A markov chain based distributed queue state prediction synchronization system

Abstract

The application relates to the technical field of deep learning, and discloses a distributed queue state prediction synchronization system based on a Markov chain, which comprises a calculation node arrival rate and a node service rate, blind area differential drift generation, blind area echo gain generation, effective arrival rate generation, closed-loop transfer matrix generation, future state distribution generation, and full-network prediction vector generation and output. The application unifies modeling of a message aging time length of a self-service terminal queue state and a user walking migration time length, constructs a quantifiable migration echo characteristic index, integrates the index into a Markov evolution process of the queue state, realizes synchronization prediction of a future queue state of the distributed self-service terminal, adapts to an operation characteristic of an asynchronous state reported by the distributed self-service terminal, restores an influence process of a user migration behavior on the queue state, outputs a full-terminal queue prediction result, and improves operation stability and service bearing capacity of the distributed self-service system.

Description

Technical Field

[0001] This invention relates to the field of deep learning technology, and more specifically, to a distributed queue state prediction and synchronization system based on Markov chains. Background Technology

[0002] Currently, large public venues, including hospital outpatient clinics, government service halls, and bank branches, widely deploy distributed self-service terminals for handling related business. Each terminal forms an independent local service queue, and queue information for each terminal is displayed on public information screens within the venue, guiding users to choose the appropriate terminal to complete their business. With the popularization of self-service models, the management and guidance of distributed queues has become a key factor affecting the service efficiency and user experience of these venues. Existing queue management solutions mostly rely on real-time queue length data reported by the terminals to guide users in selecting the shortest queue.

[0003] In actual operation, existing technologies have fixed time intervals for terminal status reporting, and the queue information displayed on wayfinding screens has an inherent time delay. Users also need physical movement time from observing the wayfinding information to walking to the target terminal. During this time, the queue status of each terminal continuously changes as the user arrives and the service is completed. What the user perceives as a short queue may become a long queue by the time the user reaches the target terminal, creating a directional echo of migrating people that continuously impacts the queue operation status of the target terminal.

[0004] Current technologies fail to perform structured analysis and quantitative modeling of this migration echo phenomenon, relying solely on static queue length comparisons to design the guidance logic. This approach cannot predict the impact of user migration behavior on future queue states, leading to a systematic deviation between queue predictions and the actual terminal operation. This deviation can further trigger secondary user migrations, causing frequent movement of people and queue oscillations within the venue, ultimately reducing the overall service efficiency and capacity of distributed self-service terminals. Summary of the Invention

[0005] This invention provides a distributed queue state prediction and synchronization system based on Markov chains, which solves the technical problems mentioned in the background.

[0006] This invention provides a distributed queue state prediction and synchronization system based on Markov chains, comprising: The first module acquires node queuing data, spatiotemporal state data, and synchronization cycle span. It converts the node queuing data into node arrival rate and node service rate, extracts message aging time and terminal walking time from the spatiotemporal state data, and reads the observation queue length. The second module generates the observation queue difference by subtracting the observation queue length, and calculates the blind zone differential drift based on the node arrival rate, node service rate, message aging time and terminal walking time. The third module generates blind zone echo gain based on the observation queue difference and blind zone differential drift; The fourth module calculates the node selection probability based on the observation queue length, node service rate, and terminal walking time. It then calculates the additional arrival rate by combining the node selection probability, node arrival rate, and blind zone echo gain. Finally, it incorporates the additional arrival rate into the node arrival rate to generate the effective arrival rate. The fifth module generates a state transition matrix based on the effective arrival rate, node service rate, and synchronization cycle span. It accumulates the overflow transition probability exceeding the node number limit to the boundary state corresponding to the node number limit to generate a closed-loop transition matrix. The sixth module maps the observation queue length to an initial state vector, and multiplies the initial state vector with the closed-loop transition matrix to generate the future state distribution. The seventh module converts the future state distribution into the prediction queue length, concatenates the prediction queue lengths of all nodes to generate the full network prediction vector, and outputs it.

[0007] The beneficial effects of this invention are as follows: By uniformly modeling the message aging time and user walking migration time of the self-service terminal queue status, this invention constructs a quantifiable migration echo characteristic index. This index is then integrated into the Markov evolution process of the queue status, enabling synchronous prediction of the future queue status of distributed self-service terminals. This invention is adaptable to the asynchronous reporting characteristics of distributed self-service terminals, reconstructs the impact of user migration behavior on the queue status, and outputs a unified future queue prediction result for all terminals. This invention can provide data support for queue guidance at self-service terminals in public places, optimize the distribution of pedestrian flow within the venue, and improve the operational stability and service capacity of distributed self-service systems. Attached Figure Description

[0008] Figure 1 This is a flowchart of the computation of a distributed queue state prediction and synchronization system based on Markov chains according to the present invention. Figure 2 This is a comparison chart of the entire network queue prediction in this invention; Figure 3 This is the migration echo effect diagram of the present invention; Figure 4 This is a future state distribution diagram of the present invention. Detailed Implementation

[0009] The subject matter described herein will now be discussed with reference to exemplary embodiments. It should be understood that these embodiments are discussed only to enable those skilled in the art to better understand and implement the subject matter described herein, and changes may be made to the function and arrangement of the elements discussed without departing from the scope of this specification. Various processes or components may be omitted, substituted, or added as needed in the examples. Furthermore, features described in some examples may be combined in other examples.

[0010] It should be noted that, unless otherwise defined, the technical or scientific terms used in one or more embodiments of the present invention should have the ordinary meaning understood by one of ordinary skill in the art to which this invention pertains. The terms "first," "second," and similar terms used in one or more embodiments of the present invention do not indicate any order, quantity, or importance, but are merely used to distinguish different components. Terms such as "comprising" or "including" indicate that the element or object preceding the term encompasses the elements or objects listed following the term and their equivalents, without excluding other elements or objects. Terms such as "connected" or "linked" are not limited to physical or mechanical connections, but can include electrical connections, whether direct or indirect. Terms such as "upper," "lower," "left," and "right" are used only to indicate relative positional relationships; when the absolute position of the described object changes, the relative positional relationship may also change accordingly.

[0011] like Figures 1-4 As shown, a distributed queue state prediction and synchronization system based on Markov chains includes: The first module acquires node queuing data, spatiotemporal state data, and synchronization cycle span. It converts the node queuing data into node arrival rate and node service rate, extracts message aging time and terminal walking time from the spatiotemporal state data, and reads the observation queue length. The second module generates the observation queue difference by subtracting the observation queue length, and calculates the blind zone differential drift based on the node arrival rate, node service rate, message aging time and terminal walking time. The third module generates blind zone echo gain based on the observation queue difference and blind zone differential drift; The fourth module calculates the node selection probability based on the observation queue length, node service rate, and terminal walking time. It then calculates the additional arrival rate by combining the node selection probability, node arrival rate, and blind zone echo gain. Finally, it incorporates the additional arrival rate into the node arrival rate to generate the effective arrival rate. The fifth module generates a state transition matrix based on the effective arrival rate, node service rate, and synchronization cycle span. It accumulates the overflow transition probability exceeding the node number limit to the boundary state corresponding to the node number limit to generate a closed-loop transition matrix. The sixth module maps the observation queue length to an initial state vector, and multiplies the initial state vector with the closed-loop transition matrix to generate the future state distribution. The seventh module converts the future state distribution into the prediction queue length, concatenates the prediction queue lengths of all nodes to generate the full network prediction vector, and outputs it.

[0012] In one embodiment of the present invention, node queuing data, spatiotemporal state data, and synchronization cycle span are acquired; the node queuing data is converted into node arrival rate and node service rate; message aging duration and terminal walking time are extracted from the spatiotemporal state data; and the observation queue length is read, including: The formula for calculating the node arrival rate, based on the number of nodes arriving and the synchronization cycle span, is as follows:

[0013] in, For node arrival rate, The number of people who arrived at the node. For the synchronous period span; The formula for calculating the node service rate, based on the number of users served by the node and the synchronization cycle span, is as follows:

[0014] in, For node service rate, The number of people served by the node. For the synchronous period span.

[0015] It should be noted that node queuing data is the full operational data generated by the self-service terminal during its operation, directly related to the queue's operational status. Spatiotemporal state data describes the temporal and spatial attributes of the self-service terminal's status. The synchronization cycle span is the unified time scale for the system to execute a complete queue status prediction synchronization process. The number of nodes arriving is the total number of new entrants to the corresponding self-service terminal's local queue within the most recent synchronization cycle span. The number of nodes completing service is the total number of entrants who completed service and left the queue within the most recent synchronization cycle span. The node arrival rate is the average number of arrivals at the corresponding self-service terminal per unit time, calculated by dividing the number of node arrivals by the synchronization cycle span. The node service rate is the average number of service completions at the corresponding self-service terminal per unit time, calculated by dividing the number of service completions by the synchronization cycle span. The message aging time is the time interval from the generation time of the most recently reported status data message from the corresponding self-service terminal to the current system prediction time. The terminal walking time is the average time required for a user to walk from the location of the source self-service terminal to the location of the target self-service terminal. The observation queue length corresponds to the real-time number of people queuing in the local queue, as reported most recently by the self-service terminal.

[0016] It should be noted that the value of the synchronization period span is based on the following factors: First, the average time for users to migrate across terminals in the target scenario, which is typically between 5 and 15 seconds. The synchronization period span needs to cover this range to ensure that the predicted time window matches the user's migration behavior. Second, the information refresh frequency of the wayfinding screens. In public places, users typically need to refresh the wayfinding information every 1 to 30 seconds. The synchronization period span needs to match the refresh frequency to ensure the real-time nature of the wayfinding information. Finally, the terminal's status reporting frequency. Self-service terminals typically report status data every 5 to 20 seconds. The synchronization period span needs to match the reporting frequency to ensure stable input data. The preferred range for the synchronization period span is 5 to 20 seconds, with a preferred value of 10 seconds. When the scene is during peak hours, the synchronization period span can be adjusted to 5 seconds to improve the prediction and refresh frequency. When the scene is during off-peak hours, the synchronization period span can be adjusted to 20 seconds to reduce the system's computational load.

[0017] It should be noted that the node queuing data is collected every second, executed by the local controller of the self-service terminal. The collected data is in fixed-length structured data format, including four fields: terminal unique ID, collection timestamp, current queue length, number of arrivals within the period, and number of people who completed service within the period. The collection frequency of spatiotemporal status data is divided into two parts: the message aging time is calculated in real-time by the central scheduling node by synchronizing the system time every second; the terminal walking time is collected once from a preset location map database by the central scheduling node during system initialization, and can be manually updated later based on on-site conditions. The collected data is in structured data format, including two fields: terminal unique ID and walking time. The reporting method uses the Message Queuing Telemetry (MQTT) protocol. The self-service terminal reports node queuing data to the central scheduling node every 5 seconds. The central scheduling node receives the reported data through a fixed IP address and port. Data transmission uses the Transport Layer Security (TLS) protocol for encryption to ensure the security and stability of data transmission.

[0018] It should be noted that there are two methods for collecting and calculating terminal walking time, which can be selected according to the actual scenario: The first method is the on-site measurement and statistical method; testers are arranged to simulate the user's walking speed and walk back and forth between each pair of terminals 10 times, recording the walking time each time, and taking the average of the 10 measurements as the terminal walking time for that terminal pair; this method closely matches the actual situation on site, the calculation results are accurate, and it is suitable for scenarios where the terminal locations are fixed. The second method is the map calculation formula method; based on a planar map of the location, the walking path length between each pair of terminals is measured, and the walking time is calculated in combination with the preset user walking speed; the calculation formula is that the terminal walking time equals the walking path length divided by the user walking speed; the preferred value for the user walking speed is 1.2 meters per second, which is the average walking speed of an adult in an indoor public place; for example, if the walking path length between two terminals is 12 meters, then the calculated terminal walking time is 10 seconds; this method does not require on-site measurement, can quickly complete the walking time calculation for all terminal pairs, and is suitable for the initial planning stage of the system.

[0019] Specifically, in its original form, node queuing data is merely the cumulative number of people within a certain statistical interval. It cannot be directly used for horizontal comparisons between different statistical windows. Therefore, the number of people arriving at a node and the number of people served by a node must first be converted into node arrival rate and node service rate before being included in subsequent blind zone differential drift and state transition calculations. For example, if one node receives 5 people in 10 seconds and another node receives 8 people in 20 seconds, directly comparing the cumulative number of people would misjudge the load level. After conversion, the former is 0.5 people per second and the latter is 0.4 people per second, making them comparable. Furthermore, because the queue states reported by distributed nodes are not generated simultaneously, the latest reported value itself has a time lag. Therefore, the message aging time cannot be omitted. Including the message aging time in the subsequent blind zone duration calculation allows for the compensation of the natural evolution that has occurred between the state's generation and its participation in prediction. For example, if a node displays a queue of 3 people, but this data has been aged for 12 seconds, then these 3 people are not equivalent to the current real-time state.

[0020] Specifically, since users do not instantly join the target node queue after seeing the navigation information, but need to undergo a physical movement from the source node to the target node, the terminal walking time directly affects the queue state when the user actually arrives. Considering both terminal walking time and message aging time allows for the unified inclusion of information lag and spatial migration delays into a single prediction framework. Furthermore, nodes in a distributed system typically employ periodic or event-triggered reporting mechanisms. The management end can often directly read the latest reported value at the current moment, rather than the absolute real-time measurement value. Therefore, the observed queue length must be understood as the visible state rather than the actual current value, which will not be elaborated upon here.

[0021] In one embodiment of the present invention, the observation queue length is subtracted to generate an observation queue difference value, and the blind zone differential drift is calculated based on the node arrival rate, node service rate, message aging time, and terminal walking time, including: The formula for calculating the difference in observation queue lengths, based on the lengths of the target node's observation queue and the source node's observation queue, is as follows:

[0022] in, The difference between the observation queues (representing the target node) Relative to the source node (Observation queue difference) The length of the observation queue for the target node (representing the target node). (length of the most recently seen or reported observation queue). The length of the observation queue of the source node (representing the source node) (length of the most recently seen or reported observation queue). As a superscript, it is used to indicate the state attribute of "most recent report" or "most recent visible observation", rather than an independent node number or a summation variable; The formula for calculating blind zone differential drift is as follows, based on node arrival rate, node service rate, packet aging time, and terminal walking time:

[0023] in, Indicates the target node. Indicates the source node, For blind zone differential drift, The node arrival rate of the target node. The node service rate for the target node. The message aging time for the target node (representing the target node's message aging time). (message aging time). Terminal walking time (representing the terminal walking time associated with the user migration path between the target node and the source node). The node arrival rate of the source node. The node service rate of the source node. This is the message aging time for the source node.

[0024] It should be noted that the observation queue difference is the difference in queue length between the target self-service terminal and the source self-service terminal in the most recent visible state, calculated as the observation queue length of the target node minus the observation queue length of the source node. Blind zone differential drift is the natural relative drift between the queue difference of the target self-service terminal and the source self-service terminal within the blind migration interval from user observation to arrival, calculated as the target drift amount minus the source drift amount. Target node net traffic is the net change in the number of people in the queue at the target self-service terminal per unit time, calculated as the node arrival rate of the target node minus the node service rate of the target node. Target node blind zone duration is the sum of the state aging time and migration time experienced by the target self-service terminal before user arrival, calculated as the packet aging time of the target node plus the terminal walking time. Target drift amount is the natural change in the number of people in the queue at the target self-service terminal within the blind migration interval, calculated as the target node net traffic multiplied by the target node blind zone duration. Source node net traffic is the net change in the number of people in the queue at the source self-service terminal per unit time. It is calculated by subtracting the source node's service rate from its arrival rate. Source node blind zone duration is the sum of the state aging time and migration time experienced by the source self-service terminal before user arrival. It is calculated by adding the terminal's walking time to the source node's packet aging time. Source drift is the natural change in the number of people in the queue at the source self-service terminal during the blind migration interval. It is calculated by multiplying the source node net traffic by the source node blind zone duration.

[0025] It should be noted that the pairing of target nodes and source nodes follows a full combinatorial traversal rule, specifically consisting of three steps: 1. Determine the list of terminals to be included in the pairing scope. All normally online self-service terminals in the system that can provide services are included in the pairing scope. 2. Generate a full set of terminal pair combinations. For the N terminals included in the scope, generate all terminal pairs where i is not equal to j, where i is the target node number and j is the source node number, ensuring that every terminal can be a target node and every other terminal can be a corresponding source node. 3. Perform traversal calculations in the pairing order. For each target node, traverse all other source nodes to complete the calculation of parameters such as the observation queue difference and blind zone differential drift. For example, if there are 3 online terminals in the system, numbered 1, 2, and 3, then the generated terminal pair combinations are 1 and 2, 1 and 3, 2 and 1, 2 and 3, 3 and 1, and 3 and 2, covering all possible user cross-terminal migration directions and ensuring that no possible echo effect sources are missed.

[0026] Specifically, the arrival rate of a single node can only indicate the entry speed, and the service rate of a single node can only indicate the processing speed. Neither of them can independently characterize whether the queue will grow or shorten naturally. Therefore, it is necessary to define the net flow of a node. When the net flow of a node is positive, it indicates that the queue has a growing trend. When the net flow of a node is negative, it indicates that the queue has a shortening trend. This directional information directly determines the sign of the drift.

[0027] In one embodiment of the present invention, generating blind zone echo gain based on observation queue difference and blind zone differential drift includes: The formula for calculating the blind zone echo gain based on the observation queue difference and the blind zone differential drift is as follows:

[0028] in, For blind zone echo gain, This represents the absolute value of the blind zone differential drift. This represents the absolute value of the difference between the observation queues.

[0029] It should be noted that the blind zone echo gain is a characteristic value representing the strength of the natural drift intensity of the queue difference between two terminals within the blind migration interval relative to the user's initial perceived short queue advantage. It is calculated by dividing the drift intensity by the advantage stability term. The drift intensity is the absolute value of the blind zone differential drift, representing the strength at which the relative queue advantage of the two terminals may be rewritten in the blind zone. It is calculated by taking the absolute value of the blind zone differential drift. The original advantage intensity is the absolute value of the observed queue difference, representing the magnitude of the queue advantage difference between the two terminals initially perceived by the user. It is calculated by taking the absolute value of the observed queue difference. The advantage stability term is a correction term that ensures the numerical stability of the calculation formula; it is calculated by adding a constant one to the original advantage intensity.

[0030] It should be noted that the constant 1 is used to ensure the numerical stability of the calculation formula, i.e., to avoid the denominator being 0. Secondly, it does not change the original physical meaning that the smaller the advantage, the easier it is to be swallowed up by blind zone drift; when the absolute value of the observation queue difference is smaller, the value of the advantage stability term is closer to 1, and the value of the blind zone echo gain is closer to the drift intensity, indicating a higher risk of short queue reversal; when the absolute value of the observation queue difference is larger, the value of the advantage stability term is larger, and the value of the blind zone echo gain is smaller, indicating a lower risk of short queue reversal. There is no need to adjust the value according to peak and off-peak traffic; a fixed value of 1 is sufficient to ensure the stability of the formula and the accuracy of its physical meaning.

[0031] In one embodiment of the present invention, the node selection probability is calculated based on the observation queue length, node service rate, and terminal walking time; an additional arrival rate is calculated by combining the node selection probability, node arrival rate, and blind zone echo gain; and the additional arrival rate is incorporated into the node arrival rate to generate the effective arrival rate, including: The formula for calculating the target selection cost, based on the observation queue length, node service rate, and terminal walking time, is as follows:

[0032] in, Choose a cost for the target (representing the cost from the source node). Departure point selection target node (Target selection cost) The length of the observation queue for the target node (representing the target node). (the length of the most recently reported observation queue). Node service ratio for the target node (representing the target node's service ratio) (node ​​service rate) The terminal walking time (representing the source node) To the target node (Terminal walking time); Based on the cost of target selection, the formula for calculating the probability of node selection is as follows:

[0033] in, Choose a probability for each node. It is a natural constant. Choosing a cost for a goal This represents the total number of nodes in the system. For candidate node indexing, Cost of candidate selection; The formula for calculating the total number of echo arrivals is as follows, based on node arrival rate, synchronization cycle span, node selection probability, and blind zone echo gain:

[0034] in, The total number of echo arrivals. The node arrival rate of the source node. For the synchronization period span, Choose a probability for each node. This represents the blind zone echo gain from the source node to the target node. For the source node index, Index the target node; The formula for calculating the additional arrival rate, based on the total number of echo arrivals and the synchronization period span, is as follows:

[0035] in, For additional arrival rate, The total number of echo arrivals. For the synchronous period span; Based on the additional arrival rate and the node arrival rate, the formula for calculating the effective arrival rate is as follows:

[0036] in, For effective arrival rate, The node arrival rate of the target node. Additional arrival rate.

[0037] It should be noted that the target selection cost is the comprehensive cost incurred by a user when choosing a target self-service terminal, starting from the location of the source self-service terminal. It is calculated by dividing the observation queue length of the target node by the node service rate of the target node, plus the terminal walking time. Single-point attractiveness is a numerical value characterizing the attractiveness of the target self-service terminal to users at the source location. It is calculated by exponentiation of the natural constant e to the negative of the target selection cost. The total number of nodes in the system refers to the total number of all online self-service terminals included in this prediction synchronization process, preferably the number of all normally operating self-service terminals in the system. The candidate node index is the sequential number used to traverse all serviceable self-service terminals in the system, preferably a consecutive integer from 1 to the total number of nodes in the system. The candidate selection cost is the comprehensive cost incurred by a user when choosing a corresponding candidate self-service terminal, starting from the location of the source self-service terminal. It is calculated by dividing the observation queue length of the candidate node by the node service rate of the candidate node, plus the terminal walking time from the source terminal to the candidate terminal. Candidate attractiveness is a numerical value that characterizes the ability of a corresponding candidate self-service terminal to attract users at the source location. It is calculated by exponentiation of the natural constant e as the base and the negative of the candidate selection cost as the exponent.

[0038] It should be noted that global attractiveness is the sum of the attractiveness of all candidate self-service terminals in the system to users at the source location, calculated by summing the attractiveness of all candidates. Node selection probability is the probability that a user at the source location will choose the corresponding target self-service terminal, calculated by dividing the single-point attractiveness by the global attractiveness. Echo adjustment denominator is a correction term used to map the blind zone echo gain to a proportional value between 0 and 1, calculated by adding a constant one to the blind zone echo gain from the source node to the target node. Echo influence ratio is a value representing the proportion of migrating traffic affected by the echo mechanism to the total migrating traffic, calculated by dividing the blind zone echo gain from the source node to the target node by the echo adjustment denominator. Source node index is the sequential number used to traverse all self-service terminals in the system except the target node, preferably a consecutive integer from 1 to the total number of nodes in the system, excluding the target node index. Target node index is the unique sequential number used to identify the target self-service terminal in this calculation, preferably a unique integer from 1 to the total number of nodes in the system.

[0039] It should be noted that the expected arrival number of the source node within a given period is the total number of people expected to arrive at the source self-service terminal within a synchronization period span. It is calculated by multiplying the source node's arrival rate by the synchronization period span. The one-way echo arrival number is the additional echo arrival number of a single source self-service terminal migrating to the target self-service terminal within a synchronization period span. It is calculated by multiplying the expected arrival number of the source node within a given period by the node selection probability, and then by the echo impact ratio. The total echo arrival number is the sum of the additional echo arrival numbers of the target self-service terminal migrating from all other self-service terminals within a synchronization period span. It is calculated by summing the one-way echo arrival numbers corresponding to all source nodes. The additional arrival rate is the average number of additional echo arrivals of the target self-service terminal per unit time. It is calculated by dividing the total number of echo arrivals by the synchronization period span. The effective arrival rate is the total average number of arrivals of the target self-service terminal per unit time, including both local original arrivals and additional echo arrivals. It is calculated by adding the target node's arrival rate to the additional arrival rate.

[0040] It's important to note that the exponential choice model based on node selection probability is suitable for scenarios where users can freely choose any terminal—that is, scenarios without physical fences or access restrictions, where users can freely migrate across terminals based on navigation information. It is also suitable for scenarios where user selection behavior follows a utility-maximizing logic, meaning users will prioritize terminals with lower overall costs, while also exhibiting some random selection behavior. Furthermore, it is suitable for scenarios with 2 to 50 terminals, where the computational cost of the exponential model is controllable and it accurately simulates user selection behavior. In addition, when the scenario exceeds its applicable boundaries, two alternative choice models exist: the first is the deterministic shortest queue model, suitable for scenarios where users can only choose the terminal with the lowest overall cost, meaning the probability of choosing the terminal with the lowest overall cost is 1, and the probability of choosing other terminals is 0. The second is the linear proportional choice model, suitable for scenarios where the user's selection probability is linearly inversely proportional to the overall cost. The selection probability is calculated by dividing the reciprocal of the overall cost by the sum of the reciprocals of the overall costs of all terminals. This model has lower computational cost and is suitable for large-scale scenarios with more than 50 terminals.

[0041] It should be noted that the mapping logic for the echo impact ratio is to smoothly map the blind zone echo gain, ranging from 0 to positive infinity, to a proportional value ranging from 0 to 1, used to characterize the proportion of migrating pedestrians affected by the echo effect. The mapping formula uses the form of blind zone echo gain divided by 1 plus the blind zone echo gain. This form has the following advantages: First, monotonicity: the mapped echo impact ratio monotonically increases with the increase of the blind zone echo gain, which conforms to the physical logic that the larger the blind zone echo gain, the higher the proportion of affected pedestrians. Second, boundedness: no matter how large the blind zone echo gain is, the mapped echo impact ratio is always less than 1, avoiding the unreasonable situation where the proportion of pedestrians exceeds 100%. Finally, smoothness: the mapping process is continuous and smooth, without numerical jumps, ensuring the stability of the system's prediction results. Furthermore, when the blind zone echo gain is 0, the echo impact ratio is 0, indicating no risk of short-line reversal and no affected pedestrians. When the blind zone echo gain is 1, the echo impact ratio is 0.5, meaning that half of the migrating pedestrian flow will be affected by the echo effect. When the blind zone echo gain is 9, the echo impact ratio is 0.9, meaning that 90% of the migrating pedestrian flow will be affected by the echo effect, which perfectly matches the rule that the higher the risk of short queue reversal, the greater the proportion of affected pedestrian flow.

[0042] Specifically, node selection behavior is usually not a hard switch that selects only the node with the lowest cost. Instead, the lower the cost, the stronger the attraction, but suboptimal nodes are still allowed to be partially selected. Therefore, an exponential mapping is needed to convert the selection cost into attractiveness. This mechanism can smoothly express the relative attractiveness of different nodes to users, avoiding the abrupt changes caused by the simple minimum value rule. In addition, the next state transition of the target node is not affected by static indicators, but by the actual rate at which the number of people entering the node queue per unit time. Therefore, the original node arrival rate and the additional arrival rate must be superimposed to obtain the effective arrival rate. Only in this way can the echo mechanism truly enter the subsequent Markov state transition modeling, which will not be elaborated here.

[0043] In one embodiment of the present invention, a state transition matrix is ​​generated based on the effective arrival rate, node service rate, and synchronization cycle span. The overflow transition probability exceeding the node limit is accumulated to the boundary state corresponding to the node limit to generate a closed-loop transition matrix, including: Based on the effective arrival rate, node service rate, and synchronization cycle span, the formula for calculating the corresponding elements of the state transition matrix is ​​as follows:

[0044] in, For the corresponding elements of the state transition matrix, This represents the current queue state. For the future queue state, To reach the required number of people, To complete the traversal of the lower bound for the number of people served, To complete the traversal of the upper bound of the number of people served, It is a natural constant. For node service rate, For the synchronization period span, To achieve the factorial of the number of people served, For effective arrival rate, To accommodate the additional arrivals, The factorial of the number of newly arrived people; The formula for calculating the boundary states corresponding to the closed-loop transition matrix based on the corresponding elements of the state transition matrix is ​​as follows:

[0045] in, The boundary states corresponding to the closed-loop transition matrix. This represents the maximum number of people allowed per node. For the future queue state, These are the elements corresponding to the state transition matrix.

[0046] It should be noted that the node's maximum number of users is the maximum value modeled for the queue state of the corresponding self-service terminal, preferably 20. The current queue state is the actual observed queue number of users at the start of the prediction process for the corresponding self-service terminal, preferably the observed queue length of the corresponding terminal. The future queue state is the possible queue number of users the corresponding self-service terminal will be in at a future synchronization time, ranging from 0 to the node's maximum number of users. The average number of services completed is the expected total number of users completed within a synchronization period for the corresponding self-service terminal, calculated as the node service rate multiplied by the synchronization period span. The average number of valid arrivals is the expected total number of valid arrivals within a synchronization period for the corresponding self-service terminal, calculated as the valid arrival rate multiplied by the synchronization period span. The lower bound of the number of users completed is the starting value when iterating through the valid number of users completed, calculated as the larger of 0 and the difference between the current queue state and the future queue state. The upper bound of the number of users completed is the ending value when iterating through the valid number of users completed, and is the value of the current queue state. The valid number of people who complete the service is the number of terminals that complete the service within a synchronization period. The value range is a continuous integer from the lower bound of the number of people who complete the service to the upper bound of the number of people who complete the service.

[0047] It should be noted that the service probability term is the probability that the self-service terminal will complete a specified number of services within a synchronization period. It is calculated by multiplying the power of the negative of the average number of services completed (base e) by the power of the number of people served (base e) and the factorial of the number of people served. The new arrivals term is the total number of people newly entering the queue at the self-service terminal within a synchronization period. It is calculated by subtracting the current queue state from the future queue state and adding the number of people served. The arrival probability term is the probability that a specified number of new arrivals will occur at the self-service terminal within a synchronization period. It is calculated by multiplying the power of the negative of the average effective arrivals (base e) by the power of the new arrivals (base e) and the factorial of the new arrivals. The joint transfer probability is the joint probability that a specified number of services will be completed and a specified number of new arrivals will occur at the self-service terminal within a synchronization period. It is calculated by multiplying the service probability term by the arrival probability term.

[0048] It should be noted that the state transition matrix is ​​a two-dimensional matrix recording all probabilities of a self-service terminal transitioning from its current queue state to a future queue state. The value of each element in the matrix is ​​the sum of the joint transition probabilities corresponding to all legally completed service users for the corresponding state combination. The internal transition sum is the sum of probabilities in the state transition matrix for transitioning from the current state to all future states that have not reached the node's maximum number of users. It is calculated by summing the elements of the state transition matrix corresponding to all future states from 0 to the node's maximum number of users minus 1. The overflow transition probability is the total probability of transitioning from the current state to a state exceeding the node's maximum number of users. It is calculated by subtracting the internal transition sum from a constant. The boundary state is the maximum numerical state modeled for the corresponding self-service terminal queue state, i.e., the state corresponding to the node's maximum number of users. The closed-loop transition matrix is ​​the complete Markov transition probability matrix after correcting for the boundary state probabilities. The values ​​of the elements corresponding to the boundary states in the matrix are the overflow transition probabilities, and the values ​​of the remaining elements are consistent with the state transition matrix.

[0049] It's important to note that the maximum number of users allowed per node is determined based on several factors. First, the maximum capacity of the queuing area on a single self-service terminal. In public places, this typically means 15 to 25 people per queuing area. Exceeding this limit will lead users to perceive the queue as too long and avoid that terminal. Second, the computational complexity of Markov chain modeling is crucial. A higher maximum node size results in a higher dimensionality of the state transition matrix and increased computational load. A balance must be struck between modeling accuracy and computational complexity. Finally, the actual queue length distribution in the scenario is critical. In public places, the probability of a queue exceeding 20 people on a single self-service terminal is extremely low. Queues exceeding 20 people do not require further subdivision; simply including them in the maximum limit ensures modeling accuracy. The optimal range for the maximum node size is 15 to 25, with 20 being the preferred value. During peak hours, the maximum node size can be adjusted to 25 to improve modeling accuracy. During off-peak hours, the maximum node size can be adjusted to 15 to reduce computational load.

[0050] It's important to note that the logic for determining the upper bound of the number of people served is based on the premise that within a synchronization cycle, a terminal can only serve everyone currently in the queue, and cannot exceed the current queue size. For example, if the current queue size is 3 people, then at most 3 people can be served within a cycle. Therefore, the upper bound is taken as the value of the current queue size. Physically, this represents the maximum possible number of people served, ensuring that no physically impossible situations arise during the traversal process.

[0051] It's important to note that determining the lower bound of the number of people served requires satisfying two constraints: First, the number of people served cannot be negative, so the lower bound cannot be less than 0. Second, the number of newly arrived people cannot be negative. The number of newly arrived people equals the future queue state minus the current queue state plus the number of people served, so the number of people served cannot be less than the difference between the current queue state and the future queue state. The value that satisfies both constraints is the larger of 0 and the difference between the current queue state and the future queue state. For example, if the current queue state has 3 people and the future queue state has 5 people, the difference is -2, so the lower bound is 0. If the current queue state has 5 people and the future queue state has 2 people, the difference is 3, so the lower bound is 3. Physically, this represents the minimum possible number of people served, ensuring that the number of newly arrived people will not be negative during the traversal, and that all traversal values ​​are physically achievable.

[0052] Specifically, this invention requires the unified propagation of the current state of nodes to future synchronization moments. Therefore, it cannot simply provide a single predicted value, but rather a probability structure for a node to transition from its current queue state to its future queue state. Using a finite-state Markov transition matrix, the system can probabilistically represent all possible state changes of each node within a synchronization period. Since service completion events and new arrival events are both counting processes occurring within a fixed time window, they can be represented by Poisson counting terms based on the average number of service completions and the average number of effective arrivals, respectively. The service probability term corresponds to the probability of completing a certain number of services within a fixed period, and the arrival probability term corresponds to the probability of adding a certain number of people within a fixed period. Multiplying these two terms yields the joint transition probability. Furthermore, the actual modeling uses a finite state space. If, theoretically, a state exceeding the node's maximum number of members occurs, the matrix dimension cannot be infinitely expanded. Therefore, all tail probabilities exceeding the node's maximum number of members need to be uniformly incorporated into the boundary states to form a closed-loop transition matrix. This ensures both finite-state implementation and maintains a probability sum of 1 for each row.

[0053] In one embodiment of the present invention, mapping the observation queue length to an initial state vector and multiplying the initial state vector by the closed-loop transition matrix to generate the future state distribution includes: The formula for calculating the initial state vector, based on the observation queue length, is as follows:

[0054] in, These are the initial probability values ​​corresponding to the discrete state variables in the initial state vector. For discrete state variables, The length of the observation queue; The formula for calculating the future state distribution based on the initial state vector and the closed-loop transition matrix is ​​as follows:

[0055] in, For the distribution of future states, Let be the initial state vector. This is the closed-loop transition matrix.

[0056] It should be noted that the discrete state variable is used to traverse all possible queue states of the corresponding self-service terminal, and its value ranges from 0 to the maximum number of users in the node, being a continuous integer. The initial probability value is the probability value of the corresponding discrete state variable in the initial state vector. When the discrete state variable is equal to the observed queue length, it takes the value of a constant one; when the discrete state variable is not equal to the observed queue length, it takes the value of a constant zero. The initial state vector is a one-dimensional vector recording the probability distribution of the queue state of the corresponding self-service terminal at the prediction starting point, where each element is the initial probability value of the corresponding discrete state variable. The future state distribution is the probability distribution of the queue state of the corresponding self-service terminal at the future synchronization time, calculated as the matrix multiplication result of the initial state vector and the closed-loop transition matrix.

[0057] It should be noted that the one-hot encoding logic of the initial state vector is suitable for scenarios where the current queue state of the terminal can be accurately observed, that is, scenarios where the self-service terminal can count the exact number of people in the local queue in real time, and there is no observation error. Secondly, it is suitable for scenarios where the state of the prediction starting point is not uncertain, that is, the system can obtain the accurate queue length reported by the terminal most recently, without needing to perform probability estimation of the current state. Finally, it is suitable for scenarios where the synchronization period is shorter than the terminal state reporting period, that is, between two reports, the system will not receive new state data and can only use the definite state of the last report as the prediction starting point.

[0058] Specifically, the state propagation of a Markov chain is essentially the process of obtaining the state distribution at the next moment from the current state distribution through the transition matrix. Therefore, the current definite state is projected to the future synchronization moment through matrix multiplication. The result is not a single point value, but a future state distribution that covers all discrete states.

[0059] In one embodiment of the present invention, the future state distribution is converted into a prediction queue length, and the prediction queue lengths of all nodes are concatenated to generate a network-wide prediction vector, which is then output. The formula for calculating the predicted queue length based on the future state distribution is as follows:

[0060] in, To predict queue length, For future synchronization moments, For discrete state variables, This represents the maximum number of people allowed per node. This represents the probability value corresponding to the future state distribution; The formula for calculating the prediction vector of the entire network, based on the prediction queue length of all nodes, is as follows:

[0061] in, This is the predicted vector for the entire network. For future synchronization moments, to The predicted queue length for all nodes. The total number of nodes in the system, indicated by the superscript. This is the matrix transpose symbol.

[0062] It should be noted that the future synchronization time is a unified projection of the queue states of all terminals by the system. The target future time is the current system time plus the synchronization cycle span. The expected component is the product of the corresponding discrete state variable and the probability value of that state in the future state distribution. It is calculated by multiplying the discrete state variable by the corresponding probability value in the future state distribution. The predicted queue length is the expected number of people in the queue for the corresponding self-service terminal at the future synchronization time. It is calculated by summing the expected components of all discrete state variables from 0 to the maximum number of people per node. The node numbering order is the preset arrangement order of all self-service terminals in the system, preferably ordered by the physical location of the terminals from near to far, or by the device number of the terminals from small to large. The network-wide prediction vector is a one-dimensional column vector recording the predicted queue lengths of all self-service terminals in the system at the future synchronization time, where each element of the vector is the predicted queue length of the corresponding terminal.

[0063] It should be noted that since the future state distribution represents the probability of a node being in each discrete state, while the wayfinding system and management system actually need a clear output of the queue length, it is necessary to multiply the discrete state variables by their corresponding probabilities and then sum them to obtain the expected value of the predicted queue length; this can convert the probability results into numerical values ​​that can be directly displayed and compared.

[0064] Specifically, the deployment and implementation of this invention consists of three key stages: hardware deployment, data collection, and process operation. In the hardware deployment stage, self-service terminals are deployed in different functional areas of the target location. Taking a hospital outpatient setting as an example, self-service terminals can be deployed in the first-floor lobby, second-floor corridor, and examination appointment area. These terminals only handle non-medical services such as check-in, ticket issuance, and receipt printing. In the data collection stage, each self-service terminal collects local queue operation data every second, including the current queue length, the number of people arriving within a period, and the number of people completing services within a period. Data is reported to the central scheduling node every 5 seconds. The reporting uses the Message Queuing Telemetry Transport Protocol (MQTT), and data transmission uses the Transport Layer Security (TLS) encryption protocol. The time synchronization module of the central scheduling node synchronizes the system time every second and calculates the aging time of each reported data message in real time. The walking time between terminals is calculated during system initialization using a location map and the average indoor walking speed of adults (1.2 meters per second), or it can be calibrated through on-site measurement and statistics. During the process execution, the central scheduling node performs a complete prediction process with a fixed synchronization period span. The preferred range for the synchronization period span is 5 to 20 seconds, with a preferred value of 10 seconds. After completing the entire process calculation, the system outputs a network-wide prediction vector, which is the set of prediction queue lengths for all terminals at the future synchronization time.

[0065] For example, the system deploys three self-service terminals, numbered 1, 2, and 3, with a synchronization period of 10 seconds. At system time 0, the central scheduling node receives data reported by the three terminals. Terminal 1 has an observation queue length of 2 people, a node arrival rate of 4 people per minute, a node service rate of 6 people per minute, and a message aging time of 2 seconds. Terminal 2 has an observation queue length of 4 people, a node arrival rate of 5 people per minute, a node service rate of 6 people per minute, and a message aging time of 3 seconds. Terminal 3 has an observation queue length of 3 people, a node arrival rate of 4 people per minute, a node service rate of 6 people per minute, and a message aging time of 1 second. The walking time between terminals is 8 seconds. After the system performs full-process calculations, the output network-wide prediction vector is: Terminal 1's predicted queue length after 10 seconds is 2.7 people, Terminal 2's is 4.2 people, and Terminal 3's is 3.1 people.

[0066] It should be noted that the predicted queue length output by this invention is a probabilistically weighted mathematical expectation of the number of people in the queue at a future synchronization time, and not an integer count of the actual physical queue size. This value is used to characterize the queue congestion level and queuing pressure of the corresponding terminal, and is a standardized indicator for queue guidance, not an absolute prediction of the actual queue size at a future time. In the user-facing guidance display, this value can be rounded up or down to show the user the estimated queue size in integer form, ensuring that the displayed content conforms to the user's conventional understanding of queue size; this will not be elaborated upon here.

[0067] It should be noted that the network-wide prediction vector output by this invention can be implemented in hospital outpatient guidance scenarios in three ways. First, it can be used for displaying queue information on public guidance screens in outpatient departments. The screens can sort and display terminals in ascending order of predicted queue length, prioritizing terminals with lower predicted queuing pressure to guide users to choose the appropriate terminal and avoid incorrect selections based on outdated queue information. Second, it can be used for displaying local guidance screens around terminals. At key nodes along a user's walking path, it can display the predicted queue length of nearby terminals, providing users with a reference for choosing the nearest terminal and shortening their walking distance and overall waiting time. Third, it can be synchronized with the hospital's terminal operation and maintenance management system. Maintenance personnel can use the trend of predicted queue length changes to replenish printing supplies and inspect the equipment status of high-load terminals in advance. During peak periods, backup terminals can be temporarily added to alleviate service pressure caused by concentrated queuing.

[0068] It should be noted that, as Figure 2 As shown, the network-wide queue prediction comparison chart illustrates the comparison of queue lengths for all terminals across the network at the same prediction time. The horizontal axis represents the terminal node number, arranged in order from terminal 1 to terminal 10; the vertical axis represents the queue length, in units of people. The observed queue represents the visible queue size at the time of the most recent report from each terminal; the predicted queue represents the future predicted value calculated by the system based on arrival rate, service rate, message aging time, walking time, and echo mechanism; and the actual queue represents the actual queue size at the corresponding future time. Higher values ​​in each bar indicate greater queuing pressure for that terminal. This chart is used to verify whether the predicted queue length output by this invention is closer to the actual queue size, thus proving the usability and engineering guidance significance of the synchronous prediction results, which will not be elaborated upon here.

[0069] It should be noted that, as Figure 3As shown, the migration echo effect diagram illustrates the relationship between blind zone echo gain and additional arrival rate on the node migration path. The horizontal axis represents the path number of the user migration from the source terminal to the target terminal, sorted by echo gain from smallest to largest. The left vertical axis represents the blind zone echo gain, which reflects the strength of the natural drift rewriting of the queue advantage within the blind zone; the larger the value, the higher the risk of short queue reversal. The right vertical axis represents the additional arrival rate, in person-seconds, which reflects the magnitude of the new inflow rate to the target terminal caused by the echo effect. This diagram is used to verify whether the additional arrival rate increases synchronously when the echo gain increases, thus proving that the migration echo mechanism can actually affect the effective arrival rate and enter the subsequent queue state prediction process, which will not be elaborated here.

[0070] It should be noted that, as Figure 4 As shown, the future state distribution diagram illustrates the probability distribution of the queue state of a single terminal at future synchronization moments. The horizontal axis represents discrete queue states, with values ​​ranging from 0 to 9, indicating the possible number of people queuing at future moments. The vertical axis represents the probability of the corresponding state occurring, with values ​​ranging from 0 to 1. The height of each bar represents the likelihood of the terminal falling into that queue state after the system's closed-loop transition matrix calculation. The probability peaks are concentrated in the middle states, indicating that the terminal is more likely to be in a medium queue level at future moments. This diagram is used to verify that the present invention does not directly output a single value, but first generates a complete future state distribution, and then obtains the predicted queue length through probability weighting, thereby ensuring that the prediction result has a clear Markov chain probability basis, which will not be elaborated here.

[0071] It should be noted that the interval and threshold sizes are set for ease of comparison. The size of the threshold depends on the amount of sample data and the base number set by those skilled in the art for each set of sample data, as long as it does not affect the proportional relationship between the parameter and the quantized value. Furthermore, the above formulas are all dimensionless calculations, and the formulas are derived from software simulations using a large amount of collected data to obtain the most recent real-world results. The preset parameters in the formulas are set by those skilled in the art according to the actual situation.

[0072] The embodiments of this example have been described above. However, this example is not limited to the specific implementation methods described above. The specific implementation methods described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms based on the guidance of this example, and all of them are within the protection scope of this example.

Claims

1. A distributed queue state prediction and synchronization system based on Markov chains, characterized in that, include: The first module acquires node queuing data, spatiotemporal state data, and synchronization cycle span. It converts the node queuing data into node arrival rate and node service rate, extracts message aging time and terminal walking time from the spatiotemporal state data, and reads the observation queue length. The second module generates the observation queue difference by subtracting the observation queue length, and calculates the blind zone differential drift based on the node arrival rate, node service rate, message aging time and terminal walking time. The third module generates blind zone echo gain based on the observation queue difference and blind zone differential drift; The fourth module calculates the node selection probability based on the observation queue length, node service rate, and terminal walking time. It then calculates the additional arrival rate by combining the node selection probability, node arrival rate, and blind zone echo gain. Finally, it incorporates the additional arrival rate into the node arrival rate to generate the effective arrival rate. The fifth module generates a state transition matrix based on the effective arrival rate, node service rate, and synchronization cycle span. It accumulates the overflow transition probability exceeding the node number limit to the boundary state corresponding to the node number limit to generate a closed-loop transition matrix. The sixth module maps the observation queue length to an initial state vector, and multiplies the initial state vector with the closed-loop transition matrix to generate the future state distribution. The seventh module converts the future state distribution into the prediction queue length, concatenates the prediction queue lengths of all nodes to generate the full network prediction vector, and outputs it.

2. The distributed queue state prediction and synchronization system based on Markov chains according to claim 1, characterized in that, Extract the number of people arriving at a node and the number of people completing services at a node from the node queuing data; The node arrival rate is generated by dividing the number of people arriving at the node by the synchronization cycle span. The node service rate is generated by dividing the number of users who completed the service by the synchronization cycle span. Extract the time interval from the generation of the data packet in the spatiotemporal state data to the current system time as the packet aging duration; Extract the physical movement time from the spatiotemporal state data as the terminal walking time; The latest reported number of people in the queue is extracted from the node queuing data as the observation queue length.

3. The distributed queue state prediction and synchronization system based on Markov chains according to claim 1, characterized in that, The observation queue difference is generated by subtracting the observation queue length of the source node from the observation queue length of the target node. Subtract the node service rate of the target node from the node arrival rate of the target node to generate the net traffic of the target node; Add the target node's message aging time to the terminal's walking time to generate the target node's blind zone time. Multiply the net flow of the target node by the blind spot duration of the target node to generate the target drift amount; Subtract the node service rate of the source node from the node arrival rate of the source node to generate the net traffic of the source node. Add the terminal walking time to the packet aging time of the source node to generate the source node blind zone time; Multiply the net flow of the source node by the dead zone duration of the source node to generate the source drift amount; Subtract the source drift from the target drift to generate the blind zone differential drift.

4. The distributed queue state prediction and synchronization system based on Markov chains according to claim 1, characterized in that, Extract the absolute value of the blind zone differential drift to generate the drift intensity; Extract the absolute value of the difference between the observation queues to generate the original dominance strength; Adding a constant to the original advantage strength generates an advantage stabilizing term; Divide the drift intensity by the dominant stability term to generate the blind zone echo gain.

5. A distributed queue state prediction and synchronization system based on Markov chains according to claim 1, characterized in that, Divide the length of the observation queue of the target node by the node service rate of the target node, and add the terminal walking time to the division result to generate the target selection cost; Calculate the single-point attraction by raising the power of the natural constant to the opposite of the cost of target selection. Extract the terminal walk time from the source node to all candidate nodes, the observation queue length of all candidate nodes, and the node service rate of all candidate nodes; Divide the observation queue length of all candidate nodes by the node service rate of all candidate nodes, and add the terminal walk time from the source node to all candidate nodes to generate the candidate selection cost.

6. A distributed queue state prediction and synchronization system based on Markov chains according to claim 5, characterized in that, Calculate the candidate attractiveness by raising the power of the natural constant to the negative of the candidate selection cost, sum all the candidate attractivenesses to generate the global attractiveness, and divide the single-point attractiveness by the global attractiveness to generate the node selection probability. Add a constant one to the dead zone echo gain from the source node to the target node to generate the echo adjustment denominator; The echo influence ratio is generated by dividing the blind zone echo gain from the source node to the target node by the echo adjustment denominator. Multiply the arrival rate of the source node by the synchronization period span to generate the expected periodic arrival rate of the source node. The expected arrival period of the source node, the node selection probability, and the echo influence ratio are multiplied together to generate the unidirectional echo arrival amount. Sum the unidirectional echo arrivals of all source nodes except the target node to generate the total number of echo arrivals; The additional arrival rate is generated by dividing the total number of echo arrivals by the synchronization cycle span. The effective arrival rate is generated by adding the node arrival rate of the target node to the additional arrival rate.

7. A distributed queue state prediction and synchronization system based on Markov chains according to claim 1, characterized in that, Multiply the node service rate by the synchronization cycle span to generate the average number of service completions; Multiply the effective arrival rate by the synchronization period span to generate the average effective arrivals; The service probability term is generated by multiplying together a power of the natural constant as the base and the opposite of the average number of services completed as the exponent, a power of the average number of services completed as the base and the number of people served as the exponent, and the reciprocal of the factorial of the number of people served. Subtract the current queue state from the future queue state and add the number of people who have completed the service to generate the new arrival number; The arrival probability term is generated by multiplying together a power of the natural constant (base) with the opposite of the average effective arrivals as the exponent, a power of the average effective arrivals with the new arrivals as the exponent, and the reciprocal of the factorial of the new arrivals.

8. A distributed queue state prediction and synchronization system based on Markov chains according to claim 7, characterized in that, Multiply the service probability term by the arrival probability term to generate the joint transition probability; Iterate through all the valid number of people who have completed service during the transition from the current queue state to the future queue state, sum the joint transition probabilities corresponding to all valid number of people who have completed service, and generate the corresponding elements of the state transition matrix; Sum the corresponding elements in the state transition matrix that have not reached the maximum number of nodes to generate the internal transition sum; Subtract the internal transition and the overflow transition probability from the constant 1, and use the overflow transition probability as the boundary state corresponding to the upper limit of the number of nodes to generate the closed-loop transition matrix.

9. A distributed queue state prediction and synchronization system based on Markov chains according to claim 1, characterized in that, Establish discrete state variables and determine whether the discrete state variables are equal to the length of the observation queue; When the discrete state variable is equal to the length of the observation queue, the initial probability value is assigned the value of constant one; otherwise, the initial probability value is assigned the value of constant zero. The initial probability values ​​are combined according to the order of the discrete state variables to generate the initial state vector; Perform matrix multiplication between the initial state vector and the closed-loop transition matrix to generate the future state distribution.

10. A distributed queue state prediction and synchronization system based on Markov chains according to claim 1, characterized in that, The discrete state variable is multiplied by the corresponding probability value in the future state distribution to generate the expected component. The predicted queue length is generated by summing all expected components from constant zero to the maximum number of nodes. Extract the prediction queue length of all nodes in order of node number, combine the prediction queue lengths of all nodes into a column vector to generate the prediction vector of the entire network and output it.

Images