Risk Modelling for Passages in Approach Channel

Abstract

Methods of multivariate statistics, stochastic processes, and simulation methods are used to identify and assess the risk measures. This paper presents the use of generalized linear models and Markov models to study risks to ships along the approach channel. These models combined with simulation testing are used to determine the time required for continuous monitoring of endangered objects or period at which the level of risk should be verified.

1. Introduction

Logistic models are arguably one of the most widely used data analysis techniques. They are the most studied discriminative models [1–3]. Logit models appear in a variety of forms in applications in biostatistics, epidemiology, economics, marketing research, and sociology. They are used to model the relationship between covariates and various types of discrete outcomes from the ubiquitous binary logit model for a two-level response to the conditional logit and multinomial (generalized) logit models concerning polytomous responses. Nested logit models allow for modeling the sequence of the decision process faced by the grouping alternatives at each stage into nests [4].

Logistic regression was applied to estimate the likelihood of mortality and severe injury in pedestrian casualties by considering the associations of such factors as demographic characteristics, injury characteristics, crash time, location, road environment, traffic control, and traffic conditions [5, 6]. Logit models were used for analyzing traffic crash severities aim at identifying and quantifying the effects of the factors which affect different crash injury severities [7, 8]. Ordered logit model were used in past studies about the great variability in conflict judgments by Air Traffic Controllers [9]. Polynomial and logistic regression and maritime transportation simulation were basis to developed an oil outflow model for collision and grounding accidents of tankers [10]. Navigation safety assessment can be carried out with the use of risk metrics analysis, multivariate statistical methods, stochastic models, and simulation methods, [11–15].

In this paper we present the possibility of using the category of ordered logit models to examine threats to vessels. Stochastic models were also proposed to allow, among other things, classification of ships with reference to the degree of risk of collision. Safe navigation requires knowledge of the whole picture of navigational, hydrometeorological situations and ship’s maneuverability. An important factor affecting the safety of the vessel is the proper execution of maneuvers, [15]. Weather conditions affect safe performance of ship’s maneuvers during entering the port and berthing, especially the vessels with a big surface exposed to wind. Wind direction and force are very important factors.

2. Safety of Navigation in the Approach Channels

Safety of navigation is a state of a system related to performing certain maneuvers without collisions, accidents in restricted waters. Among all the hazards we can distinguish [12](i)collision with another vessel or seamark in the fairway,(ii)running aground or hitting the bottom.

Probability of activating any of these risks can be represented by the following formula: Pr=𝑓𝐴_𝑖,𝑆_𝑖,𝐻_𝑖,𝑀_𝑖,𝐼_𝑖,𝑅_𝑖, (1) (see [12]) where 𝑖 is the index of hazard type, 𝐴_𝑖 are sea area parameters, 𝑆_𝑖 are vessel’s parameters, 𝐻_𝑖 are hydrometeorological parameters, 𝑀_𝑖 are parameters of the performed maneuver, 𝐼_𝑖 are parameters of traffic density, and 𝑅_𝑖 are parameters of traffic control system.

It is a dependent function; the variables 𝐴_𝑖, 𝑆_𝑖, 𝐻_𝑖, 𝑀_𝑖, 𝐼_𝑖, 𝑅_𝑖, which reflect a number of factors, describe certain states of the system the vessel—water area—the environment.

Hydrometeorological conditions have a significant impact on the safety of navigation in restricted areas. You can, amongst other weather factors, include wind, waves, and currents.

Nature of the fluctuations in water level depends on what the cause is. They can be characterized as follows [16, 17]:(i)short-term fluctuations in water level caused by(a)tides,(b)positive or negative surges,(ii)seasonal changes in water levels caused by long-term hydrometeorological changes in the sea area (region).

Water levels lowered or raised by wind are taken into consideration by adopting characteristic water level when determining fairways. Basing on these assumptions, procedures meeting the conditions of safe navigation are established to use a given sea area. Wind affects safe navigation in restricted waters and is always described by means of wind direction and force.

Sea currents influence the safety of the ship’s maneuver as they act on the part of the hull which is below water. There are three important types of currents and streams in restricted waters [12]:(i)tidal streams,(ii)sea currents,(iii)wind current.

Wind sea is an important factor in determining the safety of maneuvering in restricted waters. Parameters of waves affecting the size of the safe maneuvering area for a vessel are the following:(i)height, length, and period of the waves,(ii)directions of waves flow in relation to the available navigable water,(iii)distributions of direction and wave height in available navigable waters.In order to determine the effect of variables 𝐴_𝑖, 𝑆_𝑖, 𝑁_𝑖, 𝐻_𝑖, 𝑀_𝑖, 𝐼_𝑖, and 𝑅_𝑖 on the safety of navigation, it is necessary to adopt quantitative indicators. They are different for different types of sea areas on which navigation takes place (or maneuver is performed). The limit values of​ these indicators are the basis for assessing the safety of navigation.

There are two types of evaluation criteria of safety of navigation in restricted waters:(i)basic criteria for assessing the safety of navigation determined by a single parameter,(ii)complex evaluation criteria of safety of navigation which take into account the effects of an accident and which are a function of a number of variables.Navigational risk is a complex criterion of assessing safety of navigation, [13, 14, 18]. In this work a deviation from the approach channel is assumed to be the hazard which could result in hitting the bottom or in collision. The level of risk depends on the distance from the centerline of the fairway.

Let us assume that the classification of vessel’s states in terms of the degree of risk will be carried out basing on the value of a random variable 𝑌 using vector of limits. Elements of limits (thresholds) 𝜹^𝑇=[𝛿₀,𝛿₁,…,𝛿_𝑟−1] vectors are monotonic, nondecreasing sequence 𝛿₀≤𝛿₁≤𝛿₂,…,𝛿_𝑟−2≤𝛿_𝑟−1.

The different levels of risks and therefore states of the system are determined by 𝑦≤𝛿₀𝛿,then𝑧=0,_𝑘−1<𝑦≤𝛿_𝑘,then𝑧=𝑘for𝑘=1,…,𝑟−1,𝑦>𝛿_𝑟−1then𝑧=𝑟. (2)

3. Logit and Probit Models of Ordered States

Generalized linear models extend classical linear models. Logistic regression is an alternative to ordinary linear regression especially when we have discrete variables which describe categories in a given classification or states which may hold [19]. Sequential logit and probit models are often sequences of binary outcome models [20].

Let us assume that for a vessel at the time 𝑡 value of a variable 𝑦_𝑖(𝑡) deciding about assessing the degree of the 𝑖th threat to be defined for the moment 𝑡+𝑑𝑡 depends linearly on the vector of features 𝐱_𝑖(𝑡) [16], 𝑦_𝑖(𝑡+𝑑𝑡)=𝐱^𝑇_𝑖(𝑡)𝜷+𝜀_𝑖(𝑡), (3) where 𝜷 is a vector of unknown structural parameters of the model (3), 𝜀 has a distribution independent of 𝑖, and 𝑡 of the expected zero value and of a constant variance and is defined with distribution function of the logistic distribution. Write the model as 𝑌=1if𝑦^∗_𝑖≥00if𝑦^∗_𝑖<0. (4)

Let us divide the features vector into two ‘‘subvectors’’ 𝐱^𝑇_𝑖𝑡=𝐱_1𝑖(𝑡),𝐱_2𝑖(𝑡), (5) where 𝐱_1𝑖(𝑡) is a vector of features defining the level of threat for a vessel 𝑖 at a given moment 𝑡 and 𝐱_2𝑖(𝑡) is a vector of quantifiable characteristics.

In this situation, it describes only the value of the unobservable variable 𝑦^∗, that if the random variable/component has a continuous distribution, (3), it can be written as 𝑃𝑦_𝑖=1=1−𝐹_𝜀−𝐱^′_𝑖𝜷, (6) where 𝐹_𝜀 is the cumulative distribution function of the random variable.

An important issue is the way in which the various explanatory variables affect the dependent/explained variable and the extreme results in case of no unit variance of variable 𝜀.

Let us consider a situation in which the ship enters the approach channel and we do not have any information about this channel. What is the probability that “the vessel makes an error” (navigational error)?

Option 1. Let us assume that the vessel is typical and for which the value of an individual effect (resulting from a vessel type-operating characteristics (5)) is zero.

Then 𝑃𝑦_𝑖𝑡=1=Φ𝐱^′_𝑖𝑡𝜷+𝛼_𝑖, (7) for a standard distribution 𝜀, or generally 𝑃𝑦_𝑖𝑡𝐱=1=Φ^′𝑖𝑡𝜷+𝛼_𝑖𝛿_𝜀. (8)

Option 2. In normal and independent distribution 𝜀 and 𝛼 their sum 𝑣_𝑖𝑡=𝛼_𝑖+𝜀_𝑖𝑡 is normally distributed with parameters (0;𝛿²_𝜀+𝛿²_𝛼). This implies that we can write 𝑃𝑦_𝑖𝑡=1=𝑃𝑦^∗_𝑖𝑡𝐱≥0=1−Φ^′𝑖𝑡𝜷𝛿²_𝜀+𝛿²_𝛼. (9)

Of course, the parameter values and the variances are unknown, so you can take advantage of their assessment.

Let us consider a vessel about which we know how far it deviated from the centerline of the fairway in the approach channel in the past and what corrective maneuvers were performed. What is the probability that the vessel will make navigational error in the next period of time (𝑇)?

Knowing the vessel’s history and being able to determine the current level of risk, we get information enabling to identify the individual effects. Let us write the probability in the form of conditional probability (3): 𝑃𝑦_𝑖𝑡=1∣𝑦_𝑖1,…,𝑦_{𝑖,𝑡−1},𝐱_𝑖𝑡=𝑃𝑦,𝜷_𝑖𝑡=1,𝑌_𝑖1=𝑦_𝑖1,…,𝑌_{𝑖,𝑡−1}=𝑦_{𝑖,𝑡−1}∣𝐱_𝑖𝑡,𝜷𝑃𝑌_𝑖1=𝑦_𝑖1,…,𝑌_{𝑖,𝑡−1}=𝑦_{𝑖,𝑡−1}∣𝐱_𝑖𝑡.,𝜷 (10) We use the probability 𝑃 calculated in accordance with Options 1 or 2, respectively, and then we take a threshold value (𝑝_𝑝). If 𝑃>𝑝_𝑝 we forecast an error and otherwise lack of an error in the next period of time, [16].

Let us consider probit model with delayed response variable as explanatory variable [4]: 𝑦_𝑖𝑡=𝐱^′_𝑖𝑡𝜷+𝛾𝑦_{𝑖,𝑡−1}+𝛼_𝑖+𝜀_𝑖𝑡,𝑦_𝑖𝑡=1if𝑦_𝑖𝑡≥00if𝑦_𝑖𝑡<0. (11)

Since the probabilities in the subsequent periods for the same vessel are not independent, the model cannot be estimated with the presented method. However, conditional probabilities are independent with respect to the current value of the explanatory variable and to the individual effects.

To solve the initial conditions problem we can use the Heckman solution, [21], noting the model (11) in the form: 𝑦_𝑖𝑡=𝐱^′_𝑖𝑡𝜷+𝛾𝑦_{𝑖,𝑡−1}+𝛼_𝑖+𝜀_𝑖𝑡𝑦for𝑡≥2,_𝑖1=𝐳^′_𝑖𝑡𝜆+𝜇𝛼_𝑖+𝜀_𝑖1,𝑦_𝑖𝑡=1if𝑦_𝑖𝑡≥00if𝑦_𝑖𝑡<0. (12) In this model, for the first period an additional static equation defining the value of 𝑦_𝑖1 was used. The explanatory variables in the equation for the first period need not be the same as in subsequent periods.

An assumption that the random variable has a logistic distribution leads to a logit model in which the probability of adopting the value 1 by the dependent variable with given values of explanatory variables is given as 𝑃𝑦_𝑖=1=𝑃𝐱^′_𝑖𝑡𝜷≤𝜀_𝑖𝑡=𝐹_𝐿𝐱^′_𝑖𝑡𝜷=1+𝑒^{−𝐱′𝑖𝑡𝜷−1}. (13)

Elements of the vector of structural parameters of the model and vector of limit values can be estimated by maximum probability method using numerical procedures. Let us consider ordered model in which the observable dependent variable can take of 𝑛 different values: 𝑦_𝑖𝑡=𝐱^′_𝑖𝑡𝜷+𝛼_𝑖+𝜀_𝑖𝑡,𝑦_𝑖𝑡=0if𝛿₀≤𝑦_𝑖𝑡<𝛿₁1if𝛿₁≤𝑦_𝑖𝑡<𝛿₂⋮𝑛if𝛿_𝑛−1≤𝑦_𝑖𝑡<𝛿_𝑛. (14)

The above model is not appropriate if the variable 𝑦_𝑖𝑡 has values which cannot be arranged according to a particular scale.

Probability of accepting by the variable values of 𝑚 can be written as 𝑃𝑦_𝑖𝑡=𝑚=𝑃𝐱^′_𝑖𝑡𝜷+𝛼_𝑖+𝜀_𝑖𝑡<𝑘_𝑚+1−𝑃𝐱^′_𝑖𝑡𝜷+𝛼_𝑖+𝜀_𝑖𝑡<𝑘_𝑚 (15) or by means of cumulative distribution of the random variable in the form of 𝑃𝑦_𝑖𝑡=𝑚=𝐹_𝜀𝑘_𝑚+1−𝐱^′_𝑖𝑡𝜷−𝛼_𝑖−𝐹_𝜀𝑘_𝑚−𝐱^′_𝑖𝑡𝜷−𝛼_𝑖, (16) where 𝐹_𝜀 is the cumulative distribution function of the random variable.

In this way we obtain an ordered probit, and the estimation is based on maximizing the probability function. Estimation of the model requires the scale, taking a specific value of the variance of variable 𝜀 and finding the maximum of the probability function.

4. Examples of Models Used for Examining Risks in the Approach Channel

The methodology and logic of ordered logit model can be presented using a following example. Let there be 𝑁 ships in the channel, and assume that each ship’s navigator “degree of risk acceptance” affects the probability of being in state 𝑧. Sometimes the outcomes in a response variable are perceived as a sequence with stages (2), states 𝑧. The related probabilities can be written as [20] 𝑃₀=𝐹_𝑖𝐱_𝑖0𝜷,𝑃₁=1−𝐹_𝑖𝐱_𝑖0𝜷𝐹_𝑖𝐱_𝑖1𝜷,⋮𝑃_𝑟−1=^𝑟−2_𝑘=01−𝐹_𝑖𝐱_𝑖𝑘𝜷𝐹_𝑖𝐱_𝑖𝑟𝜷,𝑃_𝑟=^𝑟−1_𝑘=01−𝐹_𝑖𝐱_𝑖𝑘𝜷. (17) In the logit model the 𝐹 is logistic cumulative distribution function and subscripts 𝑖𝑘, for 𝑘=0,1,…,𝑟 indicate the sets of x variables included in states 𝑧=0,𝑧=1,…,𝑧=𝑟, respectively. The parameters 𝜷 can be estimated by dividing the sample into groups according to levels of risk.

4.1. Prioritizing Vessels due to the Risk That They Generate

Let us examine the case of two vessels along the approach channel, and let us ask the question which of them generates the greater threat?

Let us assume that the risk of the first ship is described by the random variable 𝑋 and the other by random variable 𝑌. Without any information, we assume that both these variables have the same probability distribution with cumulative distribution function 𝐹(𝑥) and independent (if ships are not interacting). In addition, without changing the generality, we can assume that both variables are continuous type, 𝑃(𝑋=𝑌)=0.

Let us calculate the probability 𝑃(𝑋>𝑌)𝑃(𝑋>𝑌)=^∞_−∞𝑃()(𝑥)=𝐹𝑥>𝑌𝑑𝐹²(𝑥)2^∞_−∞=0.5. (18) We obtained the result stating that without knowledge (for any distribution 𝐹_𝑋(𝑥)) the probability of analyzed events for each vessel is the same and is 0.5.

Let us consider the following algorithm, Figure 2:(i)let us assume a certain level of 𝑑,(ii)assessing the ship for example, the first one, during the remaining period of time, we check if 𝑋<𝑑,(iii)if so, we predict that 𝑋<𝑌,(iv)otherwise, we have 𝑋>𝑌.Let us calculate probability 𝑃(𝑑) of a correct assessment using the adopted assumptions about the type of random variables and probability properties we obtain: 𝑃(𝑑)()()==𝑃𝑋<𝑑,𝑋<𝑌+𝑃𝑋>𝑑,𝑋>𝑌_𝑑−∞𝑃(𝑥<𝑌)𝑑𝐹(𝑥)+^∞_𝑑𝑃(𝑥>𝑌)𝑑𝐹(𝑥)=−𝐹²(𝑑)+𝐹(𝑑)+0.5. (19) Since 𝐹(𝑥) is the cumulative distribution then for each level 𝑑 value of 𝐹(𝑥)∈[0,1], so let us examine the function 𝑔(𝑤)=−𝑤²+𝑤+0.5, Figure 1. When calculating we have 𝑔′(𝑤)=−2𝑤+1=0 for 𝑤=0.5,𝑔′′(𝑤)=−2.

Figure 1 

Graph of the function 𝑔(𝑤).

Figure 2 

The block diagram for the presented algorithm.

This function has the following maximum value for 𝑤=0.5. This means that an optimal level 𝑑_opt is the median of the distribution 𝐹(𝑥), 𝐹(𝑑_opt)=0.5. Then 𝑃(𝑑)=0.75.

4.2. Generalized Linear Model (GLM) of Leaving the Approach Channel

In the present example we will consider the following distance ranges 𝛿₀=30, 𝛿₁=50, 𝛿₂=90, 𝛿₃=120, 𝛿₄=150, and 𝛿₅=150. Range below 30 therefore corresponds to the level of threat 𝑧=0.

The vessel’s domain is defined by a rectangle of a length and width corresponding to the length and breadth of the vessel, where 𝛾-heading, 𝑑-maximum distance of the domain point from the centerline of the approach channel, Figure 3.

Figure 3 

Determination of the maximum distanced for the ship along the approach channel, [15].

The center of the domain, which is described around the ship, coincides with the geometric center of the vessel. A 𝑑[𝑚] was assumed as the maximum distance which is the maximum distance measured from the domain vertices to the fairway centerline.

Simulation studies were carried out in the laboratories of Gdynia Maritime University, using navigational-maneuvering simulator. This simulator in a very realistic way that represents sea areas and behaviour of vessels which are almost identical with real vessels. It is possible because these models use six degrees of freedom. The model of the vessel used was the model of a loaded LNG carrier of the following characteristics: 𝐿=315 m, 𝐵=50 m, and 𝑇=12 m. The ship was on an even keel. Simulation of the ship’s behaviour was carried out in the fairway and was influenced by interfering factors such as wind and swell—constant NE wind, period: 9 s., height of the swell: 1.5 m.

The value of 6 and 8 knots was taken as the nominal speed of the vessel. The wind direction adopted in simulations changed every 45 degrees starting from the 𝑁. The simulation scenario of the behaviour of the model under way along the approach channel assumes that the initial position of the vessel is in the centerline of the existing fairway in its northern most part and that the line of symmetry of the model and the course over ground overlaps with the centerline of the fairway. The parameters of the generated waves were determined, and the simulation results were obtained for winds (no squalls) of speed up to 15,0 m.

For the assumed parameters of the fairway the distanced is given by ((𝑥𝑑=tan99.8𝜋/180))_LONG−𝑦_LAT+3254.384⋅1852tan(99.8(𝜋/180))²𝛾𝜋+1+sin𝛾𝜋180+𝑎tan+18050⋅3153152, (20) where 𝑦_LAT and 𝑥_LONG Cartesian coordinates converted from geographic ones, [15].

In the simulation, in particular, the course of the function of the distance from the model vessel to the centerline of the fairway with given parameters of influencing factors (hydrometeorological conditions) was examined.

Thanks to the research on the simulator some information was received according to which the following variables were determined:(i)𝑌_𝑖 taking value of 1 (𝑧=𝑖) if the vessel at the time 𝑡 was found in 𝑖 class of the distance from the centerline of the fairway and 0 otherwise, 𝑖=0,1,…,5(ii)𝑋₁ vessel’s speed,(iii)𝑋₂ wind speed,(iv)𝑋₃ wind direction.For 𝑋₁,𝑋₂, and 𝑋₃ variables logistic models were obtained ()𝑌=expeta(),1+expeta (21) where eta is given in Table 1.

Table 1 

Models obtained for 𝑋₁, 𝑋₂, and 𝑋₃ variables.

Test of compatibility 𝜒² determines whether logistic function adequately fits the observed data. Since the value of 𝑝 is greater than or equal to 0.05, there is no reason to reject the adequacy of the fitted model at least 95.0% of confidence level.

4.3. Markov Model of Leaving the Approach Channel

Let us consider birth-death process (BD) with finite space of states {0,1,2,…,𝑟}. Graph of state of this process is shown in Figure 4.

Figure 4 

Graph of the system states.

Birth and death processes is a continuous-time Markov chain for which transitions may take place only between neighbouring states (i.e., 𝑖 to 𝑖+1 or 𝑖−1 only).

We use the following notation:(i)𝜆_𝑖—birth rate, 𝑖=0,1,2,3,…,𝑟−1;(ii)𝜇_𝑖—death rate, 𝑖=1,2,3,4,…,𝑟.

State transition probability:(i)𝑝₀₁=1(ii)birth before death: 𝑝_𝑖,𝑖+1=𝜆_𝑖𝜆_𝑖+𝜇_𝑖 (22)(iii)death before birth: 𝑝_{𝑖,𝑖−1}=𝜇_𝑖𝜆_𝑖+𝜇_𝑖. (23)Consider a system in which arrivals and departures occur one at a time. Let 𝑋(𝑠) be the risk level of the system at time 𝑠. We now consider fitting a BD model to data collected over an interval [0,𝑡]. Let 𝜆_𝑖(𝑡) and 𝜇_𝑖(𝑡) be direct estimates of the birth rates and death rates based on sample averages over the time interval [0,𝑡]. Similarly, let 𝛾_𝑖(𝑡) be estimates of the stationary distribution based on sample averages over the time interval [0,𝑡].

Moreover let 𝐴_𝑖(𝑡) be the number of up state changes (from 𝑧=𝑖 to 𝑧=𝑖+1) during the interval [0,𝑡] when the system is in state 𝑖; let 𝐷_𝑖(𝑡) be the number of down state changes (from 𝑧=𝑖 to 𝑧=𝑖−1) during the interval [0,𝑡] when the system is in state 𝑖; and let 𝑇_𝑖(𝑡) be the total time during the interval [0,𝑡] in which the system is in state 𝑖. Then, [22] we have 𝜆_𝑖(𝑡)=𝐴_𝑖(𝑡)𝑇_𝑖(𝑡),𝜇_𝑖(𝑡)=𝐷_𝑖(𝑡)𝑇_𝑖(𝑡),𝛾_𝑖(𝑡)=𝑇_𝑖(𝑡)𝑡,𝑡≥0. (24) This estimation procedure need not produce an irreducible BD process, because there can be initial and final transient states. However, under the simplifying assumption of irreducibility, this estimated BD process has the unique stationary probability distribution. For example if there are some constants 𝑎₁,𝑎₂,𝑑₁, and 𝑑₂ such that 𝜆_𝑖(𝑡)>0 for 𝑎₁≤𝑖≤𝑎₂ with 𝜆_𝑖(𝑡)=0 otherwise and 𝜇_𝑖(𝑡)>0 for 𝑑₁≤𝑖≤𝑑₂. From [22], this estimated BD process has the unique stationary probability distribution 𝛾^𝑒_𝑖(𝑡)=𝑠_𝑖(𝑡)∑^𝑟_𝑗=1𝑠_𝑗(𝑡),0≤𝑖≤𝑟, (25) where 𝑠₀(𝑡)=1 and 𝑠_𝑗(𝑡)=∏^𝑗−1_𝑖=0𝜆_𝑖/∏^𝑗_𝑖=1𝜇_𝑖,0≤𝑗≤𝑟.

4.4. System with Blocking

Let 𝛼_𝑖 be the probability that a ship will enter the higher class 𝑖+1 if it is already at class 𝑖. 𝜆_𝑖=𝛼_𝑖𝜇𝜆,𝑖≥0;_𝑖=𝜇,𝑖≥1. (26) Transition probability function is given by 𝑝_𝑖,𝑖+1=𝛼_𝑖𝜆𝛼_𝑖𝜆+𝜇,𝑝_{𝑖,𝑖−1}=𝜇𝛼_𝑖.𝜆+𝜇 (27) Transition time 𝑇_𝑖, the time required to go from state 𝑖 to state 𝑖+1 has the mean given by [13] 𝐸𝑇₀=1𝛼₀𝜆,𝐸𝑇_𝑖=1𝛼_𝑖𝜆+𝜇𝛼_𝑖𝜆𝐸𝑇_𝑖−1,Var𝑇₀=1𝛼₀𝜆²,Var𝑇_𝑖=1𝛼_𝑖𝜆𝛼_𝑖+𝜇𝜆+𝜇𝛼_𝑖𝜆Var𝑇_𝑖−1+𝜇𝛼_𝑖𝐸𝑇𝜆+𝜇^𝑖−1+𝐸𝑇_𝑖². (28) From (25) we have stationary probability distribution 𝛾^𝑒_𝑖(𝑡)=∏^𝑖−1_𝑘=0𝛼_𝑘𝜆/𝜇^𝑖∑^𝑟_𝑗=1∏^𝑗−1_𝑖=0𝛼_𝑖𝜆/𝜇^𝑗,0≤𝑖≤𝑟. (29) The properly selected sets of explanatory variables in the model (3) allow you to define Markov chains with different transition probability matrix. If we assume that the set of variables was limited to two variables that as a result of the estimation of the logit model, we obtain the assessment of probabilities that can be interpreted as the transition probability of a one threat category to another category. The use of the full set of explanatory variables of the form (5) gives us the transition probabilities for any ship 𝑖 and a time 𝑡.

5. Conclusion

The study proves suitability of logit models used to estimate the hazards to navigation. They can be used in the development of guidelines for traffic safety management system and large ships manoeuvring in restricted waters, to ensure obtaining the assumed level of safety. The approach proposed in this work can be used in Markov models by using in estimations the transition matrix of other variables than just defining the transition between states. This will allow the use of semi-Markov models to estimate safety of navigation.

Immediate work in simulation follows better evaluation measures and improvement of duration modeling, model system, and model confidence levels. The presented approach has to be further developed by more comprehensive experimental evaluations, examples of applications, and analytical models relating selected simulation responses with model parameters.

Discrete event models allow inclusion of individual variables without creating compound states, which could improve the precision of the model. Some interesting questions are still open. For example possible questions can relate to correlation between random variables.