## 3 Analysis Method of iNterference on DME
### Calculation for P(I)
#### 3.2.2 Calculation for \(P(M|j)\)  
When there are \(n\) aircraft installed DME, 95% of the aircraft are operating in track mode and the others in search mode [10]. So the number of DME in track mode is \(L_{t}=\lfloor 95\%\cdot n\rfloor\), the number of DME in search mode is \(L_{s}=\lfloor 5\%\cdot n\rfloor\), where \(\lfloor x\rfloor\) is the floor function operated on \(x\). So interrogation pulse density \(\lambda_{i}\) can be calculated as follows:  
\[\lambda_{i}=L_{t}\cdot f_{t}+L_{s}\cdot f_{s} \tag{5}\]  
where \(f_{s}\) is PRF of interrogation operating in search mode and \(f_{t}\) is PRF of interrogation operating in track mode. * When \(n\) is small: Since PRF and duty cycle of DME interrogation are all small, the collision probability of interrogations is small accordingly. It can be assumed that dead time is a part of pulse duration, thus, the desired pulse duration equals \(\tau_{m}\) plus \(t_{d}\). All the interrogations are assumed to be independently at the same time, according to [10], the probability of DME interrogations not interfered \(P_{s}\) in search mode is given as follows: \[P_{s}=[1-f_{s}\cdot(2\tau_{m\_i}+t_{d})]^{L_{t}-1}\cdot[1-f_{t}\cdot(2\tau_{m \_i}+t_{d})]^{L_{t}}\] (6) Similarly, the probability of DME interrogations not interfered \(P_{t}\) in track mode is given as follows: \[P_{t}=[1-f_{t}\cdot(2\tau_{m\_i}+t_{d})]^{L_{t}-1}\cdot[1-f_{s}\cdot(2\tau_{m \_i}+t_{d})]^{L_{s}}\] (7) Combining Equations (5)-(7), the average probability of multi-path DME interrogations not interfered \(P(M)\) can be obtained as follows: \[P\left(M\right)=\frac{P_{t}\cdot L_{t}\cdot f_{t}+P_{s}\cdot L_{s}\cdot f_{s} }{\lambda_{i}}\] (8) Event \(M\) and event \(J\) are assumed to be statistically independent, that is to say, inter-system and intra-system interference are independent of each other. Substituting Equations (4) and (8) into (3), we obtain \[P\left(I\right)=P\left(J\right)\cdot P\left(M\left|J\right.\right)=P\left(J \right)\cdot P\left(M\right)=\left(1-\frac{3}{85}\cdot\frac{\tau_{j}+\tau_{m\_ i}}{T_{j}}\right)\cdot[1- \lambda_{i}\cdot(2\tau_{m\_i}+t_{d})]^{n-1}\] (9) The collision probability between two or more DME interrogations will increase when \(n\) is large. The result based on Equation (7) is smaller than actual value. Moreover, intra-system interference is correlative with inter-system interference, the error using Equation (9) is large if \(n\) is large. Equation (9) can be apply to a small quantity of aircraft only. * When \(n\) is large: We can adopt simplifying assumption that these interrogations are randomly distributed with respect to time forming a Poisson process [5], considering that interrogations will be abandoned if its arrival time overlaps with dead time generated by front interrogation, and hence the receiving process of DME interrogations can be regarded as a quasi M/D/1/0 theory model. In such a birth-and-death model M/D/1/0, the interrogation pulses are customers, transponder is service facility, \"M\" represents that the arrivals occur from an infinite source in accordance with a Poisson process with parameter \(\lambda_{2t}\) defined in Figure 4--that is, the inter-arrival times are independent exponential with mean \(1/\lambda_{2t}\), \"D\" represents service times that are deterministic and equivalent to dead time \(t_{d}\) plus \(\tau_{m\_i}\), \"1\" represents single server,and \"0\" means that customer will depart when its arrival time overlaps with service time [16]. Service rate \(\mu\) is given as follows:  
\[\mu=\frac{1}{\tau_{m\_i}+t_{d}} \tag{10}\]  
According to [11], the probability of customers serviced \(Q\) can be calculated as follows:  
\[Q=\frac{\mu}{\lambda_{i2t}+\mu}=\frac{1}{\lambda_{i2t}\cdot(\tau_{m\_i}+t_{d}) +1} \tag{11}\]  
where the density of interrogation pulse-stream \(\lambda_{i2t}=P\left(J\right)\cdot\lambda_{i}\) is subject to \(P\left(J\right)\), \(Q\) is condition probability.