Short packet wireless communication error rate-time delay cross-layer analysis method

Abstract

The invention discloses a short packet wireless communication bit error rate-time delay cross-layer analysis method, which is oriented to a short packet URLLC scene under a single-user Nakagami-m block fading channel, considers the random arrival and queuing time delay of a service, and realizes the cross-layer analysis of the bit error rate-time delay of the short packet wireless communication based on a finite block length theory and an effective bandwidth / effective capacity theory. Under the condition of limited block length coding, the reachable rate under the bit error rate constraint is embedded into the service process, and the approximate relation between the bit error rate and the delay violation probability is deduced by combining statistical QoS analysis means such as effective bandwidth / effective capacity and the like. According to the method, an executable calculation process of system parameters-reachable service capability / maximum bearable arrival rate-delay violation probability is established, so that the dependence on large-scale simulation is reduced while the evaluation accuracy is ensured, and a basis is provided for engineering test calculation and parameter configuration of the short packet URLLC system under strict delay and ultrahigh reliability requirements.

Description

Technical Field

[0001] This invention relates to the field of wireless communication technology, and in particular to a cross-layer analysis method for bit error rate-delay in short packet wireless communication based on finite block length theory and effective bandwidth and effective capacity theory. Background Technology

[0002] With the deep integration of 5G / 6G and the Industrial Internet, key short-packet services such as industrial wireless control, tactile Internet, and remote control exhibit typical characteristics of "short data packets, frequent interactions, and strong real-time requirements." Systems need to simultaneously meet extremely low bit error rates and strict latency constraints. Maintaining an extremely low bit error rate under strict latency constraints is a core requirement for ensuring the stability of the control loop, making the coupling problem between the physical layer bit error rate and the network layer queuing delay crucial. However, in actual wireless links, there are both instantaneous random fluctuations in channel quality caused by Nakagami-m fading and queue backlogs caused by random service arrivals, making it difficult to establish a calculable unified relationship between "physical layer bit error rate - reachable rate" and "network layer queuing delay / latency violation probability."

[0003] In existing short-packet URLLC bit error rate-delay analysis and design methods, a common approach is to approximate the radio link service capability as Shannon capacity or average rate, or to establish a relationship of "bit error rate - reachable rate - delay violation probability" only under AWGN / simplified fading conditions. This approach tends to overlook the impact of finite block length dispersion on reachable service capability in scenarios with finite block length and extremely low target bit error rate, leading to biased assessments of reachable strength, queue delay distribution, and delay violation probability. Furthermore, when the channel is in a more general Nakagami-m fading environment, the service process exhibits significant time-varying randomness. Existing solutions often lack a computationally feasible framework that can directly adapt to the statistical characteristics of this type of fading. In engineering, this necessitates relying on extensive simulations or numerical iterations to indirectly evaluate bit error rate-delay performance, resulting in high computational costs and low parameter tuning efficiency, thus affecting rapid verification and reliable configuration in mission-critical scenarios.

[0004] For example, the literature (Lintao Li, Wei Chen, Khaled B. Letaief, “Short Packet Communications with Random Arrivals: An Effective Bandwidth Approach,” 2021 IEEE Global Communications Conference (GLOBECOM), Madrid, Spain, 7–11 Dec. 2021, pp. 1–6, doi: 10.1109 / GLOBECOM46510.2021.9685605.) proposes a bit error rate-delay analysis method for randomly arriving short packet services. This scheme takes low-latency services such as the Industrial Internet of Things (IIoT) as its background. It considers that short packets arrive randomly at the sending end and enter the buffer queue, and are then scheduled for transmission sequentially in the time-slotted link. The receiving end adopts a finite block length encoding / decoding mechanism. Therefore, each short packet transmission is not only constrained by the block length but also fails with a certain probability of decoding error. Thus, the “service process” is simultaneously affected by channel conditions and the short packet bit error rate constraint. The goal of this scheme is to characterize the fundamental trade-off between bit error rate and latency violation probability in a short packet system under given hard latency constraints, and to evaluate the achievable performance boundary of the system accordingly.

[0005] In terms of methodology, this scheme uses effective bandwidth to address the queuing delay problem caused by random arrivals: First, the arrival process is abstracted into a computable statistical arrival model, which is used to obtain the arrival-side characterization required to satisfy statistical quality of service (QoS) constraints; then, combined with the short packet transmission model under the AWGN channel, the decoding error probability caused by finite block length coding is introduced into the service process characterization, thereby deriving the expression of the delay violation probability under hard delay constraints; on this basis, the scheme further provides an "approximate but closed" QoS index to connect the "delay violation probability" and the "finite block length decoding error probability" in analytical form, thus forming a bit error rate-delay bridging path that can be used for theoretical analysis and parameter evaluation. However, the derivation of this scheme is mainly based on the AWGN channel assumption. It does not characterize the random service characteristics caused by time-varying fading in the more general Nakagami-m fading environment in industrial fields, nor does it form a unified calculation process that can be directly used for engineering verification and configuration determination under the conditions of "Nakagami-m fading + finite block length (including dispersion effect) + random arrival queue delay constraint". Therefore, it is difficult to directly cover the cross-layer evaluation requirements of short packet URLLC under complex fading channels.

[0006] In summary, there is a need to explore a new method for the bit error rate-delay tradeoff applicable to Nakagami-m fading channels and oriented towards short-packet services with random arrival. Summary of the Invention

[0007] This invention addresses the shortcomings of existing technologies in addressing Nakagami-m fading and random arrival queues under finite block length coding conditions. It proposes a cross-layer analysis method for bit error rate and delay in short packet wireless communication, constructs and solves an executable cross-layer modeling and calculation method, and achieves accurate evaluation of reachability and delay violation probability (or equivalent delay constraint) under bit error rate constraints.

[0008] To achieve the above objectives, the present invention provides the following technical solution:

[0009] On one hand, the present invention provides a method for cross-layer analysis of bit error rate-delay in short packet wireless communication, comprising the following steps:

[0010] S1. Establish a discrete time-slot short packet communication channel and queuing model for single-user, Nakagami-m block fading channel;

[0011] S2. Under the framework of finite block length information theory, establish a short packet transmission model under the Nakagami-m channel and give the normalized service volume of a single time slot;

[0012] S3. Based on the large deviation theory, an effective bandwidth and effective capacity model is introduced to map the random arrival process and random service process into an equivalent rate related to the quality of service (QoS) index.

[0013] S4. Using the central limit theorem, a second-order expansion of the logarithmic moment generating function of the arrival and service processes is performed to obtain an explicit approximate expression for the QoS index.

[0014] S5. Based on the explicit approximate expression of the QoS index, derive the approximate analytical relationship between the bit error rate and the probability of delay violation.

[0015] Further, in step S1, the discrete time-slot short packet communication channel and queuing model are specifically as follows:

[0016] Time is divided into equal-length transmission blocks, each containing N symbols;

[0017] The channel model for the i-th transport block is: y i [k] = h i x i [k]+z i [k], k = 1, ..., N, where x i [k] represents the transmitted signal, satisfying the average power constraint NP = ||x i || 2 , z i [k] is additive white Gaussian noise with variance N0, h i The channel gain is independent and identically distributed across all blocks, and its squared amplitude is |h|.i | 2 It follows a Nakagami-m distribution;

[0018] Data arrival process Let a[i] be the cumulative arrival amount over the first n time blocks, and a[i] be the amount of data arriving in the buffer within the i-th time block. Modeling this as a discrete Bernoulli process, a[i] = a0X i ,X i ~Bernoulli(p), where a0 is the amount of data carried each time a packet arrives, and p is the probability of a packet arriving in that time block; the mean and variance of the arrival process are μ a =a0p,

[0019] Service process This represents the cumulative service volume over the first n time blocks, where s[i] represents the amount of data successfully transmitted within the i-th time block;

[0020] The queue evolution model is: Q[n] = max{A[n] - S[n], 0};

[0021] The probability of a time delay violation is defined as: δ=Pr{D[∞]≥D max}, D[n] is the steady-state queuing delay, D max This is the time delay threshold.

[0022] Further, in step S2, the expression for the single-slot normalized service quantity s[i] is:

[0023] s[i]=R(γ i )

[0024]

[0025] Where ε is the target block bit error rate, Q -1 (·)for The inverse function, C(γ) i V(γ) represents the instantaneous channel capacity. i The channel dispersion (m) under the Nakagami-m channel is expressed as:

[0026]

[0027] Where, γ i Let m be the instantaneous signal-to-noise ratio in the i-th time block, and m be the shape parameter.

[0028] Furthermore, step S3 specifically includes:

[0029] The effective bandwidth function for the arrival process is defined as:

[0030]

[0031] Define the effective capacity function of the service procedure as follows:

[0032]

[0033] Where θ is the Quality of Service (QoS) index, obtained by solving the balance equation: α A (θ)=α S (θ);

[0034] For a given delay threshold D max The steady-state time delay violation probability is approximately:

[0035]

[0036] Where η is a constant related to the initial state.

[0037] Furthermore, step S4 specifically includes:

[0038] Using the central limit theorem, the cumulative arrival quantity A[n] is approximated as having a mean of nμ. a variance is Gaussian random variables, and the effective bandwidth function α A (θ) Performing a second-order Taylor expansion as θ approaches 0 yields an approximate expression:

[0039]

[0040] Calculate the mean and variance of the normalized service volume s[i] per time slot:

[0041]

[0042]

[0043] in,

[0044] Using the central limit theorem, the cumulative service quantity S[n] can be approximated as having a mean of nμ. s (ε), variance is Gaussian random variables, and the effective capacity function α S (θ) Performing a second-order Taylor expansion as θ approaches 0 yields an approximate expression:

[0045]

[0046] α A (θ) and α S Substituting the approximate expression for (θ) into the equilibrium equation α A (θ)=α S(θ), thus obtaining an explicit approximate expression for the QoS exponent θ:

[0047]

[0048] Furthermore, the mean and second moment of the single-slot service volume are calculated through integration:

[0049]

[0050] Where Γ(m) is the probability density function of the instantaneous signal-to-noise ratio γ. The average signal-to-noise ratio;

[0051] The integral is approximated using the Gauss-Laguerre numerical quadrature method.

[0052] Furthermore, step S5 specifically includes:

[0053] Substituting the approximate expression for the QoS exponent θ into the approximate formula for the delay violation probability:

[0054]

[0055] The mean and variance of the single-slot service volume are considered as functions of the target block bit error rate ε;

[0056] The variance is linearized near the target operating point ε0, while μ is preserved. s (ε) is a first-order dependency of ε;

[0057] By performing an algebraic transformation on the approximate formula for the delay violation probability, we obtain an approximate analytical relationship between the block error rate ε and the delay violation probability δ:

[0058]

[0059] Furthermore, the approximate analytical relationship between the block error rate ε and the delay violation probability δ is ultimately expressed as:

[0060]

[0061] in,

[0062]

[0063] System parameters include average signal-to-noise ratio. Nakagami-m fading parameters m, block length N, mean and variance of arrival process, and delay threshold D. max .

[0064] Furthermore, in the high signal-to-noise ratio region, the approximate analytical relationship between the block bit error rate ε and the delay violation probability simplifies to:

[0065]

[0066] On the other hand, the present invention also provides a communication system parameter configuration method, which applies the short packet wireless communication bit error rate-delay cross-layer analysis method described in any of the above claims, and determines the target block bit error rate according to the target delay violation probability requirement by establishing an approximate analytical relationship between the bit error rate ε and the delay violation probability P_delay, or evaluates the delay performance that the system can achieve based on the target block bit error rate, thereby guiding the configuration of at least one parameter among the system coding rate, block length, transmission power, or scheduling strategy.

[0067] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0068] This invention provides a cross-layer analysis method for bit error rate-delay in short packet wireless communication, targeting short packet URLLC scenarios under single-user, Nakagami-m block fading channels. Considering random service arrival and queuing delay, and based on finite block length theory and effective bandwidth / effective capacity theory, it embeds the achievable rate (including dispersion effects) under bit error rate constraints into the service process under finite block length coding conditions. Combined with statistical QoS analysis methods such as effective bandwidth / effective capacity, it derives an approximate relationship between bit error rate and delay violation probability, establishing an executable calculation process of "system parameters (block length, bit error rate, channel statistics, arrival model, delay constraints) → achievable service capacity / maximum bearable arrival rate → delay violation probability (or equivalent statistical delay determination)". This reduces reliance on large-scale simulations while ensuring evaluation accuracy, providing a basis for engineering verification and parameter configuration of short packet URLLC systems under strict delay and ultra-high reliability requirements. Attached Figure Description

[0069] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in this invention. For those skilled in the art, other drawings can be obtained based on these drawings.

[0070] Figure 1 This is a diagram of the discrete time-slot short packet communication channel and queuing model architecture provided in an embodiment of the present invention.

[0071] Figure 2 A graph showing the relationship between average service load per slot and block bit error rate for Nakagami-m channels (comparing different SNRs, the average service load μ is given). s Alternatively, the curve of E[S1] versus ε can be used, with SNR values ​​of 0 / 5 / 10 / 15dB, and a magnified view of the local area can be provided to verify the consistency of the high-reliability region.

[0072] Figure 3The graph shows the relationship between the delay violation probability and the delay threshold under the Nakagami-m channel (with a fixed SNR, comparing different ε, and giving δ(D)). max At ε = 10 -5 10 -4 10 -3 The theoretical and simulation curves are shown below.

[0073] Figure 4 A trade-off between bit error rate and delay violation probability in Nakagami-m channels (fixed SNR, compared with different D... max Give the trade-off curve between -lnε and -lnδ, in D max Theoretical and simulation curves at values ​​of 5, 10, and 15. Detailed Implementation

[0074] To better understand this technical solution, the method of the present invention will be described in detail below with reference to the accompanying drawings.

[0075] (1) Terminology Explanation and Standardized Expression

[0076] A. Short Packet / Finite Block Length: Each transport block contains a fixed number of symbols N (block length). Given a target block bit error rate ε, the achievable coding rate R(γ,ε,N) of a single block is approximated using a finite block length (including a dispersion term).

[0077] B. Block Fading Nakagami-m Channel: The channel coefficients remain unchanged within each transmission block and are independent between blocks; the instantaneous signal-to-noise ratio is γ, and its distribution is determined by the shape parameter m and the average signal-to-noise ratio. Decide.

[0078] C. Random arrival + buffered queuing: Services arrive in discrete time slots and enter the sender's queue; service capacity is determined by the reachability rate of short packets in the time slot, resulting in queuing delay D.

[0079] D. Delay Violation Probability (DVP): δ = Pr(D > D) max ), where D max As the delay threshold, D max It can be represented by "time slot number" or corresponding to physical time T. max =D max ×T b (The duration of a transport block).

[0080] E. Effective bandwidth / effective capacity: used to map random arrivals and random services to a statistical QoS index θ, and to approximate the delay tail probability through a large deviation; this invention can use either a linearized closed-form approximation or a numerical solution based on the moment generating function.

[0081] (2) Symbol and parameter comparison table

[0082] N: Block length (number of symbols in each transport block)

[0083] i: Discrete time slot / transmission block index

[0084] x i y i z i : Send / receive / noise signal

[0085] h i : Fading coefficient of the i-th block (Nakagami-m)

[0086] γ i : Instantaneous SNR of the i-th block; For average SNR

[0087] m: Nakagami-m shape parameter

[0088] ε: Target block error rate

[0089] Q(g), Q -1 (g): Gaussian Q-function and its inverse function

[0090] R(γ,ε,N): The achievable rate for a finite block length (or equivalently, the amount of information that can be carried per block).

[0091] s i Service volume in the i-th time slot

[0092] a i Arrival quantity in the i-th time slot (number of bits / number of packets)

[0093] Q[i]: Length of the queue at the end of the i-th time slot

[0094] D: Steady-state queuing delay random variable;

[0095] D max Delay threshold

[0096] δ: Probability of delay violation

[0097] θ: Statistical QoS Index

[0098] α A (θ): Effective bandwidth reached;

[0099] α S (θ): Effective service capacity

[0100] This invention provides a method for cross-layer analysis of bit error rate and delay in short packet wireless communication, comprising the following steps:

[0101] S1. Establish a discrete time-slot short packet communication channel and queuing model for single-user, Nakagami-m block fading channel;

[0102] S2. Under the framework of finite block length information theory, establish a short packet transmission model under the Nakagami-m channel and give the normalized service volume of a single time slot;

[0103] S3. Based on the large deviation theory, an effective bandwidth and effective capacity model is introduced to map the random arrival process and random service process into an equivalent rate related to the quality of service (QoS) index.

[0104] S4. Using the central limit theorem, a second-order expansion of the logarithmic moment generating function of the arrival and service processes is performed to obtain an explicit approximate expression for the QoS index.

[0105] S5. Based on the explicit approximate expression of the QoS index, derive the approximate analytical relationship between the bit error rate and the probability of delay violation.

[0106] The steps described above will be explained in detail below.

[0107] In step S1, the discrete time slot short packet communication channel and queuing model are as follows: Figure 1 As shown, it consists of two parts. The upper part presents a point-to-point short packet communication system under Nakagami-m fading channel: data is generated from the data source on the left, first entering the transmitter's queue / buffer Q[n], and then a finite code length encoder packages the data in the queue into codewords of length N symbols in each time slot, which are then transmitted through the Nakagami-m fading channel, experiencing a fading coefficient h. n and noise z n After action, a received signal y is formed. n The data is then decoded and written to the data pool on the right. The lower half shows the corresponding discrete-time queuing process along the time axis: data packets arrive at the queue at times n-1, n, and n+1 respectively. On the queue row, a continuous interval represents the duration of their stay in the queue, and on the data pool row, the time they leave the queue is marked with symbols such as D[n-1] and D[n], thus visually depicting the queuing delay of each arriving data packet from arrival, queuing to successful reception.

[0108] Specifically, this invention considers a point-to-point short packet communication system. The system employs a discrete time slot structure, where time is divided into equal-length transmission blocks, each containing N symbols. The baseband channel model within the i-th time block is as follows:

[0109] y i [k] = h i x i [k]+z i[k], k = 1, ..., N,

[0110] Where x i [k] represents the transmitted signal, satisfying the average power constraint NP = ||x i || 2 , z i [k] is Gaussian white noise with variance N0; channel gain h i It is independently and identically distributed among the blocks, and its squared amplitude |h i | 2 It follows a Nakagami-m distribution, therefore the instantaneous signal-to-noise ratio of a single block can be written as: {γ i This constitutes a series of random processes with iid values. The corresponding instantaneous SNR probability density function is:

[0111]

[0112] Where m is the shape parameter. This represents the average signal-to-noise ratio.

[0113] This invention models the link as a discrete-time queuing system. Let a[i] be the amount of data arriving in the buffer within the i-th time block, and a discrete Bernoulli random arrival model is adopted.

[0114] a[i]=a0X i ,X i ~Bernoulli(p)

[0115] Where a0 is the amount of data carried in each packet arrival, p is the probability of a packet arriving in that time block, and the mean and variance of the arrival process are μ. a =a0p, remember and Let be the cumulative arrivals and cumulative service volume of the first n time blocks, respectively, where s[i] represents the amount of data successfully transmitted in the i-th time block. Let Q[n] be the queue length at the end of the n-th time block. Given a sufficiently large buffer and a first-come, first-served (FIFO) strategy, we have:

[0116] Q[n] = max{A[n] - S[n], 0}

[0117] Let D[n] be the queuing delay experienced by the data arriving in the nth time block. If a steady-state distribution of D[n] exists, then the probability of delay violation is defined as:

[0118] δ=Pr{D[∞]≥D max}

[0119] Where D max is the given delay threshold.

[0120] In step S2, the short packet transmission model is established as follows.

[0121] Suppose that each time block contains N channels in use, and the instantaneous signal-to-noise ratio in the i-th time block is γ. i Given that the target block error rate is ε, based on finite block length coding theory, the transmission rate of this block can be expressed as:

[0122]

[0123] in The inverse function, C(γ) i V(γ) represents the instantaneous channel capacity. i (m) represents the channel dispersion in the Nakagami-m channel:

[0124]

[0125] Define s[i] as the normalized service quantity within the i-th time block, i.e., s[i] = R(γ). i If {s[i]} constitutes an independent identically distributed random process determined by Nakagami-m fading and finite block length encoding, its mean and variance will be used for subsequent effective capacity and time delay performance analysis.

[0126] In step S3, the process of establishing the effective bandwidth and effective capacity model is as follows.

[0127] To statistically characterize the impact of random arrivals and random service on queuing delay, this paper introduces the theories of effective bandwidth and effective capacity. Based on the large deviation method, this theory transforms random rates into equivalent rates directly related to QoS constraints by characterizing the exponential moments of cumulative arrivals and cumulative service volume.

[0128] For the arrival process, record The cumulative arrivals over the first n time blocks, whose logarithmic moment generating function, normalized to unit time, can be written as:

[0129]

[0130] Based on this, the effective bandwidth of the arrival process is defined as:

[0131]

[0132] This function is used to statistically characterize the minimum constant transmission rate of a random arrival process under a given QoS constraint parameter θ. When θ→0, we have α A (θ)→μa As θ increases, the effective bandwidth increases monotonically, reflecting the increased demand for system resources during arrival under stricter statistical QoS constraints.

[0133] For the service process, the cumulative service volume is recorded as follows: Its normalized logarithmic moment generating function is:

[0134]

[0135] The effective capacity of a service process is defined as:

[0136]

[0137] This function reflects the equivalent service rate that a random service process can provide when the statistical QoS constraint is θ. As θ→0, we have α S (θ)→μ s As θ increases, the effective capacity gradually decreases due to channel fading and uncertainties caused by finite block length.

[0138] Within the effective bandwidth / effective capacity framework, the statistical QoS constraints of a stable queue are typically characterized by a unique QoS exponent θ, which satisfies the balance equation:

[0139] α A (θ)=α S (θ)

[0140] For a given delay threshold D max The steady-state time delay violation probability is approximately:

[0141]

[0142] Where η is a constant related to factors such as the initial state, and is approximately 1 under high SNR. This model links arrival statistics, service statistics, and delay tail distribution, providing a foundation for subsequent analysis of QoS metrics in Nakagami-m short packet transmission scenarios.

[0143] In step S4, the effective capacity linearization process based on the central limit theorem is as follows.

[0144] Under the constraints of Nakagami-m channels and finite block lengths, the distribution of service volume in a single time slot is quite complex, making it difficult to obtain a closed expression in general. Solving for the QoS exponent θ often relies on numerical searches. To obtain approximate analytical results with clear physical meaning and to facilitate the analysis of the impact of parameters such as block error rate ε and block length N on statistical QoS, this paper adopts a linearization method based on the central limit rule to perform second-order approximations of effective bandwidth and effective capacity as θ→0.

[0145] For the arrival process, under the condition that {a[i]} are independent and identically distributed and have finite second moments, by the central limit theorem, when n→∞, A[n] approximately follows a mean of nμ. a variance is The Gaussian distribution is used as an approximation. Substituting this approximation into the definition of effective bandwidth and performing a second-order expansion with respect to θ, we obtain a linear expression for the arrival process:

[0146]

[0147] For the service process, it is necessary to first calculate the mean and variance of the normalized service volume s[i] per time slot. Let:

[0148]

[0149] The mean and second moment of a single time slot service volume are:

[0150]

[0151]

[0152] To obtain a form that facilitates calculation, SNR is substituted. in To achieve the average SNR, the above integral can be uniformly written as Gamma weight t. m-1 e -t form:

[0153]

[0154] For the shape of The integral can be approximated by the Gauss-Laguerre quadrature with parameter α = m⁻¹. Let... Let K be the nodes and weights of the Gaussian-Laguerre product at point K, then we have:

[0155]

[0156]

[0157] Under the conditions that {s[i]} are independent and identically distributed and have finite second moments, by the central limit theorem, when n→∞, S[n] approximately follows a mean of nμ. s (ε), variance is The Gaussian distribution is used as an approximation. Substituting this approximation into the definition of effective capacity and performing a second-order expansion with respect to θ, we obtain a linear expression for the arrival process:

[0158]

[0159] α A (θ) and α SSubstituting the linearized result of (θ) into the QoS exponential balance equation α A (θ)=α S (θ), thus obtaining an approximate expression for the QoS index.

[0160]

[0161] The result is expressed in μ a , μ s (ε) Using this as the core parameter, the statistical characteristics of the arrival process and the Nakagami-m finite block length service process are unified into the same QoS metric, providing a basis for subsequent analysis of the balance between latency violation probability and block error rate.

[0162] Figure 2 The relationship between the average service load per slot and the block error rate (BER) in a Nakagami-m channel is presented. The horizontal axis represents the block BER ε (logarithmic scale, from 10^6 to 10^6). -5 Up to 10 -1 The vertical axis represents the average service load E[S1] per time slot (unit: bit). The figure shows curves for SNR values ​​of 0dB, 5dB, 10dB, and 15dB. The marked solid lines represent simulation results, while the dashed lines of different line types represent the corresponding theoretical calculations. It can be seen that, for a given SNR, the average service load monotonically increases as the block error rate (ε) increases; under the same bit error rate constraint, the higher the SNR, the larger the average service load, exhibiting a roughly parallel distribution. The enlarged view in the lower left corner provides a detailed view of the extremely low bit error rate region, showing that the theoretical curve and the simulation curve are highly consistent in this region, thus verifying the accuracy of the average service load approximation calculation method of this invention.

[0163] In step S5, the process of deriving the approximate analytical relationship between the bit error rate and the probability of delay violation is as follows.

[0164] Based on the finite block length theory, approximate expressions for the mean, variance, and QoS index of a single time slot service volume under the Nakagami-m channel are given:

[0165]

[0166] Where μ C =E[C(γ)]、 The above constants are only related to the Nakagami-m channel statistics. This is related to the block length N. In this scenario, the mean and variance of the service volume per time slot are both functions of the bit error rate ε. To obtain a QoS approximation with a simple form, this paper adopts fixed-point linearization of the variance near the target operating point ε0, that is, letting... And retain μ s (ε) has a first-order dependence on ε. Therefore, the delay violation probability can be approximated as:

[0167]

[0168] make Δ(ε)@μ s (ε)-μ a Press Q -1 Solving for (ε), we obtain:

[0169]

[0170] Further definition:

[0171]

[0172] You will then receive:

[0173]

[0174] This result shows that, under Nakagami-m channel and finite block length constraints, the Q-value corresponding to the bit error rate is... -1 (ε) and The relationship between them is approximately linear, and its structure is consistent with the conclusions already obtained in the AWGN scenario.

[0175] In high reliability regions, Q -1 Since (ε) is relatively large, the Mills approximation of the Q function can be used to obtain an explicit ε-δ relationship:

[0176]

[0177] When the average SNR is high and the system is operating in an extremely high reliability region... When the value is large, the logarithmic and constant terms in the above formula can be ignored relative to the square terms, thus obtaining a simpler high SNR approximation form:

[0178]

[0179] The above equation gives the bit error rate-delay tradeoff between the bit error rate ε and the delay violation probability δ in a random arrival scenario under the constraints of a Nakagami-m channel and finite block length.

[0180] In summary, this invention establishes a complete integrated cross-layer analysis framework for "arrival statistics - finite block length service - queuing delay" under single-user, Nakagami-m fading channel conditions.

[0181] Figure 3The relationship between the delay violation probability and the delay threshold (SNR = 5dB) in a Nakagami-m channel is given. The horizontal axis represents the discrete delay threshold D. max (In time slots), the vertical axis represents the corresponding time delay violation probability δ(D) max The figure shows the block error rate ε = 10. -5 10 -4 10 -3 The curves are shown for three different scenarios, where dashed lines with different line types represent theoretical analysis results, and solid lines with different markings represent simulation results. It can be seen that as the time delay threshold D... max As the bit error rate (BER) increases, the probability of delay violation decreases monotonically and rapidly; for the same delay threshold, the lower the BER, the higher the corresponding probability of delay violation. The theoretical curve and the simulation curve closely match throughout the entire range, verifying the accuracy of the approximate expression for the delay violation probability given in this invention.

[0182] Figure 4 The trade-off between bit error rate and delay violation probability under Nakagami-m channel is presented (SNR = 5dB). The horizontal axis represents -lnδ(D max The vertical axis is -lnε, corresponding to the logarithmic scales of the delay violation probability and the block error rate, respectively. The delay threshold D is shown in the figure. max The theoretical and simulated curves for the three cases of 5, 10, and 15 are presented, with dashed lines representing theoretical values ​​and solid lines with different markings representing simulated values. It can be seen that the curves on logarithmic coordinates approximate a smoothly descending curve, and the curves corresponding to different delay thresholds are parallel or approximately parallel: the larger the delay threshold, the higher the allowable bit error rate under a given delay violation probability constraint, reflecting the reliability-delay trade-off characteristic established in this invention. Furthermore, the time-domain simulation results are highly consistent with the analytical approximation.

[0183] The method provided by this invention has the following innovations and advantages:

[0184] (1) A refined statistical modeling method for the service process was established: Under the Nakagami-m block fading channel, a single-slot normalized service quantity was constructed using a finite block long rate expression containing the channel dispersion term. The Nakagami-m distribution of instantaneous SNR was jointly modeled with the finite block long rate, and the mean and variance of the service quantity were efficiently calculated by Gauss-Laguerre quadrature, avoiding the errors caused by the traditional approximation based on Shannon capacity or simple average SNR.

[0185] (2) An analytical solution method for the QoS index based on linearized effective bandwidth / effective capacity was established: the central limit theorem was used to perform a second-order expansion of the logarithmic moment generating function of the random arrival process and the finite block length service process, and linearized expressions for effective bandwidth and effective capacity were given respectively. Based on this, α was constructed and solved.A (θ)=α S The equilibrium equation of (θ) yields an approximate closed-form solution for the QoS index θ(ε), realizing a unified characterization of "arrival statistics - service statistics" under a single QoS index.

[0186] (3) A cross-layer closed-form solution was established between the bit error rate and the probability of delay violation: Based on the above QoS index solution, the statistical delay tail probability expression and the finite block length bit error rate parameter Q were combined. -1 By combining (ε), we can derive an approximate analytical relationship (including an approximate linear relationship in the high reliability region) between bit error rate and delay violation probability under the conditions of Nakagami-m channel fading, finite block length coding, and random arrival queuing.

[0187] In summary, this invention addresses short-packet URLLC scenarios under Nakagami-m fading channels by explicitly introducing the reachable rate expression (including dispersion terms) of finite block length coding into the service process modeling. It unifies this with the random arrival queuing model within the effective bandwidth / effective capacity framework, enabling a more accurate evaluation and characterization of system cross-layer performance under strict delay constraints and ultra-high reliability requirements. Compared to existing methods based solely on Shannon capacity or average rate approximations, this invention significantly reduces the deviation in estimating reachable service capabilities and delay violation probabilities under low bit error rates and small block lengths, thereby improving the reliability of parameter design and verification. Furthermore, this invention utilizes central limit theorem linearization and Gauss-Laguerre quadrature to transform the computational process, which previously relied heavily on Monte Carlo simulations or high-dimensional numerical integration, into a structured and programmable computational flow. This reduces computational complexity and parameter tuning costs while maintaining accuracy, facilitating the rapid generation of bit error rate-delay violation probability trade-off curves. Furthermore, the analytical mapping relationship and step-by-step process obtained by this invention can be directly embedded into engineering software or link planning tools for online / offline evaluation and configuration determination of critical tasks such as industrial wireless control, demonstrating good feasibility and reusability. Compared to approximations based solely on Shannon capacity or ignoring dispersion terms, this invention exhibits smaller deviations under low bit error rate / small block length conditions, making it particularly suitable for the high-reliability operating range of URLLC.

[0188] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. However, these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for short packet wireless communication error rate-latency cross-layer analysis, characterized in that, The method comprises the following steps: S1, establishing a discrete time slot short packet communication channel and a queuing model in a single-user-oriented Nakagami-m block fading channel; S2, establishing a short packet transmission model in a Nakagami-m channel under a limited block length information theory framework, and giving a single time slot normalized service amount; S3, based on a large deviation theory, introducing an effective bandwidth and an effective capacity model, mapping a random arrival process and a random service process into an equivalent rate related to a quality of service (QoS) index; S4, using a central limit theorem to perform a second-order expansion on a logarithmic moment generating function of the arrival process and the service process, and obtaining an explicit approximate expression of the QoS index; S5, based on the explicit approximate expression of the QoS index, deducing an approximate analytical relationship between a bit error rate and a delay violation probability.

2. The short packet wireless communication error rate - latency cross layer analysis method of claim 1, wherein, In step S1, the discrete time slot short packet communication channel and the queuing model are specifically as follows: Time is divided into equal-length transmission blocks, and each block contains N symbols; The channel model for the ith transmission block is: y i [k] = h i x i [k] + z i [k], k = 1,..., N, where x i [k] is the transmitted signal satisfying the average power constraint NP = ‖x i ‖ 2 , z i [k] is additive white Gaussian noise with variance N0, h i is the channel gain, which is independent and identically distributed across blocks, whose magnitude squared |h i | 2 obeys Nakagami-m distribution; Data arrival process The cumulative arrival amount for the first n time blocks, a[i] is the amount of data arriving at the buffer in the ith time block, is modeled as a discrete Bernoulli process, a[i] = a0X i ,X i ~ Bernoulli(p), where a0 is the amount of data carried by each packet arriving, and p is the probability of a packet arriving in the time block; the mean and variance of the arrival process are μ a = a0p, Service process Cumulative service volume for the first n time blocks, where s[i] represents the amount of data successfully transmitted in the ith time block; A queue evolution model is: Q[n] = max{A[n]-S[n], 0}; The delay violation probability is defined as: δ = Pr{D[∞] ≥ D max}, D[n] is the steady-state queuing delay, and D max is the delay threshold.

3. The short packet wireless communication error rate - latency cross layer analysis method of claim 2, wherein, In step S2, an expression of the single time slot normalized service amount s[i] is: s[i] = R(γ i ) where ε is the target block error rate, Q -1 (·) is the inverse function of i C(γ i ,m) is the channel dispersion under Nakagami-m channel, which is expressed as:​ where γ i is the instantaneous signal-to-noise ratio in the ith time block, and m is a shape parameter.

4. The short packet wireless communication error rate - latency cross layer analysis method of claim 3, wherein, Step S3 specifically includes: An effective bandwidth function of the arrival process is defined as: An effective capacity function of the service process is defined as: where θ is the quality of service (QoS) index, obtained by solving the balance equation: a A (θ) = a S (θ); For a given latency threshold D max The steady-state latency violation probability is approximated as: Where η is a constant related to an initial state.

5. The short packet wireless communication error rate - latency cross layer analysis method of claim 4, wherein, Step S4 specifically includes: Using the central limit theorem, the cumulative arrival amount A[n] is approximated as a Gaussian random variable with mean nμ a and variance The second-order Taylor expansion of the effective bandwidth function α A (θ) at θ→0 gives the approximate expression: The mean and the variance of the single time slot normalized service amount s[i] are calculated: wherein, Using the central limit theorem, the cumulative service S[n] is approximated as a Gaussian random variable with mean nμ s (ε) and variance S (θ) is approximated as a second-order Taylor expansion around θ = 0, yielding the approximation​ Substituting the approximate expressions for α A (θ) and α S (θ) into the balance equation α A (θ) = α S (θ) gives an explicit approximate expression for the QoS index θ:

6. The short packet wireless communication error rate - latency cross layer analysis method of claim 5, wherein, The mean and the second-order moment of the single time slot service amount are calculated by integration: where Γ( m ) is the probability density function of the instantaneous signal-to-noise ratio γ, is the average signal-to-noise ratio; Gaussian-Laguerre numerical quadrature is used for approximate calculation of the integration.

7. The short packet wireless communication error rate - latency cross layer analysis method of claim 6, wherein, Step S5 specifically includes: The approximate expression of the QoS index θ is substituted into an approximate formula of the delay violation probability: The mean and the variance of the single time slot service amount are regarded as functions of a target block bit error rate ε; The linearization is performed around the target operating point ε0, keeping μ s (ε) first order dependence on ε; The approximate formula of the delay violation probability is algebraically transformed, and an approximate analytical relationship between the block bit error rate ε and the delay violation probability δ is obtained:

8. The short packet wireless communication bit error rate - latency cross layer analysis method of claim 7, wherein, The approximate analytical relationship between the block bit error rate ε and the delay violation probability δ is finally expressed as: Where, System parameters include average signal-to-noise ratio Nakagami-m fading parameter m, block length N, arrival process mean and variance, and delay threshold D max .

9. The short packet wireless communication error rate - latency cross layer analysis method of claim 8, wherein, In a high signal-to-noise ratio region, the approximate analytical relationship between the block bit error rate ε and the delay violation probability is simplified as:

10. A method of configuring parameters of a communication system, characterized by The short packet wireless communication bit error rate-delay cross-layer analysis method according to any one of claims 1-9 is applied, the approximate analytical relationship between the bit error rate ε and the delay violation probability P_delay is established, the target block bit error rate is determined according to a target delay violation probability requirement, or the delay performance that can be achieved by the system is evaluated according to the target block bit error rate, so as to guide configuration of at least one parameter in a system coding rate, a block length, a transmission power or a scheduling strategy.

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