Application of stochastic process modeling in the prediction of emergency response time for public emergencies

The city is a multifunctional building cluster with a large number of integrated buildings, and a large number of the public have been accustomed to shopping, leisure, food, accommodation, entertainment and other “one-stop” activities in urban complexes, which have become an important part of the daily life of urban residents [1–3]. The acceleration of urbanization process makes the development of urban complexes in China has experienced from nothing to something, and then to the current development trend in full swing, but with the outbreak of public emergencies is also gradually increasing the potential danger [4–6]. There are many factors that cause emergencies, including man-made, natural, equipment and many other factors, which cannot be completely avoided [7]. When the emergency management capacity can not keep up with the development needs of urbanization, the emergence of serious public safety accidents has exposed the many loopholes and shortcomings in the emergency management of public emergencies.

The introduction of computers and information technology to provide new means for the development of information decision-making, the solution of complex decision-making problems rely more and more on a variety of artificial intelligence engineering systems, including decision support systems [8–10]. Intelligent algorithm-assisted calculation of the decision-making results can not only respect the subjective will of the decision-maker, but also help the relevant emergency decision-making departments to generate timely and scientific evacuation programs, improve data analysis capabilities, reduce the decision-making “limited rationality”, to ensure that the evacuation program and decision-making objectivity and rationality [11–14]. In addition, the parameters of the intelligent model and its related application methods can help determine the data sources, reduce the limited access to data, solve the problem of insufficient access to some emergency evacuation information, improve the enforceability of the emergency plan, assist in the successful completion of the evacuation work, and minimize the loss of the accident [15–18].

To study the construction of an emergency response time prediction model based on a stochastic process for effective emergency management of public emergencies. The process requires first building a GM(1,1) model to derive the predicted values of the original series. Then a Markov prediction model is built to predict the residual values. Finally, the predicted values of the GM(1,1) model are residual corrected for public emergency response time prediction. Compare the prediction accuracy of this paper’s model with the gray model, differential integration moving average autoregressive model, fuzzy time series prediction, and long and short-term memory network, and test the effectiveness of the prediction model by predicting the emergency response time of the simulated public emergencies.

2

Method

Public emergencies include natural disasters, accidents, accidents, public health events, and social security events. This paper considers that the basic connotation of public emergencies has the following characteristics: firstly, it is sudden and unpredictable, once the event occurs suddenly, it is difficult to predict comprehensively. The second is the seriousness of the consequences. The incident on society can cause or cause different degrees of damage. Thirdly, it is urgent and imperative to take appropriate emergency measures within a short time to avoid serious consequences. Therefore, the construction of an emergency response time prediction model is of great significance to the emergency management of public emergencies.

2.1

Stochastic processes

2.1.1

Definition and classification of stochastic processes

If a single observation of the whole process of change of a thing, the result obtained is a function of time t, but the process of change of the same thing independently repeated many times, the results obtained are not the same, then the process of change is called a stochastic process [19], i.e., let E be a randomized trial, S is its sample space, if for each e∈S, we can always determine a function of time t according to some rule: 1 $X (e, t), t \in T$

Corresponds to it (T is the range of variation of time t). Thus, for all e ∈ S, a set of functions of time t is obtained, and we call this set of functions of time t a stochastic process, and each function in the set is a sample function of this stochastic process.

Stochastic processes can be categorized according to their characteristics as smooth processes Gaussian processes Markov processes, second order moment processes, purely stochastic processes, and so on.

2.1.2

Definition and properties of Markov processes

In practice, one often encounters stochastic processes that have the property that the future evolution of an event depends on its past evolution, provided that the present state of the event is known. This property of no posteriority that the “future” is independent of the “past” under the condition that the “present” is known is called Markovianity, and a stochastic process with this property is called a Markov process The stochastic process with this property is called a Markov process [20].

A Markov process is described mathematically as follows: suppose {X(t),t ∈ T} is a stochastic process if for any positive integers n and t₁ < t₂ < … < t_n, P(X(t₁) = x₁, X(t₂) = x₂,…, X(t_n) = x_n) > 0, its conditional distribution satisfies the following conditions: 2 $\begin{array}{l} P {X (t_{n}) & < x_{n}) ∣ X (t_{n - 1}) = x_{n - 1}, X (t_{n - 2}) = x_{n - 2} \dots, X (t_{1}) = x_{1},)} \\ = P {X (t_{n} < x_{n}) ∣ X (t_{n - 1}) = x_{n - 1}} \end{array}$

Then {X(t), t ∈ T} is said to be a Markov process. Equation (2) is called the Markovianity or non-sequitur of the process and indicates that if the present state of the event is known, the probabilistic regularity of the future state in which the event will be located is already determined, regardless of how the event arrived at the present state. In other words, if t_n+1 is considered to be “now”, then t_n represents “future”, t₁, t₂ through t_n–2 represent “past”, “X(t_i) = x_i” means that the event is in state x_i at time t_i, and the above equation shows that the state in which the event is in the future is independent of the state it was in in the past under the condition that the present state is known.

The state space Iand time parameter T of a Markov process can be continuous or discrete, and according to this criterion, Markov processes can be classified into three categories: 1)

Markov processes with continuous time and state.

2)

Markov processes that are continuous in time and discrete in state, called continuous time Markov chains.

3)

Markov processes with discrete time and state, called Markov chains.

The mathematical description of a Markov chain is as follows:

Let a stochastic process {X(n), n ∈ T}, where the parameter set T is a discrete set of times, i.e., T = {0,1, 2,…}, and the geometry of the state space consisting of the totality of the possible values of its corresponding X(n) is a discrete state I = {i₀, i₁, i₂, i₃,…} if its conditional probability is satisfied for any nonnegative integer n ∈ T and any i₀, i₁,…,i₊₁ ∈ I : 3 $\begin{array}{l} P {X (n + 1) & = i_{n + 1} ∣ X (n) = i_{n}, X (n - 1) = i_{n - 1}, \dots X (0) = i_{0}} \\ = P {X (n + 1) = i_{n + 1} ∣ X (n) = i_{n}} \end{array}$

Then {X (n), n ∈ T} is called Markov chain.

Markov chain theory is a theory that describes the transfer law between the state and the state of an event, i.e., the transfer probability between different states can be obtained by Markov chain, so as to predict the future trend of the state of the event, and to achieve the purpose of predicting the future [21].

2.1.3

Markov forecasting methods

Markov forecasting is a method of predicting the probability of an event occurring. It is a forecasting method based on the Markov chain, which predicts the movement of events at various moments (or periods) in the future based on their current status. Markov model is suitable for the prediction of time series with large stochastic volatility, which can reveal the stochastic nature of events affected by various complex factors, and can portray the overall trend of the time series from the macro level [22].

When Markovianity is introduced into the stochastic process {X (t), t ∈T}, T is the finite set of moments, X (t) denotes the state at the moment t, and the sequence of states is a stochastic vector satisfying Markovianity with a time factor. The mathematical expression (3) for the non-sequitur of the Markov chain gives: 4 $\begin{array}{l} P {X (n) = i_{n}, X (n - 1) = i_{n - 1} ..., X (0) = i_{0}} \\ = P {X (n) = i_{n} | X (n - 1) = i_{n - 1}, X (n - 2) = i_{n - 2} ..., X (0) = i_{0}} \\ \cdot P {X (n - 1) = i_{n - 1}, X (n - 2) = i_{n - 2} ..., X (0) = i_{0}} \\ = P {X (n) = i_{n} | X (n - 1) = i_{n - 1}} \cdot P {X (n - 1) = i_{n - 1},, X (n - 2) = i_{n - 2} ... X (0) = i_{0}} \\ = ... \\ = P {X (n) = i_{n} | X (n - 1) = i_{n - 1}} \cdot P {X (n - 1) = i_{n - 1} | X (n - 2) = i_{n - 2}} \dots \\ \cdot P {X (1) = i_{1} | X (0) = i_{0}} \end{array}$

It can be seen that the statistical properties of the Markov chain are completely determined by the conditional probability P{X (n + 1) = i_n+1|X(n) = i_n}, which intuitively means the probability that an event is in state j at moment n + 1 under the condition that it is in state i at moment n. The state transfer probability matrix P is naturally obtained from the probabilities of transitions between all states, and this matrix reflects the statistical regularity of transitions between states. From the initial state probability vector and transfer probability matrix P, it is possible to determine the probability distribution of the future state, so as to make predictions about the future state. 1)

State transfer probability matrix

Often called the conditional probability p_ij (n) = P{X_n+1 = j | X_n = i}, i, j ∈ I for the Markov chain {X (n), n ∈ T} at the moment n of the one-step transfer probability, on the state space I = {1, 2,…}, the event of all the one-step state transfer probability p_ij constructed into a matrix, notated as: 5 $P = (\begin{matrix} p_{11} & p_{12} & \dots & p_{1 n} & \dots \\ p_{21} & p_{22} & \dots & p_{2 n} & \dots \\ \dots & \dots & \dots & \dots & \dots \end{matrix})$

The matrix is a one-step state transfer probability matrix for the event state, which has the following properties: I:

p_ij ≥ 0, i, j ∈ I.

II:

$\sum_{j \in I} p_{i j} = 1, i \in I$

The matrix satisfying the above properties is a random matrix, and the conditional probability $p_{i j}^{(n)} = P {X_{m + n} = j | X_{m} = i}$ ,i, j ∈ I, m ≥ 0, n ≥ 1 is the n-step transfer probability of the Markov chain {X (n), n ∈ T}, which can be constructed as a matrix of all the n-step state transfer probabilities of the event, noting $P^{(n)} = (p_{i j}^{(n)})$ as the n -step state transfer probability matrix of the Markov chain. where $p_{i}^{(n)} \geq 0; \sum_{i \in I} p_{i j}^{(n)} = 1$ , i.e., P⁽ⁿ⁾ is also a stochastic matrix.

From the Chepman-Kolmogorov equation (Eq. 6), the n step transfer probability is the n multiplier of the one-step transfer probability: 6 $p_{i j}^{(n)} = \sum_{k \in I} p_{i k}^{(l)} p_{k j}^{(n - l)}$

2)

Initial probability vector

For Markov chain {X (n), n ∈ T}, if the moment n is the initial state, then p_j = P{X_n = j} is called its initial probability, and remember {p_j, j ∈ I} is the initial distribution of the Markov chain, and A^T (n) = ( p₁, p₂,…) is called the initial probability vector.

3)

Prediction results

The product of the initial probability vector A^T (n) and the one-step state transfer probability matrix P is the probability distribution of the state of the event at the moment n + 1, denoted as B⁽ⁿ⁺¹⁾ = (b₁, b₂,…), according to which the probability of the occurrence of the future state can be known. That is, if the largest element in B is b_k, then the event will be in state i_k at the moment of n + 1 with probability b_k, i.e., i_k is the most probable state.

2.2

Emergency response time prediction based on improved Markov process

Gray prediction model modeling requires less data, does not require data with a typical probability distribution, short-term prediction accuracy is higher, and its prediction graph is a smoother curve, which portrays the trend of the development of things, but does not take into account the random volatility of the development of things [23]. For these weaknesses of the gray prediction model, we can compensate for them with the Markov prediction model.

In this paper, we use the gray model to predict the short-term, local trend of the sequence change, and use the Markov model to portray the overall fluctuation law of the sequence, and predict the emergency response time of public emergencies through the organic combination of the two models.

2.2.1

Predicted values of the original series based on the GM(1,1) model

1)

Given a sequence

Let the time series x⁽⁰⁾ of data samples have n sample values: 7 $x^{(0)} = {x^{(0)} (1), x^{(0)} (2), x^{(0)} (3), \dots, x^{(0)} (n)}$ x⁽¹⁾ is a one-time cumulative generating sequence of x⁽⁰⁾, i.e: 8 $x^{(1)} = {x^{(1)} (1), x^{(1)} (2), x^{(1)} (3), \dots, x^{(1)} (n)}$

Among them: 9 $x^{(1)} (k) = \sum_{i = 1}^{k} x^{(0)} (i), (k = 1, 2, \dots, n)$

2)

Modeling

Assuming the exponential nature of the process of change in the x⁽¹⁾ series, the following whitened differential equation can be developed: 10 $\frac{d x^{(1)}}{d t} + a x^{(1)} = μ$ where α, μ is the parameter to be solved, where α is the development coefficient and μ is the gray coefficient.

By discretizing Eq. (3), the differential equation becomes a difference equation: 11 $x^{(0)} (k + 1) + \frac{α}{2} [x^{(1)} (k + 1) + x^{(1)} (k)] =_{μ} (k = 1, 2, \dots, n)$ written in the form of a matrix: 12 $Y = B \hat{α}$ where the vector coefficients: 13 $\hat{α} = {[\begin{matrix} α, μ \end{matrix}]}^{T}$ 14 $B = (\begin{matrix} - \frac{1}{2} (x^{(1)} (1) + x^{(1)} (2)) & 1 \\ - \frac{1}{2} (x^{(1)} (2) + x^{(1)} (3)) & 1 \\ ... & ... \\ - \frac{1}{2} (x^{(1)} (n - 1) + x^{(1)} (n)) & 1 \end{matrix})$ 15 $Y = {[\begin{matrix} x^{(0)} (2), x^{(0)} (3), \dots, x^{(0)} (n) \end{matrix}]}^{T}$

Using the least squares method to find: 16 $\hat{α} = {[\begin{matrix} α, μ \end{matrix}]}^{T} = {(B^{T} B)}^{- 1} B^{T} Y$

3)

Establish the prediction formula

Substituting parameters α and μ obtained from Eq. (15), the time response function is solved: 17 ${\hat{x}}^{(1)} (k + 1) = (x^{(0)} (1) - \frac{μ}{α}) e^{- a k} + \frac{μ}{α} (k = 1, 2, \dots, n)$

Doing a cumulative subtraction of ${\hat{x}}^{(1)} (k + 1)$ yields the reduced value of the initial data series, i.e., the GM (1, 1) predictive model expression: 18 ${\hat{x}}^{(0)} (k) = {\hat{x}}^{(1)} (k) - {\hat{x}}^{(1)} (k - 1) (k = 2, 3, \dots, n)$

To wit: 19 ${\hat{x}}^{(0)} (k) = (1 - e^{a}) (x^{(0)} (1) - \frac{μ}{α}) e^{- a (k - 1)} (k = 2, 3, \dots, n)$

2.2.2

Predicted residual values based on Markovian prediction models

1)

State division

Based on the sequence of original data and the sequence of fitted values of the GM (1, 1) model, we can obtain the residual sequence, which is a non-smooth random sequence with the characteristics of Markov chain. In order to define the state transfer matrix of Markov chain, the residual sequence is divided into t state.

2)

Construct the state transfer probability matrix

In general, the state transfer probability matrix in step m is defined as follows: 20 $R^{(m)} = (\begin{matrix} P_{11}^{(m)} & P_{12}^{(m)} & \dots & P_{1 k}^{(m)} \\ P_{21}^{(m)} & P_{22}^{(m)} & \dots & P_{2 k}^{(m)} \\ ... & ... & ... & ... \\ P_{k 1}^{(m)} & P_{k 2}^{(m)} & ... & P_{k k}^{(m)} \end{matrix})$

Here $P_{i j}^{(m)}$ is the m -step transfer probability from state i to state j, defined as: 21 $P_{i j}^{(m)} = \frac{M_{i j}^{(m)}}{M_{i}^{(m)}} (i, j = 1, 2, 3, \dots, n)$

Where, $M_{i}^{(m)}$ is the number of transfers from state i to state j via m steps, $M_{i}^{(m)} = \sum_{j} M_{i j}^{(m)}$ .

3)

Preparation of the prediction table

Examine the states at r step prior to the predicted moment, and fill in the table with the probabilities of transferring them to each state via r, r–1,…,1 steps respectively, and let the sum of the probabilities of which the outcome is state 1, 2,…, l be ω₁, ω₂,…,ω_l respectively. Let the midpoint values of the l state intervals be ν₁, ν₂,…,ν_l in order. Let the weights of the l states be λ₁, λ₂,…, λ_l respectively.

Definition: 22 $λ_{i} = \frac{ω_{i}}{ω_{1} + ω_{2} + \dots + ω_{l}}, (i = 1, 2, \dots, l)$

Then the residual prediction value z at the predicted moment is: 23 $z = λ_{1} ν_{1} + λ_{2} ν_{2} + \dots + λ_{l} ν_{l}$

2.2.3

Residual corrections to the predicted values of the GM(1,1) model

The formula for the final prediction value ${\tilde{x}}^{(0)} (k)$ is: 24 ${\tilde{x}}^{(0)} (k) = {\hat{x}}^{(0)} (k) + (λ_{1} ν_{1} + λ_{2} ν_{2} + \dots + λ_{l} ν_{l})$

3

Results and discussion

3.1

Presentation of cases

Huainan City is located in the north of Anhui Province, the middle reaches of Huaihe River, the jurisdiction of the Huaihe River main stream length of 76.13km, the general channel width of 400m or so, 250-300m in the dry season, and 400-800m in the flood season, the Huaihe River is the main source of water for the production of agriculture and industry and the life of the people in Huainan City. There are 5 water plants in Huainan City, mainly located in the south bank of Huaihe River and the south bank of Dongfang River, of which: except for Pingshantou Water Plant, whose water source is taken from Wabu Lake, the water source of Lizizuzi Water Plant, Municipal Water Plant No. 4, Municipal Water Plant No. 3 and Municipal Water Plant No. 1 are all taken from the Huainan section of Huaihe River. As the water source of the four water plants in Huainan City, the water quality and safety of the Huaihe River Huainan section is closely related to the normal social production and living water safety of the 2,425,000 people in Huainan City. In recent years, with the rapid social and economic development of Huainan City, the potential risk of environmental accidents in the Huaihe River has increased due to factors such as chemical enterprises near the banks of the Huaihe River and increasingly busy land and water transportation routes, which seriously threaten the safety of water supply in Huainan City.

The Huainan section of the Huaihe River is also the main sewage receiving water body in Huainan City. There is a Kongli mine outfall 700m upstream of the secondary protection area of the water source of Lizuzi Water Plant, and there is a Lizuzi mine outfall 80m downstream in the primary protection area. There is an outfall called Yingtaizi in the secondary protection area, 950m upstream of the water source of the former Wangfenggang Water Plant, and there is also an outfall called Shijian Lake outside the secondary protection area. In the city of a water plant, there is a water source upstream of 150 meters. The primary protection area is 150 meters. In the secondary protection area, there is a small station, harbor all the way, Longwanggou, Yaojiawan, and other four outfalls. Sewage discharged into the river through the outfalls into the water source protection area includes three categories: industrial wastewater, urban domestic sewage, and nonpoint source wastewater (including urban surface runoff and rural surface runoff).

This paper takes the Huainan section of the Huaihe River as the study area, and uses the emergency response time prediction model based on the improved Markov process to estimate the emergency response time of each water plant, with a view to providing a basis for the development of the water supply emergency plan in Huainan City in case of sudden pollution accidents.

3.2

Model Validation

3.2.1

Prediction accuracy metrics

Applying equation (25), the prediction error of the prediction result is calculated. Considering the error tolerance of the predicted variable is ±0.1, when the prediction error is less than the error tolerance, the prediction result is considered correct. Count the number of correctly predicted events in the total number of steps of prediction m_η<0.1, and calculate the prediction accuracy rate RATE of the corresponding method as the accuracy measure of the event prediction result: 25 $η = | \frac{i t_{p r e} - i t_{o b s}}{i t_{o b s}} |$

Eq:

it _pre - predicted value.

it _obs - Sample value.

3.2.2

Analysis of forecast results

According to the methodology proposed in this paper, the data collected from the monitoring points of Lizuzi water plant, city water plant 1, city water plant 3, and city water plant 4 are predicted for the year 2023. Let the amount of data from 2021 to 2023 be ni – 2021, ni – 2022, and ni – 2023, and i is the serial number of the monitoring device. In each data prediction, for monitoring point i, the first j–1 data of 2021, 2022 and 2023 were used as model training samples, and the number of training samples was (ni – 2021 + ni – 2022 + j – 1) for the j th data of 2023. The results of the predicted series obtained are shown in Fig. 1. (a)~(d) represent the comparison between the real and predicted sequences of LiZhouZi water plant, City 1 water plant, City 3 water plant, City 4 water plant, respectively, and it can be seen that the trend of the predicted curve of the emergency response time of the four water plants for the sudden pollution event in 2023 basically overlaps with the trend of the real curve, and the η mean values are 0.076, 0.1141, 0.006, and 0.008, respectively, which indicates that the method proposed in this paper suggests that the method proposed in this paper can reasonably predict the emergency response time.

3.2.3

Comparison of accuracy

The experiment selects the current mainstream prediction methods: gray model (GM), differential integration moving average autoregressive model (ARIMA), fuzzy time series prediction (FTSP), and long-short time memory network (LSTM) to compare the prediction accuracy with the method proposed in this paper. The accuracy of single event prediction was used as a precision measure to predict the emergency response time for a sudden pollution event in 2023 for four water plants. The accuracy comparisons of the five prediction methods are shown in Figure 2. The mean single-event prediction accuracy of this paper’s method is 83.93%, which is 26.5%, 23.53%, 21.35, and 11.59% higher than GM, ARIMA, FTSP, and LSTM, respectively.

3.3

Emergency response time projections

3.3.1

Setting up the situation of sudden water pollution accident

In the simulation calculation of sudden water pollution accident in the Huainan section of the Huaihe River, according to the characteristics of seasonal difference in the flow rate of the upper reaches of this river section, it is divided into the abundant water period and the dry water period for simulation and analysis. The simulation conditions of sudden water pollution were set: a serious pollutant leakage accident was initiated near a river-crossing bridge near the upstream inflow section of the Huainan section of the Huaihe River under the design flow rate in the dry and abundant water periods, and a certain type of organic pollutant with a concentration of 5,000 mg/L (TOX) was continuously leaked for 6 hours.

In different periods corresponding to different inflow flows, the change process of pollutant TOX concentration at the water intake of each water plant under the sudden water pollution accident is shown in Fig. 3. (a)-(b) represent the changes of pollutant concentration at the intake of each water plant under sudden water pollution accident in dry and abundant water periods, respectively. During the dry water period, the highest pollutant concentrations were reached in 3d, 6.6d, 7d, and 7.6d in LiZhouZi water plant, the city’s fourth water plant, the city’s third water plant, and the city’s first water plant, respectively. During the abundant water period, the highest concentration of pollutants was reached at 0.5d at Lizuzi water plant, and the highest concentration was reached at 1.5d at the city’s fourth water plant, the city’s third water plant, and the city’s first water plant.

3.3.2

Predicted response time to water pollution emergencies

Using the method of this paper can know the time of the water intake of each water plant to respond to the occurrence of the upstream pollution load and the time of the water intake of each water plant to be affected, the prediction results are shown in Table 1. After the pollution accident at the water intake of the pollutant concentration reduced to its normal condition concentration of 110% of the moment as the pollutant cloud leaving time, the time difference between the two (i.e., the pollution cloud began to affect the time and the time interval between the time of the pollution cloud leaving time) that is, the time of the water intake affected by the water pollution accident. The results show that under the condition of the same scale of pollutant leakage accident, there exists a certain regularity in the affected time and pollution peak value of the intake of the downstream four major water plants. During the dry season, the emergency response time for the accident at the water intake of LiZhouZi water plant, which is 13 km downstream of the accident site, is 2.59d. The emergency response time to the accident at the City’s 4, 3, and 1 water plant intakes, which are 32 to 39 km downstream from the accident site, is 5.98 to 6.98d. The emergency response time of each water plant in the abundant water period is between 0.34 and 0.66d. Comparison with the results of the analysis in Figure 3, based on the method of this paper predicts that the response time of each water plant pollution accident is less than the time when the pollutant reaches the maximum concentration, and each water plant has sufficient time for emergency management of pollution accidents.

Table 1.

Emergency time and impact duration of each water plant

	Dry season		Flood season
	Emergency time/d	Impact duration/d	Emergency time/d	Impact duration/d
Li Zuizi Water Plant	2.59	1.42	0.43	0.34
Fourth Municipal Water Plant	5.98	2.96	1.16	0.64
Third Municipal Water Plant	6.73	2.74	1.23	0.66
First Municipal Water Plant	6.98	2.98	1.35	0.53

4

Conclusion

In this study, we successfully constructed an emergency response time prediction model for public emergencies based on the stochastic process model and verified its effectiveness in real cases. In the simulation calculations of sudden water pollution accidents in the Huainan section of the Huaihe River, during the dry water period, the pollutants reached the highest concentration at LiZhouzi Water Plant, City 4 Water Plant, City 3 Water Plant, and City 1 Water Plant in 3d, 6.6d, 7d, and 7.6d, respectively. During the abundant water period, the highest concentration of pollutants was reached at 0.5d at Lizuzi water plant, and the highest concentration was reached at 1.5d at the city’s fourth water plant, the city’s third water plant, and the city’s first water plant. The model predicts the emergency response time of each water plant accident in the dry water period as 2.59 d, 5.98 d, 6.73 d, 6.98 d. The emergency response time of each water plant in the abundant water period is 0.43 d, 1.16 d, 1.23 d, 1.35 d. The response time of each water plant pollution accident predicted by the model of this paper is less than the time of the pollutant reaching the maximum concentration, which indicates that the model of this paper can provide the accuracy of the emergency response time.

Lingua:: Inglese

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Application of stochastic process modeling in the prediction of emergency response time for public emergencies

Runhan Zhang

Pubblicato online: 19 mar 2025

Ricevuto: 09 ott 2024

Accettato: 06 feb 2025

DOI: https://doi.org/10.2478/amns-2025-0382

Parole chiaveStochastic process, Markov process, Gray prediction model, Emergency response time

© 2025 Runhan Zhang, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Parole chiave
Stochastic process, Markov process, Gray prediction model, Emergency response time