**4. Feasibility study of physical, datalink, and network layers**

In order to confirm the feasibility of the protocol discussed in the prior section, the authors developed the model described below and conducted a simulation experiment.

Assume that *N* aircrafts *ai* are existing at positions *pi* , respectively. Position *pi* may or may not be a three-dimensional orthonormal coordinate system. The primary point in this discussion is that the distance *pi* − *pj* ∨ between position *pi* and position *pj* is defined.

Let *xij* be the information the authors want to send to aircraft *aj* from aircraft *ai* . The information received by aircraft *ai* is, nonetheless, different from *xij*—the authors denote it as *yij*.

When there's no error in the communication route, the information *xij* originating from aircraft *ai* corresponds to *yj* received by aircraft *aj* . This can be described as

$$\mathcal{Y}\_{\vec{j}} = \mathsf{U}\mathcal{H}\mathcal{X}\_{\vec{\eta}} \tag{1}$$

Let us denote the transmission rate of the communication route from aircraft *ai* to aircraft *aj* by *cij*. The transmission rate *cij* refers to the packet data transmitted by aircraft *ai* that is actually received by aircraft *aj* , that is, the probability of correct information transmission. Let *cij* be a function of position *pi* and position *pj* . That is, *cij* = *e*(*pi*, *pj*). In wireless communication, we can reasonably assume that

$$\mathcal{L}\_{ij} = \mathcal{k} \,\mathrm{e}^{-K[p\_i - p\_j]^2} \tag{2}$$

for some constants *k*, *K*. These parameters may be changed according to the results of experiments. In this report, the authors adopt Eq. (2) as the transmission rate *cij*.

Taking the transmission rate into account, Eq. (1) is modified to

$$\mathbf{y}\_{\mathbf{j}} = \mathbf{U} \mathbf{i} \,\mathrm{c}\_{\mathbf{i}\mathbf{j}} \,\mathrm{x}\_{\mathbf{i}\mathbf{j}} \tag{3}$$

Until now, the authors have dealt with only one-to-one communication from aircraft *ai* to *aj* , but we can consider another aircraft *ak* relaying the communication described by *xij*. When the information that aircraft *ak* receives and retransmits is *zk*, where *zk* = *ijcikxij*, there would be an explosive increase in the amount of data if the authors do not use an artificial attenuation (decay) term *dijk*, as in

$$
\varpi\_k = \textit{U} \textit{U} \textit{j} \ c\_{ik} d\_{ij} k \,\varkappa\_{ij}. \tag{4}
$$

Here, the attenuation term *dijk* denotes the probability of intentionally discarding the packet during the packet relay. Finally, the equation that takes the relay into consideration has the form

$$\mathcal{Y}\_{\vec{j}} = \mathsf{U}\mathring{\mathsf{U}}\mathcal{L}\_{\vec{\eta}\overleftarrow{\eta}}(\mathfrak{X}\_{\vec{\eta}} \cup \mathfrak{x}\_{\vec{\eta}\overleftarrow{\eta}}).\tag{5}$$

In this report, to statistically investigate the arrival rate of data based on Eqs. (4) and (5), the authors performed computer simulation with the following parameters:

**Airspace.** Three-dimensional orthonormal space. It is a cube whose vertexes are (0, 0, 0)-(0, 0, 1)-(0, 1, 0)-(0, 1, 1)-(1, 0, 0)-(1, 0, 1)-(1, 1, 0)-(1, 1, 1). The units are arbitrary.

**Time.** The simulation is performed between 0 and 100 s with a time step of 1 s.

**Number of aircraft.** 1000 aircraft are randomly placed in the airspace.

**Probability of successful communication.** The probability of the successful communication is based on Eq. (2) according to the interval between the aircrafts. However, the authors set *k* = 1 and simulate *K* as *K* ∈ {0.1,0.316,1}. **Figure 1** shows the probability of success depending on the interval for each parameter.

For every simulation, the authors transmitted data from aircraft *a*999 to *a*0 in each time step. If the data did not reach *a*0 in a single time step, and if they reached *an* where *n* ≠ 0, then *an* attempted to send the data to *a*0 in the next time step.

Theoretically, the attenuation term *dijk* in Eq. (4) must be equal to or less than 1, and the authors assumed that *dijk* = 1 in the simulations to clarify the experimental results.

The authors have done the following two investigations.

**Simulation 1.** Investigate the degree of data transmission per time step using multi-hop communication via neighboring aircraft wherever the communication path is unstable.

**Simulation 2.** Investigate the entire range of hops for every time step using multi-hop communication via neighboring aircraft wherever the communication path is unstable.
