A modeling method of repairman multi-stage delay vacation system based on general distribution

Abstract

The application provides a modeling method of a repairman multistage delay vacation system based on general distribution, which comprises the following steps: step one, establishing a system model under the repairman multistage delay vacation strategy; step two, transforming the repairman multistage delay vacation strategy model into other models; step three, preliminarily solving the system state equation at the steady state and calculating the parameters at the steady state; step four, introducing constants and performing equivalent transformation of the equation; and step five, calculating the system state steady-state probability. The application is helpful for providing a theoretical basis for research and analysis of a system under a similar condition in practice, solving the availability, and providing an effective decision-making means for managers to make resource allocation and personnel vacation decisions.

Description

Technical Field

[0001] This invention provides a modeling and solution method for a multi-level extended leave system for repair workers under a general distribution, which is an innovative method in the field of engineering applications. Background Technology

[0002] In existing research on repairable systems, modeling not only considers the system's own design and structural characteristics but also incorporates the influence of external factors (such as spare parts, support equipment, and repairmen) into the system state modeling. By studying the impact of external factors on changes in system state, system state models under different influencing factors are established. Since system failures do not always allow for immediate repair, studying the repairman problem can make the model more closely reflect actual engineering conditions.

[0003] In the context of repairman leave, "leave" is a broad concept; it could mean the repairman is on vacation or performing other work. Many scholars have explored how to limit the time repairmen spend on auxiliary tasks or how to optimally schedule their leave frequency, resulting in numerous different leave strategies and the establishment of corresponding state models based on these strategies. However, existing single and multiple leave strategies mostly assume that the repairman begins their leave at the initial moment, which is clearly inconsistent with reality. In practice, whether in the initial state or immediately after completing repairs, the repairman needs to remain in the system for a period of time to ensure everything is functioning correctly before taking leave. Summary of the Invention

[0004] (1) Objective of this invention

[0005] The purpose of this invention is to provide a modeling and solution method for a multi-level extended leave system for repair workers under a general distribution, which helps to provide a theoretical basis for the research and analysis of systems in similar situations in reality and the solution of availability, and provides managers with an effective decision-making tool for resource allocation and personnel leave decisions.

[0006] (2) Technical solution of the present invention

[0007] This invention proposes a multi-level delayed leave strategy for repairmen under certain assumptions. By describing the system operation process and making conditional assumptions, the state transition process of the system under this leave strategy is obtained. The supplementary variable method is used to describe the system state transition process as a generalized Markov process. The model is built according to Markov process theory and solved using the solution method under the general distribution.

[0008] This invention presents a system modeling and solution method considering multi-level delayed leave for spare parts and repairmen under a general distribution, implemented through the following five steps:

[0009] Step 1: Establish a system model under the multi-level extended leave strategy for repairmen;

[0010] A vacation strategy for repairmen is proposed and described. The vacation strategy is analyzed, and a system model under the multi-level extended vacation strategy for repairmen is established based on the assumptions.

[0011] Step 2: Transformation of the multi-level extended leave strategy model for repairmen into other models;

[0012] Adjust the parameters in the model and use the multi-level delayed leave strategy model for repairmen to represent the single leave model, multiple leave model, single delayed leave model, multiple delayed leave model, and multi-level adaptive leave model;

[0013] Step 3: Preliminary solution of the system state equations and steady-state parameter calculation;

[0014] By applying the supplementary variable method, a state dwell time is introduced to represent the time the component spends in its current state. The non-Markov process is then transformed into a generalized Markov process using the properties of the regeneration point. Expressions for the steady-state probabilities of each system state are obtained, thus completing the solution for the system's steady-state probability.

[0015] Step 4: Introduce constants and perform equivalent transformations of the equations;

[0016] Combining knowledge such as the mean value theorem of generalized integrals, constants are introduced to perform equivalent transformations on the state equations of the system.

[0017] Step 5: Calculate the steady-state probability of the system;

[0018] Solve the homogeneous linear equations to obtain the steady-state probabilities of the system under different states.

[0019] Advantages of the present invention

[0020] The model is characterized by its compatibility with five different leave models for repairmen: single leave, multiple leave, single extended leave, multiple extended leave, and multi-level adaptive leave. Under certain conditions, it can be transformed into these five models.

[0021] Most current models assume that a certain variable in the system follows an exponential distribution for ease of solution. However, in reality, the transition time distribution between states in a system is uncertain. Therefore, for modeling purposes, choosing a model with a fully generalized distribution would be more practical. This invention will assume that the repairman's leave rate follows a generalized distribution, which is more consistent with the actual situation of equipment use. Attached Figure Description

[0022] Figure 1 This is a diagram showing the state changes during the system's working cycle.

[0023] Figure 2 State transition diagram for the K-level extended leave system for repairmen.

[0024] Figure 3 This is a diagram showing the transition between multiple delayed leave statuses. Detailed Implementation

[0025] Step 1: Establish a system model under the multi-level extended leave strategy for repair workers

[0026] (1) First, define a multi-level delayed leave strategy for repairmen. When the system first starts running, the repairman plans to take K leave sessions. Initially, the repairman stays on duty for a period of time, i.e., a delay time T before the leave. During this time, if no component in the system fails, the repairman enters leave mode; if a component fails, the repairman performs repairs. Each time a repairman completes their leave or restores the system to its original condition, they remain on duty for a period to ensure no system failure occurs before entering the next leave mode. After each leave session, the number of leave sessions taken is checked. If the number of leave sessions is less than K, the repairman moves to the next leave session after the remaining time; if the number of leave sessions equals K, the repairman does not take any more leave and remains on duty until a system failure occurs, at which point the repair work is completed, and the next K-level delayed leave session with K leave sessions begins. Based on the above description, the system's state changes during the work cycle as follows: Figure 1 As shown:

[0027] Figure 1 In the diagram, the X-axis represents the absolute time from the initial state to the start of operation, the Y-axis represents the state of the system during operation, and T is the delay before each repairman's leave.

[0028] (2) Define all possible states of the system at time t;

[0029] Status S indicates that the repairman is on duty but not in use, and the system is working normally;

[0030] State N indicates that the repairman is on vacation and the system is working normally;

[0031] Status M indicates that a repairman is on duty and the system is malfunctioning;

[0032] Status H indicates that the repairman is on vacation and the system is malfunctioning;

[0033] State S0: N(t) = S0, the repairman is on duty but not in use, the system is working normally, and the repairman has taken 0 days off at this time;

[0034] State S j N(t) = S jj = 1, 2, 3…K. The repairman is on duty but not in use, the system is working normally, and the repairman has already taken j days of leave.

[0035] State N j N(t) = N j j = 1, 2, 3…K. The repairman is on vacation, the system is working normally, and the repairman has taken vacation j times.

[0036] State M0: N(t) = M0; The repairman is on duty, the system malfunctions, and the repairman has already taken 0 days off.

[0037] State M j N(t) = M j j = 1, 2, 3…K. A repairman is on duty repairing the system when a malfunction occurs. At this point, the repairman has already taken j days of leave.

[0038] State H j N(t) = H j Let j = 1, 2, 3…K. The repairman is on vacation, the system malfunctions, and the repairman has already taken j vacations.

[0039] Where N represents the number of planned vacation days.

[0040] (3) Establish the state transition diagram and the state equation of the N-level delayed leave system for repairmen;

[0041] Based on the system's state and assumptions, the state transition diagram of the system at any given time is as follows: Figure 2 As shown:

[0042] Figure 2 In the diagram: λ(x) represents the degradation rate of components in the system; r(x) represents the rate at which a repairman completes his leave; μ(x) represents the repair rate of faulty components; z 1j , z 2j It is the cumulative time during which the system has not experienced a fault when entering the current state; 3j It is the vacation time that the repairman has completed when entering the current state (j = 1, 2, 3...K).

[0043] From the above analysis, the steady-state state equation of the system can be obtained as follows:

[0044]

[0045] Among them, P i (t, x) represents the probability that a system component is in state i (i = S, N, M, H) at time t and stays in state i for a time of x (0 < x < t).

[0046] Step Two: Transformation of the multi-level extended leave strategy model for repairmen into other models.

[0047] (1) Changed to a single leave

[0048] If we set the number of planned leave days K for the repairman to be 1 and the delay time T before the leave to be 0, then the model is the same as the single leave model.

[0049] The state equation of the system at this time is as follows:

[0050]

[0051] (2) Become multiple leave

[0052] If we set the number of planned leave days K for the repairman to be ∞ and the delay time T before the leave to be 0, then the model is the same as the multiple leave model.

[0053] The system state equations are as follows:

[0054]

[0055] (3) Changed to a single extended leave

[0056] If we set the number of planned leave days K for the repairman to be 1, then the model is the same as the single-delay leave model. The system's state equations are as follows:

[0057]

[0058] (4) Become multiple extended leave

[0059] If we set the number of planned leave days K for repairmen to be ∞, then the model is the same as that of multiple delayed leave.

[0060] The state equation of the system at this time is as follows:

[0061]

[0062] (5) Transform into a multi-level adaptive leave system

[0063] If we set the delay time T before the repairman takes leave to 0, then the model is the same as the multi-level adaptive leave model.

[0064] The state equation of the system at this time is as follows:

[0065]

[0066] in: This indicates that the system enters N. K The cumulative time during which no faults occurred.

[0067] Step 3: Preliminary solution to the system state equations and steady-state parameter calculations

[0068] (1) When the system is in steady state, the absolute time t of the system in the model approaches infinity, then the partial derivative of t is 0. At this time, the boundary conditions and state equations are no longer affected by t, and the state equations of the system can be rewritten as:

[0069]

[0070] in: This represents the probability that a system component is in state S0 at steady state time t→∞ and stays in that state for a time ; This indicates that the system component is in state S at steady state t→∞. j The probability of entering a state and remaining in that state for a time of x; This indicates that the system component is in state N at steady state t→∞. j The probability of entering a state and remaining in that state for a time of x; This indicates that the system component is in state M at steady state t→∞. j The probability of entering a state and remaining in that state for a time of x; This indicates that the system component is in state H at steady state t→∞. j The probability of entering a state and staying in that state for a time of x.

[0071] The system boundary conditions are:

[0072]

[0073] Where: parameters This represents the probability that a system component is in state S0 at steady state time t→∞; This indicates that the system component is in state S at steady state t→∞. j The probability of a state; This indicates that the system component is in state S at steady state t→∞. K The probability of a state; This represents the probability that a system component is in state N1 at steady state t→∞; This indicates that the system component is in state N at steady state t→∞. j The probability of the state; This represents the probability that a system component is in state M0 at steady state time t→∞; This indicates that the system component is in state M at steady state t→∞. j The probability of a state; This indicates that the system component is in state M at steady state t→∞. K The probability of a state; This indicates that the system component is in state H at steady state t→∞. j The probability of a state; This indicates that the system component is in state S at steady state t→∞. (j-1) The probability of a state; Is the system entering M? K The cumulative time during which no faults occurred; The repairman entered M K The status indicates the vacation time that has been completed.

[0074] Solving the differential equation in the above formula yields:

[0075]

[0076] (2) In the formula, the variable z 1j , z 2j , z 3j The expression is as follows:

[0077] Assume the repairman's pre-work delay is T, and the expected vacation time is T. r The expected repair time for the faulty component is T. μ The system starts from N j Status transition to H j The time taken for each state is T. λ2 ,

[0078] When the system reaches steady state, the expected values ​​at each time step are as follows:

[0079]

[0080]

[0081]

[0082] z 1j , z 2j , z 3j The expression is as follows:

[0083]

[0084] in: The z-value represents the system in steady state. 1j , z 2j , z 3j Value, k S，j，Input For S j The proportion of states transitioned, k N，S，j For the system from N j The state returns to S j The probability of a state, k s，N，j For the system from S j State transition to N j+1 The probability of a state.

[0085] Step 4: Introduce constants and perform equivalent transformations of the equations;

[0086] Define coefficients α r α μ The output rate of the previous state between adjacent states:

[0087]

[0088] Will α r α μ Substituting these values ​​into the expression obtained after solving the differential equation, we can transform it into:

[0089]

[0090] To continue the equivalent transformation system of equations, we define λ 1h * 、(r+λ 2j ) * , μ * The input rate of the next state between adjacent states.

[0091]

[0092] The probability of system components in each steady state Integrating both sides of the system of equations (1), we get:

[0093]

[0094] Step 5: Calculation of the steady-state probability of the system

[0095] When the system is in steady state, the coefficients in the expression relating the probabilities of its states are... α r α μ λ represents the output rate of the previous state between adjacent states. 1j * 、(r+λ 2j ) * , μ * This represents the input rate of the next state between adjacent states. For a single-input, single-output system, this represents the output rate between adjacent states. α μ With input rate λ 1K * r 3j * μ * satisfy α μ =μ * For a system with multiple outputs (including discrete outputs), the output rate between adjacent states in the system is... With input rate λ 1j* satisfy For a single-input multiple-output system, the output rate α between adjacent states in the system is... r , With input rate (r+λ) 2j ) * satisfy

[0096] When the system is in steady state, the output of a single-output state node per unit time is equal to the sum of the inputs of all its previous states; the output of a multi-output state node to each output state per unit time is equal to the sum of the inputs of all its previous states, but the output to each output state is affected by the corresponding output rate and there is a certain proportional relationship.

[0097] The transition ratios for each multi-output state in the system are defined as follows:

[0098]

[0099] Taking the output ratio of state N into account, we can obtain:

[0100] α r =k N，S，j (r+λ 2j ) * ,

[0101] Based on this, and combining it with the law of total probability, we can obtain the following homogeneous linear equation system:

[0102]

[0103] Solving this homogeneous linear system of equations yields the steady-state probability of the system in each state.

[0104] Example

[0105] This invention presents a method for modeling and solving a multi-level delayed leave system for repair workers under a general distribution, which can be applied to practical engineering problems involving modeling and solving repair worker leave issues. Taking an n-component cascaded repairable system in an unattended communication base station in the field, where a certain repairable device is replaceable and repair workers can have multiple delayed leave periods, as an example, the component lifespan distribution, the repair worker delayed leave time distribution, and the repair device lifespan distribution all follow exponential distributions. Simultaneously, the faulty component repair time distribution, the repair worker leave time distribution, and the repair device replacement time distribution all follow general distributions, and under steady-state conditions, the mean faulty component repair time, the mean repair worker leave time, and the mean repair device replacement time are respectively... The method of this invention can solve for its steady-state availability.

[0106] The system has a failure rate λ(x) = 1, a vacation rate r(x) = 0.5, a pre-vacation delay rate ε(x) = 1, and a maintenance rate μ(x) = 1.

[0107] In this model, let the number of planned leave days K = ∞, simplifying the system model to a multiple-delay leave model for repairmen, resulting in the system state transition diagram as follows: Figure 3 As shown:

[0108] The state equation of the system at this time is:

[0109]

[0110] The boundary conditions and initial conditions are as follows:

[0111]

[0112] From the above equation, the differential equation can be solved to obtain:

[0113]

[0114] Let α be a constant. λ1 α ε α λ2 α r α r3 α μ :

[0115]

[0116] α λ1 α ε α λ2 α r α λ3 α μ Substitute into the formula:

[0117]

[0118] Define the following constant (λ1+ε). * (λ2+r) * r3 * μ * :

[0119]

[0120] Define the output ratio for multiple output states:

[0121]

[0122] in, for:

[0123]

[0124] When a system component reaches a steady state, the output rate between its adjacent states; α λ2 +α r =(λ²+r) * 、(α λ1 +α ε )=(λ1+ε) * ;α r3 α μ With input rate r3 * μ * They are equal. Using the law of total probability, the solution matrix is ​​as follows:

[0125]

[0126] The steady-state probabilities of the system in each state can be calculated using the matrix. Substituting the failure rate λ(x) = 1, the vacation rate r(x) = 0.5, the pre-vacation delay rate ε(x) = 1, and the maintenance rate μ(x) = 1, the results are obtained by substituting the inputs into the formula using Matlab: P S =0.2143, P N =0.1429, P M =0.3571, P H =0.2857.

Claims

1. A modeling method for a multi-level extended leave system for repair workers based on a general distribution, characterized in that, Includes the following steps: Step 1: Establish a system model under the multi-level extended leave strategy for repairmen; A vacation strategy for repairmen is proposed and described. The vacation strategy is analyzed, and a system model under the multi-level extended vacation strategy for repairmen is established based on the assumptions. Step 2: Transformation of the multi-level extended leave strategy model for repairmen into other models; Adjust the parameters in the model and use the multi-level delayed leave strategy model for repairmen to represent the single leave model, multiple leave model, single delayed leave model, multiple delayed leave model, and multi-level adaptive leave model; Step 3: Preliminary solution of the system state equations and steady-state parameter calculation; By applying the supplementary variable method, the state dwell time is introduced to represent the time the component stays in the current state. The non-Markov process is transformed into a generalized Markov process by utilizing the property of the regeneration point. The expressions for the steady-state probabilities of each state of the system are obtained, thus completing the solution for the steady-state probabilities of the system; Step 4: Introduce constants and perform equivalent transformations of the equations; Combining the mean value theorem of generalized integrals, we introduce constants to perform equivalent transformations on the system's state equations; Step 5: Calculate the steady-state probability of the system; Solve the homogeneous linear equations to obtain the steady-state probabilities of the system under different states; In step one, a multi-level delayed leave strategy for repairmen is defined. When the system first starts running, the repairman plans to take K leave sessions. Initially, the repairman stays on duty for a period of time, known as the pre-leave delay time T. If no component in the system malfunctions during this time, the repairman enters leave mode; if a component malfunctions, the repairman performs repairs. Each time the repairman completes their leave or restores the system to its original condition, they remain on duty for a period to ensure no system malfunctions before entering the next leave mode. After each leave session, the number of leave sessions taken is checked. If the number of leave sessions is less than K, the repairman moves to the next leave session after the remaining time. If the number of leave sessions equals K, the repairman does not take any more leave and remains on duty until a system malfunction occurs, the repair work is completed, and the next K-level delayed leave session with K leave sessions is initiated. In step one, the state equations for the system in steady state are established; ； in, Indicates system components in Always in a state of readiness and in state The length of stay is The probability of; Indicates the degradation rate of components in the system; This indicates the rate at which the repairman's leave is completed; Indicates the repair rate of faulty parts; state S This indicates that the repairman is available and the system is functioning normally. state N This indicates that the repairman is on vacation and the system is functioning normally. state M This indicates that a repairman is on duty and the system is malfunctioning. state H This indicates that the repairman is on vacation and the system is malfunctioning. state : The repairman is on duty but not in use, the system is working normally, and the repairman has taken 0 days off. state : j=1,2,3…K; The repairman is on duty but not in use, and the system is working normally. At this time, the repairman has taken j days off. state : j=1,2,3…K; The repairman is on vacation, and the system is working normally. At this point, the repairman has taken vacation j times. state : The repairman was on duty when the system malfunctioned. At this point, the repairman had already taken 0 days off. state : j=1,2,3…K; The repairman is on duty repairing the system when a fault occurs. At this point, the repairman has already taken j days off. state : j=1,2,3…K; The repairman is on vacation, the system malfunctions, and the repairman has already taken j vacations. in, Indicates the state when the system is in steady state value; This indicates that the system enters a state under steady-state conditions. The total time without a failure; This indicates that the system enters a state under steady-state conditions. The total time without a failure; This indicates that, under steady-state conditions, the repairman is in state... The vacation time that has been completed, j=1,2,3…K.

2. The modeling method for a multi-level extended leave system for repairmen based on a general distribution, as described in claim 1, is characterized in that: In step two, the single leave model is as follows: let the number of planned leave K for the repairman be 1, and the delay time T before the leave be 0, then the model is the same as the single leave model; The state equation of the system at this time is as follows: ； in, This indicates that the system enters a state under steady-state conditions. The total time without a failure; This indicates that the system enters a state under steady-state conditions. The total time without a failure; This indicates that, under steady-state conditions, the repairman is in state... Completed vacation time; The multiple vacation model is as follows: If we set the number of planned leave days K for the repairman to be ∞ and the delay time T before the leave to be 0, then the model is the same as the multiple leave model. The system state equations are as follows: ； in, This indicates that the system enters a state under steady-state conditions. The total time without a failure; This indicates that, under steady-state conditions, the repairman is in state... Completed vacation time; The single-delayed leave model is as follows: If we set the number of planned leave days K for repairmen to be 1, then the model is the same as the single-delay leave model; The state equation of the system at this time is as follows: ； in, This indicates that the system enters a state under steady-state conditions. The total time without a failure; The multiple delayed leave model is as follows: If we set the number of planned leave days K for repairmen to be ∞, then the model is the same as that of multiple delayed leave; The state equation of the system at this time is as follows: ； The multi-level adaptive vacation model is as follows: If we set the delay time T before the repairman takes leave to 0, then the model is the same as the multi-level adaptive leave model. The state equation of the system at this time is as follows: ； in: Indicates that the system is entering The cumulative time during which no faults occurred.

3. The modeling method for a multi-level extended leave system for repairmen based on a general distribution, as described in claim 2, is characterized in that: In step three, when the system is in steady state, the absolute running time of the system in the model is... If the partial derivative of t is 0, then the boundary conditions and state equations are no longer affected by t, and the state equations of the system can be rewritten as: ； in: This indicates that the system components are in a steady state. Always in The state and the time spent in that state is The probability of; This indicates that the system components are in a steady state. Always in The state and the time spent in that state is The probability of; This indicates that the system components are in a steady state. Always in The state and the time spent in that state is The probability of; This indicates that the system components are in a steady state. Always in The state and the time spent in that state is The probability of; This indicates that the system components are in a steady state. Always in The state and the time spent in that state is The probability of; The system boundary conditions are: ； Where: parameters This indicates that the system components are in a steady state. Always in The probability of a state; This indicates that the system components are in a steady state. Always in The probability of a state; This indicates that the system components are in a steady state. Always in The probability of a state; This indicates that the system components are in a steady state. Always in The probability of a state; This indicates that the system components are in a steady state. Always in The probability of the state; This indicates that the system components are in a steady state. Always in The probability of a state; This indicates that the system components are in a steady state. Always in The probability of a state; This indicates that the system components are in a steady state. Always in The probability of a state; This indicates that the system components are in a steady state. Always in The probability of a state; This indicates that the system components are in a steady state. Always in The probability of a state; Is the system entering? The cumulative time during which no faults occurred; It was the repairman who entered. The status indicates the duration of vacation that has already been completed; This indicates that the system enters a state under steady-state conditions. The cumulative time without failure. This indicates the rate of degradation of components in the system.

4. The modeling method for a multi-level extended leave system for repairmen based on a general distribution, as described in claim 3, is characterized in that: In step three, solving the differential equation yields: ； variables in the formula The expression is as follows: Assume the repairman's pre-delay time is T, and the expected vacation time is... The expected repair time for the faulty component is The system from Status transition The time taken for each state is , When the system reaches steady state, the expected values ​​at each time step are as follows: ； The expression is as follows: ； in: It is the cumulative time during which the system has not experienced a fault when it enters the current state; This refers to the leave time that the repairman has already taken when entering the current state; Indicates the state when the system is in steady state value, for The proportion of states transitioned into a new state. For the system from Status returned to The probability of a state. For the system from State transition The probability of a state.

5. The modeling method for a multi-level extended leave system for repairmen based on a general distribution, as described in claim 4, is characterized in that: In step four, the coefficients are defined. , , , , The output rate of the previous state between adjacent states: ； Will , , , , Substituting these values ​​into the expression obtained after solving the differential equation, we can transform it into: 。 6. The modeling method for a multi-level extended leave system for repairmen based on a general distribution, as described in claim 5, is characterized in that: In step four, to continue the equivalent transformation equation system, we define... The input rate of the next state between adjacent states; ； The probability of system components in each steady state Integrating both sides of the system of equations (1), we get: ； in, The input rate of the next state between adjacent states.

7. The modeling method for a multi-level extended leave system for repairmen based on a general distribution, as described in claim 6, is characterized in that: In step four, when the system is in steady state, the coefficients in the expression relating the probabilities of its various states are... , , , , The output rate of the previous state between adjacent states. The input rate of the next state between adjacent states; for a single-input, single-output state in the system, the output rate between adjacent states. With input rate satisfy , , For a system with multiple output states, the output rate between adjacent states in the system. With input rate satisfy For a single-input multiple-output (SMILE) system, the output rate between adjacent states in the system is... With input rate satisfy .

8. The modeling method for a multi-level extended leave system for repairmen based on a general distribution, as described in claim 7, is characterized in that: In step four, when the system is in steady state, the output amount of a single-output state node per unit time is equal to the sum of the input amounts of all its previous states; the output amount of a multi-output state node to each output state per unit time is equal to the sum of the input amounts of all its previous states, but the output amount to each output state is affected by the corresponding output rate and there is a certain proportional relationship. The transition ratios for each multi-output state in the system are defined as follows: ； Taking the output ratio of state N into account, we get: ； Based on this, and combining it with the law of total probability, we obtain the following homogeneous linear equation system: ； Solving this homogeneous linear system of equations yields the steady-state probability of the system in each state; where, For the system from Status returned to The probability of a state.

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