An adaptive sampling method for risk-sensitive state update service

Abstract

The application belongs to the technical field of wireless state updating, and relates to a self-adaptive sampling method for risk-sensitive state updating services, which comprises the following steps: step 1, introducing the concept of statistical information age; step 2, minimizing the statistical AoI of key state updating in a wireless fading channel, and denoting the problem as P0; and step 3, finding the sampling and power allocation solution of the P0 problem, so that the minimum statistical AoI can be achieved under a given peak age violation probability; the application can always achieve the minimum statistical information age, which remains unchanged at the beginning and then decreases with the increase of the violation probability; therefore, the application can achieve the minimum statistical information age and better meet the demand of risk-sensitive state updating.

Description

Technical Field

[0001] The present invention belongs to the technical field of wireless status update, and relates to an adaptive sampling method for risk-sensitive status update services. Background Art

[0002] Wireless state update systems are a typical time-sensitive application. In particular, some state update systems are risk-sensitive. Key state information on the target side should always be as fresh as possible. Otherwise, even a small amount of outdated information can lead to undesirable system behavior or even damage.

[0003] In wireless state update systems, evaluating the freshness of state information is crucial. The Age of Information (AoI) is a recently emerging and intriguing concept in this field. It is defined as the time elapsed since the source generated the last received state information. Based on this concept, AoI exhibits a sawtooth-like function characteristic over time. Unlike traditional latency metrics, AoI is affected not only by end-to-end network latency but also by the source's sampling rate. By fine-tuning the source's sampling rate, the freshness of information can be significantly improved.

[0004] Average AoI is widely used to evaluate the long-term performance of information freshness in various state update systems. However, due to the coupling between the arrival interval of state packets and system time, the derivation of average AoI is often very complex. In contrast, because AoI is a sawtooth function of time, peak AoI is able to capture the core characteristics of the aging process. Therefore, average peak AoI can be used as an alternative long-term performance metric to circumvent the complex derivation caused by the correlation between arrival interval and system time, but it cannot meet the requirements of critical state update systems. Using minimum maximum peak AoI can mitigate the worst effects of outdated information, but minimizing maximum peak AoI often introduces bias, which in turn degrades overall performance. The minimum peak Age of Information (AoI) achievable at a given violation probability can meet the requirements of various critical state update systems. The level of violation probability can be correlated with the risk sensitivity of the state update system.

[0005] Existing technologies have the following drawbacks: average AoI or average peak AoI are insensitive to risk and are therefore unsuitable for use in wireless status update systems. Maximum peak AoI is overly sensitive to certain critical status update systems, potentially allocating excessive resources to worst-case scenarios, significantly reducing overall system performance. The probability distribution function for the minimum peak age achievable under a given violation probability often lacks a closed-form expression, making it difficult to derive the minimum peak AoI achievable under a given violation probability, and even more difficult to optimize.

[0006] Therefore, an adaptive sampling method is needed to solve the above technical problems, which can achieve the minimum statistical information age and better meet the needs of risk-sensitive state updates. Summary of the Invention

[0007] The technical solution adopted by the present invention to solve the technical problem is: an adaptive sampling method for risk-sensitive status update services, the method comprising the following steps:

[0008] Step 1: Introduce the concept of statistical information age, which is a tight upper bound on the average peak age of the tail under a given violation probability. Because the statistical age information considers the average value of the peak age of the tail, it can better reflect the risk level. The statistical information age is defined as Δ(ρ), which is expressed as:

[0009]

[0010] In formula (5), A represents the peak information age, θ represents the age information index, and M A (θ) represents the moment generating function MGF of peak age, which is e θA The mathematical expectation of ; 1 / λ(γ,ρ) represents the sampling interval, γ represents the channel power gain;

[0011] Step 2: Minimize the statistical AoI of the key state update in the wireless fading channel, denoted as P0:

[0012]

[0013] stτBlog2(1+P(γ,ρ)γ)≥D, (5b)

[0014]

[0015] τλ(γ,ρ)≤1, (5d)

[0016] Tλ(γ,ρ)≥1 (5e)

[0017] Among them, the constraint condition formula (5b) is obtained based on Shannon capacity, where the noise power is assumed to be 1, B represents the bandwidth, P(γ,ρ) represents the transmission power that varies with γ and ρ, and D represents the number of bits of a state data packet;

[0018] The constraint condition (5c) is the constraint condition for the average transmission power. Indicates the average transmission power of the source;

[0019] Constraint (5d) ensures that the sampling interval is greater than the transmission time, thus ensuring that each state data packet can be transmitted in time after it is generated;

[0020] The constraint condition (5e) indicates that a channel must be sampled at least once within the coherence time;

[0021] Step 3: Find the sampling and power allocation solution for the P0 problem to achieve the minimum statistical AoI under a given peak age violation probability.

[0022] Preferably, in step 3, P0 is equivalently converted to P1:

[0023]

[0024] τλ(γ,ρ)≤1,(8c)

[0025] Tλ(γ,ρ)≥1(8d)

[0026] The objective function of P1 is represented by f(θ,λ(γ,ρ)). The objective function of P1 consists of two parts, which can be expressed as:

[0027]

[0028] More preferably, in step 3, a two-step approach is used to solve problem P1: in the first step, the AoI index is considered a constant and an optimal sampling scheme is found. In the second step, based on the optimal sampling scheme obtained in the first step, an optimal AoI index is searched for.

[0029] More preferably, in step 3, the sub-problem of the first step is expressed as problem P2:

[0030]

[0031] τλ(γ,ρ)≤1, (10c)

[0032] Tλ(γ,ρ)≥1 (10d)

[0033] In problem P2, the objective function is the first part of f(θ,λ(γ,ρ)) P2 is a standard convex-concave fractional programming problem, where the objective function follows a convex-concave form and the constraints are convexity.

[0034] More preferably, in step 3, P2 is converted to P3 by Dinkelbach transformation:

[0035]

[0036] τλ(γ,ρ)≤1, (11c)

[0037] Tλ(γ,ρ)≥1 (11d)

[0038] In formula (11a), β represents a newly introduced auxiliary variable; the auxiliary variable is iteratively updated by formula (12):

[0039]

[0040] In formula (12), i represents the i-th iteration, and i = 1, 2, 3, etc. The initial value of the auxiliary variable β is set to e by selecting λ(γ,ρ)[0] = 1 / T. θT , by iteratively updating β and solving P3 until its value converges, at which point the optimal solution of P3 is also the optimal solution of P2.

[0041] More preferably, in step 3, the optimal solution of P3 is:

[0042]

[0043] in, and The two thresholds representing the channel power gain are equal to η / (1-βe θT (1-θT)) and η / (1-βe θτ (1-θτ)), is the Lambert W function, and η is given by Sure.

[0044] Preferably, in step 3, the second step is specifically: find the optimal AoI index under a given peak age violation probability, and use a binary search method to find the optimal AoI index; assuming that in the binary search, the interval of parameter θ is [θ left ,θ right ], in (θ left +θ right ) / 2, and if the first derivative of the function f(θ,λ(γ,ρ)) is less than zero, then θ left Update to (θ left +θ right ) / 2, otherwise θ right Update to (θ left +θ right ) / 2; According to the optimal AoI index derived above, substituting it into formula (16) can obtain the optimal sampling scheme.

[0045] The beneficial effects of the present invention are:

[0046] The present invention can always achieve a minimum statistical information age, which remains unchanged at the beginning and then decreases as the probability of violation increases, and can better meet the needs of risk-sensitive status updates. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 The AoI process diagram and peak age impact diagram of the adaptive sampling method for risk-sensitive status update services of the present invention are shown in FIG. Figure 1 (a) is the AoI process at the source and destination ends, Figure 1 (b) is the effect of sampling rate on peak age;

[0048] Figure 2 The present invention and the changing trend of f(θ,λ(γ,ρ)) with AOI index;

[0049] Figure 3 : This is a probability distribution comparison diagram of the peak age of the sampling schemes of the Avg-PAoI-O sampling scheme, the Max-PAoI-O sampling scheme, and the scheme proposed in the present invention under different risk sensitivity requirements;

[0050] Figure 4 2 is a performance comparison chart of the present invention, the Avg-PAoI-O sampling scheme, and the Max-PAoI-O sampling scheme. DETAILED DESCRIPTION

[0051] The following will provide a clear and complete description of the relevant technologies in the present invention in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the embodiments described are only part of the embodiments of the present invention, not all of them. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0052] refer to Figures 1-4 This embodiment provides an adaptive sampling method for risk-sensitive status update services. Taking into account the insensitivity of average AoI and average peak AoI to risk, the concept of statistical information age is introduced. The statistical information age is a tight upper bound for the average peak age of the tail under a given violation probability. Because the statistical information age considers the average of the peak age of the tail, it can better reflect the risk level.

[0053] Mathematically, Δ(ρ) is defined as

[0054]

[0055] Where A represents the peak information age, θ is the age information index, M A (θ) is the moment generating function MGF of peak age, which is e θA The mathematical expectation of .

[0056] Consider Figure 1The age variation process at the source and destination shown in (a) assumes that the channel coherence time is long enough, consistent with typical real systems, so that multiple sampling steps can be performed within the channel coherence interval. As shown in the figure, t n represents the time of the nth sampling at the source. According to the definition of AOI, the age of the source is set to zero at each sampling moment and gradually increases between two adjacent sampling intervals. Since the transmission time is τ, once a new status packet is received, the age of the destination is set to τ and gradually increases between the arrival of two adjacent status packets. Then, the peak age A of the destination is expressed as:

[0057]

[0058] like Figure 1 As shown in (b), the higher the sampling rate, the more age peaks there are in a channel coherence interval. Since the duration of a channel coherence time is T and the sampling interval is 1 / λ(γ,ρ), the number of age peaks in the time interval [t, t+T) is approximately equal to Tλ(γ,ρ). In addition, the probability that the channel power gain is equal to γ ​​is f γ (γ)dγ, where f γ (γ) is the probability density function of the channel power gain γ. Therefore, the probability that the peak age is equal to A can be expressed as:

[0059]

[0060] According to equations (2) and (3), the moment generating function MGF of A can be deduced as:

[0061]

[0062] Then, according to the definition of statistical AoI in formula (1), the statistical information age of A can be expressed as

[0063]

[0064] The statistical AoI minimization problem of the key state update in the wireless fading channel is denoted as P0:

[0065]

[0066] stτBlog2(1+P(γ,ρ)γ)≥D, (5b)

[0067]

[0068] τλ(γ,ρ)≤1, (5d)

[0069] Tλ(γ,ρ)≥1 (5e)

[0070] Constraint (5b) is obtained based on the Shannon capacity, where the noise power is assumed to be 1, B represents the bandwidth, P(γ,ρ) represents the transmit power that varies with γ and ρ, and D represents the number of bits in a state packet.

[0071] Constraint (5c) is the constraint on the average transmit power, Indicates the average transmission power of the source.

[0072] Constraint (5d) ensures that the sampling interval is greater than the transmission time, thus ensuring that each state data packet can be transmitted in time after it is generated.

[0073] Constraint (5e) indicates that a channel is sampled at least once within the coherence time.

[0074] Under the above constraints, finding the sampling and power allocation solution for the P0 problem can achieve the minimum statistical AoI under a given peak age violation probability.

[0075] Since the statistical AoI of A specified in equation (4) decreases with the increase of sampling rate λ(γ,ρ), the feasible solution with a larger sampling rate is closer to the optimal solution under the conditions that satisfy the constraints of equations (5b) to (5e). In order to obtain a larger sampling rate, the corresponding transmit power in equation (5c) should be smaller. To this end, the optimal solution should make the equal sign "=" in equation (5b) hold, and then equation (5b) can be simplified to

[0076]

[0077] in = Dlog 2 / τB. Then, substituting Equation (6) into Equation (5c), the average transmit power constraint can be rewritten as follows:

[0078]

[0079] in Combining constraints (5b) and (5c) converts P0 to P1 equivalently as follows:

[0080]

[0081] τλ(γ,ρ)≤1, (8c)

[0082] Tλ(γ,ρ)≥1 (8d)

[0083] Let f(θ,λ(γ,ρ)) represent the objective function of P1, which consists of two parts and can be expressed as:

[0084]

[0085] Given that λ(γ, ρ) and θ are highly coupled in the function f(θ, λ(γ, ρ)), and that its derivatives with respect to them are complex, jointly optimizing the sampling rate and the Age of Information (AoI) index is extremely challenging. To address this, a two-step approach is proposed to solve Problem P1. In the first step, the optimal sampling scheme is considered constant and the AoI index is searched for. In the second step, based on the optimal sampling scheme obtained in the first step, the optimal AoI index is searched for.

[0086] Based on this idea, the sub-problem of the first step can be expressed as problem P2:

[0087]

[0088] τλ(γ,ρ)≤1, (10c)

[0089] Tλ(γ,ρ)≥1 (10d)

[0090] In problem P2, the objective function is the first part of f(θ,λ(γ,ρ)) Because θ is considered a constant, it is omitted Operation. P2 is a standard convex-concave fractional programming problem, where the objective function follows a convex-concave form and the constraints are convex. P2 is transformed into P3 through Dinkelbach transformation as follows:

[0091]

[0092] τλ(γ,ρ)≤1, (11c)

[0093] Tλ(γ,ρ)≥1 (11d)

[0094] Among them, β is a newly introduced auxiliary variable, which is iteratively updated through formula (12).

[0095]

[0096] Where i represents the i-th iteration, and i=1,2,3..... The initial value of the auxiliary variable β is set to e by choosing λ(γ,ρ)[0]=1 / T θT By iteratively updating β and solving P3 until its value converges, the optimal solution of P3 is also the optimal solution of P2.

[0097] P3 is a convex problem and is solved using the Lagrange multiplier method. The Lagrange function of P3 is constructed as follows:

[0098]

[0099] Among them, η,η τ ,η Tare the Lagrange multipliers associated with constraints (11b), (11c) and (11d) respectively. Taking the first-order derivative about λ(γ,ρ), we get

[0100]

[0101] The Karush-Kuhn-Tucker (KKT) conditions for P3 are as follows:

[0102]

[0103] By solving the above equations, the optimal solution of P3 can be obtained as follows:

[0104]

[0105] in, and are two thresholds of channel power gain, which are equal to η / (1-βe θT (1-θT)) and η / (1-βe θτ (1-θτ)), is the Lambert W function, and η is given by Sure.

[0106] The second step is to find the optimal AoI index under a given peak age violation probability. Figure 2 The f(θ,λ(γ,ρ)) in equation (9) and the two parts that constitute the function are plotted. and It can be seen that as θ increases, Increasing, and Decreasing, together determine the trend that f(θ,λ(γ,ρ)) first decreases and then increases with θ. The binary search method can be used to find the optimal AoI index. Assume that in the binary search, the interval of parameter θ is [θ left ,θ right ], in (θ left +θ right ) / 2, and if the first derivative of the function f(θ,λ(γ,ρ)) is less than zero, then θ left Update to (θ left +θ right ) / 2, otherwise θ right Update to (θ left +θ right ) / 2. Substituting the optimal AoI index derived above into Equation (16) yields the optimal sampling scheme. Since the binary search for the optimal AoI index is used and the closed-form solution for the sampling rate has been derived, the corresponding computational complexity is relatively low.

[0107] The effectiveness of the proposed algorithm is verified by numerical simulation. The simulation parameters are set as follows: average transmission power P is 0.1W, bandwidth B is 1MHz, state packet size D is 100 bits, channel power gain γ follows Rayleigh distribution, average channel power gain is 1, the transmission time τ of a status packet is 1ms, and the channel coherence time T is 0.1s.

[0108] Figure 3 The probability distributions of peak age are shown for the average peak information age-oriented (Avg-PAoI-O) sampling scheme, the maximum peak information age-oriented (Max-PAoI-O) sampling scheme, and the sampling schemes proposed in this paper based on different risk sensitivity requirements. The Avg-PAoI-O sampling scheme and the Max-PAoI-O sampling scheme are derived based on the average peak information age and the maximum peak information age, respectively. As can be seen from the figure, the Avg-PAoI-O sampling scheme exhibits excessively large peak ages due to the insensitivity of the average peak information age to risk. However, the Max-PAoI-O sampling scheme stabilizes the peak age at approximately 0.02 seconds. With the proposed sampling scheme and setting ρ = 0.7, the peak age has a probability of 0.7 of being distributed around 0.002 seconds and a probability of 0.3 of being distributed between 0.02 and 0.05 seconds. Furthermore, the maximum peak age is not very large. Setting ρ = 0.5 further concentrates the probability distribution of the peak age.

[0109] Figure 4 The performance of the sampling scheme proposed in this embodiment, the Avg-PAoI-O sampling scheme, and the Max-PAoI-O sampling scheme is demonstrated. It can be seen that the statistical information age of the Avg-PAoI-O sampling scheme gradually decreases as the violation probability increases. In contrast, the statistical information age of the Max-PAoI-O sampling scheme remains unchanged regardless of how the violation probability changes. This phenomenon is due to the constant sampling rate in the Max-PAoI-O sampling scheme. The scheme proposed in this embodiment consistently achieves the minimum statistical information age, which initially remains constant and then decreases as the violation probability increases.

[0110] In summary, the present invention introduces the statistical information age, denotes P0 as the problem of minimizing the statistical AoI of key state updates in wireless fading channels, and finds a sampling and power allocation solution for the P0 problem. This achieves the minimum statistical AoI under a given peak age violation probability. Therefore, the present invention can always achieve the minimum statistical information age. This statistical information age initially remains constant and then decreases as the violation probability increases, which can better meet the needs of risk-sensitive state updates.

[0111] It should be emphasized that the above are only preferred embodiments of the present invention and do not limit the present invention in any form. Any simple modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention are still within the scope of the technical solution of the present invention.

Claims

1. An adaptive sampling method for risk-sensitive status update services, characterized in that: The method comprises the following steps: Step 1: Introduce the concept of statistical information age, which is a tight upper bound on the average peak age of the tail under a given violation probability. The statistical information age is defined as Δ(ρ), which is expressed as: In formula (5), A represents the peak information age, θ represents the age information index, and M A (θ) represents the moment generating function MGF of peak age, which is e θA The mathematical expectation of ; 1 / λ(γ,ρ) represents the sampling interval, γ represents the channel power gain; Step 2: Minimize the statistical AoI of the key state update in the wireless fading channel, denoted as P0: stτBlog2(1+P(γ,ρ)γ)≥D,(5b) τλ(γ,ρ)≤1,(5d) Tλ(γ,ρ)≥1(5e) In the constraint equation (5b), the noise power is assumed to be 1, B represents the bandwidth, P(γ,ρ) represents the transmission power that varies with γ and ρ, and D represents the number of bits in a state data packet. In the constraint equation (5c), Indicates the average transmission power of the source; Step 3: Find a sampling and power allocation solution for the P0 problem that can achieve the minimum statistical AoI under a given peak age violation probability; In step 3, P0 is converted equivalently to P1: τλ(γ,ρ)≤1,(8c) Tλ(γ,ρ)≥1(8d) The objective function of P1 consists of two parts, expressed as: In step 3, a two-step approach is used to solve problem P1. In the first step, the AoI index is considered a constant and the optimal sampling scheme is found. In the second step, based on the optimal sampling scheme obtained in the first step, the optimal AoI index is searched. In step 3, the sub-problem of the first step is expressed as problem P2: τλ(γ,ρ)≤1,(10c) Tλ(γ,ρ)≥1(10d) In problem P2, the objective function is the first part of f(θ,λ(γ,ρ)) P2 is a standard convex-concave fractional programming problem, where the objective function follows a convex-concave form and the constraints are convexity.

2. The adaptive sampling method for risk-sensitive status update services according to claim 1, characterized in that: In step 3, P2 is transformed into P3 by Dinkelbach transformation: τλ(γ,ρ)≤1,(11c) Tλ(γ,ρ)≥1(11d) In formula (11a), β represents a newly introduced auxiliary variable; the auxiliary variable is expressed by formula (12) Perform iterative updates: In formula (12), i represents the i-th iteration, and i = 1, 2, 3, etc. The initial value of the auxiliary variable β is set to e by selecting λ(γ,ρ)[0] = 1 / T. θT , by iteratively updating β and solving P3 until its value converges, at which point the optimal solution of P3 is also the optimal solution of P2.

3. The adaptive sampling method for risk-sensitive status update services according to claim 2, characterized in that: In step 3, the optimal solution of P3 is: in, and The two thresholds representing the channel power gain are equal to η / (1-βe θT (1-θT)) and η / (1-βe θτ (1-θτ)), W(·) is the Lambert W function, and η is given by Sure.

4. The adaptive sampling method for risk-sensitive status update services according to claim 1, characterized in that: In step 3, the second step is specifically: find the optimal Aol index under a given peak age violation probability, and use a binary search method to find the optimal Aol index; assuming that in the binary search, the interval of parameter θ is [θ left ,θ right ], in (θ left +θ right ) / 2, and if the first derivative of the function f(θ,λ(γ,ρ)) is less than zero, then θ left Update to (θ left +θ right ) / 2, otherwise θ right Update to (θ left +θ right ) / 2; According to the optimal Aol index derived above, substituting it into formula (16) can obtain the optimal sampling scheme.

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