2.1 Fujii's method

Fujii et al. (Reference Fujii, Yamanouchi and Mizuki1974) developed a method to estimate the number of Ship_2s in the area where collisions with a Ship_1 occur. This is the hatched area in Figure 1. The shape of each ship is approximated as an ellipse. The important value is the collision diameter (W) in Figure 1. Ship_2s that collide with a Ship_1 have a relative velocity $\overrightarrow {{V_2}} - \overrightarrow {{V_1}}$. The number of Ship_2s that collide with a Ship_1 in a unit time can be obtained by multiplying the hatched area of Figure 1 by the density of Ship_2s. The value of the shaded area can be obtained by adding $W \cdot |{\overrightarrow {{V_2}} - \overrightarrow {{V_1}} } |$ to the area made by sliding Ship_2 around Ship_1.

Figure 1. Fujii's method for counting the number of ships potentially colliding with Ship_1

This method is regarded as the basis for the geometric estimation of ${N_{a }}$. However, Fujii did not present an analytical form of W. Further, he did not include the number of Ship_2s in the area created by graphically sliding Ship_2 around Ship_1. This area is called CP, which is mentioned in the nomenclature and is explained in Section 3.

2.2 Pedersen's method

Based on Fujii's method, Pedersen (Reference Pedersen1995, Reference Pedersen2010) developed the analytical solution (Equation (2.1)) for ${D_{\text{COL}}}$ using a rectangle to represent the shape of a ship. This was the first occurrence of rectangle approximation.

(2.1)\begin{equation}{D_{\text{COL}}} = \frac{{{L_1}{V_2} + {L_2}{V_1}}}{{|{\overrightarrow {{V_2}} - \overrightarrow {{V_1}} } |}}\sin (\theta )+ {B_1}\sqrt {1 - {{\left( {\frac{{{V_2}}}{{|{\overrightarrow {{V_2}} - \overrightarrow {{V_1}} } |}}\sin (\theta )} \right)}^2}} + {B_2}\sqrt {1 - {{\left( {\frac{{{V_1}}}{{|{\overrightarrow {{V_2}} - \overrightarrow {{V_1}} } |}}\sin (\theta )} \right)}^2}} \end{equation}

Pedersen's method obtained the number of collision candidates $({N_{CC}})$ in the intersection area of two waterways. This is schematised as a risk area in Figure 2, using the collision diameter. Equation (2.2) is supplemented with $1/\sin (\theta )$, which was not included in the original formulation. This sinusoidal form first appeared in Friis-Hansen (Reference Friis-Hansen2008). The variables in Equation (2.2) include suffixes i and j. These suffixes denote that there are groups of ships with the same characteristics, such as velocity and ship size. The suffixes i and j also denote the i-th and j-th groups of Ship_1s and Ship_2s, respectively.

(2.2)\begin{equation}{N_{CC}} = \mathop \sum \limits_i \mathop \sum \limits_j \int_{z_{i - \text{min}}^{(1 )}}^{z_{i - \text{max}}^{(1 )}} {\int_{z_{j - \text{min}}^{(2 )}}^{z_{j - \text{max}}^{(2 )}} {\frac{{{Q_{1i}}{Q_{2j}}}}{{\sin (\theta ){V_{1i}}{V_{2j}}}}f_i^{(1 )}f_j^{(2 )}{D_{\text{COL} - ij}}{V_{ij}}d{z_j}d{z_i}\Delta t} }\end{equation}

Figure 2. Definition of the intersection area of two waterways (Pedersen, Reference Pedersen1995)

Some variables in Equation (2.2), which are not included in the nomenclature of this study, have the following meanings:

• ${N_{CC}}$: Number of collision candidates in the intersectional area of waterways 1 and 2.

• V _1i: Velocity of ships in the i-th group of Ship_1.

• V _2j: Velocity of ships in j-th group of Ship_2.

• Q _1i: Mean appearance ratio of ships in the i-th group of Ship_1.

• Q _2j: Mean appearance ratio of ships in the j-th group of Ship_2.

• ${D_{\text{COL} - ij}}$: Collision diameter between a ship in the i-th group of Ship_1 and a ship in the j-th group of Ship_2.

• V_ij: Relative velocity of a ship in the j-th group of Ship_2 as seen from a Ship_1 in group i.

• $z_{i - \text{max}}^{(1 )}$: The right-most appearance position of Ship_1 in the i-th group in waterway 1.

• $z_{i - \text{min}}^{(1 )}$: The left-most appearance position of Ship_1 in the i-th group in waterway 1.

• $z_{j - \text{max}}^{(2 )}$: The right-most appearance position of Ship_2 in the j-th group in waterway 2.

• $z_{j - \text{min}}^{(2 )}$: The left-most appearance position of Ship_2 in the j-th group at the end of waterway 2.

• $f_i^{(1 )}$: Probabilistic distribution function of the appearance position of Ship_1 in group i in waterway 1.

• $f_j^{(2 )}$: Probabilistic distribution function of the appearance position of Ship_2 in group j in waterway 2.

• Δt: Considered time.

It is assumed that Δt is sufficiently longer than the time it takes for ships in waterways 1 and 2 to pass through the intersection. When the intersection angles are 0° and 180°, Equation (2.2) diverges, and therefore, different treatments are required. In these cases, ${N_{CC}}$ can be obtained using Equation (2.3):

(2.3)\begin{equation}{N_{CC}} = \mathop \sum \limits_i \mathop \sum \limits_j \frac{{\Delta t{D_W}{Q_{1i}}{Q_{2j}}|{{V_{1i}} \pm {V_{2j}}} |}}{{{V_{1i}}{V_{2j}}}}\int_{{z_{i - \text{min}}}}^{z_{j - \text{max}}^{(2 )}} {\int_{z_i^{(1 )} - ((B_i^{(1 )} + B_j^{(2 )})/2)}^{z_i^{(1 )} + ((B_i^{(1 )} + B_j^{(2 )})/2)} {f_i^{(1 )}f_j^{(2 )}dz_i^{(1 )}dz_j^{(2 )}} } \end{equation}

where

• $B_i^{(1 )}$: Breadth of Ship_1 in group i.

• $B_j^{(2 )}$: Breadth of Ship_2 in group j.

• $z_i^{(1 )}$: Lateral position of Ship_1 in group i.

• $z_j^{(2 )}$: Lateral position of Ship_2 in group j.

If $f_i^{(1 )}$ and $f_j^{(2 )}$ are assumed to be probabilistic distribution functions of normal distribution, then Equation (2.3) can be approximated by Equation (2.4).

(2.4)\begin{align}{N_{CC}} & = \frac{{{D_W}\mathrm{\Delta }t}}{{\sqrt {2\pi } }}\mathop \sum \limits_i \mathop \sum \limits_j \frac{{{Q_{1i}}{Q_{2j}}|{V_i^{(1 )} \pm V_j^{(2 )}} |}}{{V_i^{(1 )}V_j^{(2 )}}}({B_i^{(1 )} + B_j^{(2 )}} )\notag\\ & \quad \times \frac{1}{{\sqrt {{{({\sigma_i^{(1 )}} )}^2} + {{({\sigma_j^{(2 )}} )}^2}} }}\exp\left( { - \frac{{{{({\mu_i^{(1 )} - \mu_j^{(2 )}} )}^2}}}{{2({{{({\sigma_i^{(1 )}} )}^2} + {{({\sigma_j^{(2 )}} )}^2}} )}}} \right)\end{align}

where

• $\mu _i^{(1 )}$: Mean value of the horizontal coordinate of the appearance position of a ship in the i-th group of Ship_1s.

• $\mu _j^{(2 )}$: Mean value of the horizontal coordinate of the appearance position of a ship in the j-th group of Ship_2s.

• $\sigma _i^{(1 )}$: Standard deviation of the horizontal coordinate of the appearance position of a ship in the i-th group of Ship_1s.

• $\sigma _j^{(2 )}$: Standard deviation of the horizontal coordinate of the appearance position of a ship in the j-th group of Ship_2s.

  \[|{V_i^{(1 )} \pm V_j^{(2 )}} |= \left\{ {\begin{array}{*{20}{c}} {V_i^{(1 )} + V_j^{(2 )};\theta = 180\;\text{deg}}\\ {|{\; V_i^{(1 )} - V_j^{(2 )}} |;\theta = 0\; \;\text{deg}} \end{array}} \right.\]

2.3 COWI's method

Both Fujii and Pedersen represented the collision of Ship_2s with a Ship_1 in a coordinate system fixed to the Ship_1 and modelled the collision, whereas the method devised by COWI engineering consultants (COWI, 2008) used a space fixed coordinate system. As shown in Figure 3, ships in the diagonal narrow waterway, whose breadth is the same as the breadth of a Ship_2, will collide with Ship_1s that travel from right to left. Here, the intersection angle between waterways 1 and 2 is expressed as $\; \theta$. The diagram on the left-hand side of Figure 3 is the earliest situation for a Ship_1 to collide with a Ship_2, and the diagram on the right-hand side is the latest. Assuming that the time difference between the two situations is Δt and the mean appearance ratio of Ship_2 is ${Q_2}$, then the number of Ship_2 s that collide with Ship_1 can be obtained by Equation (2.5). For the same reason that Pedersen's formulation cannot be used when $\theta$ is 0° or 180°, Equation (2.5) cannot be used.

Figure 3. Representation made by COWI (2008) of collision between a Ship_1 and Ship_2 using a space fixed coordinate system

The COWI's formulation appears to be different from Pedersen's formulation. However, these two formulations produced the same results. Therefore, in Section 6, where the comparison with the other methods is shown, only Pedersen's method is used.

(2.5)\begin{equation}{N_{CC}} = {Q_2}\Delta t = {Q_2}\frac{1}{{{V_1}{V_2}}}\left( {{B_2}\left|{\frac{{{V_2}}}{{\sin (\theta )}} - \frac{{{V_1}}}{{\sin (\theta )}}} \right|+ {B_1}\left|{\frac{{{V_1}}}{{\sin (\theta )}} - \frac{{{V_2}}}{{\sin (\theta )}}} \right|+ {L_1}{V_2} + {L_2}{V_1}} \right)\end{equation}

2.4 Kaneko's methods

2.4.1 First method

Kaneko (Reference Kaneko2002) developed a method for calculating ${N_{CC}}$ when a Ship_2 appeared on the border of a circular area representing the Poisson distribution and travelled inside the area at optional progress direction and a same velocity while a Ship_1 travelled toward the centre of the area (Figure 4). ${N_{CC}}$ for Ship_1 is obtained using Equation (2.6).

(2.6)\begin{equation}{N_{CC}} = \frac{{4\rho VrT}}{\pi }({1 + \alpha } )E\left( {\frac{{2\sqrt \alpha }}{{1 + \alpha }}} \right)\end{equation}

where

• V: Velocity of Ship_2 which appears on the border of the circular area.

• r: Radius of a circle which approximates the shape of Ship_1 and of Ship_2.

• ρ: Density of Ship_2s.

• α: Ratio of V to the velocity of Ship_1.

• T: Considered time.

• E(x): Complete elliptic integral of the second kind of x.

Figure 4. The hatched area is where Ship_2 in the hatched area will collide with a Ship_1 at the center of the circular area (Kaneko, Reference Kaneko2002)

Kaneko (Reference Kaneko2002) also developed a method for the case of a rectangular area.

3.1 Case 1: ${D_{\textit{WL}}}\tan ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )< {D_{\rm{COL}}}$

Figure 9 shows three regions made by dividing CA by the longest CS, which passes through one of vertices and the shorter side of CA, and has a length of ${D_{\text{COL}}}$. This case is likely to occur when the intersection angle is small, such that $\theta$ is close or equal to 0° (head-on) or 180° (same direction). It also tends to occur when the velocity of Ship_1 is close to zero. Friis-Hansen (Reference Friis-Hansen2008) suggests that Pedersen's equation should be used only in the range of intersection angles between 10° and 170°. In this case, the value of Pedersen's equation becomes very large. In Pedersen (Reference Pedersen1995), ${N_{CC}}$ is calculated using Equation (2.4) when the intersection angles are 0° and 180°. Until now, intersection angles outside the range of 10° to 170° have not been considered, except in the case of 0° and 180°, but the solution was obtained by the method described by Kaneko (Reference Kaneko2013). In this case, regions I and III in Figure 9 cannot be ignored. When ${\theta _{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }} = 0$, $\cos ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )= 1$ not zero. In addition, if $\tan ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )< {D_{\text{COL}}}$, then ${\theta _{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}$ is not 90°. Therefore, ${N_{CC}}$ can be determined using the method introduced in Section 4.

3.2 Case 2: ${D_{\textit{WL}}}\tan ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )\ge {D_{\rm{COL}}}$

Figure 10 shows three regions made by dividing CA with the longest CS, which passes through one of the vertices and the longer side of CA, and has a length of D_WL. In this case, $\tan ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )$ does not become zero, that is, $\sin ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )$ does not become 0, and ${\theta _{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }} \le \theta$, therefore $\sin (\theta )$ does not become 0, and Pedersen's method can be used in this case. If the length of the longer side of the CA is longer than the width of waterway 2, that is, if Ship_1 completely passes through the intersection area, the integration of CS is necessary only in region II (Figure 10). Thus far, discussions on how to obtain ${N_{CC}}$ have mainly focused on region II. However, if the width of waterway 2 is wide or Ship_1 stops in the intersection area, it is necessary to consider regions I and III.

4.1 Case 1

Based on Figure 11, the estimation of ${N_{CC}}$ in this case is calculated as follows. Assuming the intersection of the line segments E ₁F ₃ and E ₁E ₄ of the CA is Q, the length ${L_{\text{Q}{\text{E}_\text{3}}}}$of the line segment QE₃ can be obtained by Equation (4.1).

(4.1)\begin{equation}{L_{\text{Q}{\text{E}_\text{3}}}} = \frac{{{D_{\textit{WL}}}}}{{\cos ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} ) }}\end{equation}

In case V ₂ is constant, the number of Ship_2s that collide with Ship_1 while Ship_1 is navigating the distance D_W can be obtained by integrating the quotient of CS in regions I, II, and III divided by Ship_2's velocity (V ₂), with x ₂.

${N_{CC}}$ in region I, ${N_{CC}}(\text{I})$, was calculated using Equation (4.2).

(4.2)\begin{equation}{N_{CC}}(\text{I} )= \int_{{F_4}({{x_1}} )}^{{F_3}({{x_1}} )} {\frac{1}{{{V_2}}} \cdot \frac{{{D_{\textit{WL}}}}}{{\cos ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )\; }} \cdot \frac{{{x_2} - {F_4}({{x_1}} )}}{{{F_3}({{x_1}} )- {F_4}({{x_1}} )}}{\text{Q}_2}{f_2}({{x_2}} )d{x_2}} \end{equation}

In the same manner, ${N_{CC}}$ in regions II and III, that is, ${N_{CC}}$ (II) and ${N_{CC}}$ (III), is determined in Equations (4.3) and (4.4), respectively.

(4.3)\begin{align}{N_{CC}}({\text{II}} )& = \int_{{F_3}({{x_1}} )}^{{F_1}({{x_1}} )} {\frac{1}{{{V_2}}} \cdot \frac{{{D_{\textit{WL}}}}}{{\cos ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )\; }}{\text{Q}_2}{f_2}({{x_2}} )d{x_2}}\end{align}

(4.4)\begin{align}{N_{CC}}({\text{III}} )& = \int_{{F_1}({{x_1}} )}^{{F_2}({{x_1}} )} {\frac{1}{{{V_2}}} \cdot \frac{{{D_{\textit{WL}}}}}{{\cos ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )\; }} \cdot \frac{{{F_2}({{x_1}} )- {x_2}}}{{{F_2}({{x_1}} )- {F_1}({{x_1}} )}}{\text{Q}_2}{f_2}({{x_2}} )d{x_2}} \end{align}

The sum of the results is the total ${N_{CC}}$s (N _CC(I, II, III)) resulting from one Ship_1 travelling the distance D_W when Ship_1s and Ship_2s have the same constant velocity, length, and breadth respectively. Moreover, by integrating ${N_{CC}}$ (I, II, III) over the entire waterway 1 for a certain time (T), the number of collision candidates that are caused by Ship_1s and Ship_2s with the same constant velocity, ship length, and breadth respectively, while travelling for the duration (T), can be obtained. Furthermore, when the velocity, ship length, and breadth of Ship_1 and Ship_2 are probabilistically distributed, ${N_{CC}}$ can be obtained by integrating them with these random variables as in Equation (4.5).

(4.5)\begin{align} & {N_{CC}}\ ({{Q_1},{Q_2},{V_1},{V_2},{L_1},{L_2},{B_1},{B_2}} )\notag\\ & \quad = \int_{{B_{2 - \text{min}}}}^{{B_{\text{2 - max}}}} \int_{{B_{1 - \text{min}}}}^{{B_{1 - \text{max}}}} \int_{{L_{2 - \text{min}}}}^{{L_{2 - \text{max}}}} \int_{{L_{1 - \text{min}}}}^{{L_{1 - \text{max}}}} \int_{{V_{2 - \text{min}}}}^{{V_{2 - \text{max}}}} \int_{{V_{1 - \text{min}}}}^{{V_{1 - \text{max}}}} \int_{{x_{1 - \text{min}}}}^{{x_{1 - \text{max}}}}\notag\\ & \qquad \times \left( {\int_{{F_4}({{x_1}} )}^{{F_3}({{x_1}} )} {\dfrac{1}{{{V_2}}}} } \cdot \dfrac{{{D_{\textit{WL}}}}}{{\cos ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )\; }} \cdot \dfrac{{{x_2} - {F_4}({{x_1}} )}}{{{F_3}({{x_1}} )- {F_4}({{x_1}} )}}{\text{Q}_2}{f_2}({{x_2}} )d{x_2}\right.\notag\\ & \qquad + \int_{{F_3}({{x_1}} )}^{{F_1}({{x_1}} )} {\dfrac{1}{{{V_2}}} \cdot \dfrac{{{D_{\textit{WL}}}}}{{\cos ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )\; }}{\text{Q}_2}{f_2}({{x_2}} )d{x_2}} \notag\\ & \qquad \left. { + \int_{{F_1}({{x_1}} )}^{{F_2}({{x_1}} )} {\dfrac{1}{{{V_2}}} \cdot \dfrac{{{D_{\textit{WL}}}}}{{\cos ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )\; }} \cdot \dfrac{{{F_2}({{x_1}} )- {x_2}}}{{{F_2}({{x_1}} )- {F_1}({{x_1}} )}}{\text{Q}_2}{f_2}({{x_2}} )d{x_2}} } \right)T\notag\\ & \qquad \cdot {Q_1}{f_1}(x ){v_1}({{V_1}} ){v_2}({{V_2}} ){l_1}({{L_1}} ){l_2}({{L_2}} ){b_1}({{B_1}} ){b_2}({{B_2}} )d{x_1}d{V_1}d{V_2}d{L_1}d{L_2}d{B_1}d{B_2} \end{align}

Equation (4.5) becomes Equation (4.6) if the variables, except for the appearance position, are constant.

(4.6)\begin{align} & {N_{CC}}({{Q_1},{Q_2}} )= \left( {\dfrac{{T{Q_1}{Q_2}{D_W}|{\overrightarrow {{V_2}} - \overrightarrow {{V_1}} } |}}{{{V_1}{V_2}\cos ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )\; }} + \dfrac{{2T{Q_1}{Q_2}d}}{{{V_2}\cos ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )\; }}} \right)\notag\\ & \int_{{x_{1 - \text{min}}}}^{{x_{1 - \text{max}}}} \left( \int_{{F_4}({{x_1}} )}^{{F_3}({{x_1}} )} \dfrac{{{x_2} - {F_4}({{x_1}} )}}{{{F_3}({{x_1}} )- {F_4}({{x_1}} )}}{f_2}({{x_2}} )d{x_2} + \int_{{F_3}({{x_1}} )}^{{F_1}({{x_1}} )} {f_2}({{x_2}})d{x_2}\right.\notag\\ & \quad +\left. \int_{{F_1}({{x_1}} )}^{{F_2}({{x_1}} )} {\dfrac{{{F_2}({{x_1}} )- {x_2}}}{{{F_2}({{x_1}} )- {F_1}({{x_1}} )}}{f_2}({{x_2}} )\; d{x_2}} \right){f_1}({{x_1}} )d{x_1} \end{align}

When the intersection angle is 0° or 180°, Equation (4.7) is obtained.

(4.7)\begin{equation}{N_{CC}}({{Q_1},{Q_2}} )= \left( {\frac{{T{Q_1}{Q_2}{D_W}|{\overrightarrow {{V_2}} - \overrightarrow {{V_1}} } |}}{{{V_1}{V_2}}} + \frac{{2T{Q_1}{Q_2}d}}{{{V_2}}}} \right)\int_{{x_{1 - \text{min}}}}^{{x_{1 - \text{max}}}} {\int_{ - (({B_1} + {B_2})/2)}^{(({B_1} + {B_2})/2)} {{f_2}({{x_2}} )d{x_2}{f_1}({{x_1}} )d{x_1}} }\end{equation}

Comparing Equation (4.7) with Equation (2.3) for the case where the intersection angle is 0° or 180° by Pedersen's method, Equation (4.7) contains the term $2T{Q_1}{Q_2}d/{V_2}$ which is not included in Equation (2.3). This shows that ${N_{CC}}$ at both ends of the CA is included in Equation (4.7), which shows that the method developed here is more accurate than Pedersen's method.

4.2 Case 2

Based on Figure 12, the estimation of ${N_{CC}}$ in this case is calculated as follows. First, let the intersection of the line segments E ₁F ₃ and E ₁E ₄ of CA be Q, then the length ${L_{\text{Q}{\text{E}_\text{3}}}}$of the line segment QE₃ can be obtained by Equation (4.8). The length of CS in region II is constant.

(4.8)\begin{equation}{L_{\text{Q}{\text{E}_\text{1}}}} = \frac{{{D_{\text{COL}}}}}{{\sin ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )\; }}\end{equation}

Therefore, ${N_{CC}}$ of Ship_1 with Ship_2s in regions I, II, and III are calculated using Equations (4.9)–(4.11), respectively.

(4.9)\begin{align}{N_{CC}}(\text{I} )& = \int_{{F_4}({{x_1}} )}^{{F_1}({{x_1}} )} {\frac{1}{{{V_2}}} \cdot \frac{{{D_{COL}}}}{{\sin ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )}} \cdot \frac{{{x_2} - {F_4}({{x_1}} )}}{{{F_3}({{x_1}} )- {F_4}({{x_1}} )}}{\text{Q}_2}{f_2}({{x_2}} )d{x_2}}\end{align}

(4.10)\begin{align}{N_{CC}}({\text{II}} )& = \int_{{F_1}({{x_1}} )}^{{F_3}({{x_1}} )} {\frac{1}{{{V_2}}} \cdot \frac{{{D_{\text{COL}}}}}{{\sin ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )}}{\text{Q}_2}{f_2}({{x_2}} )d{x_2}} \end{align}

(4.11)\begin{align}{N_{CC}}({\text{III}} )& = \int_{{F_3}({{x_1}} )}^{{F_2}({{x_1}} )} {\frac{1}{{{V_2}}} \cdot \frac{{{D_{\text{COL}}}}}{{\sin ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )}} \cdot \frac{{{F_2}({{x_1}} )- {x_2}}}{{{F_2}({{x_1}} )- {F_1}({{x_1}} )}}{Q_2}{f_2}({{x_2}} )d{x_2}} \end{align}

In Case 2, $\sin ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )\;$ does not become zero, so a solution always exists.

Furthermore, when all the variables concerned are random variables, the expected value of the total number of collision candidates $({N_{CC}}({Q_1},{Q_2},{V_1},{V_2},{L_1},{L_2},{B_1},{B_2}))$ in the duration (T) is calculated using Equation (4.12).

(4.12)\begin{align} & {N_{CC}}({{Q_1},{Q_2},{V_1},{V_2},{L_1},{L_2},{B_1},{B_2}} )\notag\\ & \quad = \int_{{B_{2 - \text{min}}}}^{{B_{2 - \text{max}}}} {\int_{{B_{1 - \text{min}}}}^{{B_{1 - \text{max}}}} {\int_{{L_{2 - \text{min}}}}^{{L_{2 - \text{max}}}} {\int_{{L_{1 - \text{min}}}}^{{L_{1 - \text{max}}}} {\int_{{V_{2 - \text{min}}}}^{{V_{2 - \text{max}}}} {\int_{{V_{1 - \text{min}}}}^{{V_{\text{1 - max}}}} {\int_{{x_{1 - \text{min}}}}^{{x_{1 - \text{max}}}} {\left( {\int_{{F_4}({{x_1}} )}^{{F_3}({{x_1}} )} {\dfrac{1}{{{V_2}}}} } \right.} } } } } } } \notag\\ & \qquad \cdot \dfrac{{{D_{\text{COL}}}}}{{\sin ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )}} \cdot \dfrac{{{x_2} - {F_4}({{x_1}} )}}{{{F_3}({{x_1}} )- {F_4}({{x_1}} )}}{Q_2}{f_2}({{x_2}} )d{x_2} + \int_{{F_3}({{x_1}} )}^{{F_1}({{x_1}} )} {\dfrac{1}{{{V_2}}}} \cdot \dfrac{{{D_{\text{COL}}}}}{{\sin ({{\theta_{\overrightarrow {{V_2}} \cdot \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )}}{Q_2}{f_2}({{x_2}} )d{x_2}\notag\\ & \qquad \left. { + \int_{{F_4}({{x_1}} )}^{{F_3}({{x_1}} )} {\dfrac{1}{{{V_2}}}} \cdot \dfrac{{{D_{\text{COL}}}}}{{\sin ({{\theta_{\overrightarrow {{V_2} \cdot } \overrightarrow {{V_2}} - \overrightarrow {{V_1}} }}} )}} \cdot \dfrac{{{x_2} - {F_4}({{x_1}} )}}{{{F_3}({{x_1}} )- {F_4}({{x_1}} )}}{Q_2}{f_2}({{x_2}} )d{x_2}} \right)T\notag\\ & \qquad \cdot {Q_1}{f_1}({{x_1}} ){v_1}({{V_1}} ){v_2}({{V_2}} ){l_1}({{L_1}} ){l_2}({{L_2}} ){b_1}({{B_1}} ){b_2}({{B_2}} )d{x_1}d{V_1}d{V_2}d{L_1}d{L_2}d{B_1}d{B_2} \end{align}

Equation (4.12) becomes Equation (4.13) if the variables, except for the position of occurrence, are constant.

(4.13)\begin{align} & {N_{CC}}({{Q_1},{Q_2}} )= \left( {\dfrac{{T{Q_1}{Q_2}{D_{\text{COL}}}|{\overrightarrow {{V_2}} - \overrightarrow {{V_1}} } |}}{{{V_1}{V_2}\sin (\theta )\; }}} \right)\int_{{x_{1 - \text{min}}}}^{{x_{1 - \text{max}}}}\notag\\ & \left( {\int_{{F_4}({{x_1}} )}^{{F_3}({{x_1}} )} {\dfrac{{{x_2} - {F_4}({{x_1}} )}}{{{F_3}({{x_1}} )- {F_4}({{x_1}} )}}{f_2}({{x_2}} )d{x_2}} + \int_{{F_3}({{x_1}} )}^{{F_1}({{x_1}} )} {{f_2}({{x_2}} )d{x_2} + } \int_{{F_1}({{x_1}} )}^{{F_2}({{x_1}} )} {\dfrac{{{F_2}({{x_1}} )- {x_2}}}{{{F_2}({{x_1}} )- {F_1}({{x_1}} )}}{f_2}({{x_2}} )\; d{x_2}} } \right)\notag\\ & \quad \times {f_1}({{x_1}} )d{x_2} \end{align}

If Ship_1 completely passes through the intersection, that is, if the width of waterway 2 is completely included in Ship_1's CA, Equation (4.13) becomes Equation (4.14), which is the same as the equation by Pedersen (Equation (2.2)).

(4.14)\begin{equation}{N_{CC}}({{Q_1},{Q_2}} )= \left( {\frac{{T{Q_1}{Q_2}{D_{\text{COL}}}|{\overrightarrow {{V_2}} - \overrightarrow {{V_1}} } |}}{{{V_1}{V_2}\sin (\theta )\; }}} \right)\int_{{x_{1 - \text{min}}}}^{{x_{1 - \text{max}}}} {\int_{{F_3}({{x_1}} )}^{{F_1}({{x_1}} )} {{f_2}({{x_2}} ){f_1}({{x_1}} )d{x_2}d{x_1}} } \end{equation}