Cellular Automata Model on AIS-based for Variable Two-way Waterway

Purpose: We aim at “heavy traffic direction” and “light traffic direction” in two-way waterway traffic and attempt to promote the transit capacity of two-way waterway system. Design/methodology/approach: We propose overtaking rules, head-on rules and a cellular automaton model for variable two-way waterway on AIS-based on the basis of NaSch (Nagel-Schreckenberg) model. Findings: By numerical simulation to the two situations which allow changing lane and prohibit changing lane, we obtain fundamental functions between traffic flux (speed) and density and find that changing lane can promote traffic flux and average speed of two-way waterway system under the premise of no impact to the traffic order, and when waterway ship traffic is dense, flux of waterway system has a visible promotion, and when traffic is sparse, average speed of waterway system adds significantly.


Introduction
In general, two-way waterway is usually composed by inbound lane, outbound lane and separation zone (or separation line).For the safety of navigation, Rules 9 and Rules 10 of Convention on the International Regulations for the Preventing Collisions at sea, 1972 (COLREGs), in fact divides the two-way waterway into two isolated and independent navigable lanes.However ship traffic is always in disequilibrium, and there exist "heavy traffic direction" and "light traffic direction" (Tingting, Qin & Chaojian, 2013) in the waterway.The performance of "traffic disequilibrium" is that, ship traffic is dense in one lane (or some sections of the lane), but sparse in the other (or the others of the lane).As a result, traffic jam incurs in one lane (or some sections of the lane), but in the other lane ship traffic is disengaged.Therefore, we can make full use of the disengaged lane (or some sections of the lane) to promote the capacity of overall waterway.
In marine traffic field, the original research of two-way waterway focused on calculating the transit capacity and waterway width by empirical formulas (Fenghua & Xuefeng, 2007;Zhibang & Xin, 2011).During the process of calculation, the two-way waterway is usually treated as two independent waterways, which are hardly to embody the effect of human, machine, environmental and management factors to the waterway capacity.
For its virtues which effectively reflect the ship's response to human and environmental factors in the trajectory (waterway width), ship handing simulator is widely used in waterway research fields shortly after its birth (Inoue, 2000;Kobylinski, 2011;Seong, Jeong & Park, 2012;Yuezong & Hongbo, 2014).However, single-ship handling simulator fails to carry out largescale, real-time and parallel simulation in hardware and software.So empirical formulas and single-ship handling simulator are helpless in the aspect of complex systems problems.
Considering the integrity and systematicness of two-way waterway, Tingting et al. (2013)  forward the concept of "Tidal Reversible Channel", but the effect of changing lane hasn't been solved and discussed in their works.
For the virtues of discretization in space, time and state, and easy implementation in algorithm on a computer, cellular automata (CA) model has been widely developed and used in traffic flow study.Feng (2013) presented a ship traffic CA model which took marine characteristics into account.In order to simulate ship traffic from the micro view and reveal the effect of lanechanging to waterway transit capacity, the paper establishes a variable two-way waterway CA model on the basis of Feng (2013).The work can be applied to the optimization, organization and management of ship traffic.

Cellular Automata Model on AIS-Based for Variable Two-way Waterway
The model is formed by an inbound lane and an outbound lane.Each lane is divided into n equal cells.Each cell is either occupied by a ship or empty.The velocity of ship i takes Vi, and Vi  {0,1,2,…, Vmax}, where Vmax is the maximum speed of some type of ships.In the discrete lane, ship i occupies Li cells from i to i + Li -1 (when inbound) or to i -Li + 1 (when outbound), and Li  {1,2,…, Lmax}, where Lmax is length of the largest ship in waterway.

Navigational Rules
Under normal conditions, ships navigate in their lanes in compliance with "COLREGs 1972" and other related traffic regulations, and no ships shall normally enter a separation zone or cross a separation line.At this time, ships evolve in accordance with the NS ship traffic CA model in Feng (2013).

Overtaking Rules
Figure 1 is the sketch-map for overtaking.Suppose at time t, there is ship f1 locating in front of ship i, and (1) Where d1(t) denote the distance from ship i to ship f1, dsafe1 denote the safety distance from ship i to ship f1.
-676-Journal of Industrial Engineering and Management -http://dx.doi.org/10.3926/jiem.1347 For the purpose of collision avoidance, ship i has to slow down or stop engine.At this time, if there are enough navigable waters in front of ship f1 and neighbor lane is disengaged, ship i can change to her neighbor lane and overtake ship f1, then goes back to her original lane.
Obviously the overtaking action would not impact the safety and traffic order of her neighbor lane, and the transit capacity of overall waterway system promotes.(2) Where dsafe2 and dsafe3 respectively denote the safety distances between ship i and her fore ship f2 and aft ship f1 when overtaking is finished; dsafe4 and dsafe5 respectively denote the safety distance between ship i and her fore ship f1_0 and aft ship b1_0 which locate in her neighbor lane; d12 denote the distance between the two nearest ships in front of ship i; d1_0 and db1_0 respectively denote the distances between ship i and her fore ship and aft ship which locate in her neighbor lane; tovertake denote the time to overtake ship i's fore ship f1; theadon denote the time to head on ship i's fore ship f1_0 in her neighbor lane.

Reverse Navigational Rules for Overtaking
In mariner's practice and in order to reduce the overtaking time, overtaking ship usually accelerates and the overtaken ship decelerates.Therefore, we make the reverse navigational rules for overtaking.
Overtaking ship accelerates at each time step, and then navigates at the maximum speed when reaches the maximum.For the overtaken ship, she shall maintain her speed and navigate with caution, so there is no random slowing-down during overtaking. (6) Where pi a n d pf1, respectively, denote random slow probabilities of overtaking ship i and overtaken ship f1.

Algorithm
The algorithm of above-mentioned cellular automata model on AIS-based for variable two-way waterway can describe as in Figure 3.

Simulation
Considering the actual traffic situation, suppose there are 3 types of ships sailing along the

Safety Distance
According to the discussion in paper (Feng, 2013), the minimum safety distances between ship i and other ships are determined by the following formulas:

Characteristic of flux (speed) when inbound traffic and outbound traffic is unsymmetrical
In the marine practice, the common traffic density in two-way waterway is dynamic and unsymmetrical.So the discussion of unsymmetrical ship density is more significant.

The Relationship Between Traffic Flux and Ship Arrival Rate when Inbound and Outbound Traffic is Unsymmetrical
Figures 10 and 11 are respectively the relationship between the flux of waterway system and outbound ship arrival rate at different inbound ship arrival rate when allow changing and prohibit changing.We can find that the flux of waterway system decrease as ship arrival rate increase; when ship traffic flow in waterway is in a free status (ship arrival rate is 4, 5 or 6), traffic flux decreases monotone; and when ship traffic flow is dense, a phenomenon in which ship arrival rate is low and however the traffic flux is low too, occurs in some areas of the relational diagraphs.The phenomenon is caused by low-speed ship.When there are a large amount of low-speed ships locating in waterway, or when two or more low-speed ships sail one after the other, it's hard for fast-speed ships to overtake.

Changing and Prohibit Changing
Figures 12 to 17 are respectively the relationship between traffic flux and outbound ship arrival rate when inbound ship arrival rate is determined.In the diagram, changing-lane would promote the traffic flux significantly when inbound or outbound ship arrival rate is both low (ship arrival rate is 1 or 2); what's more, the bigger inbound and outbound lanes differ, the more ship flux promotes.In the simulation, when ship arrival rate of one lane is 1 and the other is 6, ship traffic of waterway system would promote 11.4% at most.In the diagram, average speed of waterway system increase as ship arrival rate increases; when ship traffic in waterway is dense (ship arrival rate is 1, 2 or 3), average speed increases monotone; and when ship traffic is in a free status (ship arrival rate is 4, 5 or 6), the regularity of average speed and ship arrival rate is unfirm and irregular in the diagram; explanation to the phenomenon is that the interaction of ships is weak when waterway traffic is in a free status.

Conclusions
This paper has proposed a cellular automaton model for variable two-way waterway on AIS-based.By numerical simulation to the two situations which allow changing lane and prohibit changing lane, fundamental functions between traffic flux (speed) and density are obtained.That is, the flux of waterway system decreases and the average speed increases as ship arrival rate increases.We so find that changing lane can promote traffic flux and average speed of two-way waterway system under the premise of no impact to the traffic order.When waterway ship traffic is dense, flux of waterway system has a visible promotion, and when traffic is sparse, average speed of waterway system adds significantly.
As an implication, we can reach a compromise between traffic efficiency and safety.When no collision risk incurred, the marine administrations should allow involved ships to change lane for overtaking; and as a suggestion, Rule 9 and Rule 10 of COLREGs should make some adjustments correspondingly.

Figure 2
Figure 2 is the schematic diagram of overtaking vessel sailing back to her original traffic lane.When overtaking ship i passes and keeps clear the overtaken ship f1, she shall sail back to her original lane.At this time, the precaution and obligations of overtaking ship and overtaken ship dismiss.Safety conditions to sailing back are as follows:

Figure 3 .
Figure 3. Cellular automata model for variable two-way waterway on AIS-based

Where
Lf1 and Lf2 respectively denote the lengths of her two nearest ships in front of overtaking ship; Lf1_0 and Lb1_0 respectively denote the lengths of fore ship and aft ship which locate in her neighbor lane; and K = {0,1} denote the running direction of overtaking ship relative to another ship, 0 means inbound and 1 outbound.

Figure 4 .
Figure 4. Inbound spatial-temporal spot diagram when allow changing

Figure 7 .
Figure 7. Outbound spatial-temporal spot diagram when prohibit changing

Figure 8 .
Figure 8. Traffic flux and ship arrival rate as a function when inbound and outbound ship arrival rates are symmetrical

Figure 10 .
Figure 10.Traffic flux of waterway system and outbound ship arrival rate as a function at variable inbound ship arrival rate (when allow changing)

Figure 12 .
Figure 12.Traffic flux of waterway system and ship arrival rate as a function (inbound ship arrival rate = 1)

Figure 18 .
Figure 18.Average speed of waterway system and outbound ship arrival rate as a function at variable inbound ship arrival rate (when allow changing)

Figure 20 .
Figure 20.Average speed of waterway system and ship arrival rate as a function (inbound ship arrival rate = 1) Journal of Industrial Engineering and Management -http://dx.doi.org/10.3926/jiem.1347