Age-Critical Long Erasure Coding-CCSDS File Delivery Protocol for Dual-Hop S-IoT

The upcoming Satellite Internet of Things (S-IoT) can provide status updates relaying for ground user equipment (UE). However, the S-IoT cannot utilize the conventional hybrid automatic retransmission request (HARQ) for reliable transmission due to the high bit error rate (BER) and long propagation latency. The consultative committee for space data systems (CCSDSs) has proposed the CCSDS file delivery protocol (CFDP) to relieve the long propagation latency, and the CFDP utilizes retransmission to guarantee the reliability. In this article, we propose two age-critical long erasure coding-CFDP (LEC-CFDP) schemes to realize dual-hop timely status updates in S-IoT via a relay satellite over shadowed Rician (SR) fading channel, where the satellite and destination can select the deferred or asynchronous mode to adjust the number of inserted long erasure codes (LECs) packets, called D-LEC CFDP and A-LEC CFDP, respectively. Furthermore, the satellite can select half-duplex or full-duplex relay mode, i.e., LEC-h CFDP or LEC-f CFDP to forward packets to the destination. We derive a closed-form expression for the Peak Age of Information (PAoI) and an approximation expression for the expected end-to-end delay for the D-LEC-f CFDP scheme. Moreover, we propose an A-LEC-f CFDP scheme to further improve the PAoI, and model it as a partially observable Markov decision process (POMDP) problem, which can be solved by a low-complexity point-based informed bound (PIB) algorithm. Simulation results verify the accuracy of the theoretical derivations and illustrate that the A-LEC-f CFDP scheme can achieve lower end-to-end delay and PAoI in comparison with the existing schemes.

Abstract-The upcoming Satellite Internet of Things (S-IoT) can provide status updates relaying for ground user equipment (UE).However, the S-IoT cannot utilize the conventional hybrid automatic retransmission request (HARQ) for reliable transmission due to the high bit error rate (BER) and long propagation latency.The consultative committee for space data systems (CCSDSs) has proposed the CCSDS file delivery protocol (CFDP) to relieve the long propagation latency, and the CFDP utilizes retransmission to guarantee the reliability.In this article, we propose two age-critical long erasure coding-CFDP (LEC-CFDP) schemes to realize dual-hop timely status updates in S-IoT via a relay satellite over shadowed Rician (SR) fading channel, where the satellite and destination can select the deferred or asynchronous mode to adjust the number of inserted long erasure codes (LECs) packets, called D-LEC CFDP and A-LEC CFDP, respectively.Furthermore, the satellite can select halfduplex or full-duplex relay mode, i.e., LEC-h CFDP or LEC-f CFDP to forward packets to the destination.We derive a closedform expression for the Peak Age of Information (PAoI) and an approximation expression for the expected end-to-end delay for the D-LEC-f CFDP scheme.Moreover, we propose an A-LEC-f CFDP scheme to further improve the PAoI, and model it as a partially observable Markov decision process (POMDP) problem, which can be solved by a low-complexity point-based informed bound (PIB) algorithm.Simulation results verify the accuracy of the theoretical derivations and illustrate that the A-LEC-f CFDP scheme can achieve lower end-to-end delay and PAoI in comparison with the existing schemes.
Index Terms-Age of Information (AoI), consultative committee for space data system (CCSDS) file delivery protocol, long erasure coding, partially observable Markov decision process (POMDP), Satellite Internet of Things (S-IoT).

I. INTRODUCTION
W ITH the commercial applications of the fifth-generation (5G) mobile communication and the in-depth research of the sixth generation (6G), the Satellite Internet of Things (S-IoT), which can provide seamless communication coverage and instant response for a wide geographical area and extreme terrain, has attracted considerable attention [1].Moreover, with the upcoming applications for S-IoT, higher information freshness becomes an urgent need for many emerging timeliness applications [2], such as environmental monitoring, smart grid, and voyage navigation [3].Consider the information freshness is a destination-centric measurement that only focuses on the latest successfully received status update, which implies that traditional communication indicators, such as delay, cannot fully characterize it [4].Therefore, to evaluate information freshness, a timeliness metric named Age of Information (AoI) is introduced in this article, which is defined as the time elapsed since the most recently received status update [5].
Moreover, low Earth orbits (LEOs) high-throughput satellite (HTS) can collect the status updates and forward them over the inter-or intrasatellite links to the ground station [6], [7].However, the satellite-territory link (STL) have nontrivial propagation delay and high bit error rate (BER), which would lead to additional retransmission rounds and larger propagation delay, and severely deteriorating the freshness of the received status updates [8], [9], [10].Note that the consultative committee for space data systems (CCSDSs) has proposed the CCSDS file delivery protocol (CFDP), which can relieve the long propagation latency and poor timeliness in space communications by reducing the feedback rounds [11].In addition, two reliable transmission modes are proposed in CFDP, named the deferred and asynchronous modes, where the receiver feeds back the acknowledgment (ACK) or non-ACK (NAK) after receiving all the packets of status update in the deferred mode, and sends the NAK according to the predetermined trigger in the asynchronous mode.Yang et al. [12] proposed a CFDPbased two-hop relaying protocol (CTRP), where the relay satellite immediately forwards the received protocol data unit (PDU) packets in a full-duplex mode, which can significantly reduce the end-to-end delay.Furthermore, CCSDS released the long erasure codes (LECs) specification [13] to perform a packet-level LEC at the transport layer, and the receiver could recover the lost PDUs by utilizing the LEC PDU packets.Therefore, to fulfil the requirement of reliable transmission and avoid retransmission to achieve lower AoI, we propose two age-critical long erasure coding-CFDP (LEC-CFDP) schemes to realize dual-hop timely status updates in S-IoT.

A. Related Works
AoI is a time-sensitive performance metric, which is different from the end-to-end delay in wireless communications [14].Jiao et al. [15] studied the Average AoI (AAoI) of a two-hop system with packet arrivals only at the first node and zero-waiting policy at the second node.The analysis and optimization of Peak AoI (PAoI) is studied for the twohop tandem exponential queues with multiple sources in [16].Moradian and Dadlani [17] analyzed and optimized the AoI with static link scheduling policies through the Markovian jump linear systems in two wireless relay networks.However, their works do not take the reliable transmission into account, which are not suitable for satellite communication.Moreover, there are multiple packets that may be generated at the same time in a status update, and AAoI cannot exactly characterize the freshness in this case.Since the PAoI can capture the recovery of the last packet in each status update transmission [18], it is introduced into our scheme to measure the freshness.
Nevertheless, the bit-level forward error correction (FEC) approaching the Shannon limit still cannot guarantee the reliable transmission due to the high BER in STL [19], [20].Cohen et al. [21] proved that the LEC in the transport layer can effectively alleviate the packet lost in the physical layer to reduce the retransmission rounds in space communication.The random linear network coding (RLNC) is an excellent LEC scheme to provide high-efficiency transmission in downlink satellite-to-ground scenario [22], where the original n information packets can be recovered with a large probability by successfully receiving n NC packets, thus completing the status update.In [23], the RLNC scheme has been proved that it can achieve lower delay and AoI than several classical hybrid automatic retransmission request (HARQ) schemes in S-IoT.
However, the satellite cannot know the precise channel status information (CSI) over the widely used shadowed Rician (SR) channel in satellite communications [24], [25].Thus, the satellite is hard to select an appropriate number of LEC packets, which would lead extra delay and energy consumption to guarantee reliability.Note that the partially observable Markov decision process (POMDP) can model this unprecise CSI and minimum-AoI communication problems [10], [26], [28], Yao et al. [26] analyzed the transmission scheduling of a status update system without channel sensing, and apply POMDP to minimize the long-run AoI under the energy constraint.Ding et al. [27] investigated an end-to-end NC HARQ transmission scheme with delayed CSI through POMDP and making transmission decision to lower PAoI.In [28], POMDP is used to formulate the minimum-age scheduling problem in a wireless network, where the real-time AoIs of all sensors are unknown.Moreover, since POMDP is polynomial space hard to solve as it requires exponential computational complexity and memory, several algorithms such as point-based methods [29] have been proposed to solve it.
Therefore, to take both timeliness and reliability into consideration, we propose two age-critical LEC-CFDP schemes in the dual-hop S-IoT transmission scenario based on the deferred and asynchronous CFDP modes, which can adjust the LEC PDU packets insertion after or during the transmission of status update, respectively.Moreover, we propose two relay modes inspired by the CFDP and CTRP, and apply them to our proposal schemes.Then, according to the difference between uplink and downlink transmission, we formulate the dual-hop transmission as two appropriate POMDP models, respectively.

B. Contributions
The main contributions of this article can be summarized as follows.
1) We introduce a deferred-LEC CFDP (D-LEC CFDP) scheme for the dual-hop S-IoT transmission scenario, where the sender and relay satellite insert an LEC PDU after every L − 1 information PDUs to accelerate the recovery of lost information PDUs, and L can be adjusted via the ACK/NAK after each transmission of status update.Moreover, we propose two D-LEC CFDP schemes based on two relay modes, named D-LEC CFDP with half-duplex relay mode (D-LEC-h CFDP) and D-LEC CFDP with full-duplex relay mode (D-LEC-f CFDP), respectively.2) We analyze the D-LEC-f CFDP scheme, since it can lower the PAoI and expected end-to-end delay than that of the D-LEC-h CFDP scheme, when the channel at moderate and high signal-to-noise ratio (SNR) regions, while the latter can only achieve slightly better performance with lower complexity at low SNR.In addition, to yield the insight into the performance of the D-LEC-f CFDP, we derive a closed-form expression for PAoI and an approximate expression for the end-toend expected delay by using the renewal process, which also provides a theoretical fundament to estimate the appropriate insert position of LEC PDU to guarantee the freshness.3) We further propose an asynchronous LEC CFDP with full-duplex relay mode (A-LEC-f CFDP) scheme to improve the PAoI of D-LEC-f CFDP scheme at low SNR, where L can be adjusted via the ACK/NAK during the transmission of status update with a predetermined trigger at the receiver.Moreover, we introduce the POMDP to model the transmission and minimize the PAoI and end-to-end delay.Furthermore, we design a low-complexity optimization algorithm called pointbased informed bound (PIB) to solve this POMDP problem.Finally, simulation results validate the accuracy of our theoretical derivations and show that the A-LEC-f CFDP scheme can outperform the D-LEC-f CFDP scheme and state-of-the-art scheme.

C. Outline
The remainder of this article is organized as follows.In Section II, we introduce the system model, the D-LEC CFDP dual-hop transmission schemes and the channel model.In Section III, we analyze the PAoI and expected end-to-end delay in the D-LEC-f CFDP scheme.In Section IV, we propose the A-LEC-f CFDP scheme based on POMDP, and solve it by a low-complexity PIB algorithm.Simulation results are presented in Section V. Finally, we conclude this article in Section VI.

II. SYSTEM MODEL
We consider a dual-hop S-IoT transmission scheme as illustrated in Fig. 1, where a user equipment (UE) S collects status updates and transmits them to a relay satellite R, which is a LEO HTS and covers a wide remote area.Then, the received PDUs are forwarded by R to the ground station D as soon as possible.The channel from S to R and R to D are both modeled as the widely used SR fading channel [24], [25].
In the S-IoT dual-hop transmission scenario, we assume that there are n information PDUs and [n/(L − 1)] LEC PDUs in a status update, and these PDUs are temporarily buffered at S and then transmitted in order.Note that S can get the realtime CSI, while R only has delayed CSI because of the long STL propagation latency over time-varying SR fading channel.When these PDUs are relayed to R, R selects an appropriate relay mode and determines L to forward them to D as shown in Fig. 1.At the receiver side, if the ith information PDU is successfully received and all previous transmitted PDUs have been uploaded, it can be uploaded to the application layer.Otherwise, it is backlogged until the previous lost PDUs all be recovered by the later received LEC PDUs.Additionally, R and D are equipped with a single buffer with infinite capacity.
Consider the asymmetric link in satellite communications, we assume that the duration of 1.5 time slot equals the uplink transmission delay t s 1 for a 1-kB PDU specified by CFPD [11], while the downlink transmission delay t s 2 is one time slot for a PDU, and t r denotes the propagation delay.Moreover, since the LEO HTS has much larger Doppler shift than GEO/MEO due to the high mobility, the method to handle it is one of the main issues that being studying in the Third Generation Partnership Project (3GPP), which is called the Doppler compensation [30], [31], [32].We can first precompensate the Doppler shift roughly by exploiting the predictability of the satellite's movement [30].After that, a refine precompensation of the oscillator offsets should be performed by utilizing a phase-locked loop (PLL) to track the reference tone [31].Furthermore, Consider the bandwidth of HTS is about 800 MHz-2 GHz, a guard band that double than the Doppler shifts can be utilized to relieve the influence of Doppler shifts on the system [32].
As shown in Fig. 2, in our D-LEC CFDP scheme, each status update includes n/(L − 1) LEC PDUs generated via RLNC from a sufficiently large field to combat the erasures of information PDU, where an LEC PDU C k j (j = 1, 2, . . ., n/(L − 1) , k = 1, 2) is encoded from all the preceding information PDUs d i (i = 1, 2, . . ., (L − 1)j) in the kth hop, and is periodically sent after the transmission of L − 1 information PDUs [33].Hence, the lost PDUs can be successfully recovered when the number of LEC PDUs collected by the receiver is equal to that of the lost information PDUs.Moreover, the precise SNR of CSI is estimated from the receiving signals at the receiver [34], and feeds back to the transmitter within 2-3 bits in the ACK or NAK at the end of receiving the last PDU, the transmitter adjusts L or retransmits according to the feedback [11], and the related notations are summarized in Table I.

A. D-LEC CFDP Scheme and Relay Mode
In this section, we propose the D-LEC CFDP scheme with two relay modes in dual-hop S-IoT to satisfy higher freshness requirement.Since the basic idea of D-LEC CFDP scheme has been introduced above, we focus on two relay forward schemes, named the D-LEC-h CFDP transmission scheme as shown in Fig. 2(a), and the D-LEC-f CFDP transmission scheme as shown in Fig. 2(b).
In the D-LEC-h CFDP scheme, both R and D feed back ACK or NAK until receive all PDUs of the status update, and R would not forward any PDU until the first-hop transmission is completed at t = 15 as shown in Fig. 2(a).Then, R selects L 2 = 3 according to the last received CSI from D, and transmit PDUs as same as the first-hop transmission.Moreover, L 2 = 3 is not suitable for R due to the CSI is worse in the second hop, which means that D needs retransmission from R. Therefore, the HTS sends three LEC PDUs C 2 4 , C 2 5 , and C 2 6 according to the NAK at t = 27, and D utilizes them to recover four lost PDUs at t = 30.5.Finally, all information PDUs are uploaded to the application layer in order.
On the other hand, R would start transmission immediately when the first PDU could be sent in the D-LEC-f CFDP scheme, i.e., each successfully received or recovered PDU is immediately forwarded, if there is no more PDU with smaller sequence number than itself, such as d 1 and d 2 .Otherwise, the PDUs are backlogged at the transport layer in R, i.e., d 4 recovered at t = 8 cannot be forwarded before d 3 is recovered by C 1 2 at t = 10.The transmission is finished at t = 22.5 in the D-LEC-f CFDP scheme, which means the D-LEC-f CFDP can achieve about 28% lower PAoI compared with D-LEC-h CFDP in the same channel condition, without considering the influence of self interference (SI) [35].
It is worth noting that unlike S, which can get realtime CSI and make correct decisions on L, R can only receive the delayed CSI.Therefore, when the CSI of R to D turns to bad, D always needs retransmission to recover the lost information PDUs, which evidently deteriorates the performance of expected end-to-end delay and PAoI.Thus, we design the A-LEC CFDP with full-duplex relay mode (A-LEC-f CFDP) to solve the problem.

B. A-LEC-f CFDP Scheme
We propose the A-LEC-f CFDP scheme to further improve the timeliness over SR fading channel in this section, where S or R can receive NAKs from R or D in each T time slots, and reselect an appropriate L according to the NAK to avoid retransmission.Next, we describe the example A-LEC-f CFDP scheme as shown in Fig. 3, where the channel condition is the same as that in Fig. 2.
In the first-hop transmission, it is identical to the D-LEC-f CFDP scheme since S can get the real-time CSI and make proper decision.But in the second-hop transmission, when d 3 and d 4 fail to be received at D, and the LEC PDU C 2 2 received at t = 14 cannot recover both d 3 and d 4 .Thus, R reselects a new L 2 by the NAK received at t = 15 and immediately transmits three LEC PDUs.Thus, D utilizes them to recover the lost information PDUs at t = 20.5, and all PDUs can be uploaded at this time.Else if there are no new PDUs, and the receiver needs LEC PDUs to recover the lost PDUs, the sender would set L = 1, and calculate the number of LEC PDUs n L /(1 − P e ) + v L (v L ≤ n − n L ) that needs to be transmitted, where n L is the number of lost information PDU at that moment, P e is the erasure probability and v L is the redundancy of LEC PDUs.
To sum up, the A-LEC-f CFDP can reduce the consumption of time slots than the D-LEC-f CFDP in the same channel condition by sending extra NAKs.Note that an accurate block fading SR channel is modeled in the following, i.e., P e is accurate enough to make R calculates the required number of LEC PDUs to recover all lost PDUs via the feedback CSI, there is no more than one retransmission round in our D-LEC CFDP and A-LEC CFDP schemes.Furthermore, to select a proper L with less frequency NAK feedback, we can design an optimal POMDP transmission policy base on the SR channel as follows, and the details are given in Section IV.

C. Channel Model
The channel of S-IoT is modeled as the widely used SR fading channel in this article.We assume that the CSI remains constant during a status update, and randomly changes when the next status update comes.Thus, the probability density function (PDF) with instantaneous SNR r of SR fading channel can be written as follows [36]: where 1 F 1 (., ., .) is the confluent hypergeometric function, γ denotes the average value of SNR, is the average power of LoS component, 2b 0 is the average power of the multipath component, and m(m ≥ 0) is the parameter of channel fading severity.
The closed-form expression of BER with single antenna in SR fading channel is given in [23].Moreover, to relieve the deep fading in SR channel, we introduce the multiple-input multiple-output (MIMO) system to improve BER [37].Thus, the BER with MIMO system in SR fading channel can be derived as follows [26]: where N = N r × N t is the diversity gain, depending on the number of receiving and transmitting antennas, and is the Gamma function.
However, since the SR fading channel is quite volatile, the reliable transmission cannot be fully guaranteed in MIMO system [38].Thus, we propose the A-LEC-i CFDP scheme to further reduce PAoI, where S or R can reselect a proper L via a POMDP optimization with feedback delay T and propagation delay t r .The detailed analysis is introduced in Section IV.

III. THEORETICAL ANALYSIS OF D-LEC-F SCHEME
In this section, we mainly analyze the expected end-to-end delay and PAoI of D-LEC-f CFDP scheme, since the derivation of these two indicators in the D-LEC-h CFDP scheme are similar to the successive transmissions of existing end-to-end schemes.

A. End-to-End Delay
Without loss of generality, we assume that S generates one update with n(n > 0) information PDUs at t = 0.According to the transmission processes as shown in Fig. 2(b), the end-to-end delay D i of an information PDU d i (i = 1, 2, . . ., n) can be divided into four parts.
1) Queueing Delay D q : The time elapses for the information PDU from its generation until its transmission at S. 2) Forwarding Delay D f : The time elapses for the information PDU since it is received by R until forwarded to D. 3) Service Delay D s : The time elapses for the information PDU since it is transmitted from R until it could be uploaded at D. 4) Propagation Delay t r : The time elapses for the information PDU transmits from S to R, or R to D. Hence, we have ( Then, when d i is to be transmitted, S has already sent (i−1) information PDUs and N t = ( i/[L 1 − 1 ] − 1) LEC PDUs.Therefore, D q (i) can be written as where t s is the transmission delay and we assume that t s 1 = 1.5t s 2 = 1.5t r .Therefore, the average queueing delay (1/n) n i=1 D q (i) can be further expressed as Thus, by combining (3) and ( 5), the expected end-to-end delay D = (1/n) n i=1 D i for the D-LEC-f CFDP scheme can be written as We can find that the first and second terms of ( 6) are fixed in this scenario, while the third and fourth term depends on the channel status.Therefore, we introduce the concept of "busy/idle stage" in our transmission scheme for further analysis [33].
First, the "busy stage" refers to the time elapses since the transmission is suspended due to the lost information PDUs until that the received PDUs are all be recovered, e.g., as the time interval from t = 11 to t = 22 in the second-hop transmission as shown in Fig. 2(b).Then, we define X q (q = 1, 2, . ..) as the number of transmitted LEC PDUs during the qth busy stage.We also define the "idle stage" as the period of sequential upload without waiting in the queue at the receiver.Note that X q cannot accurately describe where the busy stage begins, because any information PDU can be the first lost PDU in an L PDUs transmission interval as shown in Fig. 2(b).Thus, we only use X q to measure the length of the busy stage started from the second L-interval, i.e., X q ≥ 1, and denote B q as the length of the first L-interval in the qth busy stage.Moreover, the idle stage is an exceptional case where there is only one Linterval in a busy stage.Therefore, X q follows independent and identically distributed (i.i.d.) due to the PDU lost is following i.i.d., and can be characterized by the probability distribution refers to [33] where p is the block error rate (BLER), which can be calculated by the BER P e and PDU size B as follows: Accordingly, it is easy to find the expectations that which indicates that the lost PDUs can be recovered by the previous transmitted LEC PDUs, i.e., it could avoid retransmission.Moreover, let p k and L k denote the BLER and inserted gap in the kth hop transmission, respectively.If the channel condition deteriorates and R dose not predict it because of the delayed CSI, it will lead to L 2 •p 2 ≥ 1.In this case, the received LEC PDUs cannot recover the lost information PDUs and the retransmission is necessary in the second-hop transmission as shown in Fig. 2. Thus, we provide the theoretical analysis for denote the queue length of d i at the kth hop receiver, which indicates the number of PDUs in a backlogged queue containing d i , and Q k m denotes the maximum queue length at the kth hop receiver, and X m denotes the number of transmitted LEC PDUs during the largest X q .Thus, the expected of (1/n) n i=1 Q k (i) and Q k m in each hop can be expressed as follows: and where β k 1 , β k 2 and β k 3 are three constants, which represent the impact of the first lost PDU on average queue length and maximum queue length in the kth hop transmission, respectively, and can be written as and and Their detailed derivation are given in Appendices A and B, respectively.Note that the forwarding delay D f (i) of an information PDU d i in the qth busy stage is affected both by X q and the number of PDUs in all previous busy stage in R. In addition, in the transmission of a status update with n information PDUs, if d M has the maximum queue length Q 1 m , all subsequent PDUs d i (n ≥ i ≥ M) after d M are the same D f , which can be expressed as Therefore, we can assume that X 1 in R is the largest number in X q to obtain a upper bound of average forwarding delay as (1/n) n i=1 D f (i).Furthermore, n information PDUs can be divided into (n/[L − 1]) groups, and the probability of X q starts from the ith information PDU in the (j+1 ) without considering the extra LEC PDUs, and the total number of lost information PDUs in Then, the expected of (1/n) n i=1 D f (i) can be expressed as ( 14), shown at the bottom of the page, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where P re denotes the probability that there is only X 1 in R as follows: Similarly, the expected service delay D s is not only affected by X q in the ground station D but also affected by in R, which depends on the distribution of X m as shown (16) at the bottom of the previous page.Let X denote the lower bound of X m , and we can obtain the service delay without considering D f through the derivation similar to (9), which is the first term in (16).Thus, the expected of (1/n) n i=1 D s (i) can be expressed in ( 16) at the top of this previous page, where β 4 and β 5 are constants and can be expressed as and 14)-( 16) are inaccurate since E(X q ) = i (X q = i) • P X q (i) is infinite.Then, two cases can be distinguished in the following.
1) L k • p k > 1, which implies the preselected L cannot provide enough LEC PDUs to recover the lost information PDUs.Once the status update enters busy stage, the lost information PDUs will backlog at the receiver until the sender starts retransmission.In this case, only X 1 exists, and the probability of X 1 starts from the ith information PDU in the (j+1)th L-interval equals P D,1 = p k (1 − p k ) j(L k −1)+i−1 , which represents the second term in (25) and the second and fourth terms in (26), both shown at the bottom of the page.Moreover, there are R lr intervals in X 1 , which can be expressed as R lr = n/(L k − 1) − j − 1, and the total number of PDUs R tp and the number of information PDUs R ip in X 1 are Thus, the number of extra LEC PDUs in the retransmission to recover the lost PDUs is Therefore, the total delay time for all information PDUs in R can be written as The delay time in D without and with considering D f in R can be expressed as follows: and 2) L k • p k = 1, which means that the number of lost PDUs is exactly equal to that of LEC PDUs in the long-term average.Since the PDU lost follows i.i.d., the transmission is actually the same as (i) unless there exactly only one PDU is lost, i.e., X q = 1.Thus, if X q = 1, the probability of X q starts from the ith information PDU in the (j+1)th which is the first term in (25) and the first and third terms in (26), respectively.Furthermore, the delay time in R when X q = 1 can be written as The delay time in D when X q = 1 with and without considering D f in R are given as follows: and Therefore, we can assume that there are at most two denote the position of PDUs at D, i.e., the boundary of whether D S is affected by D f , and the approximate expectation of forwarding delay and service delay can be derived in (25) and (26).
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Finally, the expected end-to-end delay of D-LEC-f CFDP scheme can be calculated by substituting ( 14), ( 16), (25), and ( 26) into (6).

B. Peak AoI
Assume that S generates a status update in each T, let A 0 = T denote the initial AoI of a status update at S, and t denotes the time elapses from the generation of a status update to upload all the information PDUs at D, respectively.We define the generation time of each information PDU d i is t i = t 0 , and t i is the time when d i is successfully uploaded at D. Therefore, D i can be written as D i = t i − t 0 , and D n = t is the delay time of the last information PDU.
Moreover, to achieve higher timeliness, we assume that S, R and D are ruled by last-come-first-served (LCFS) policy that discarding the current status update if it has not been uploaded to the application layer in D when the next status update has been collected by S [27], and the PAoI of the last status update is 2 T and resetting AoI to A 0 .Therefore, the AoI evolution of each status update transmission in the D-LEC-f CFDP scheme is shown in Fig. 4, which can be divided into two cases.
If the first information PDU d 1 can be uploaded before all PDUs at D as shown in Fig. 4(a), we set this AoI evolution case as State 1. Otherwise, if all information PDUs are uploaded at t n as shown in Fig. 4(b), we denote this AoI evolution case as State 2.Then, the expression of PAoI A for the D-LEC-f CFDP scheme can be written as Therefore, the expectation of A can be derived as Let Nk and T k fe denote the total number of transmitted PDUs and feedback delay in the kth hop transmission, respectively.Note that D n depends on the total number of PDUs transmitted by S, the number of PDUs remain ought to be transmitted by R after the first hop and the feedback delay between R and D, which are denoted as N1 , Nr , and T 2 fe , respectively.Thus, D n can be further expressed as D n = 2t r + N1 + Nr + T 2 re , and the expectation of N2 and T 2 fe are and Moreover, it can be easily concluded that Nr equal to the sum of the queue length caused by X m in R and the number of extra LEC PDUs transmitted in the second hop retransmission, i.e., Since the second term of ( 28) depends on the value of X 1 , we define a random variable X a as the number of transmitted LEC PDUs in X 1 that failed to receive the first PDU at D. Therefore, the probability distribution of X a is given as follows [33]: Thus, the probability of State 2 can be expressed as Finally, by substituting ( 10), ( 29), (30), and ( 32) into ( 28), the closed-form expectation of PAoI can be further written as follows:

IV. OPTIMIZATION OF A-LEC-F CFDP
In this section, we formulate a POMDP problem to achieve further lower PAoI and end-to-end delay, and solve it with a lower complexity PIB algorithm for the A-LEC-f CFDP scheme.

A. POMDP Formulation
The POMDP problem in our A-LEC-f CFDP scheme can be described by a tuple <S, Z, A, T r , R, O, b>, and each parameter is introduced as follows.
S: S is the state space, where that m(0 ≤ m ≤ n) information PDUs have been transmitted, and δ(0 ≤ δ < m) information PDUs in D need to be recovered at t in the second hop.Z: Z is the set of observations, where z t−2t r = δ implies that the system can observe δ lost information PDUs in the receiver at t − 2t r .
A: A is the set of possible actions to be taken by the system, where A = {L Note that we can replace s t and s t+T with s and s , respectively, by ignoring the subscripts t and t + T, which can also be used for a, z and b.Then, the next belief state b can be calculated as follows: Denote V * (b) is the optimal value of belief state b, which can represent the maximum long-term reward based on the following optimal policy π * , and can be expressed as [39]: where τ (b, a, z ) is the updated belief state function in (34).Moreover, the optimal policy π * that can obtain the maximum expected cumulative return for the POMDP problem is also necessary.Thus, given the optimal value function V * , the optimal policy π * can be expressed as [40] π Furthermore, in order to calculate V * (b) with large belief space, (35) can be further expressed by the point-based value iteration (PBVI) as follows [29]: where {φ i t } ∈ is a set of linear vectors that is the maximizing element of V * (b) for fixed action a t , observation o and belief state b t , which can be written as However, its complexity still increases dramatically with the state space, and hence is not suitable for the code design problem considered in this POMDP.Thus, we design a lowcomplexity algorithm called the PIB that measures the optimal value function with the upper bound directly based on PBVI for the considered POMDP in the following section.

B. PIB Algorithms
In this section, some details of PIB are described.First, we set the reward function according to (33).Note that when T = 1, the time between generation and forwarding of a PDU in the first hop equals the end-to-end delay of first hop, which means the reward function of this hop can be measured by it.Considering the function is smaller than real delay when T > 1, we can increase the weight of the impact of backlogged PDUs.Meanwhile, we only need to consider the number of PDUs that remain ought to be transmitted by R at time t in the second hop, if the initial AoI is ignored.Therefore, the immediate reward function of both two hops can be, respectively, expressed as Considering the optimal belief value of POMDP is difficult to solve, and it is related to the optimal state values V * (s) as If the POMDP turns to fully observable, i.e., Markov decision process (MDP), the optimal state value of the first hop V first * MDP and second hop V sec * MDP can be easily determined by immediate reward function and transition function as follows: and Thus, the upper bound of belief value can be found by taking ( 41) and ( 42) into (40), which can be used to approximate the optimal belief value.
Furthermore, since the probability of State 2 needs to be minimized, the policy at the initialization stage must be conservative, i.e., the lower bound of the optimal value function can be used for initialization procedure.Therefore, given the belief state b 1 t , all the subsequent b i t can be determined by L, until transmitting all information PDUs, i.e., reaching b n t , and its optimal value V L (b n t ) can be expressed as Mathematically, the lower bound of the belief value is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Algorithm 1: A-LEC-f CFDP Scheme
Input: b, S, Z, V, V, T, p, n, π , Output: The optimal policy of the first or second hop transmission 12 while m ≤ n do  , p, T, π, δ, ω, d) p, T, π, δ, ω, d Output: The optimal policy of next feedback interval 1 initialization:calculate s with δ or ω, d, and update b with (s, a, z); 11 return a i Therefore, we utilize the above policy to design an PIB algorithm and solve the POMDP problem in our A-LEC-f CFDP scheme, which is summarized in Algorithms 1 and 2 in detail.The complexity analysis of our PIB algorithm is given in Section V.

V. SIMULATION RESULTS AND DISCUSSIONS
In this section, we present the simulation results to evaluate the performance of our proposed D-LEC-h, D-LEC-f, and A-LEC-f CFDP schemes, and the main parameters are shown in Table II.Moreover, the BER performance is significantly different under the infrequent light shadowing (ILS), average shadowing (AS), and frequent heavy shadowing (FHS) SR fading parameters, depending on different elevation angle, altitude, etc., which is summarized in Table III.
The BER of our LEC-CFDP scheme with different diversity gains N under FHS, ILS and AS SR fading parameters are shown in Fig. 5    in the following simulations to reflect the effectiveness of our proposed schemes.
We simulate the D-LEC-h CFDP and D-LEC-f CFDP with L k = 2, 3, 5, 11 to evaluate the performance of two schemes, and compare with an age-optimal transmission scheme for dual-hop called NC HARQ with one-time retransmission (NCor-HARQ) [23] in Fig. 6.Specifically, the D-LEC-h CFDP scheme has lower expected end-to-end delay and PAoI than the D-LEC-f CFDP scheme when SNR ≤ 8.5 dB as shown in Fig. 6(a) and (b).However, their performance will reverse if SNR ≥ 8.5 dB.The reason is that when SNR is higher, R can receive more PDUs successfully, which enable itself to forward faster, and the reception and transmission of the D-LEC-f CFDP are carried out at the same time, while that of the other are separate.In the low SNR region, the BLER of the D-LEC-f CFDP is little higher than the D-LEC-h CFDP due to the high SI, which can be represented as αP, where P is the normalized transmit power of the satellite and α = 0.1 is the SI cancelation quality parameter [35].Moreover, it can observe that our D-LEC CFDP scheme always outperform the NCor-HARQ scheme in different SNR regions.The main reason is that the LEC construction in the D-LEC CFDP can accelerate the recovery of lost PDUs, particularly when L k is small, while the LEC PDUs in the NCor-HARQ are transmitted after all information PDUs have been sent.Furthermore, when SNR ≥ 11 dB, the PAoI of NCor-HARQ gradually lower than the D-LEC-h CFDP with L k < 3, i.e., L 1 = L 2 < 3 as shown in Fig. 6, since the receivers need less LEC PDUs to recover the lost PDUs in a high SNR region.However, benefiting by the full-duplex relay mode, the D-LEC-f CFDP scheme can still outperform the NCor-HARQ scheme in this case.
Then, we simulate the D-LEC-f CFDP scheme with three different parameters of L 1 and L 2 to validate the accuracy of our derivations as shown in Fig. 7.We can observe that the theoretical derivations of expected end-to-end delay in (6) and PAoI in (33) are agreed well with the simulation results in Fig. 7(a) and (b), respectively.Moreover, we can observe that there is an optimal group (L 1 , L 2 ) that enables the D-LEC CFDP scheme to achieve the best performance in different SNR regions.For example, When (L 1 = 3, L 2 = 2) and SNR = 9 dB, both two parameters fall to the floor region.Thus, if L k , especially L 2 , is too large or too small in a specific SNR region, the PAoI and endto-end delay will both increase due to the retransmission or redundant LEC PDUs.This also requires us to design the A-LEC-f CFDP to update the optimized L k to improve the timeliness.
In Fig. 8(a) and (b), we can observe that the A-LEC-f CFDP scheme greatly outperforms the D-LEC-f CFDP scheme in both PAoI and end-to-end delay.The main reason is the A-LEC-f CFDP can reselect an appropriate L k to adapt to the varying CSI according to the delayed feedback NAK, especially in a low SNR region.Thus, R can effectively predict Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.CSI to avoid retransmission, which greatly improves the performance of the D-LEC-f CFDP in bad channel condition.Furthermore, the performance of these two schemes is similar when SNR ≥ 11 dB, because the minimum value of these two parameters mainly depends on the number of information PDUs when channel is always in good state.Moreover, we also compare the difference of throughput between two schemes and the performance versus SNR as shown in Fig. 8(c where It can be observed that the throughput of two schemes is similar when SNR≤ 10 dB.Moreover, the A-LEC-f CFDP can obviously achieve higher throughput when SNR ≥ 10 dB.However, if T is too large, L k is adjusted by the sender may deviate from the optimal policy to a certain extent, which may lead to obviously decreasing of throughput.As shown in Fig. 8(c), when the D-LEC-f CFDP scheme approaches the maximum throughput, it is better than that of the A-LEC-f CFDP scheme when T = n.Finally, we simulate the expected end-to-end delay and PAoI of the A-LEC-f CFDP scheme by optimizing via PIB, and compare with the D-step search algorithm that can achieve near-optimal solutions for the POMDP problem [41] as shown in Fig. 9(a) and (b), and take the illustrative case of 2-step search.We can observe that both two performance parameters of POMDP solved by two algorithms is similar, and the smaller the feedback interval T leads to better performance of our schemes.Moreover, the tight upper bound of V * (b) is given in ( 41) and ( 42

VI. CONCLUSION
In this article, we modeled an age-critical dual-hop LEC-CFDP transmission scheme in S-IoT, in which a UE S transmitted the collected status updates with multiple PDUs to the relay satellite R, then R selected an appropriate relay mode to forward the re-encoded PDUs to the ground station D.Then, we proposed two D-LEC CFDP schemes to improve the PAoI and expected end-to-end delay with half-duplex and full-duplex relay modes, respectively, and introduced them in detail.Furthermore, we further analyzed the D-LEC-f CFDP scheme, and derived an approximated expression for expected end-to-end delay and a closed-form expression for PAoI in the D-LEC-f CFDP scheme.Moreover, we utilized the results to preselect an appropriate L to improve the timeliness.Furthermore, we proposed the A-LEC-f CFDP scheme and formulated a POMDP problem to achieve the lowest PAoI, and solved it through our low-complexity PIB algorithm.Simulation results validated the accuracy of our theoretical derivations of expected end-to-end delay and PAoI, and showed that the A-LEC-f CFDP scheme can greatly lower the PAoI and end-to-end delay than the D-LEC-f CFDP scheme and state-of-the-art scheme, as well as higher throughput.Finally, it is worthy of noting that our work will shed important light on the timely multihop transmission design for the mega-constellation LEO satellite networks, such as Starlink and OneWeb.APPENDIX A DERIVATION OF THE AVERAGE BACKLOGGED QUEUE Consider N k t PDUs are transmitted in the k-hop transmission at time t, which can be divided into N k t /L L-intervals.According to the definition of X q , we assume that the qth backlogged queue is consisted of X q L-intervals.Since {X 1 , X 2 , . . ., X q } is a sequence of random positive integers in (7), the recovering process of a status update can be modeled as a renewal process, where X q is the qth renewal interval.Define T q as T q = q i=1 X i , q > 0 and the renewal interval [T q , T q+1 ] is the number of L-interval in a recovering process.Then the random variable (N k t ) t≥0 given by Y t = sup{q : T q ≤ t} represents the number of L-intervals in a status update transmission until time t.
In each recovering process of the k-hop transmission in a status update, define Q k 1 , Q k 2 , . . ., Q k q as a sequence of i.i.d.random variables denoting the sum-length of the backlogged queue (X q − 1)L k + B q for all PDUs, i.e., Q k q is the sum of the number of PDUs in the qth backlogged queue.Thus, the transmission can be divided into two cases for analysis in the following.
1) X q = B, which means the backlogged queue only depends on B q .2) X q ≥ 1, and the queue length of each information PDU is the same as (X q − 1)L k + B q .
Therefore, the sum-length of (X q − 1)L k + B q for all information PDUs in a backlogged queue can be expressed as Let Z k t = Y t j=1 Q k q denote a renewal-reward process, and its expectation is the total length of the backlogged queues for all information PDUs until time t.According to the elementary renewal theorem for renewal-reward processes [33], we have lim t→∞ (1/t)E(Z k t ) = [E(Q k q )/E(X q )].Moreover, base on the above analysis of Q k q , we have E(Q k q ) = ∞ i=1 E(Q k q |X q = i) Pr(X q = i) = (E(X 2 q )−2E(X q )+1)(L k −1) L k + E(X q )β 1 + β 2 .Since we assumed that N k t = tL k , we have lim APPENDIX B DERIVATION OF THE LONGEST BACKLOGGED QUEUE If the number of transmitted PDUs is limited, the closeform expectation of the longest backlogged queue length can be derived.In this case, the maximum value of X q is n/(L k −1), and let X denote X q with a upper bound n/(L k − 1), which also represents the qth renewal interval of a truncation renewal process.Therefore, the probability distribution of X can be expressed as P X (x) = X q , for 0 < x < n/(L k − 1) n/(L k − 1), for x ≥ n/(L k − 1). (49) Then, we have In this case, the rounds of the renewal process is also finite, and its expectation is E(Y t ) = m(t) = ∞ s=1 P s , where P s equals the convolution of s distribution function of X.Moreover, m(t) can be expressed by a simple form M = [n/(L k − 1)/E( X)] based on the renewal theorem.
According to the definition of X m , we can find that all x that satisfy M • Pr(x) ≥ 1 must exist during a status update on average, and the maximum value xmax of them equals the lower bound X of X m .On the other hand, if x meets M • Pr(x) < 1 and all x m ≤ x in this status update, M •Pr(x) is the probability of x m = x.As a result, the probability distribution of X m can Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.(X m = i) • P X m (i) can be directly calculated.Therefore, through the similar derivations to Appendix A, we can get the expectation of the longest backlogged queue length without retransmission as follows: Meanwhile, the retransmission time T k re in the k-hop transmission depends on the number of extra LEC PDUs and feedback delay, which implies its expectation can be obtained by combining ( 29) and (30) as follows: Finally, the longest backlogged queue for each hop transmission can be calculated by substituting ( 13), (52), and (53) into (10).

Fig. 2 .
Fig. 2. Dual-hop illustration of our D-LEC CFDP scheme, where L 1 and L 2 are the LEC PDU inserted interval in the first and second hop, respectively.(a) D-LEC-h CFDP transmission scheme, where n = 6 and L 1 = L 2 = 3.(b) D-LEC-f CFDP transmission scheme, where n = 6 and L 1 = L 2 = 3.

Fig. 3 .
Fig. 3. Dual-hop illustration of a status update transmission in the A-LEC-f CFDP scheme, where n = 6 and T = 3.

Fig. 4 .
Fig. 4. AoI evolution model of a status update transmission.(a) AoI evolution with the first PDUs has been upload before the transmission is completed.(b) AoI evolution with all PDUs upload together at the end.
PDUs have been transmitted, ω(d < ω ≤ m) information PDUs are backlogged in the queue at R, which needs d(0 ≤ d < ω) successfully received PDUs to be recovered at t in the first hop, while s 2 t = (m, δ) denotes Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

13 a
i = PIB(m, p, T, π, δ, ω, d), update m, π, δ or ω, d. 14 end 15 return π Algorithm 2: Function: a i = PIB(m 3 for every a i ∈ A do 4 for every z j ∈ Z do 5 update b with the selected group (b, a i , z j ), 6 compute V U ij under the belief state b ; (a), and validate the accuracy of our SR fading channel model.Moreover, the result also shows that the SR channel with FHS needs multiple antennas to guarantee reliable transmission compared to the ILS and AS fadings.Then, as shown in Fig. 5(b), we simulate the BLER performance with N = 8, B = 200, 500, 1000 bytes under FHS and N = 1, B = 200 bytes under ILS and AS.We can observe that

Fig. 5 .
Fig. 5. BER with QPSK and different diversity gains N under ILS, AS, and FHS, and the BLER performance for N and SR fading parameters under different PDU size.(a) BER versus transmission SNR.(b) BLER versus transmission SNR.

Fig. 8 .
Fig. 8.Comparison of D-LEC-f CFDP and A-LEC-f CFDP schemes, the number of information PDUs is n = 60.(a) End-to-end delay versus transmission SNR.(b) PAoI versus transmission SNR.(c) Throughput versus transmission SNR.
) and show that the complexity of our PIB is low.Specifically, the number of states is O(|S 1 |) = O(n 3 ) in the first-hop transmission, and O(|S 2 |) = O(n 2 ) in the secondhop transmission.Since the belief state b is restricted by S, the maximum number of greedy search is the same as S k in kth hop transmission, while the number of actions is O(|A|) = O(T) and the number of observations is O(|Z|) = O(T) in each hop for POMDP.Therefore, the complexity of PIB algorithm is O(T 2 • n 6 ).On the other hand, the complexity of D-step search algorithm is O(T 2D • n 12 ) [42].

TABLE I RELEVANT NOTATION
× A × S → T denotes the state transition function, where T r (s t , a t , s t+T ) denotes the probability of transiting to the next state s t+T ∈ S under the current state s t ∈ S and action a t ∈ A at t. R: S × A → R is the reward function, with R(s t , a t ) representing the immediate reward for taking action a t in state s t .O: S × A × Z → O represents the observation function, where O(s t , a t−T , z t−2t r ) gives the probability for observing z t−2t r at S or R, when it reaches s t under action a t−T .b: b is the belief state in each state, where b t (s t ) denotes the probability distribution of being in each state s t ∈ S, and it satisfies s t ∈S b t (s t ) = 1.