is defined as the probability that an RF link transmits successfully in the presence of interference. Based on the above assumption, for a Rayleigh fading model,

Therefore, the additive interference model of the fading physical model which is denoted as RSet [4] differs from that of the non-fading physical model, the only modification is to use the following criterion instead of Eq. (12)

where ρ is a given requirement.

As shown in Fig. 1, traffic demands are generally fixed to be between gateway MRs in the backbone of WMNs, Other MRs in the network only act as routers. Each traffic demand which we denote as F^(b) is identified by its source and destination nodes (s, d), where b =1,2,..., B. If we denote the flow of F^(b) on link or link by , the throughput of F^(b) is . We define the throughput of the entire network as the sum flow for all demands served by the RF and FSO sub-networks. Based on the above definitions and notation, we model our optimization problem as an instance of MILP as follows, which extends from [4]. The input of the MILP includes the RSets, the physical topology of entire network and the power of RF and FSO links. For the physical model, the power is determined by RSet, as discussed in Section V.

The objective can then be formally expressed as:

The objective function (16) is attained subject to certain constraints, which are divided into several classes:

a. Flow Constraints

To ensure that the (s, d) connection pair is routed correctly, based on the flow conservation law, we require, all j ≠ s or d , ∀b:

and ∀b, i, j :

b. Scheduling Constraints

Our scheduling scheme is a time-division conflict-free scheduling [25]. All RSets include all feasible FSO links, for FSO links do not interfere with RF links. We schedule the RSets by breaking up one unit of time into fractions, which are denoted by λ_k, k =1,2,...K, and we schedule one RSet in each time fraction. Therefore, the network route and schedule is determined by λ_k. Let RSets be represented by I = {I₁, I₂,..., I_k}, to ensure proper scheduling, we require ∀k:

c. FSO Constraints

Supposing M FSO links are to be installed,

where is an indicator of an FSO link from node i to node j, and if there is a directed FSO link from node i to node j, , otherwise .

d. Capacity Constraints

An RSet can only be active for a time fraction, so the sum flow on each link is limited. Different RF and FSO links have different link availability, we require that ∀i, j :

where is the link capacity of FSO links.

In this section we provide some experimental results based on the RSet, channel fading model of FSO links, MILP formulation, CPLEX and Gurobi. We compare our simulation results with [4] to measure the merits of our scheme.

a. The first experiment investigates how to choose the appropriate transmitted power of the mesh nodes in our scheme. The parameters used in this paper are shown in Table 1.

The feasible link is determined by Eq. (11). Table 2 provides the numbers of ISets for the protocol model of quasi-static network with rectangular 4 × 4 node grid network which is illustrated in Fig. 2. As shown in Fig. 2, the distances of horizontal and vertical neighbor nodes represented as d_node are 2 km.

Comparing the numbers of ISets of the same network with [4], it is concluded that the results in [4] are not accurate. For instance, the nodes in the interference range of the two nodes of each link are identical for and , therefore, the links which can communicate at the same time are identical, i.e., the ISets are the same. However, the numbers of ISets of [4] for and are not the same, and the number of ISets for the physical model are not quite accurate as well.

The number of RSets versus the transmitted power of mesh nodes is presented in Table 3. To clearly visualize our experimental results, a randomly generated fading network topology with 15 nodes, which is illustrated in Fig. 3, is considered. As shown in Fig. 3, 2 nodes are distributed at the top right corner and the bottom left corner in a square area of 10 × 10 km, the other 13 nodes are randomly distributed within the square. We assume that there is one traffic demand diagonally across the network from bottom left to top right.

Table 3 shows that the network throughput is determined by the number of the RSets with 2 links, that is, scheduling flexibility is related to the number of the RSets with 2 links and throughput increases with it as well. Thus, we draw a more accurate conclusion than [4] that network throughput is only related to the proportion of the number of the RSets with multiple links rather than the total number of RSets. Based on this, we choose appropriate transmitted power which can generate a maximum number of RSets with 2 links and throughput.

b. The second experiment investigates the fading nature of the FSO links. Figure 4 describes link availability versus link distance and the transmitted power of mesh node based on gamma-gamma atmospheric turbulence channel.

The transmitted power of the FSO transmitters is also chosen in this experiment. As Fig. 4 illustrated, for higher transmitted power, the link availability rises rapidly, however, there is a transmitted power for which increasing the transmitted power cannot improve the link availability. Thus, the assumption made in [4] that each FSO transmitter is operated at its maximum power is not necessary.

c. Next, the relationship between network throughput and the number of FSO links in our system is investigated. Figure 5 shows the optimal throughput obtained by solving the MILP using Gurobi with Python interface for various numbers of FSO links, and compares the optimal throughput of our system with the system in [4]. The same rectangular 4 × 4 node grid network topology as shown in Fig. 2 is considered in this experiment, and we assume that there are two traffic demands, top right to bottom left and top left to bottom right. Optimal throughput of our system and [4] are obtained for different ρ when . From Fig. 5 we can see that the optimal throughput of our system is much better than [4], and our MILP solver can obtain an exact solution for larger M than [4]. In our system, the optimal throughput of the adjacent value of M is approximate, therefore, network administrators can choose the appropriate M to achieve the desired optimal throughput and do not waste FSO links. Figure 5 also shows, the throughput improvement at the cost of adding FSO link.

Then we investigate the relationship between the network throughput and the number of FSO links for the randomly generated network topology. Figure 6 provides the average throughput for 100 randomly generated network topologies with 15 nodes as shown in Fig. 3. The source and destination are fixed and the traffic demand is assumed diagonally across the network from bottom left to top right. We draw a conclusion from Fig. 6 that there is an M for which increasing the number of FSO links cannot improve the optimal throughput of the networks.

d. Next, how to choose the appropriate MILP solver by comparing and evaluating the quality of MILP solver is discussed. We consider the randomly generated network topology with 15 nodes as shown in Fig. 3, and we also assume 1 traffic demand diagonally across the network.

We compare the computing time of CPLEX and Gurobi, which is shown in Table 4. The computing time of CPLEX is much shorter than Gurobi. Therefore, CPLEX can be selected if the overall time is specifically required. However, Gurobi with Python interface is convenient to solve the network flow problems and find the optimal solutions from medium to large-scale networks. The relationship between the optimal network throughput and the number of FSO links for the randomly generated network topology with 60 nodes is shown in Fig. 7, the network map of the randomly generated network topology with 60 nodes is illustrated in Fig. 8. As shown in Fig. 8, 12 nodes are distributed at the edges and vertices of a square area of 15km×15km and the other nodes are randomly distributed within the square. We assume that there are 24 traffic demands between the source-destination pairs consisting of these 12 nodes with uniform probability. As Fig. 7 illustrates, the performance of our modeling and MILP solver is much better than [4]. Because the exact solution of the topology for more than 16 nodes with 12 traffic demands cannot be computed in [4].

In this paper, we address the joint transmission slot assignment, FSO links allocation and power control problem for hybrid RF/FSO wireless mesh networks. Considering the fading nature of realistic hybrid RF/FSO networks, we provide a mathematical formulation to maximize overall throughput. The formulation transforms the joint optimizing problem for transmission slot assignment, FSO links allocation and power control into an instance of MILP. Moreover, we examine the theoretical results by CPLEX and Gurobi and show how to choose the appropriate solver by comparing their performances. Numerical results are provided to show the advantages of our system.

Multi-channel allocation has not been considered in the system of this paper. In future work, we will extend our model to include the channel allocation problem for hybrid RF/FSO networks and develop a new approach based on an intelligent bionic algorithm for larger scale problems.