Age of Information Minimization for Frameless ALOHA in Grant-Free Massive Access

In this paper, we focus on the optimal problem of average age of information (AAoI) in grant-free massive access, and propose an age-critical frameless ALOHA (ACFA) random access protocol, where the AAoI is implicitly reduced by banning the transmission of activated user equipments (UEs) recovered successfully in the last frame. In particular, we analyze the dense and sparse access models according to the activation probability, and present these scenarios with time-stamped sampling either at the beginning of the frame or in the first slot transmitting the packet. In order to qualify the AAoI of proposed protocol, we define two virtual rates and establish an iterative framework to analyze the access successful probability (ASP) of the protocol in asymptotic regime, and derive the closed-form expressions of AAoI as a function of ASP and virtual rate in all cases. Further, we formulate the optimal problems of normalized AAoI in all cases, and obtain the selection of access parameters by asymptotic analysis and simulations, respectively. Finally, we compare our protocol with state-of-the-art schemes, and the simulation results show that the ACFA random access protocol outperforms these benchmark schemes, and has great potential of access-banned policy in minimizing AAoI for frame-based protocols.


I. INTRODUCTION
M ASSIVE machine-type communication (mMTC) is the most important scenario for future Internet of Things (IoT), and aspires to support the sporadic burst access, where massive number of uesr equipments (UEs) activate and transmit packets to the base station (BS) over a common shared Yuhao Huang, Jian Jiao, Xingjian Zhang, Shaohua Wu, and Qinyu Zhang are with the Guangdong Provincial Key Laboratory of Aerospace Communication and Networking Technology, Harbin Institute of Technology at Shenzhen, Shenzhen 518055, China, and also with the Peng Cheng Laboratory, Shenzhen 518055, China (e-mail: 20s152118@stu.hit.edu.cn;jiaojian@hit.edu.cn;x.zhang@hit.edu.cn;hitwush@hit.edu.cn;zqy@hit.edu.cn).
Color versions of one or more figures in this article are available at https://doi.org/10.1109/TWC.2023.3273531.
Digital Object Identifier 10.1109/TWC.2023.3273531channel [1], [2].For the IoT UEs with low computing capabilities, the traditional 4-step access scheme (e.g.NR) needs to be simplified to adapt to mMTC service, and the grant-free protocols are proposed with the 2-step access process and simpler configurations to reduce the complexity and delay [3], [4].
A type of grant-free protocol is represented by the slotted ALOHA protocol [5], where the processing complexity and delay are reduced.Since there is no conflict detection, there is a high collision probability between packets, which can only reach 1/e of normalized throughput under high load conditions.In order to break through the performance bottleneck of the above protocols, a new frame-based grantfree access protocol using successive interference cancellation (SIC) technology is proposed, which is called code slotted ALOHA (CSA) protocol [6], [7], [8], [9], [10], [11].Among them, the frameless ALOHA (FA) protocol is a simple CSA protocol, where UEs transmit the replicas of the packets with a constant probability (called access probability) on each slot in a frame, and the BS utilizes the SIC to decode the packets, which is similar to the belief propagation (BP) decoding of the LT code in binary erasure channel (BEC) [12].The finitelength performance of FA protocol is analyzed in [13], showing that it is a simple but throughput-efficient protocol.While the performance of CSA protocol, especially the FA protocols are mainly determined by the access parameters, so the performance of active UE detection is also critical [14], [15].
On the other hand, in some typical IoT scenarios, e.g., in wireless sensor network (WSN), sensor nodes transmit monitoring indicators to information fusion nodes in real time, which puts forward higher requirements for the system freshness [16].How to measure and improve the freshness of the system has become a key problem that need to be solved urgently [17].At present, researchers use a new metric called age of information (AoI) to describe the freshness of packets monitored by UEs in mMTC scenarios, which originated from [18].AoI is related to the time-stamp interval between two update packets transmit by UE, and BS calculates the average AoI (AAoI) to get the freshness description of the system by fairly measuring the AoI of all UEs [19].It should be noted that AoI is different from other traditional indicators, e.g., delay, which has been verified in some works [17], [18], [19].
A class of fundamental works has gained attention, deriving the closed expressions of AAoI in CSA protocols [20], [21], [22], [23].In [23], the authors further discuss the gain of AAoI brought by the SIC process early in the frame, and obtain the corresponding optimal degree distribution based on the asymptotic regime analysis.However, these works focus on giving closed expressions of AAoI in the CSA protocols without redesigning a new access policy based on the AoI characteristics.
Another category of works is based on the slotted ALOHA protocol, which discussed the impact of the arrival rate and packet error rate (PER) to the system AoI [24], and redesign the age-related random access policy [25], [26], [27], [28].The authors in [25] find that the AAoI can be effectively reduced, when the UE can access only if the AoI is greater than a certain threshold.The authors in [26] and [27] conduct a comprehensive steady-state analysis of the threshold ALOHA protocol, and obtain the corresponding AoI expressions, which can optimize the access parameters of the protocol.The authors in [28] study the slotted ALOHA protocol with a fixed and adaptive thresholds, respectively, and derive the lower bounds.
Therefore, we propose an access-banned policy with a threshold policy to improve the AAoI performance of the CSA protocol [29], and extend to the dense and sparse access models with two sampling methods.We reveal how the access-banned policy affects the AAoI performance of the FA protocol, further supporting its use in mMTC scenarios.Then, we analyze the theoretical performance of the corresponding access models and AoI sampling methods, and innovatively propose the age-critical frameless ALOHA (ACFA) random access protocol to ensure higher throughput while minimize the AAoI.Our contributions are summarized as follows.
• We propose the ACFA random access protocol, which improves the overall AAoI performance by banning the access of UEs that have been decoded successfully in the previous frame.According to different access requirements of mMTC scenarios, we subdivided into dense and sparse access models according to the active probability of UEs.Then, we introduce the AoI to measure the information freshness of the whole system, and two AoI sampling methods are subdivided at the sender according to the characteristics of the mMTC service.
• In order to analyze the AAoI performance of the proposed protocol, we distinguish the active UE and the access UE, and define two virtual code rates to track the trend of the number of allowed-access UEs in each frame, and then establish an iterative framework for the access successful probability (ASP) of the ACFA random access protocol in the asymptotic regime in two access models, respectively.Note that this iterative framework can analyze the ASP of all CSA protocol under the access-banned policy.We derive the closed-form expressions of the AAoI as a function with virtual code rates and ASP in two access models and two sampling methods, respectively.• Finally, we propose four normalized AAoI minimization problems according to the closed-form expressions of AAoI, and give the selection of optimal parameters according to the analysis and simulation.Simulation results verify the accuracy of the theoretical analysis, and show that the AAoI performance of the proposed ACFA protocol is better than that of the FA protocols, and outperforms the REARLY-5 scheme [23] in both dense and sparse access models.The remainder of the paper is organised as follows.We start our work in Section II by introducing the access model and the AAoI sampling method.To emphasize the freshness of proposed protocol, we first describe the standard frameless ALOHA protocol and our access-banned policy in Section III.We mainly utilize the itereator method to analyze the asymptotic behavior of protocol, and obtain the normalized AAoI formulas in Section IV.The optimal problem and numerical results are given in Section V. Finally, conclusions are drawn in Section VI.

II. SYSTEM MODEL
We consider a typical synchronous slot uplink random access scheme [29], where M consecutive slots are organized in medium access control (MAC) frames, and M is the frame length.Moreover, all UEs and the BS are frame and slot synchronous with each other, and a UE schedules at most one packet transmission in each frame, and the channel is modeled as collision channel [29], [30].Accordingly, the BS receives the superimposed signal from multiple UEs in each slot, and the BS can identify the following three situations according to the received signal power: a) Silence, there is no activated UE transmitting in this slot; b) Singleton, there is a single UE transmitting replica in this slot, where the slot is called singleton slot; c) Collision, multiple UEs transmit replicas in this slot, and a collision occurs.Without loss of generality, when a singleton slot is detected by BS, it will always be successfully decoded if the received signal-to-noise ratio is larger than a predetermined threshold [31].When a collision is detected, the BS will perform SIC to recover the information in the slot at the end of this frame.We call the above scheme defined on collision channels a "frame-based" scheme, for the following reasons, 1) For any UE participating in the access, although multiple replicas of data packets can be transmitted in one frame, the valid information transmitted in one frame is always one data packet.2) In the uplink mMTC scenario, the minimum unit of data transmission is a slot, but the minimum unit of performance parameters analysis is the MAC frame.3) In a frame, the transmission slots selected by N t UEs on M slots constitute a N t ×M transmission matrix, which is related to the corresponding parity check matrix of LDPC code or the LT code.The BS performs the SIC process on this transmission matrix, which is equivalent to performing BP decoding for the LDPC code or the LT code.Then, we introduce the access model and the sampling method of AoI in detail, which are related to the specific performance analysis of the ACFA random access protocol.

A. Access Model
In order to limit the randomness of UEs and facilitate subsequent discussions, we define two access models as shown Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
in Fig. 1.The access model in our paper refers to the mode of whether the UEs plans to transmit data packets in one frame.Therefore, here we mainly discuss the relationship between the number of active UEs N a and the total number of UEs N , and our measure is the average number of active UEs E[N a ].
1) Dense Access: The most common access model that all potential access UEs are activated on each frame as shown in Fig. 1(b), that is, the activation probability π = 1 and E[N a ] = N a = N .Most frame-based protocols use this access model to analysis [7], [8], [32], and we also utilize the dense access model as the starting point of our work.
2) Sparse Access: Contrary to the dense access, we suppose that each UE is activated with a small π at the beginning of the frame as shown in Fig. 1(c).In this case, the average number of activated UEs is E[N a ] = πN .The activation probability π in the sparse access model is a key parameter, and we assume that π ≤ 0.2 to ensure the sparsity.
In fact, when π gradually approaches to 1, there is no clear boundary between the two access models, so the above assumptions of π are very necessary, and we keep the E[N a ] of the two access models at the same order of magnitude to facilitate our discussions.
Similarly, no matter which access model is adopted, we assume that the total number of potential access UEs is N , and each UE in the uplink system has an identification (ID).The details are in Sec.III.Since a UE is assigned an ID, we use subscript i to indicate the UE as U i .

B. Age of Information and Sampling Method
In order to measure the freshness of information, we denote the current AoI of U i in the k-th frame as A (k) i (t), where t is the slot index in the k-th frame.We assume that AoI starts at 0, i.e.A (1) i (1) = 0, ∀i ∈ {1, 2, . . ., N }, and A (k) i (t) is incremented by one for each slot in the frame.Since the framebased protocol only performs SIC at the end of the frame, the current AoI has been increasing steadily within the frame.On the other hand, denote the slot index sampled by U i in the k-th frame as T (k) i , then the change of the current AoI at the end of the frame can be expressed as: The above formula establishes the AoI connection between frames, so that the AAoI of U i can be expressed as: Note that the AoI increases linearly in steps over time or slots when no update is recovered successfully.We follow this setting and calculate the AoI in frames.Thus, the AAoI of the system can be expressed as: Āi .
In Eq. ( 1), a key parameter is the slot index when the UE samples, which affects the updating of the AoI of the successfully decoded UEs.Similar to [23], we also define two AoI sampling methods as follows as shown in Fig. 2: , ∀i, each UE performs the sampling process at the beginning of the frame. GT: where s i 1 is the slot number of the first replica of the packet transmitted by U i .In this case, the UE samples the AoI in real time when transmits the first packet.

III. AGE-CRITICAL FRAMELESS ALOHA PROTOCOL
In this section, we mainly describe a simple access protocol from frame-based protocol family, the frameless ALOHA (FA) protocol, and describe our proposed access-banned policy.Finally, the specific access process of the proposed ACFA protocol is given.

A. Standard Frameless ALOHA Protocol
The basic settings of FA protocol are as follows [33]: • It is a protocol defined based on the MAC frame.For each slot in a frame, the active UE chooses whether to transmit a replica or not with a fixed probability, which is called the access probability.Each replica has one or more pointers pointing to the slot indexes of other replicas transmitted by the same UE.• The BS can buffer the entire MAC frame, and the BS uses the SIC for decoding after receiving a frame with the help of pointers.From the above, the FA protocol is a completely distributed, grant-free protocol.In fact, the FA protocol can be seen as a simplified version of the IRSA protocol [34].In the IRSA protocol, each UE controls the number of replicas transmitted by itself through a degree distribution, which can be represented by a polynomial [30].In related studies [35], the access probability p is generally indirectly controlled by another parameter β defined as p = β/N t , where N t means the number of accessed UEs, and β removes the influence of the number of accessed UEs.β is conducive to asymptotic analysis and also simplifies the search for optimal parameters.Therefore, in the following, we mainly discuss β rather than p.
Another aspect is the packets recovery rate (PRR) analysis of the FA protocol denoted by p r , which represents the ratio of the number of UEs successfully decoded after the SIC process in one frame to the number of accessed UEs in one frame.Similarly, we define the normalized system load g = N t /M in the FA protocol, which is expressed as the ratio of the number of accessed UEs N t to the frame length M .Further, the throughput of the protocol is denoted as gp r .At this time, p r can be seen as a function of N t , M and β.In the asymptotic regime, that is, N t → ∞ and M → ∞, p r can be regarded as a function of g and β.It is shown in [12] that, under this asymptotic regime, the PRR of the FA can be analyzed using an AND-OR tree method [32], by establishing a connection between the SIC process and the decoding of the LT code on the binary erasure channel.Based on the above, p r can be obtained by a set of iterative formulas.First, we utilize the generator polynomial form to define the UE node and slot node-respective degree distributions as follows [8], [33], where Λ i represents the probability that a UE node transmits i replicas in a frame, and P i represents the probability that a slot node has i replicas in a frame, and x is a virtual variable of polynomial.Thus, the edge-respective degree distribution can be expressed as: , and ρ(x) = P ′ (x)/P ′ (1).
For the FA protocol, the probability of each UE accessing on a slot is β/N t , and there are M slots in one frame, so Λ i obeys the binomial distribution with parameters M and β/N t , which can be approximated by a Poisson distribution in the asymptotic regime.It can be expressed as follows: where we use g = N t /M for substitution, which ensures that the above expression is only related to g and β.Thus, Λ(x) can be expressed as: and the corresponding UE node edge-respective degree distribution can be shown as: Similarly, the slot node edge-respective degree distribution can be obtained as: According to Eq. ( 8) and ( 9), the iterative expressions of the AND-OR tree analysis of the FA protocol can be expressed as follows: and where q j and r j are the average probability that there is an edge connected to the UE and slot nodes during the j-th iteration, respectively.The average probability of UEs recovered after the j-th iteration is expressed as p de r = 1 − q j .Note that in the above AND-OR tree analysis, p de r can be regarded as a function of g and β, which can be written as follows: We will use this form in the derivation in Sec.IV.

B. Access-Banned Policy
We propose the access-banned policy according to our two findings.Specifically, • In CSA protocol, UE should not perform retransmission operations, and drop the recovery unsuccessfully packets in the frame in order to achieve the minimum AoI [21], [29].• In the case of limited M slots in mMTC service with a large number of UEs, the UE with low AoI should cancel the transmission plan to provide for the access and update of other UE with high AoI, in order to make the overall AAoI lower [36], [37].A feasible policy is to give an AoI threshold, if the current AoI of each UE is greater than the threshold, the access is allowed, otherwise the access is banned.This policy has a large gain on slotted ALOHA [28], however the slotted ALOHA protocol is not a frame-based protocol.Thus, we propose the access-banned policy, regard frame as the unit of access as follows: Access-banned Policy: The UE that successfully decodes in the current frame is banned in the access of the next frame.According to whether access of UE is banned, we divide UEs into two categories: access-banned UEs and access-allowed UEs.
To support the above policy, recall that our system model has following assumptions: • UEs in the system have IDs.
• The BS can broadcast the access parameters, and there is a feedback channel to inform the UEs whether the decoding is successful on the current frame.Based on the above assumptions, our proposed ACFA protocol consists of two parts as shown in Fig. 3: • The frameless ALOHA protocol.
• Access-banned policy that does not depend on the access model and AoI sampling model.Moreover, the initialization of our ACFA random access protocol is as follows, the BS first broadcasts the beacon to establish the uplink and perform synchronization.The parameter p is broadcast by the BS through the beacon, and is set to 1 where β is predefined, and E[N t ] is the predicted number of access-allowed UEs.The UEs receive the beacon and prepare to transmit their packets in the next frame.Then, the detail procedure of the k-th frame in our ACFA random access protocol is as follows: Step 1: When the activated UE receives the feedback signal of the (k − 1)-th frame, if it is allowed to access the BS, it will access with independent probability p = β/N (k) t .If UE is not allowed to access, the UE keeps silent in the k-th frame.Note that access-allowed UEs still access according to uncoordinated contention methods, so the ACFA protocol is a grant-free scheme.
Step 2: When the BS receives the whole k-th frame, it performs the SIC process recovering packets.Then, the BS estimates 2 the number of access UEs in the next frame according to the decode status and p, and broadcasts the beacon for synchronization and updated p = β/N (k+1) t . The BS broadcasts feedback signal containing the list of accessbanned UEs for the next frame.
In the above steps of ACFA random access protocol, we omit the description related to AoI.In fact, AoI is only used as an indicator to measure the performance of the system. 1 In fact, the initial value of p does not affect the performance of the system after reaching steady state, only the time it takes to reach steady state. 2 In dense access method, the Nt can be calculated exactly with the N R .But it is difficult to estimates the Nt in sparse access, due to the complex relationship between activated UEs and banned-access UEs.The active UEs estimation can be find in [38], [39], and [40].
In our model, it can be implicitly assumed that there is a total of N column caches at the BS, each column corresponds to a potential access UE, and the AoI is obtained by calculating and accumulating the slot difference of the update time-stamp of the UE transmitted.The BS records the AoI in the cache.However, these buffers are not necessarily required in an actual BS.
Note that our proposed access-banned policy can also be extended to all frame-based protocols, such as CRDSA, IRSA, CSA and other protocols.In this paper, we take ACFA protocol as the main analysis object in Sec.V.

IV. ASP AND AAOI PERFORMANCE OF ACFA PROTOCOL
In this section, we discuss how to calculate the ASP of ACFA protocol in the asymptotic regime, and derive the closeform expressions of AAoI for ACFA protocol under two access models and two AoI sampling methods, respectively.

A. Parameters and Metrics for ACFA Protocol
Before going into the derivation of ASP, we need to review the parameters and the metrics used in this section.We define η as the ratio of the total activated UEs N to the frame length M as follows, and called the virtual code rate of the ACFA protocol, since this ratio is not equal to the actual number of UEs participating in access to the frame length.In the k-th frame, the number of activated UEs, access-allowed UEs, and decoded successfully UEs are R , respectively.Activated UEs is discussed in Sec.II, only relevant to the access model.The access-allowed UEs, who are activated and allowed to access in this frame, are affected by the access model and the accessbanned policy, which is also the key to our analysis of the ASP.The number of successfully decoded UEs is related to the SIC decoding itself, and is affected by the parameters N t , M and β.Therefore, we define the average number of UEs who are activated, access-allowed, and decoded successfully as follows, and and respectively, where the average normalized system load G a of ACFA protocol is related to E[N t ] and is defined as: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Similarly, the ASP of ACFA protocol P s is related to E[N R ] and E[N t ], and is defined as follows: Finally, we discuss these number of UEs with the given example in dense access model, where all the UEs are activated to prepare to access, i.e., N

B. ASP in ACFA Protocol
One of the contributions of this paper is to derive the iterative expression of the ASP for ACFA protocol, where we give in the following Theorem 1 and Theorem 2.
Theorem 1: The asymptotic ASP of the ACFA protocol in the dense access model P de s can be obtained through the iterative method described by the following two points: • The initial value of the iteration variable x is 1, i.e., x (1) = 1; • The iteration variable x using fixed point iteration scheme as follows: x (k+1) = ϕ(x (k) ), where ϕ(x) = 1 − xf (ηx, β), and f (ηx, β) is the PRR of the FA protocol obtained by the AND-OR tree analysis mentioned in Sec.III.
Under the above iterative method, P de R is expressed as: Proof: We first consider the evolution of the number of accessed UEs N (k) t , which is the key and difficulty to analyze the asymptotic ASP in our ACFA protocol.Note that we use • to describe the normalized number of UEs, where t /N , and R /N .Thus, we can observe from Fig. 4 that in the k-th frame, there are N UEs access-allowed, and we have, In the dense access model, the number of activated UEs is always equal to the number of total UEs in the system, which means N (k) a = N, ∀k.Thus, we can obtained the normalized number of UEs as follows, Here we can estimate the

R
with p de r and Ñ (k) t in the asymptotic regime as follows, By substituting into Eq.( 22), we have Let ϕ(x) = 1−xf (ηx, β), and the Ñt is the iteration variable.Now, we have established the relationship between the number of access-allowed UEs from frame to frame, using the fixed point iteration method described above.One question is that whether Ñt converge to a fixed point in this iterative method.In this regard, we first analyze ϕ(x).The most critical part of this function is the PRR of FA protocol f (ηx, β), but this function does not have an analytical expression, so we can only take some specific parameters η and β for rough verification.
Fig. 5 shows the PRR curves for some specific η and β.For the fixed point iteration method, the iterative process is represented as follows and shown in Fig. 5
Step 1: Make a vertical line to the y-axis and the line y = x compared to the point p 1 .The coordinates of the point are (ϕ(ϕ(1)), ϕ(1)), and x (3) is the value of x-axis.
Step 2: Make a vertical line to the x-axis at point p 1 , and intersect the curve y = ϕ(x) at the point p 2 , and x (3) is the value of y-axis.
Step 3: Repeat Step 1 and Step 2, and finally x (k) converges to the first intersection of the curve y = ϕ(x) and the straight line y = x from right to left.
To finish the proof, we first need to ensure that there is at least one intersection point between the line and the curve.In this regard, we let L(x) = ϕ(x)−x, then the function L(x) is continuous on [0, 1], and it is easy to know: Therefore, there is L(0) • L(1) < 0, i.e., L(x) must have zero point on [0, 1], so the line and the curve has at least one intersection.Now we have proved that under this iterative method, the system will eventually be in a stable state.We use • * to mark various parameters in this stable state, such as the number of UEs access-allowed in a frame is N * t , and each frame is successfully decoded with N * R UEs.According to Eq. ( 18), the ASP in steady state can be expressed as: which completes the proof.■ In the above proof, when the system is in steady state, we can also measure the normalized system load of the system as: In fact, we can only specify the virtual code rate η of the system, and cannot know the overall normalize system load G a , we can still use the above iterative method to analyze G a .
In order to verify the accuracy of asymptotic ASP via the iterative method, we carry out simulations including ASP, G a , and throughput under three sets of parameters as shown in Fig. 6, the detailed parameters are: N = [100, 500, 1000], β = 3, η in the range of [1,2] with a step of 0.05, and the corresponding M = ⌊N/η⌋.Fig. 6(a) shows the ASP versus the virtual rate η, Fig. 6(b) shows the normalized system load G a versus η and Fig. 6(c) shows the corresponding throughput performance of our ACFA protocol, we can obtain the suitable η from the simulation results.We first obtain the number of accessallowed UEs in the steady state Ñ * t via the iterative method.Then, the corresponding ASP is obtained by f (η Ñ * t , β), the corresponding G a curve is obtained by η Ñ * t , and the throughput under the corresponding η is obtained by multiplying G a with ASP.
Furthermore, the simulated ASP in Fig. 6(a) gradually approaches the asymptotic theoretical curve with the increasing of N , and when N = 500, it can agree well with the asymptotic curve.Similarly, the asymptotic throughput can also agree well with the simulation point when N = 500, and achieves 0.74.For the asymptotic theoretical curve, there is a jump point at η ≈ 1.34, and the ASP curve jumps from close to 0.98 to around 0.29.We denote η corresponding to this jump point as η τ , and divide the entire curve into three regions: 1) {η < η τ , P s ≈ 1}, there is almost no performance degradation; 2) {η > η τ , P s < 0.3}, these above two regions are called linear regions.3) The last region is {η ≈ η τ }, named the jump region, and pay more attentions to the left of the jump point, marked as η − τ .At the point η − τ , ASP is still at the a higher level, and the corresponding η is the largest.Similarly, for the simulation points, we also observed this jump phenomenon, and when N = 500, 1000, both jumping occur around η = 1.40.Note that Fig. 6(b) and 6(c) also have this jump phenomenon.
A fundamental cause of the jump phenomenon is the ϕ(x) function as shown in Fig. 5, which is determined by the PRR of FA protocol p r .It can be observed from the two curves with η = 1.3, 1.5, ∃η τ ∈ [1.3, 1.4], the curve y = ϕ(x) and the line y = x have a tangent point at about x = 0.8, which makes the convergence point jump from x ≈ 0.5 to x ≈ 0.8 in the iteration method, causing the jumping phenomenon in Fig. 6.In a physical sense, the decoding result of the previous frame determines the decoding result of the next frame due to the access-banned policy, i.e., they are coupled, which leads to a consistent decoding rate of two adjacent frames.Thus, the system performance varies rapidly after η exceeds a threshold in the FA protocol.
Then, we discuss the asymptotic ASP in the sparse access model.We also utilize the above fixed point iteration method and give the following Theorem 2: Theorem 2: The asymptotic ASP of the ACFA protocol in the sparse access model P de s can be obtained through the iterative method described by the following two points: • The initial value of the iteration variable x is π, i.e., x (1) = π; Fig. 6.The result of simulation and asymptotic curve using iterative method for ASP performance, system load estimation, and throughput performance of ACFA protocol in dense access model.
• The iteration variable x using fixed point iteration method as follows: , and f (ηx, β) is also PRR of the FA protocol obtained by the AND-OR tree analysis mentioned in Sec.III.Under the above iterative method, P de s is expressed as: Proof: We start from the number of access-allowed UEs in the (k + 1)-th frame N (k+1) t , and the remaining number of UEs who are activated but banned to access, denoted as , together constitute the total number of activated UEs in the (k + 1)-th frame N (k+1) a , as follows: where can be expressed as the product of the total number of UEs, the activation probability π and the probability of banned, i.e., N • π • p b .Because the access-banned UEs are successfully decoded in the k-th frame, so p b can be expressed as the probability of successful decoding in the k-th frame with The expectation of the above expression can be expressed as: Arranging this expression, the normalized number of accessallowed UEs in the (k + 1)-th frame can be expressed as: We can also use the PRR of FA to estimate the number of UEs decoding successfully in the k-th frames, then an approximate iterative method of the above expression can be expressed as: We follow the process used in Theorem 1 and let ψ(x) = π(1 − xf (ηx, β)), then Ñt is the iteration variable.Similarly, this iterative method will eventually reach a stable state.The proof is similar to Theorem 1.We also denote Ñt when the system reaches a steady state as Ñ * t , and the asymptotic ASP can also be obtained as: which completes the proof.■ Note that in the sparse access model, the ratio of N to M is usually fixed, and the number of UEs activated in a frame is adjusted by the activation probability π.Similar to the virtual code rate η in the dense access model, we need to define a virtual code rate ζ to characterize the activated UEs load, which is defined as the ratio of the average number of activated UEs to the frame length as In the sparse access model, we have, where ζ < 1 indicates that the number of activated UEs is less than the frame length, but ζ cannot be used as a basis for judging whether the system is overloaded, because N a ̸ = N t .Similarly, in order to verify the accuracy of asymptotic ASP, we simulated the ASP, G a and throughput versus different ζ, and the parameters are N = [10 3 , 5×10 3 , 10 4 ], η = 10, β = 3, M = N/η, π in the range of [0.5,2].We also use the iterative method to get the asymptotic performance and draw in Fig. 7, marked with a solid line.
With the same growth of N and M , the simulation point is getting closer to the asymptotic curve.Note that the ASP in Fig. 7(a) in the sparse access model still has jumps, which is related to the parameter η.When the relevance of UEs are weakened, the number of banned UEs is reduced.In this case our access-banned policy will gradually fail, but there are still some performance gains.
Unlike the dense access model, the number of accessbanned UEs is significantly reduced, making G a versus ζ in Fig. 7(b) almost close to the line y = x without obvious discontinuities.Note that the analysis in asymptotic regime is still valid, and we can select the suitable ζ from the simulation results.Fig. 7(c) shows the throughput performance of the ACFA protocol under the sparse access model, achieving a maximum of 0.85 around ζ ≈ 1 in the asymptotic regime.When N = 100, 500, the maximum throughput Fig. 7.The result of simulation and asymptotic curve using iterative method for ASP performance, system load estimation, and throughput performance of ACFA protocol in sparse access model.The jump phenomenon seems to be improved.Fig. 8. Example timeline of current age of U i .Between two recovery process, the number of frame is described as X k , and the accumulated age is illustrated by shaded area and marked by A k . of 0.763 is obtained at ζ = 0.75, 0.9, respectively.In general, its throughput performance is slightly better than that of dense access model.On one hand, fixing M makes the overall code length longer, and on the other hand, β = 3 is more suitable for sparse access model.

C. AAoI Analysis
In this subsection, we derive the closed-form AAoI expression for the proposed ACFA protocol with the following four Theorems.
Theorem 3: The AAoI of ACFA protocol in dense access model with G1 sampling method Ād,G1 is given by where P s is the ASP defined in Eq. (18).We can also get the expression of Ãd,G1 after normalized the total number of UEs N , Proof: Consider the current AoI evolution of U i as shown in Fig. 8. Let X k ∈ {2, 3, . . .} denote the number of frames between the (k − 1)-th and the k-th successful packet recovery.Thus, there are M X k slots between these two recovery process.Hence X k satisfies Pr{X = k} = (1 − P s ) k−2 P s , k = 2, 3, . . ., because there are at least two frames between two recovery process.Moreover, when P s is dependent on β and the SIC process in ACFA protocol and independent on k.Therefore, we derive the first and second moments of X k as follows, Between the (k − 1)-th and the k-th frame recovery, the accumulated AoI is expressed as A k , where the AoI starts from M and increases linearly up to M + M X k except for k = 1, marked by shaded area in Fig. 8, and we have and the sequence {A k } consists of i.i.d random variables.The AAoI in U i can be calculated by Recalling that all UEs access independently, it means Ād,G1 = Āi with Eq. ( 3), which completes the proof.■ Theorem 4: The AAoI of ACFA protocol in dense access model with GT sampling method Ād,GT is given by where the average number of indexes of the sampling slot is which also is the gain obtained by sampling in the first transmission slot under the GT method.We can also get the expression of Ãd,GT after normalized N , Proof: The proof is similar to the Theorem 3, and the sampling method only affects the calculation of the accumulated AoI A k .Specifically, for U i between the k-th and Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
(k + 1)-th frames, its AoI starts from M − T i and increases linearly to (M − T i + M X k ), and A k can be expressed as: Therefore, the AAoI in GT can be expressed as follows: Thus, we need calculate E[T ] for all access-allowed UEs.The access-allowed UEs choose whether to transmit replicas with probability p in the M slots, then the index of the first transmission slot follows the geometric distribution, which satisfies Pr{T = s 1 } = (1 − p) s1−1 p, s 1 = 1, 2, 3, . . ., M .Thus, E[T ] can be expressed as: which completes the proof.■ The above Theorem 3 and Theorem 4 describe the AAoI expression of the ACFA protocol under the dense access model.Among them, GT has one more gain term brought by sampling just in time than G1, which is related to the access probability.Since the access probability changes according to the access-banned policy in each frame in the ACFA protocol, we can use the number of access-allowed UEs in the steady state N * t to calculate the probability.Fig. 9(a) shows the AAoI performance in the above two sampling methods.We simulate with N = 100, 500, β = 3.When N = 500, the simulated Ãd basically fit Ãde d , and at η ≈ 1.35, 1.40, the ACFA protocol reaches the minimum AAoI Ãd,G1 = 1.493 for G1 and Ãd,GT = 1.312 for GT , with M = ⌊500/1.4⌋≈ 357.We have analyzed the jump phenomenon at Fig. 6(b), which can also be used to explain the jump phenomenon at η = 1.4.Corresponding to the above, we denote the point where the jump occurs as η τ , and it is the point with minimum AAoI.For η < η τ , all simulation points are very close to the theoretical curve, and there is only a deviation due to the short frame length near this point.When η > η τ , there is almost no error in the G1 case, and the gain brought by the GT case is not as much as the theoretical analysis.
Next, we will give the AAoI expression under sparse access model.
Theorem 5: The AAoI of ACFA protocol in sparse access model with G1 sampling method Ās,G1 is given by and we can also get the normalized AAoI Ãs,G1 as Proof: We also examine the behavior of X k , let p = πP s represent the average decoding rate within a frame, as follows: where X k consists of two items when k ≥ 2. When η is large, it means that UEs activation is relatively sparse, and there are very few UEs who are banned for access.We can consider that X k approximately follows the geometric distribution, satisfying Pr{X = k} = (1 − p) k−1 p, k = 1, 2, . . . .We also use Eq.(38) to calculate A k , and the first and second moments of X k can be expressed as E[X k ] = 1/p and E[X 2 k ] = (2 − p)/p 2 .Thus, we obtain the above expression and complete the proof.■ Fortunately, the relationship between G1 and GT is also applicable under the sparse access model.Therefore, we give the AAoI expression under GT sampling in the following Theorem 6, and the proof is similar to the Theorem 4 and Theorem 5, and we omit the proof.
Theorem 6: The AAoI of ACFA protocol in sparse access model with GT sampling method Ās,GT is given by and we can get the normalized AAoI Ãs,GT as Fig. 9(b) shows the AAoI performance of the ACFA protocol in two sampling methods in sparse access model.Now the key parameter is η, which describes the sparsity of the access process.The larger η is, the sparser the of active UEs in each frame.Therefore, the simulation is divided into three groups, the total number of UEs is N = [1000, 5000, 5000], and the corresponding frame length is M = [100, 500, 100].Then, η of the first two groups is guaranteed in 10.The second and third groups ensure that the total number of UEs is the same, and the impact of the difference in η on the ACFA protocol is verified.In the sparse access model, ζ is mainly used to measure the system load.
It can be seen that the sampling method does not have much influence on the normalized AAoI of the system, so we mainly discuss the comparison between the above three groups.The AAoI performance of the ACFA protocol under the three sets of parameters is convex, and it is a unimodal function about ζ.When the total number of UEs expands, the AAoI of the protocol will tend to the asymptotic curve.Specifically, when N increases from 1000 to 5000, the corresponding lowest normalized AAoI in G1 mode decreases from 1.56 to 1.27, and the corresponding lowest point also shifts from ζ = 0.75 to 0.9.We also found that when the sparsity η increases, the overall AAoI will increase.In general, the ACFA protocol also has a certain performance improvement in the sparse access model, but the gain brought by the access-banned policy will gradually fail with the increasing of the sparseness.

V. MINIMUM AAOI FOR ACFA PROTOCOL
In this section, we mainly discuss the choice of parameters β for different N to minimize the AAoI of proposed ACFA protocol, so as to improve the freshness of information.Specifically, we divide it into two parts according to the access model and propose the corresponding optimization problem, and focus on choice of the parameter β in asymptotic regime.

A. Minimum Normalized AAoI in Dense Access Model
Regardless of the access model, in the asymptotic regime, N → ∞, M → ∞, and AAoI Ā → ∞, so we mainly discuss the normalized AAoI.Taking the G1 sampling method as an example, under the dense access model, the normalized AAoI optimization problem in the asymptotic regime can be described as follows, minimize which is obtained by substituting P de s for P s , and we denote the solution of the problem as Ã * d,G1 .Similarly, for GT sampling method, there is a normalized AAoI optimization problem as follows, minimize The above two problems only differ by one item Ga β , and there are only two variables to be optimized, so we use a simple traversal method for optimization in the asymptotic regime.We fix β, and use the iterative method proposed by Theorem 1 and Eq. ( 34) to find the Ã * d,G1 under this β and the corresponding η.We show the minimum normalized AAoI that can be achieved with different β in Fig. 10.Similarly, we also simulated β versus Ã * d,G1 under the finite length of N = 100, 500, where the step size ∆β = 0.1.
Fig. 10 shows the minimum normalized AAoI under different β.Ã * d,G1 and Ã * d,GT are convex functions of β, and they are not sensitive to the change of β near the minimum value.Therefore, the requirement for the accuracy of active UEs detection is low, and a part of the computational overhead can be balanced.The minimum Ã * d,G1 is obtained around β = 2.5, and the minimum Ã * d,GT is obtained around β = 2.3 in the asymptotic regime.At N = 100, β in the range of [2, 2.5], Ã * d,G1 and Ã * d,GT have a little changes, and get 1.53 and 1.34 respectively.At N = 500, the curve in G1 sampling method is close to the asymptotic curve, while the GT sampling method has a deviation of about 5% to the asymptotic curve.
According to the approximate optimal β * obtained in Fig. 10, the AAoI performance of the optimized ACFA protocol compared to the FA protocol as shown in Fig. 11.We note that the optimal β * of two protocols are different.For the ACFA protocol, the minimum normalized AAoI is almost always obtained near the jump point η τ , and the corresponding G * a is smaller compared to FA protocol.The curve in N = 500 is already very close to the asymptotic curve.At N = 100, 500, the Ã * d,G1 of the ACFA protocol reaches 1.54, 1.36, which is lower than the 2.01, 1.87 of the FA protocol by 23%, 27%, respectively, as shown in Fig. 11(a).Fig. 11(b) shows that in GT sampling method.At N = Fig.11.The minimum Ãd,G1 and Ãd,GT versus average system load Ga for frameless ALOHA and ACFA protocol with their own β * in dense access model.100, 500, the Ã * d,GT of the ACFA protocol reaches 1.34, 1.18, which is lower than the 1.68, 1.57 of the FA protocol by 25% and 33%, respectively.For the ACFA protocol, the gain term in GT is Ga βη = 0.77 2.6×1.55 = 0.191 at N = 500, which is consistent with the simulation results.For the FA protocol, since G a = η, its gain term is always 1/β, which makes the difference between Ã * d,G1 and Ã * d,GT smaller compared with ACFA protocol.
We finally compare with the corresponding FA and the IRSA-based REARLY-5 schemes in Fig. 12, where the REARLY-5 scheme performs 5 rounds SIC after the end of each slot in the frame.The ACFA protocol is close to the result of the asymptotic analysis when the number of UEs is large in G1 sampling method, while there is approximately 7% away from the asymptotic value in GT .In G1 sampling method, REARLY-5 uses the optimal degree distribution O8 [23], and finally Ã * d,G1 reaches 1.53.The minimum normalized AAoI for the ACFA protocol reaches 1.36, which is 12.5% lower than REARLY-5.In GT sampling method, the Ã * d,GT of the ACFA protocol reaches 1.19, which is slightly lower than 1.22 achieved by REARLY-5 using the optimal degree distribution OT 8. Overall, ACFA protocol achieves a lower normalized AAoI than the REARLY-5 scheme.We note that REARLY-5 scheme obtains a timeliness gain by decoding each slot early, which increases the computation and complexity, while the ACFA protocol controls access adaptively through the access-banned policy, which only increases the cost of feedback.

B. Minimum Normalized AAoI in Sparse Access Model
In the case of sparse access, η is an uncontrollable variable used to describe the access behavior, so the optimization problem under G1 can be written as follows, minimize β,π The difference between the above two problems is still Ga βη , but in the sparse access model, the magnitude of η is larger, making the gain of GT smaller.
For the two sampling methods under sparse access model, we take η = 10, 50 as an example, obtain the curves about β, Ã * s,G1 and Ã * s,GT with the iterative method in Theorem 2 and Eq.(51-52) as shown in Fig. 13.In the asymptotic regime, β * ≈ 3, which is consistent with the growth direction of β * when M increases.Specifically, at M = 100, 500, β * is close to 2.5 and 2.8, respectively.In addition, the two curves of η = 50 have almost no difference, because the GT gain term is 0.75 2.5×50 = 0.006.At this time, due to the increasing of η, the lowest Ã * s,GT increased from 1.41 to 1.46, and the lowest Ã * s,G1 from 1.44 to 1.47.Fig. 14 shows the normalized AAoI performance of the FA and ACFA protocols under the optimal β * for two sampling methods.In both sampling methods, the asymptotic curves of the FA and ACFA protocols can effectively fit the normalized AAoI curve at high frame length.When G a increases from 0.5 to 0.9, all AAoI curves first decrease due to the high ASP when G a < 0.9 as shown in Fig. 7 (a).Then, when G a is larger than 0.9, under the combined influence of a large number of UEs perform accessing and the access-banned policy, the ASP drops sharply and most UEs cannot complete the update in time, resulting in AAoI also significantly increases.In G1 in Fig. 14(a), the ACFA protocol reaches 1.44, 1.24 at M = 100, 500, which is lower than the 1.53 and 1.38 achieved by the FA protocol by 6% and 11%, respectively.Similarly, in GT in Fig. 14(b), the ACFA scheme achieves 1.42 and 1.22, which is 7% and 10% lower than the 1.52 and 1.35 achieved by the FA protocol, respectively.Under the same frame length, the G a corresponding to the lowest point of the ACFA protocol is also almost the same as that of the FA protocol, but it shifts to the left with the increasing of M .At M = 100, the G * a of the two schemes are almost 0.75, while in the case of M = 500, the G * a corresponding to ACFA and FA are 0.83 and 0.85, respectively.We note that the sampling method hardly affects the choice of β * combined with Fig. 13, so it does not affect the overall trend of the curves (such as G * a ).

VI. CONCLUSION
In this paper, we proposed a new policy to reduce the AAoI of the FA protocol.We first proposed two typical access models and AoI sampling methods to establish the basic scenarios.We introduced two aspects of ACFA protocol in detail, i.e., standard FA protocol and access-banned policy.In order to analyze the ASP performance of the ACFA protocol in two access models, we defined some key parameters and proposed two iterative frameworks for ASP analysis in the asymptotic regime, where the results of G a v.s.η and G a v.s.ζ can provide an efficient solution for different performance requirements.Subsequently, we derived the closed expressions of AAoI in four scenarios, and established four optimization problem of AAoI, respectively.By solving the above four optimization problem, the choice of the key parameter β * was given to help adjusting the other parameters, and we showed the normalized AAoI performance of the optimized FA and ACFA protocols.In particular, in the dense access model, we compared with the REARLY-5 scheme, and the simulation results showed that the proposed ACFA protocol outperformed than the existing schemes.Our future work includes extend the access-banned policy to the multiply frames and slots cases to further optimize the energy efficiency, and also can extend our ACFA protocol for a multitype services coexistence scenario.

Manuscript received 22
November 2022; revised 11 March 2023; accepted 25 April 2023.Date of publication 11 May 2023; date of current version 12 December 2023.This work was supported in part by the National Natural Sciences Foundation of China (NSFC) under Grant 62071141, Grant 61871147, Grant 61831008, and Grant 62027802; in part by the Shenzhen Science and Technology Program under Grant JSGG20220831110801003 and Grant GXWD20201230155427003-20200822165138001; and in part by the Major Key Project of PCL Department of Broadband Communication.The associate editor coordinating the review of this article and approving it for publication was S. Zhou.(Corresponding authors: Jian Jiao; Qinyu Zhang.)

Fig. 1 .
Fig. 1.The massive access scenario and two access models.The average number of activated UEs E[Na] is N and πN in dense and sparse access models, respectively.The dotted line in (c) indicates that the UEs are activated according to the activation probability π and access to BS.

Fig. 3 .
Fig. 3.The access process of proposed ACFA random access protocol.

Fig. 4 .
Fig. 4. Example of evolution of Nt and N R in ACFA protocol with dense access model.Access operation of the UEs is reported in (a), filled rectangles indicate the access-allowed UEs.Recovery operation of the UEs is shown in (b), filled rectangles indicate the UEs with successful recovered packets.

= 1 )R = 2 .
N, ∀k.The evolution of N t and N R in the ACFA protocol is shown in Fig.4, with the simple case of N = 4 and M = 5.Firstly, all UEs are allowed to access the BS in the first frame i.e., N (1) t = N .Then, the packets transmitted by U 1 and U 3 are successfully recovered, thus N (In frame 2, U 1 and U 3 are banned to transmit packets, and U 2 and U 4 participate in access process, which means N (2) t = 2. Therefore, we get the trend of N t and N R in this example, denoted by {N (k) successfully and in the (k + 1)-th frame there are N (k+1) a UEs activated and N (k+1) t

Fig. 5 .
Fig.5.The iterative method.The iteration starts from point (1,ϕ(1)), and converges to the intersection of y = ϕ(x) and y = x closet to the start point.
ζ = E[N a ]/M .Recall that in the dense access model, we use η as this indicator, because of E[N a ] = N and ζ = E[N a ]/M = N/M = η.

Fig. 9 .
Fig. 9. AAoI performance of the ACFA protocol in two sampling method with N = 100, 500 and β = 3 versus virtual rate η and ζ in two access models.

Fig. 14 .
Fig.14.The minimum Ãs,G1 and Ãs,GT versus average system load Ga for frameless ALOHA and ACFA protocol with their own β * in sparse access model.
is the ASP in the asymptotic regime, and the solution is denoted as Ã * s,G1 .Similarly, the normalized AAoI optimization problem under GT can be formulated as follows,