Member, IEEE Wensheng Lin

Member, IEEE Lixin Li

Fellow, IEEE Jinhong Yuan

Fellow, IEEE Zhu Han

Fellow, IEEE Markku Juntti

Life Fellow, IEEE Tad Matsumoto

Cooperative Lossy Communications in Unmanned Aerial Vehicle Networks: Age-of-Information with Outage Probability

Keywords: Unmanned aerial vehicles, outage probability, Age-of-Information, cooperative communications, lossy communications

This paper analyzes the robustness and timeliness for cooperative lossy communications in unmanned aerial vehicle (UAV) networks. The analytical framework consists of two steps: 1) calculating the outage probability, and 2) characterizing the Age-of-Information (AoI) for a given outage probability. Initially, we determine the outage probability based on the Shannon's lossy source-channel separation theorem. Numerical results indicate that joint decoding reduces the outage probability, and the system can achieve higher diversity order for less stringent distortion requirement. Then, we derive a closed-form expression of the lower bound on the average AoI for the communication system where outage events are constrained to an acceptable level. Moreover, we conduct a series of simulations for verifying the lower bound on the average AoI and evaluating the impact of UAV locations on the average AoI. It is demonstrated that the lower bound is tight when the server utilization ratio is either relatively busy or idle. For the case with a relatively large outage probability, we propose an intermittent transmission scheme to refine the AoI performance without extra energy consumption. The effectiveness of the intermittent transmission scheme for reducing the average AoI is verified by both the simulations and the lower bound analysis.

I. INTRODUCTION

It is universally acknowledged that unmanned aerial vehicles (UAVs) have become an important part of Internet-of-Things (IoT) infrastructure for supporting smart society [START_REF] Wang | Survey on unmanned aerial vehicle networks: A cyber physical system perspective[END_REF]. In IoT systems, UAVs can play versatile roles, such as providing multiple access services [START_REF] Li | Millimeter-wave networking in the sky: A machine learning and mean field game approach for joint beamforming and beam-steering[END_REF]- [START_REF] Li | Optimal trajectory and downlink power control for multi-type UAV aerial base stations[END_REF], replenishing energy of IoT devices [START_REF] Li | A predictionbased charging policy and interference mitigation approach in the wireless powered internet of things[END_REF], [START_REF] Xiong | UAV-assisted wireless energy and data transfer with deep reinforcement learning[END_REF], and supporting computation offloading [START_REF] Lim | Towards federated learning in UAV-enabled internet of vehicles: A multi-dimensional contract-matching approach[END_REF].

W. Lin and L. Li are with the School of Electronics and Information, Northwestern Polytechnical University, Xi'an, Shaanxi 710129, China (e-mail: linwest@nwpu.edu.cn; lilixin@nwpu.edu.cn). J. Yuan is with the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia (e-mail: j.yuan@unsw.edu.au). Z. Han is with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX, USA (e-mail: zhan2@uh.edu). M. Juntti is with the Centre for Wireless Communications, University of Oulu, 90014 Oulu, Finland (e-mail: markku.juntti@oulu.fi). T. Matsumoto is with the School of Information Science, Japan Advanced Institute of Science and Technology, Ishikawa, Japan 923-1292, Japan, and also with the Centre for Wireless Communications, University of Oulu, 90014 Oulu, Finland (The position in University of Oulu has been frozen since January, 2018) (e-mail: matumoto@jaist.ac.jp). Corresponding author: Wensheng Lin. This work was supported in part by National Natural Science Foundation of China (NSFC) under Grant 62001387, in part by Shanghai Academy of Spaceflight Technology (SAST) under Grant SAST2020124 and Grant SAST2018045, in part by the Fundamental Research Funds for the Central Universities, in part by Key R & D plan of Shaanxi Province under Grant 2020GY-034, in part by Grant NSF EARS-1839818, Grant CNS-1717454, Grant CNS-1731424, and Grant CNS-1702850, and in part by the Academy of Finland 6Genesis Flagship (Grant 318927). When adopting UAVs in surveillance systems, compared to the devices deployed at a solid location, UAVs can carry cameras and sensors to collect richer information from various altitudes and closer to the object over a rough terrain. For example, Fig. 1 illustrates a typical implementation of UAVs in cooperative operations. A surveillance center wants to monitor the state of a remote object. Therefore, it relies on UAVs to detect the object state conveniently and economically. To obtain more information of the object and increase the robustness of the surveillance system, multiple UAVs are deployed in different locations for cooperative surveillance. Noticed that the information collected by multiple UAVs is correlated for the same object. Hence, the surveillance center can enhance the system performance by cooperative communications, i.e., jointly utilizing the information uploaded from multiple UAVs.

However, there is an inevitable problem that vehicular communications are not always reliable. Specifically, the surveillance center may fail to reconstruct the object information, if the UAVs transmit information under harsh channel conditions. Fortunately, the final goal of IoT is to make further decisions rather than recover the information itself. As long as the decision is correct, the information is not necessarily to be losslessly recovered. Thus, the lossy recovery is still acceptable if its distortion is no larger than a specified distortion requirement. Based on the above reality, cooperative lossy communications in UAV networks are promising for IoT applications.

In some cases, the IoT system may need to perform operations according to the object state. Intuitively, the appropriateness of an operation is affected by the information freshness of the object state. Conventional performance metrics for timeliness, i.e., delay and latency, are static for each packet.

Nonetheless, the information freshness of the object state changes at different time point when the system performs an operation. Therefore, we need a dynamic metric to indicate the timeliness as time elapses. Age-of-Information (AoI) [START_REF] Kosta | Age of Information: A New Concept, Metric, and Tool[END_REF] is such a performance metric that continuously characterizes the dynamics of freshness of the object information. It has been utilized in diverse communications and control systems, especially for systems with stringent real-time requirements. AoI is also becoming popular in various applications including scheduling design [START_REF] He | Optimizing freshness of information: On minimum age link scheduling in wireless systems[END_REF], [START_REF]On optimal link scheduling with min-max peak age of information in wireless systems[END_REF], updating policy design for energy harvesting devices [START_REF] Yates | Lazy is timely: Status updates by an energy harvesting source[END_REF], [START_REF] Bacinoglu | Age of information under energy replenishment constraints[END_REF], and trade-off between sensing and communication [START_REF] Zhang | Sensing and communication tradeoff design for AoI minimization in a cellular internet of UAVs[END_REF]. Hence, we select AoI as the performance metric to evaluate the timeliness of the system. Moreover, since AoI is time-variant, the average AoI can better reflect the system performance during a long time period. Consequently, this paper aims at analyzing the average AoI for cooperative lossy communications in UAV networks.

Definitely, AoI decreases as the surveillance center refresh the state information of the object. As stated above, in IoToriented cooperative lossy communications, the system decision is reliable if the information recoveries in all links satisfy the distortion requirements. Conversely, an outage event occurs once the distortion requirements cannot be all satisfied. In this case, the surveillance center fails to refresh the state information of the object, resulting in AoI keeping increasing. Thus, to analyze AoI for cooperative lossy communications in UAV networks, we need to investigate the robustness of the UAV communications by evaluating the outage probability for the first step.

In the literature, we can find a lot of work related to the calculation of the outage probability for cooperative communication systems. Laneman et al. [START_REF] Laneman | Cooperative diversity in wireless networks: Efficient protocols and outage behavior[END_REF] derived the outage probability for Rayleigh fading in a relaying system applying diverse forwarding strategies. For lossy-forward (LF) relaying systems [START_REF] He | A tutorial on lossy forwarding cooperative relaying[END_REF], the outage probabilities over Rayleigh fading and Nakagami-m fading were characterized by Zhou et al. [START_REF] Zhou | Exact and approximated outage probability analyses for decode-and-forward relaying system allowing intra-link errors[END_REF] and Qian et al. [START_REF] Qian | Performance analysis for lossy-forward relaying over Nakagami-m fading channels[END_REF], respectively. Then, Lin et al. [START_REF] Lin | Lossy-forward relaying for lossy communications: Rate-distortion and outage probability analyses[END_REF] extended the results to the LF relaying systems with lossy information reconstructed at the destination. Besides, Qian et al. compared the LF strategy with the decode-and-forward (DF) and the adaptive DF strategies for block Rayleigh fading in [START_REF] Qian | Fading correlations for wireless cooperative communications: Diversity and coding gains[END_REF]. With more than one source in the system, Lu et al. [START_REF] Lu | Outage probabilities of orthogonal multiple-access relaying techniques with imperfect source-relay links[END_REF] analyzed the outage probability of the orthogonal multiple-access relaying systems suffering from block Rayleigh fading, provided that intra-link errors occurred in the source-relay transmissions. Lin et al. [START_REF] Lin | Rate-distortion and outage probability analyses for single helper assisted lossy communications[END_REF] conducted the outage probability analysis for lossy communications with two sources and one helper over block Rayleigh fading channels.

It is noticed that the results in the literature are based on conventional fading models, e.g., original Rayleigh fading and Nakagami-m fading. However, the original Rayleigh and Nakagami-m fading models are too simple for UAV communications, and hence neither of the original fading models is sufficiently accurate to describe the UAV channel conditions. To analyze the performance of the cooperative lossy communications in UAV networks, we need to conduct derivations over UAV channels. As reported in [START_REF] Khuwaja | A survey of channel modeling for UAV communications[END_REF], there are large quantities of measurement campaigns for UAV channel modeling. Consequently, we will select the appropriate UAV channel model to calculate the outage probability in UAV cooperative lossy communication systems, and further analyze the AoI performance.

In the UAV cooperative surveillance system considered in this paper, the update of the object state starts from the object changing its state to the surveillance center recovering the object information. Certainly, the start of the update is at one time point, and so is the end of the update due to the joint decoding scheme in the surveillance center. Even though there are two links for information transmission in the system, there is only one update. Hence, from the view of the whole system, the update process can be equivalently regarded as a queueing process with only one queue and one server. If a new update arrives when the whole system is busy, it has to wait until the system finishes processing the previous update. We further assume that the change of the object state follows a Poisson process, while the service time of the whole system follows the exponential distribution. Consequently, this system is equivalent to an M/M/1 system with the first-come-firstserved (FCFS) scheme, i.e., a typical queueing model. Here, M/M/• indicates Poison arrival and exponential service time distributions.

Generally, the queueing model described by "A/S/N S /N B " is interpreted as follows. "A" and "S" represent the arrival and service processes. The symbol "M" specifies the process with Poisson arrivals or exponential service times, and the symbol "D" denotes the process with a deterministic distribution. N S stands for the number of servers. N B is the buffer size of the queue. If not specified, N B is assumed to be infinite.

To date, there are already numerous research achievements with respect to AoI. Early in 2011, Kaul et al. [START_REF] Kaul | Minimizing age of information in vehicular networks[END_REF] introduced the concept of AoI to optimize the freshness of state information in a vehicular network adopting the IEEE 802.11 protocol. Then, Kaul et al. presented the fundamental characterizations with regard to AoI in [START_REF] Kaul | Real-time status: How often should one update?[END_REF], [START_REF] Kaul | Status updates through queues[END_REF]. They determined the average AoI for M/M/1, M/D/1, and D/M/1 systems with the FCFS scheme in [START_REF] Kaul | Real-time status: How often should one update?[END_REF]. Subsequently, in [START_REF] Kaul | Status updates through queues[END_REF], the average AoI of M/M/1 systems using the last-come-first-served (LCFS) scheme was investigated for the cases that a new arrived packet can or cannot pre-empt the currently served packet. For the AoI analysis in UAV networks, Han et al. [START_REF] Han | Age of information and performance analysis for UAV-aided IoT systems[END_REF] calculated the average AoI for successfully transmitted packets in UAVaided IoT networks, which was modeled as the M/M/1 system with the FCFS scheme. Ahani et al. [START_REF] Ahani | Age-optimal UAV scheduling for data collection with battery recharging[END_REF] designed a route scheduling algorithm for a UAV with limited battery capacity to minimize the average overall AoI. In [START_REF] Abd-Elmagid | Average peak age-ofinformation minimization in UAV-assisted IoT networks[END_REF], Abd-Elmagid and Dhillon minimized the average peak AoI in an IoT network with a UAV relay.

Note that the work shown above implicitly assumes that the communication channels are all reliable in the system. Indubitably, vehicular communications are not always reliable, and the occurrence of an outage event will affect the AoI because a renewal of information is missed. Hence, the above work is not suitable for the performance analysis of cooperative lossy communications in UAV networks. In the literature, there is some research work taking packet errors into consideration for AoI analysis as exemplified below. Chen and Huang [START_REF] Chen | Age-of-information in the presence of error[END_REF] derived the peak AoI for the M/M/1 system with the FCFS or LCFS scheme. Zhang et al. [START_REF] Zhang | Age of information in a cellular internet of UAVs: Sensing and communication tradeoff design[END_REF] considered AoI with a successful sensing probability for optimizing sensing and transmission time in a cellular Internet of UAVs. Gu et al. [START_REF] Gu | Timely status update in internet of things monitoring systems: An age-energy tradeoff[END_REF] characterized both the average AoI and the average peak AoI for the IoT monitoring system with truncated automatic repeat request scheme. Gu et al. [START_REF] Gu | Minimizing age of information in cognitive radio-based IoT systems: Underlay or overlay?[END_REF] also analyzed the average peak AoI of both the primary IoT system and the secondary IoT system in an overlay or underlay scheme. Nevertheless, identifying the average AoI for the M/M/1 system with the FCFS scheme under the constraint of certain packet errors has not been finished yet.

In brief, the analytical framework of cooperative lossy communications in UAV networks consists of the following two steps: 1) calculating the outage probability for the system, and 2) determining the average AoI for a given outage probability. The contributions of this paper are summarized as follows:

• We establish the analytical framework for the outage probability and the average AoI of cooperative lossy communications in UAV networks. When calculating the outage probability, it is difficult to directly determine the signal-to-noise ratio (SNR) thresholds for satisfying distortion requirements in cooperative communications.

We solve this problem by formulating an equivalent multiterminal source coding problem based on the Shannon's lossy source-channel separation theorem [START_REF] Shannon | A mathematical theory of communication[END_REF], [START_REF]Coding theorems for a discrete source with a fidelity criterion[END_REF]. • We derive a lower bound on the average AoI for the M/M/1 system with the FCFS scheme and accepting outage events. Specifically, we start from two extreme cases, i.e., the idle and busy cases, and then combine the results from the two cases together to obtain a relatively tight lower bound on the average AoI. • We conduct a series of simulations to compare the lower bound with the exact AoI. The curves of the lower bound and the simulation results have very small differences when the system is either relatively busy or idle. • To refine the AoI performance, we propose an intermittent transmission scheme to reduce the average AoI when the outage probability is relatively large. The principle is to allocate more energy to reduce the outage probability rather than increase the updating frequency, if the relatively large outage probability becomes the dominant factor of AoI. Both the simulation results and the lower bound verify that intermittent transmission can reduce the average AoI without additional energy consumption. • We evaluate the impact on AoI due to different UAV locations based on a practical scenario. The results show that the average AoI keeps at a low level if the UAVs are located within a specific range, while it rapidly increases if the UAVs move outside of the specific range.

The rest of this paper is organized as follows. Section II describes the system model and the channel model of the UAV cooperative surveillance system in detail. Section III analyzes the outage probability, and Section IV further derives a lower bound of the average AoI. The lower bound is compared with simulation results in Section V. Finally, we conclude this work in Section VI.

Throughout this paper, we use the standard notation rules as follows. The random variables and their realizations are denoted by uppercase and lowercase letters, respectively. Calligraphic letters X and R denote the corresponding finite sets. The base of logarithm function log(•) is assumed to be 2 unless specified.

II. PROBLEM STATEMENT

We present first the details of the communication model for the UAV cooperative surveillance system in Section II-A. Then, Section II-B and Section II-C introduce the corresponding multiterminal source coding problem and the channel model, respectively, for deriving the outage probability.

A. System Model

As depicted in Fig. 1, the simplest scenario of the UAV cooperative surveillance system consists of one object, two UAVs and one surveillance center. The UAVs are deployed in different locations to monitor the same object, whose state changes randomly. Once the UAVs detect the change of the object state, they separately collect the object information and upload it to the surveillance center. Since the UAVs monitor the same object, correlation exists in the object information collected by the UAVs. Therefore, the surveillance center is able to minimize the distortion of the object information by jointly decoding after receiving the signals from multiple UAVs.

However, when suffering from the channel fading, the transmissions in UAV communication systems are not always reliable. Therefore, the recoveries of the object information may contain distortions if the channel condition is not sufficiently good. Notice that the goal of the IoT system is not only to recover the object information, but also to make some decisions and consequent operations. It means that the distortions are acceptable as long as the surveillance center can still make correct inference. The maximum acceptable distortions are given by the distortion requirements (D 1 , D 2 ) for two links. Once either of the distortion requirements cannot be satisfied, the recoveries are discarded and an outage event occurs. In this case, the surveillance center has to utilize the previous acceptable recoveries for making further decisions. Certainly, the decision may be not sufficiently appropriate due to the reduction of the information freshness for the object state.

For the purpose of dynamically evaluating the information freshness in the system, AoI is introduced and defined as the elapsed time of the latest served update from the generation of the update. In the UAV cooperative lossy communication system, AoI fails in updating and keeps increasing if an outage event occurs. In order to analyze AoI of the system illustrated in Fig. 1, we need to evaluate the outage probability for the first step, and then derive the AoI for a specified value of the outage probability.

The conventional way to calculate the outage probability is to determine the instantaneous SNR of the received signals for given channel gains. Subsequently, the outage probability is calculated by the integral with respect to the instantaneous

Encoder 1 Joint decoder Encoder 2 X n Z 1 n R 1 R 2 M 1 M 2 X 1 n Z 2 n X 2 n X 1 n X2 n Fig.
2. The multiterminal source coding problem for outage probability analysis.

SNR distribution that cannot achieve the threshold for successful decoding. However, in cooperative communication systems with correlated sources, it is not easy to directly determine the SNR thresholds according to specified distortion requirements.

To solve the problem of determining the SNR thresholds, we utilize a two-step method to equivalently derive the relationship between the instantaneous SNR and the final distortions. Initially, given an instantaneous SNR, we can calculate the corresponding achievable rate. Then, the transmitter compresses the information at the code rate satisfying the instantaneous achievable rate. The distortions can be evaluated by solving a multiterminal source coding problem. Based on the Shannon's lossy source-channel separation theorem, we will finally establish the relationship between the instantaneous SNRs and the distortions. Conversely, given the distortion requirements, we will determine the achievable link rates and further obtain the SNR thresholds.

B. Multiterminal Source Coding Problem

Fig. 2 illustrates the multiterminal source coding problem corresponding to the UAV cooperative surveillance system. There is a common source X n separately observed by two encoders while suffering from random noise processes Z n 1 and Z n 2 , where n denotes the sequence length. In practical systems, it is reasonable to assume that the source is Gaussian. Consequently, the observations X n 1 and X n 2 can be regarded as two correlated Gaussian sources 1 . Then, two distributed encoders compress the observations X n 1 and X n 2 into two codewords M 1 and M 2 at the code rates R 1 and R 2 , respectively. Since the source is Gaussian, the encoder quantizes the observation and then maps it into a codeword taking values in a finite alphabet. Therefore, the code rates can be larger than 1. After receiving two codewords M 1 and M 2 , the joint decoder recovers the estimates of two observations as Xn 1 and Xn 2 by exploiting the correlations between the observations. Constrained by the link rates, the recovered signals Xn

1 and Xn 2 may not be lossless, and hence, the distortion measure d i : X i × Xi → [0, ∞) for i ∈ {1, 2} is defined to describe the distortion degree between two symbols x i and xi . Furthermore, the average distortion of the whole sequence is defined as

d i (x n i , xn i ) = 1 n n t=1 d i (x i (t), xi (t)). (1)
In general, the distortion is evaluated by quadratic (squared error) distortion measure d Q,i (x i , xi ) = (x i -xi ) 2 for Gaussian sources. Given the distortion requirements (D 1 , D 2 ), the achievable rate-distortion region R(D 1 , D 2 ), which consists of all achievable rate pairs (R 1 , R 2 ), is defined as

R(D 1 , D 2 ) = (R 1 , R 2 ) : (R 1 , R 2 ) is admissible such that lim n→∞ E(d i (x n i , xn i )) ≤ D i + , for i = 1, 2,
and any > 0 .

(

) 2
The achievable rate-distortion region establishes the connection between the link rates and the distortions.

C. Channel Model

In order to characterize the channel conditions and further determine the constraints on link rates, we have to select the appropriate channel models. Since the distortions are defined between X n i and Xn i , we only need to consider the channels from the UAV to the surveillance center, i.e., typical air-to-ground (A2G) channels. We further assume that the A2G channel conditions depend on the elevation angle of the UAV and the impact of environment including line-of-sight (LoS) and non-LoS (NLoS) conditions. According to [36, Eq. ( 2)], given the transmission power P UAV,i of the i-th UAV, the power of the i-th signal received in the surveillance center can be expressed as

P SC,i = P UAV,i |g| 2 P L i , (3)
where |g| 2 is the fading gain, and P L i stands for the A2G propagation path-loss. The path-loss in dB is given by [START_REF] Al-Hourani | Optimal LAP altitude for maximum coverage[END_REF] P L i (dB) = 10 log 10 P L i = η LoS -η NLoS 1 + ae -b(θi-a) + ξ i , (4) where η LoS , η NLoS , a and b are the environmental parameters. θ i = arctan hi si is the elevation angle, with h i being the altitude of the i-th UAV, and s i being the horizontal distance between the i-th UAV and the surveillance center. Furthermore,

ξ i = 20 log 10 4πf i d SC,i c + η NLoS , (5)
where d SC,i = s 2 i + h 2 i is the distance between the i-th UAV and the surveillance center. f i is the carrier frequency for i-th UAV, and c is the speed of light.

We assume that two UAVs utilize different carrier frequency for communications. Let N 0 be the noise power spectral density of the thermal additive Gaussian noise in the receiver. Then, the instantaneous SNR in each link can be calculated by

γ i = P SC,i N 0 = P UAV,i |g| 2 N 0 • P L i . (6)
Hereafter, we consider the average fading power normalized to unity or E(|g| 2 ) = 1. Thereby, the average SNR in each link is

γ i = P UAV,i N 0 • P L i , (7)
and we have |g| 2 = γi γ i for the instantaneous channel power gain. As summarized in [START_REF] Khuwaja | A survey of channel modeling for UAV communications[END_REF], many measurement campaigns demonstrate that the Nakagami-m model characterizes the UAV fading channel well. Consequently, we apply the Nakagami-m model to |g| 2 , and, hence, the probability density function (PDF) of |g| 2 follows the Gamma distribution. Then, it is easy to derive the PDF of γ i as

p(γ i ) = m m γ m-1 i γ m i Γ(m) exp - mγ i γ i , (8)
where Γ(•) represents the Gamma function [START_REF] Papoulis | Probability, random variables, and stochastic processes[END_REF]].

III. OUTAGE PROBABILITY ANALYSIS

To begin with, Section III-A analyzes the condition for the occurrence of an outage event, and then presents the derivation results of the outage probability. The numerical results of the outage probability are further discussed in Section III-B.

A. Outage Probability Derivation

Consider two correlated Gaussian sources with covariance matrix

Λ = σ 2 1 qσ 1 σ 2 qσ 1 σ 2 σ 2 2 , (9)
where σ 2 i is the variance of the i-th Gaussian source, and q represents the correlation degree of two sources. The achievable rate-distortion region is the set of rate pairs (R 1 , R 2 ) such that [START_REF] Oohama | Gaussian multiterminal source coding[END_REF]

R 1 ≥ ϕ 1 (R 2 ), (10)
R 2 ≥ ϕ 2 (R 1 ), (11)
R 1 + R 2 ≥ ψ, (12)
where

ϕ i (R j ) = 1 2 log + σ 2 i D i 1 -q 2 + q 2 2 -2Rj , (13)
ψ = 1 2 log + 1 -q 2 βσ 2 1 σ 2 2 2D 1 D 2 , ( 14
)
for i ∈ {1, 2}, j = {1, 2}\i, with log + (•) = max{log(•), 0} and
β = 1 + 1 + 4q 2 D 1 D 2 (1 -q 2 ) 2 σ 2 1 σ 2 2 . ( 15
)
According to the Shannon's lossy source-channel separation theorem, the distortions remain below the given values where C(γ i ) is the Shannon capacity using the Gaussian codebook, and r i represents the end-to-end rates of joint source-channel coding. Fig. 3 illustrates the achievable rate-distortion region for two Gaussian sources. In order to clearly display the shape of the achievable rate-distortion region, we set q = 0.6, σ 1 = σ 2 = 0.8 and D 1 = D 2 = 0.2 as an example. Obviously, an outage event occurs once the rate pair (R 1 , R 2 ) falls outside of the achievable rate-distortion region, and hence the outage region is the area outside of the achievable rate-distortion region. Since the instantaneous channel capacities are random variables, the outage probability is the expected value with respect to the link rates supported by the instantaneous channel capacities. To facilitate the integral calculation of outage probability, the outage region is divided into four sub-regions as:

(D 1 , D 2 ) if [40, Theorem 3.7] R i ≤ Θ i (γ i ) = C(γ i ) r i , for i = 1, 2, ( 16
)
0 0.5 1 1.5 2 0 0.5 1 1.5 2 Achievable Region A๖ R * ๗ R * ๖ A๗ A๘ A๙ ψ -R * ๖ R๖ + R๗ = ψ R๖ = φ๖(R๗) R๗ = φ๗(R๖)
A 1 = {0 ≤ R 1 < ϕ 1 (R 2 ), R * 2 ≤ R 2 }, (17)
A 2 = {0 ≤ R 1 < ψ -R 2 , ψ -R * 1 ≤ R 2 < R * 2 }, (18)
A 3 = {0 ≤ R 1 < R * 1 , 0 ≤ R 2 < ψ -R * 1 }, (19)
A 4 = {R * 1 ≤ R 1 , 0 ≤ R 2 < ϕ 2 (R 1 )}, (20)
where

(R * 1 , ψ -R * 1 ) and (ψ -R * 2 , R * 2 )
are the corner points to the achievable rate-distortion region, and hence R * i is the solution of ψ -ϕ j (R i ) -R i = 0. Therefore, the outage probability for the UAV cooperative surveillance system can be calculated by

P out = 4 k=1 P k , (21)
where P k represents the outage probability for sub-region A k .

The derivation results of P k are

P 1 = Γ m, m γ 2 Θ -1 2 (R * 2 ) Γ(m) - m m γ m 2 [Γ(m)] 2 ∞ Θ -1 2 (R * 2 ) γ m-1 2 • exp - mγ 2 γ 2 Γ m, m γ 1 Θ -1 1 (ϕ 1 ) dγ 2 , (22)
P 2 = Γ m, m γ 2 Θ -1 2 (ψ -R * 1 ) -Γ m, m γ 2 Θ -1 2 (R * 2 ) Γ(m) - m m γ m 2 [Γ(m)] 2 Θ -1 2 (R * 2 ) Θ -1 2 (ψ-R * 1 ) γ m-1 2 exp - mγ 2 γ 2 • Γ m, m γ 1 Θ -1 1 [ψ -Θ 2 (γ 2 )] dγ 2 , (23)
P 3 =   1 - Γ m, m γ 1 Θ -1 1 (R * 1 ) Γ(m)   •   1 - Γ m, m γ 2 Θ -1 2 (ψ -R * 1 ) Γ(m)   , (24)
P 4 = Γ m, m γ 1 Θ -1 1 (R * 1 ) Γ(m) - m m γ m 1 [Γ(m)] 2 ∞ Θ -1 1 (R * 1 ) γ m-1 1 • exp - mγ 1 γ 1 Γ m, m γ 2 Θ -1 2 (ϕ 2 ) dγ 1 , (25)
where Γ(•, •) stands for the upper incomplete gamma function.

The detailed derivations of P k are presented in Appendix A.

Notice that ϕ 1 (R 2 ) defined in ( 13) is a function of R 2 and hence also a function of γ 2 . Therefore, it is difficult to further derive a closed-form solution from [START_REF] Khuwaja | A survey of channel modeling for UAV communications[END_REF]. Instead, we will show the numerical results later to discuss the impact on the outage probability due to different parameter settings.

B. Numerical Results

For the comparison of performance gain between joint decoding and separate decoding, the outage probability of separate decoding is derived as follows.

The rate-distortion function for a Gaussian source

X i ∼ N (0, σ 2 i ) is [41, Theorem 10.3.2] R i (D i ) = 1 2 log + ( σ 2 i D i ). (26)
Thus, the outage event does not occur only when the instantaneous channel capacities are no less than R i (D i ) in both links. Based on the Shannon's lossy source-channel separation theorem, the outage probability of separate decoding can be calculated as

P S out = 1 - ∞ Θ -1 1 [R1(D1)] p(γ 1 )dγ 1 • ∞ Θ -1 2 [R2(D2)] p(γ 2 )dγ 2 = 1 - Γ m, m γ 1 Θ -1 1 1 2 log + σ 2 1 D1 Γ(m) • Γ m, m γ 2 Θ -1 2 1 2 log + σ 2 2 D2 Γ(m) . (27)
Fig. 4 presents the numerical results of the outage probability for cooperative communications with two correlated Gaussian sources. Clearly, the outage probability is reduced by joint decoding for correlated sources with arbitrary correlation degree. There is no doubt that the outage event occurs less frequently if the distortion requirements become less strict, i.e., D i increases. By comparing Fig. 4(a) and Fig. 4(b), we can find that the more correlated the sources are, i.e., q is larger, the lower the outage probability can be. Interestingly, joint decoding also increases the diversity order, which is influenced by both the correlation degree and the distortion requirements. As shown in Fig. 4(a), the slopes of curves obviously change from D i = 0.7 to 0.9, and hence, the system with less strict distortion requirements can obtain more diversity gains. However, it is found that the obvious diversity gains appear from D i = 0.1 to 0.3 and also 0.3 to 0.5 in Fig. 4(b). This observation demonstrates that it is easier to obtain diversity gains with more correlated sources. In addition, we can observe that the performance gains from D i = 0.1 to 0.3 and from D i = 0.7 to 0.9 are larger than that from D i = 0.3 to 0.5 and from D i = 0.5 to 0.7. The reason for this observation

t i-1 t i t i+1 t i+k t' i-1 t' i t' i+1 t' i+k t Δ(t) Y i T i k times of outage Ỹi T i+1 Y i+1
T i+k Y i+k Ti Fig. 5. The AoI when outage events occur.

is explained as follows. The distortion requirement determines the SNR threshold for successful transmissions, and the SNR threshold further determines the outage probability. Since the rate-distortion function is convex, the required rate decreases fast as the distortion requirement becomes less stringent for a small range of D i , resulting in a7 significant reduction of the SNR threshold. Hence, the outage probability also decreases fast as a relatively small D i increases. When D i is relatively large, although the SNR threshold becomes very small and does not significantly change, the outage probability reduces fast as the small SNR threshold decreases. Consequently, the performance gain is more obvious when the distortion requirement is relatively small or relatively large.

IV. AOI ANALYSIS

Section IV-A first presents the general analytical framework of the average AoI for a given value of outage probability. Then, in Section IV-B, we consider the average AoI analysis for two extreme cases as follows:

• Busy case: The queue is never empty and the system continuously serves the updates without break. • Idle case: The queue is always empty and the system must have a break after serving an update. We find that the results of these two cases provide the lower bounds on the average AoI. Therefore, we finally obtain a relatively tight lower bound by combining the lower bounds for busy and idle cases together.

A. General Analytical Framework

As stated above, this system can be regarded as an M/M/1 System with the FCFS scheme. Fig. 5 illustrates the AoI in the general case that outage events occur k times continuously. Assume that the i-th update is generated at the time t i and the service completed at t i . Then, Y i is the interarrival between the (i -1)-th and the i-th updates, i.e., Y i = t i -t i-1 . Moreover, T i is the time for the i-th update staying at the system, i.e., the system time corresponding to the sum of the queue waiting time W i and service time S i . Therefore, we have T i = t i -t i = W i + S i , and W i = (T i-1 -Y i ) + . The update is generated and served at mean rate λ and µ, respectively, and hence, the server utilization ratio is ρ = λ µ . The AoI ∆(t) is the time elapsed from the generation of the last served update, and the average AoI for an interval (0, T ) is defined as

∆ = 1 T T 0 ∆(t)dt, (28)
where the integral can be calculated by the area under ∆(t).

In Fig. 5, the outage events occur from the i-th update at t i to the (i+k -1)-th update at t i+k-1 , and the (i+k)-th update is successful at t i+k . Obviously, the AoI keeps increasing if an outage event occurs. Given the outage probability P out , the probability of outage events occurring exactly k times continuously is P k out (1-P out ), where P k out means that the outage events occur at the previous k transmissions and (1 -P out ) means no outage event at the (k + 1)-th transmission. Let ∆ (k) represent the average AoI for outage events occurring k times continuously. For a sufficiently large number N T of transmissions, the total area under ∆(t) is

∞ k=0 N T P k out (1 -P out ) • (k + 1)E[Y ]∆ (k) , (29)
and the total time is

∞ k=0 N T P k out (1 -P out ) • (k + 1)E[Y ]. (30)
Hence, the average AoI can be calculated by the total area under ∆(t) averaged by the total time, i.e.,

∆ = ∞ k=0 P k out (1 -P out )(k + 1)∆ (k) ∞ k=0 P k out (1 -P out )(k + 1) . ( 31
)
It is clear that ∆ (0) is the same as the original AoI for completely reliable communications without outage, i.e., [START_REF] Kosta | Age of Information: A New Concept, Metric, and Tool[END_REF]Eq. (2.11)-(2.12)]

∆ (0) = E[(Y + T ) 2 ] -E[T 2 ] 2E[Y ] = 1 µ 1 + 1 ρ + ρ 2 1 -ρ . ( 32
)
For ∆ (k) with general value of k, since the i-th to the (i + k -1)-th updates are lost, we can set two new variables Ỹi = i+k j=i Y j and Ti = T i+k . Then, the calculation of AoI follows the same form as ∆ (0) , i.e., we have

∆ (k) = E[( Ỹi + Ti ) 2 ] -E[ T 2 i ] 2E[ Ỹi ] . (33)

B. Lower Bounds on AoI

Since the queue gradually changes from busy to idle as the server utilization ratio ρ decreases from 1 to 0, the busy and idle cases are two extreme cases for ρ = 1 and 0, respectively. Based on the systems in busy and idle cases, we can derive two lower bounds of ∆ (k) as follows.

Proposition 1 (busy case): ∆ (k) is lower bounded by

∆ (k) ≥ ∆ B (k) = 1 -ρ 2µρ k (k + 1)   2 (1 -ρ) 3 - k j=0 ρ j (j + 2)(j + 1)   - ρ µ(1 -ρ) 2 (k + 1) , (34)
which asymptotically approaches ∆ (k) as ρ → 1. Proposition 2 (idle case): ∆ (k) is lower bounded by

∆ (k) ≥ ∆ I (k) = 1 λ + k 2λ + 1 µ , (35)
which asymptotically approaches ∆ (k) as ρ → 0.

The proofs of Proposition 1 and Proposition 2 are presented in Appendix B and Appendix C, respectively. Now, we verify the lower bounds by numerical results. Consider the simplest case that k = 0, i.e., no outage event occurs, and then the lower bounds of ∆ (0) can be obtained as

       ∆ B (0) = ρ(2 -ρ) µ(1 -ρ) , for busy case, ( 36
)
∆ I (0) = 1 λ + 1 µ , for idle case. ( 37
)
Fig. 6(a) compares the lower bound for busy case with the exact AoI ∆ (0) presented in [START_REF] Gu | Minimizing age of information in cognitive radio-based IoT systems: Underlay or overlay?[END_REF]. Clearly, the lower bound is always below the exact AoI for arbitrary ρ and µ. Moreover, the lower bound and the exact value of AoI asymptotically converge together as ρ tends to 1. This observation matches our intuition that the system becomes busier with larger ρ, and hence the total service time in [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF] becomes closer to (ỹ i + ti ). Consequently, we can use the lower bound for busy case to approximate AoI when ρ is relatively large.

On the other hand, the lower bound for idle case is compared with the exact AoI in Fig. 6(b). It is found that the lower bound and the exact AoI start from the same point when ρ is extremely small. This observation is understood in the following way: the queue will be almost empty if ρ decreases to a relatively small value, and hence the zero waiting time exactly matches the condition for deriving ∆ I (k) . Opposite to Fig. 6(a), the gap between the lower bound and the exact AoI extends as ρ goes larger. Inspired by the different tendency between Fig. 6(a) and Fig. 6(b), we combine the lower bounds for two cases together and obtain a more accurate lower bound, i.e., Proposition 3: ∆ (k) is lower bounded by

∆ (k) ≥ ∆ LB (k) = max ∆ B (k) , ∆ I (k) . (38)
Finally, we can substitute [START_REF] Papoulis | Probability, random variables, and stochastic processes[END_REF] into [START_REF] Gu | Timely status update in internet of things monitoring systems: An age-energy tradeoff[END_REF] to calculate the lower bound of the average AoI in the communication system where outage events occur.

V. SIMULATIONS

In order to evaluate how precise the lower bound is, we have conducted a series of simulations to measure the exact AoI. Fig. 7 illustrates the simulation results and the corresponding lower bound. Although there is a gap between the lower bound and the simulation result, they have imperceptible differences when ρ ≤ 0.4 or ρ ≥ 0.9. From the comparison between Fig. 7(a) and Fig. 7(b), we can make a conclusion that the stronger the service ability, i.e., larger µ, the smaller the gap. In addition, the gap between the lower bound and the simulation result increases with larger outage probability P out . Because the outage event occurs more frequently for larger P out , and more updates are contained in a greater number of continuous outage events. Consequently, the deviation resulting from the idle time and the waiting time also increases for continuous updates. This implies that it may not be sufficiently accurate to estimate the exact AoI by the lower bound for large P out . Nevertheless, notice that the gap for P out = 0.5 is still tolerable. Actually, P out = 0.5 for practical systems is a quite large outage probability which means half of the transmissions failed. Therefore, for such an unreliable communication system with large P out , the performance refinement should focus more on how to reduce the outage probability. A simple method is to skip some transmissions and accumulate more energy to the rest of transmissions. By this means, we can reduce the outage probability for the transmitted updates and consume the same energy in total. In the following, we are going to show an interesting instance that the average AoI can be reduced with the same total energy consumption by the intermittent transmission scheme. We consider the case with the correlation degree q = 0.8 and the distortion requirements D 1 = D 2 = 0.1. From Fig. 4(b), we can find that the outage probabilities are approx- imately 0.8046 and 0.3699 for γ i equal to 6.3 dB and 9.3 dB, respectively. For two continuous updates, the UAV can use two times the power (+3 dB) for transmitting only one update and ignoring the next update. By this means, the outage probability can be reduced from 0.8046 to 0.3699, while keeping the total energy consumption unchanged. Then, the equivalent arrival rate for P out = 0.3699 reduces to half, and the equivalent server utilization ratio becomes ρ = ρ/2. Fig. 8 depicts the average AoI curves with respect to ρ and ρ for P out equal to 0.8046 and 0.3699, respectively. Clearly, the curve with ρ for P out = 0.3699 is below that with ρ for P out = 0.8046. Thus, both the simulation result and the lower bound verify that the AoI can be refined to a lower level without extra energy consumption by intermittently transmitting the updates for a relatively large outage probability. Moreover, the actual performance gain will be more if we perform refinement based on the lower bound, due to the extension of the gap between the lower bound and the exact AoI as P out increases.

Based on practical parameter settings, we evaluate the effect on the average AoI due to different UAV locations. Assume that the surveillance system is deployed at suburban areas, where the channel conditions are dominated by the impact of environment. Therefore, the average SNR is calculated by [START_REF] Lim | Towards federated learning in UAV-enabled internet of vehicles: A multi-dimensional contract-matching approach[END_REF], and the parameters for calculating path-loss at suburban are (η LoS , η NLoS , a, b) = (0.1, 21, 4.88, 0.43) [START_REF] Khuwaja | A survey of channel modeling for UAV communications[END_REF]Table V]. The altitudes h i of UAVs keep at 500 m while their distances s i from the surveillance center change from 100 m to 2000 m. In order to compare the tendencies of the average AoI and the outage probability, we plot the curves in a two-dimensional plane by setting s i at the same value. If the distances from the surveillance center to the UAVs are different, we can also obtain similar conclusions. Although the UAV locations may affect the correlation degree between the information collected by different UAVs, we assume that the correlation degree stays the same at q = 0.8 for simplicity. We set the distortion requirements D i = 0.1 and the UAV transmission power P UAV,i = 0.5 W. From [42, Eq. (4.2.4)], the thermal noise power is -174 dBm/Hz at the room temperature. The carrier frequencies for two UAVs are set at f 1 = 2 GHz and f 2 = 1.99 GHz, respectively. Let the bandwidth be 10 MHz, and we can obtain N 0 = 3.9811 × 10 -14 W. As shown in Fig. 9, the average AoI increases obviously when s i > 800 m where P out > 0.1, due to the rapid increase of outage probability. On the contrary, below the 800 m distance range, the system can keep low level of the average AoI. Moreover, the gap between the lower bound and the exact AoI enlarges only when ρ = 0.8 and P out > 0.1.

VI. CONCLUSION

We have established an analytical framework for the outage probability and the average AoI of cooperative lossy com-munications in UAV networks. The analytical framework was applied to a UAV cooperative surveillance system for instance. To begin with, we formulated the UAV cooperative surveillance system as a multiterminal source coding problem, and then calculated the outage probability based on the Shannon's lossy source-channel separation theorem. The numerical results verify that the outage probability decreases in cooperative communication systems by joint decoding, and we can observe the diversity gain for relatively large distortion requirements. Subsequently, given a specific value of outage probability, we further derived two lower bounds on the average AoI for the system dominated by the busy and idle states. By combining the lower bounds for the busy and idle cases together, we obtained a more accurate lower bound on the average AoI. Finally, we compared the lower bound with the exact value of AoI by simulations. The simulation results showed that the gap between the lower bound and the exact AoI is imperceptible for relatively busy or idle systems. Moreover, the gap still seems tolerable even for quite large outage probability values in practical systems, i.e., up to P out = 0.5, which is well sufficient for practical system performance evaluations. Therefore, we proposed an intermittent transmission scheme to reduce the outage probability while keeping the same total energy consumption. Interestingly, the system can thereby achieve a lower level of both the outage probability and the average AoI without consuming extra energy. We also implemented the analytical framework with practical parameter settings to evaluate the impact on the average AoI due to the change of the UAV locations. Besides UAV cooperative surveillance systems, the framework for the AoI analysis can be also implemented in other multiterminal communication systems accepting outage events. In the future, we are going to apply the analytical framework of the outage probability and the average AoI for correlated fading channels.

APPENDIX A DERIVATIONS OF P k

The calculations of P k are presented as follows.

P 1 = Pr{A 1 } = Pr{0 ≤ R 1 < ϕ 1 , R * 2 ≤ R 2 } = Pr{0 ≤ Θ 1 (γ 1 ) < ϕ 1 , R * 2 ≤ Θ 2 (γ 2 )} = Pr{Θ -1 1 (0) ≤ γ 1 < Θ -1 1 (ϕ 1 ), Θ -1 2 (R * 2 ) ≤ γ 2 } = Pr{0 ≤ γ 1 < Θ -1 1 (ϕ 1 ), Θ -1 2 (R * 2 ) ≤ γ 2 } = ∞ Θ -1 2 (R * 2 ) dγ 2 Θ -1 1 (ϕ1) 0 p(γ 2 )p(γ 1 )dγ 1 = ∞ Θ -1 2 (R * 2 ) dγ 2 Θ -1 1 (ϕ1) 0 p(γ 2 ) m m γ m-1 1 γ m 1 Γ(m) • exp - mγ 1 γ 1 dγ 1 = ∞ Θ -1 2 (R * 2 ) p(γ 2 )   1 - Γ m, m γ 1 Θ -1 1 (ϕ 1 ) Γ(m)   dγ 2 (39) = Γ m, m γ 2 Θ -1 2 (R * 2 ) Γ(m) - m m γ m 2 [Γ(m)] 2 ∞ Θ -1 2 (R * 2 ) γ m-1 2 • exp - mγ 2 γ 2 Γ m, m γ 1 Θ -1 1 (ϕ 1 ) dγ 2 , (40)
where [START_REF] Oohama | Gaussian multiterminal source coding[END_REF] follows according to [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF]Eq. (3.381.3)].

P 2 = Pr{A 2 } = Pr{0 ≤ R 1 < ψ -R 2 , ψ -R * 1 ≤ R 2 < R * 2 } = Pr{0 ≤ Θ 1 (γ 1 ) < ψ -Θ 2 (γ 2 ), ψ -R * 1 ≤ Θ 2 (γ 2 ) < R * 2 } = Pr{Θ -1 1 (0) ≤ γ 1 < Θ -1 1 [ψ -Θ 2 (γ 2 )], Θ -1 2 (ψ -R * 1 ) ≤ γ 2 < Θ -1 2 (R * 2 )} = Pr{0 ≤ γ 1 < Θ -1 1 [ψ -Θ 2 (γ 2 )], Θ -1 2 (ψ -R * 1 ) ≤ γ 2 < Θ -1 2 (R * 2 )} = Θ -1 2 (R * 2 ) Θ -1 2 (ψ-R * 1 ) dγ 2 Θ -1 1 [ψ-Θ2(γ2)] 0 p(γ 2 )p(γ 1 )dγ 1 = Θ -1 2 (R * 2 ) Θ -1 2 (ψ-R * 1 ) dγ 2 Θ -1 1 [ψ-Θ2(γ2)] 0 p(γ 2 ) m m γ m-1 1 γ m 1 Γ(m) • exp - mγ 1 γ 1 dγ 1 = Θ -1 2 (R * 2 ) Θ -1 2 (ψ-R * 1 ) p(γ 2 ) •   1 - Γ m, m γ 1 Θ -1 1 [ψ -Θ 2 (γ 2 )] Γ(m)   dγ 2 = Γ m, m γ 2 Θ -1 2 (ψ -R * 1 ) -Γ m, m γ 2 Θ -1 2 (R * 2 ) Γ(m) - m m γ m 2 [Γ(m)] 2 Θ -1 2 (R * 2 ) Θ -1 2 (ψ-R * 1 ) γ m-1 2 exp - mγ 2 γ 2 • Γ m, m γ 1 Θ -1 1 [ψ -Θ 2 (γ 2 )] dγ 2 . (41)
P 3 = Pr{A 3 } = Pr{0 ≤ R 1 < R * 1 , 0 ≤ R 2 < ψ -R * 1 } = Pr{0 ≤ Θ 1 (γ 1 ) < R * 1 , 0 ≤ Θ 2 (γ 2 ) < ψ -R * 1 } = Pr{Θ -1 1 (0) ≤ γ 1 < Θ -1 1 (R * 1 ), Θ -1 2 (0) ≤ γ 2 < Θ -1 2 (ψ -R * 1 )} = Pr{0 ≤ γ 1 < Θ -1 1 (R * 1 ), 0 ≤ γ 2 < Θ -1 2 (ψ -R * 1 )} = Θ -1 2 (ψ-R * 1 ) 0 dγ 2 Θ -1 1 (R * 1 ) 0 p(γ 2 )p(γ 1 )dγ 1 = Θ -1 2 (ψ-R * 1 ) 0 dγ 2 Θ -1 1 (R * 1 ) 0 p(γ 2 ) m m γ m-1 1 γ m 1 Γ(m) • exp - mγ 1 γ 1 dγ 1 =   1 - Γ m, m γ 1 Θ -1 1 (R * 1 ) Γ(m)   • Θ -1 2 (ψ-R * 1 ) 0 p(γ 2 )dγ 2 =   1 - Γ m, m γ 1 Θ -1 1 (R * 1 ) Γ(m)   t i-1 t i t i+k t Y i t i+1 t i+k-1 Y i+k n updates n+1 updates t' i+k T i+k Y i+1
n+k+2 updates served in total Ỹi Ti Fig. 10. The number of updates been served from t i-1 to t i+k .

•

  1 - Γ m, m γ 2 Θ -1 2 (ψ -R * 1 ) Γ(m)   . (42)
The calculation of P 4 is symmetric to the calculation of P 1 .

APPENDIX B PROOF OF PROPOSITION 1

We calculate [START_REF] Shannon | A mathematical theory of communication[END_REF] term by term. For E[( Ỹi + Ti ) 2 ], we have to find the distribution of ( Ỹi + Ti ). As shown in Fig. 10, a new update arrives at t j and its service is finished at t j . Assuming that there are n updates in the queue just before t i-1 , there (n + k + 2) updates have been served from t i-1 to t i+k . The total service time of (n + k + 2) updates is

u = u 1 + n+k+2 j=2 u j , (43)
where u 1 is the residual service time of an update being served and u j is the service time of the updates waiting in the queue.

If the system is always busy, it means that all updates are continuously served without break from t i-1 to t i+k , i.e., u = ỹi + ti in this case. However, the system may be idle during [t i-1 , t i+k ], and therefore, u ≤ ỹi + ti in general. Obviously, U is the lower bound of ( Ỹi + Ti ). According to [START_REF] Papoulis | Probability, random variables, and stochastic processes[END_REF]], given n, u follows the gamma distribution as

p U |n (u) = µ n+k+2 u n+k+1 (n + k + 1)! e -µu . (44)
The probability for n updates in the queue is

p n = (1 -ρ)ρ n [38, Eq. (16-88)]. Hence, the distribution of u is p U (u) = ∞ n=0 p n • p U |n (u) = ∞ n=0 (1 -ρ)ρ n µ n+k+2 u n+k+1 (n + k + 1)! e -µu = ∞ n=-k-1 (1 -ρ)ρ n µ n+k+2 u n+k+1 (n + k + 1)! e -µu - -1 n=-k-1 (1 -ρ)ρ n µ n+k+2 u n+k+1 (n + k + 1)! e -µu . (45)
Consider the first term of (45) as

∞ n=-k-1 (1 -ρ)ρ n µ n+k+2 u n+k+1 (n + k + 1)! e -µu = ∞ n =0 (1 -ρ) ρ k+1 ρ n µ n +1 u n n ! e -µu = (1 -ρ)µ ρ k+1 e -µu ∞ n =0 ρ n µ n u n n ! = (1 -ρ)µ ρ k+1 e -µ(1-ρ)u . ( 46
)
For the second term of (45), we have (53)

Then, we have

E[ T 2 i ] = E[T 2 i+k ] = E[T 2 ] = ∞ 0 t 2 p T (t)dt = ∞ 0 t 2 µ(1 -ρ)e -µ(1-ρ)t dt = µ(1 -ρ) ∞ 0 t 2 e -µ(1-ρ)t dt = µ(1 -ρ)e -µ(1-ρ)t •   2 j=0 -j! 2 j [µ(1 -ρ)] j+1 t 2-j   ∞ t=0 (54) = µ(1 -ρ) • 2 µ 3 (1 -ρ) 3 = 2 µ 2 (1 -ρ) 2 , (55)
where (54) follows according to [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF]Eq. (2.321.2)].

Regarding E[ Ỹi ], since the arrival of update follows a Poisson process with mean arrival rate λ, the arrival time Y i follows independent and identically distributed (i.i.d.) exponential distribution with parameter λ. Therefore, E[Y i ] = 1 λ , and

E[ Ỹi ] = E   i+k j=i Y j   = i+k j=i E [Y j ] = k + 1 λ . (56)
Now, we substitute the results of each term into [START_REF] Shannon | A mathematical theory of communication[END_REF] as

∆ (k) = E[( Ỹi + Ti ) 2 ] -E[ T 2 i ] 2E[ Ỹi ] ≥ E[U 2 ] -E[ T 2 i ] 2E[ Ỹi ] = 1 -ρ µ 2 ρ k+1   2 (1 -ρ) 3 - k j=0 ρ j (j + 2)(j + 1)   • λ 2(k + 1) - 2 µ 2 (1 -ρ) 2 • λ 2(k + 1) = 1 -ρ 2µρ k (k + 1)   2 (1 -ρ) 3 - k j=0 ρ j (j + 2)(j + 1)   - ρ µ(1 -ρ) 2 (k + 1) . (57)
Consequently, (57) is the lower bound of ∆ (k) for busy case, and this finishes the proof of Proposition 1.

APPENDIX C PROOF OF PROPOSITION 2

Proof : Converse to the busy case, we consider the opposite extreme case, i.e., the system is completely idle and hence the queue is empty. In this case, the waiting time of each update is 0. Consider

∆ (k) = E[( Ỹi + Ti ) 2 ] -E[ T 2 i ] 2E[ Ỹi ] = E[ Ỹ 2 i ] + 2E[ Ỹi Ti ] 2E[ Ỹi ] = E[ Ỹ 2 i ] + 2E i+k j=i Y j • T i+k 2E[ Ỹi ] = E[ Ỹ 2 i ] + 2 i+k j=i E[Y j T i+k ] 2E[ Ỹi ] = E[ Ỹ 2 i ] + 2 i+k j=i E[Y j (W i+k + S i+k )] 2E[ Ỹi ] ≥ E[ Ỹ 2 i ] + 2 i+k j=i E[Y j S i+k ] 2E[ Ỹi ] (58) = E[ Ỹ 2 i ] 2E[ Ỹi ] + 2 i+k j=i E[Y j ]E[S i+k ] 2 i+k j=i E[Y j ] = E[ Ỹ 2 i ] 2E[ Ỹi ] + E[S i+k ], (59)
where the equality of (58) holds when the system is completely idle. Therefore, the lower bound for idle case can be further derived from (59). We can calculate the first term of (59) as

E[ Ỹ 2 i ] 2E[ Ỹi ] = E i+k j=i Y j 2 2E i+k j=i Y j = E i+k j=i Y 2 j + 2 i+k-1 j=i i+k j =j+1 Y j Y j 2 i+k j=i E [Y j ] = (k + 1)E[Y 2 ] + 2 i+k-1 j=i i+k j =j+1 E[Y j ]E[Y j ] 2(k + 1)E [Y ] = (k + 1)E[Y 2 ] + k(k + 1)(E[Y ]) 2 2(k + 1)E [Y ] = E[Y 2 ] + k(E[Y ]) 2 2E [Y ] . ( 60
)
It is easy to obtain E[Y 2 ] = 2 λ 2 for exponential distribution. Hence, we have

E[ Ỹ 2 i ] 2E[ Ỹi ] = 1 λ + k 2λ . ( 61
)
By substituting (61) and E[S] = 1 µ into (59), we can obtain the lower bound of ∆ (k) for idle case as

∆ (k) ≥ 1 λ + k 2λ + 1 µ . (62)
This finishes the proof of Proposition 2.

Fig. 1 .

 1 Fig. 1. A UAV cooperative surveillance system for IoT applications.

Fig. 3 .

 3 Fig. 3. Achievable rate-distortion region and outage region.

Fig. 4 .

 4 Fig. 4. Outage probability for cooperative communications, where m = 2, σ 1 = σ 2 = 1 and r 1 = r 2 = 1.

Fig. 6 .

 6 Fig. 6. The exact value versus the lower bound of ∆ (0) .

Fig. 7 .

 7 Fig. 7. The comparison between the relatively tight lower bound and simulation results.

Fig. 8 .

 8 Fig. 8. Refinement of AoI performance.

Fig. 9 .

 9 Fig. 9. The average AoI for different UAV locations with µ = 5.

- 1 n=-k- 1 ( 1 - 0 u 2 p 0 u 2 ( 1 -

 11102021 ρ)ρ n µ n+k+2 u n+k+1 (n + k + 1)! e -µu = k n =0 (1 -ρ) ρ k+1 ρ n µ n +1 u n (n )! e -µu = (1 -ρ)µ ρ k+1 e -µu PDF of U is given by p U (u) = (1 -ρ)µ ρ k+1 e -µ(1-ρ)u -e -µu can calculate E[U 2 ] as follows E[U 2 ] = ∞ U (u)du = ∞ ρ)µ ρ k+1• e -µ(1-ρ)u -e -µu n n ! u n +2 e -µu du.

u 2 e 2 j=0-j! 2 j 1 -

 2221 -µ(1-ρ)u du = e -µ(1-ρ)u   [µ(1 -ρ)] j+1 u 2-j can finally obtain the closed-form expression of E[U 2 ] as E[U 2 ] =(the PDF of the system time T is given by [38, Eq. (16-97)] p T (t) = µ(1 -ρ)e -µ(1-ρ)t .

In essence, the system is a special case of communications with correlated sources, and hence we can design the encoder and the joint decoder for practical implemetations according to[START_REF] Garcia-Frias | Near-Shannon/Slepian-Wolf performance for unknown correlated sources over AWGN channels[END_REF].


S Song

T Matsumoto
email: tadashi.matsumoto@imt-atlantique.fr

Rate-Distortion and Outage Probability Analyses of Wyner-Ziv Systems over Multiple Access Channels Shulin Song, Jiguang He, Member, IEEE, and Tad Matsumoto, Life Fellow, IEEE Abstract-In this paper, we conduct rate-distortion and outage probability analyses for Wyner-Ziv (WZ) systems, where the information sequences from a source and a helper are transmitted to the destination over block Rayleigh fading multiple access channels (MACs). This work has been motivated by decisionmaking systems, where the aim is to make right decisions based on the observations. Hence, the most critical performance metric is the accuracy of decisions, even allowing distortion in wireless transmission. A sufficient condition for the successful transmissions is that the WZ and MAC regions intersect. For the ease of calculating the outage probability, we propose an approximated, yet accurate, WZ rate region. The outage probabilities of the WZ systems are then evaluated both in orthogonal and MAC transmissions. It is shown that the transmission efficiency of the system with MAC is significantly improved compared to orthogonal transmission. Furthermore, a helper-selection scheme is introduced to further reduce the outage probability of the WZ-MAC systems. The results show that the outage probability decreases, of which decay corresponds to the (L + 1)-order diversity with L helpers in small value range of the average signal-to-noise ratio (SNR), while it converges into L as the average SNR becomes large.

Index Terms-Wyner-Ziv problem, multiple access channel, outage probability, helper-selection, rate distortion analysis.

I. INTRODUCTION

In the fifth generation (5G) and beyond 5G (B5G) wireless communication systems, ultra-reliable and low-latency communications (URLLC) have been considered as being of crucial importance [START_REF] Li | 5G ultra-reliable and low-latency systems design[END_REF], [START_REF] Parvez | A survey on low latency towards 5G: RAN, core network and caching solutions[END_REF]. In wireless communication systems, to reasonably compromise between reliability and latency, error-control strategies such as error correction coding as well as automatic repeat request has been widely used. Nowadays, with the drastic increase in the wireless terminals and/or devices, it has been recognized that the cooperation with surrounding terminals and devices is effective to both improve the reliability and reduce the transmission latency [START_REF] Popovski | Wireless access for ultra-reliable low-latency communication: principles and building blocks[END_REF], [START_REF] He | Performance analysis of lossy decode-and-forward for non-orthogonal MARCs[END_REF]. In dense terminal environments, for example, industry Internet-of-things (IoT) systems, multiple devices track the target, and send the observations to the destination, where high reliability can be achieved by the cooperation with a large number of available devices. Furthermore, if the transmissions are over multiple access channels (MACs), the latency can be significantly reduced, and thereby, the transmission efficiency improved.

On the other hand, in decision-making systems such as autonomous vehicle driving and/or unmanned factory, ultrareliable wireless communications should not be necessarily a mandatory requirement. This is because the objective of the wireless data transfer is to provide the destination with the information required to make accurate final decisions. In this case, the destination can accept a certain level of distortion occurring in the wireless data transfer, as long as the decision results meet the reliability requirement.

Hence, we define the global objective of the study as analyzing the outage probability of the correlated multi-source transmissions over block fading MAC channels with distortion level as a quality-of-service (QoS) parameter. This research objective is classified into multi-source-multi-helper lossy communications with MAC transmission, which has several open questions in network information theory [START_REF] Gamal | Network information theory[END_REF].

In [START_REF] Zhou | Exact and approximated outage probability analyses for decode-and-forward relaying system allowing intra-link errors[END_REF], a lossy forward (LF) relaying system is investigated, where the relay forwards the information to the destination even though the information part may contain some errors. The original information may be recovered at the destination by joint decoding with the relay serving as a helper. To achieve lossless recovery, the rates supported by the source-destination (SD) and relay-destination (RD) links have to satisfy the Slepian-Wolf (SW) rate region with a helper, referred to as h-SW rate region. It is shown in [START_REF] Zhou | Exact and approximated outage probability analyses for decode-and-forward relaying system allowing intra-link errors[END_REF] that since the information sequences transmitted over the SD and RD links are correlated, even though some errors may occur in the information part at the relay, the system coverage increases significantly compared to the conventional decode-and-forward systems. Joint decoding is utilized in LF systems to take full advantage of the correlation knowledge. However, because of the fading channel variation, the rates supported by the SD and RD links do not always fall into the admissible h-SW rate region. Zhou et al. [START_REF] Zhou | Exact and approximated outage probability analyses for decode-and-forward relaying system allowing intra-link errors[END_REF] theoretically calculates the probability that the rate pair does not fall into the admissible h-SW rate region, which defines the outage probability of the system.

Various types of the LF systems have been investigated, which are summarized in a tutorial paper [START_REF] He | A tutorial on lossy forwarding cooperative relaying[END_REF]. However, as described above, with the global objective of the decisionmaking systems, lossless communication should not be necessarily required. Lin et al. [START_REF] Lin | Lossy-forward relaying for lossy communications: rate-distortion and outage probability analyses[END_REF] analyzed the outage probability of a LF system with the end-to-end distortion level as a parameter. It is shown in [START_REF] Lin | Lossy-forward relaying for lossy communications: rate-distortion and outage probability analyses[END_REF] that if the rate pair supported by the SD and RD links satisfy the Wyner-Ziv (WZ) rate region, the end-to-end distortion can be made equal to or smaller than the specified level. Furthermore, the numerical results show that compared to the lossless cases, lower outage probability can be achieved. In addition, an inner bound of the rate region with one-source-multiple-helper is derived in [START_REF] Lin | Performance analysis of distortion-acceptable cooperative communications in wireless sensor networks for internet of things[END_REF]. Obviously, the more helpers, the lower the outage probability. However, the exact rate region derivation is still an open question.

Motivated by the results of [START_REF] Lin | Lossy-forward relaying for lossy communications: rate-distortion and outage probability analyses[END_REF] and the advantage of MAC over the orthogonal transmission in LF systems [START_REF] He | A tutorial on lossy forwarding cooperative relaying[END_REF], in this paper we analyze the outage probability of one-source-onehelper lossy communication over block fading MAC channels with the end-to-end distortion level as a parameter. Even though this scenario set-up is the smallest case of the global objective, up to the authors' maximum knowledge, the paper for the first time provides the theoretical analysis towards achieving the global objective.

The results of this study can be applied to autonomous vehicle driving, as exemplified in Fig. 1, where two sensors observe the road condition, and transmit the collected data to the vehicle. The information received by the vehicle is utilized to make a decision on whether or not the vehicle should take an action to avoid the obstacles. Obviously, low latency transmission is required in this scenario to avoid the collision, for which MAC transmission is of great importance. As investigated in [START_REF] Zhou | Outage probability of correlated binary source transmission over fading multiple access channels[END_REF], it is shown that a sufficient condition for the lossless transmission for the correlated sources transmitted over MAC is that MAC and SW rate regions have an intersection. In [START_REF] Song | Outage probability of one-source-with-one-helper sensor systems in block Rayleigh fading multiple access channels[END_REF], it is shown that the successful transmission is possible when h-SW and MAC rate regions have an intersection, which is also a sufficient condition [START_REF] Rajesh | Transmission of correlated sources over a fading multiple access channel[END_REF]. The difference between [START_REF] Zhou | Outage probability of correlated binary source transmission over fading multiple access channels[END_REF] and [START_REF] Song | Outage probability of one-source-with-one-helper sensor systems in block Rayleigh fading multiple access channels[END_REF] is that in [START_REF] Zhou | Outage probability of correlated binary source transmission over fading multiple access channels[END_REF], two sources are aimed to be reconstructed, while in [START_REF] Song | Outage probability of one-source-with-one-helper sensor systems in block Rayleigh fading multiple access channels[END_REF], one of the two sources are to be reconstructed with the other serving as a helper. An obvious, but very important finding from [START_REF] Song | Outage probability of one-source-with-one-helper sensor systems in block Rayleigh fading multiple access channels[END_REF] is that one-source-one-helper systems with MAC transmission can achieve twice as large transmission efficiency as its orthogonal transmission counterpart (equivalently, a half the latency) without sacrificing outage probability.

This paper aims to extend the major results of [START_REF] Song | Outage probability of one-source-with-one-helper sensor systems in block Rayleigh fading multiple access channels[END_REF] to lossy cases, where the end-to-end distortion is regarded as a QoS parameter. The rates of the source and the helper should satisfy both the WZ and MAC rate regions for successful transmission with the distortion level equal to or smaller than the predefined QoS requirement, even though it is also a sufficient condition [START_REF] Rajesh | Transmission of correlated sources over a fading multiple access channel[END_REF]. Conversely, outage probability can be evaluated as the probability that the two regions have no intersection. In this paper, the correlation between the source and the helper is expressed by the bit-flipping model, where the flipping probability is fixed as a parameter. In this case, WZ rate region is also fixed. Since the transmission links suffer from block Rayleigh fading, the MAC rate region varies, according to the variation of the instantaneous signalto-noise ratio (SNR). Furthermore, a helper-selection scheme is introduced, where the helper-destination (HD) link with the largest instantaneous SNR is selected. The main results of this paper provide an initial solution to the global objective, and the major contributions are summarized as follows:

(1) For the ease of the calculations of outage probability, we propose an approximated, yet accurate, WZ rate region, with which outage probabilities in the orthogonal transmission can be derived, and a comparison is made to that derived from the exact WZ rate region. (2) We calculate the outage probability of the WZ system in block Rayleigh fading MAC by using the approximated rate region, where the effect of helper locations is also taken into account. We also study the trade-off between the distortion and the outage probability in the MAC transmission.

(3) We also propose a helper-selection scheme in multihelper WZ systems, of which the outage probabilities are evaluated both in orthogonal and MAC transmissions. The rest of this paper is organized as follows: The system model of the WZ system in block Rayleigh fading MAC is introduced in Section II. We discuss the exact WZ rate region, and based on the observation of the shape of the exact WZ region, an approximated, yet accurate, WZ rate region is proposed in Section III. Furthermore, the outage probability expressions are presented in the orthogonal transmission. Section IV identifies the cases where the WZ and MAC rate regions do not intersect, resulting in the outage event in the MAC transmission. Results on the outage probability calculated by the approximated WZ rate regions in MAC are then provided, where the helper locations are taken into account. In Section V, we investigate multi-helper WZ systems with helper selection, and the outage probability is evaluated both in orthogonal and MAC transmissions. Finally, we conclude this paper in Section VI.

II. SYSTEM MODEL

As shown in Fig. 2, two sensors observe the same target and send the observations to the destination over the MAC, where the information from one of the sensors, referred to as source, is aimed to be reconstructed, while the other serves as a helper. Let X and Y represent the information from the source and the helper, respectively, and

X n = (X 1 , X 2 , • • • , X n ) and Y n = (Y 1 , Y 2 , • • • , Y n ) are the corresponding length-n sequences.
Since the two sensors observe the same target, X and Y are correlated, of which the relationship is assumed to be described by the bit-flipping model [START_REF] Anwar | Accumulator-assisted distributed turbo codes for relay systems exploiting source-relay correlation[END_REF], as

X = Y ⊕ E, (1)
Fig. 2: One-source-one-helper Wyner-Ziv system model.

Fig. 3: Block diagram for encoder and decoder of Wyner-Ziv system followed by decision making process.

with E = 1, with probability P e , 0, with probability

1 -P e , (2)
P e ∈ [0, 0.5], where P e = 0 and P e = 0.5 represent X and Y are fully correlated and independent, respectively. The coded and modulated sequences from the source and the helper, denoted, respectively, by x and y with the unity average signal powers, are transmitted to the destination at the rates R 1 and R 2 . After joint decoding, the reconstructed information X of X can be used for decision-making, as shown in Fig. 3. Noticed that X should not necessary be exactly the same as the original information X. The distortion between X and X is defined as

D X = 1 n n j=1 d(X j , Xj ), (3)
where

d(X j , Xj ) = 1, if X j = Xj , 0, if X j = Xj . (4)
D X ∈ [0, 0.5] represents the QoS requirement of the wireless data transfer. The decisions may rely on some hypothesis testing [START_REF] Shimokawa | Error bound of hypothesis testing with data compression[END_REF] or statistical inference [START_REF] Mackay | Information theory, inference, and learning algorithms[END_REF]. However, as discussed in Introduction, this paper does not intend to cover the decision making process. The requirement D X = 0 refers to the lossless case and D X = 0.5 refers to the case any distortion is acceptable. The SD and HD links are assumed to suffer from block Rayleigh fading, where the channel gain stays the same within each block, and changes block-by-block. Let P i and G i denote the transmit power and the geometric gain [START_REF] Ochiai | Design and analysis of collaborative diversity protocols for wireless sensor networks[END_REF], respectively, where the index i ∈ {1, 2} refers to the SD and HD links. The received signal can be then represented as

z = P 1 G 1 h 1 x + P 2 G 2 h 2 y + n, (5)
where h i ∼ CN (0, 1) and n ∼ CN (0, I) represent the channel gain and the zero-mean complex additive white Gaussian noise (AWGN) sequences, respectively. Since shadowing is ignored for simplicity, the geometric gains can be calculated by

G 1 = d -3.52 sd , (6)
and

G 2 = d -3.52 hd , (7)
where d hd and d sd represent the distance of the SD and HD links, respectively. Under the assumption of normalized noise power, the average SNR is calculated by Γ i = P i G i , and the instantaneous SNR is given by

γ i = P i G i • |h i | 2 .
In this paper, we assume the transmit powers of x and y are the same. Under the assumption of Rayleigh fading, the probability density function (pdf ) of the received instantaneous SNR is given by [START_REF] Schwartz | Communication systems and techniques[END_REF] p

(γ i ) = 1 Γ i exp - γ i Γ i . (8)

III. WYNER-ZIV RATE REGION ANALYSIS

A. Exact Wyner-Ziv Rate Regions

The rates of two distributed correlated sources, one of which serves as a helper, have to satisfy WZ rate regions in the lossy case, while h-SW rate region in the lossless case for the successful transmission [START_REF] Slepian | Noiseless coding of correlated information sources[END_REF], [START_REF] Cheng | Outage probability of a relay strategy allowing intra-link errors utilizing Slepian-Wolf theorem[END_REF]. The admissible WZ rate region can be mathematically expressed by

R 1 ≥ I (X; U |V ) , R 2 ≥ I (Y ; V ) , (9)
where I(•; •) indicates the mutual information between the first and second arguments. As shown in Fig. 4, the admissible rate region in the lossy case is larger than that in the lossless case.

V denotes the independently decoded versions of Y , and U denotes the jointly decoded versions of X with the help of 9) can be calculated by

V 1 . Since U → X → Y → V
I (X; U |V ) = H (U | V ) -H (U | X, V ) = H (U | V ) -H (U | X) = H b (D X * P e * p ) -H b (D X ) , (10)
and

I (Y ; V ) = H (Y ) -H (Y | V ) = 1 -H b (p ) , (11)

B. Approximated Wyner-Ziv Rate Regions

As discussed in Introduction, the goal of this paper is to calculate the outage probability of lossy one-source-one-helper systems in block Rayleigh fading channels. However, the exact calculation requires multi-fold integral: three-fold integral is needed in the three-node LF relay systems [START_REF] Cheng | Outage probability of a relay strategy allowing intra-link errors utilizing Slepian-Wolf theorem[END_REF], and two-fold integral in h-SW-MAC systems [START_REF] Song | Outage probability of one-source-with-one-helper sensor systems in block Rayleigh fading multiple access channels[END_REF]. Especially, as described in Section IV, the probability has to be calculated for the event that the WZ and MAC regions have an intersection when the MAC transmission is considered. Since the exact WZ rate region has a curve part, the multi-fold integral is complex. To alleviate the computational complexity, we propose an approximated, yet accurate, WZ rate region that has only lines. This idea was first proposed in [START_REF] Cheng | Outage probability of a relay strategy allowing intra-link errors utilizing Slepian-Wolf theorem[END_REF], and its accuracy was theoretically evaluated in [START_REF] Zhou | Exact and approximated outage probability analyses for decode-and-forward relaying system allowing intra-link errors[END_REF], both for the three-node lossless LF relay systems. As shown in Fig. 5, the curve part of the exact WZ rate region is approximated only by lines, with which the approximated admissible WZ rate region is expressed by

R 1 ≥    I (X; U |Y ) , for R 2 ≥ H(Y ), 1 α R 2 + β α , for H(Y |X) ≤ R 2 ≤ H(Y ), I (X; U ) , for 0 ≤ R 2 ≤ H(Y |X), (12)
with

α = I (X; Y ) I (X; U |Y ) -I (X; U ) , (13)
and

β = I (X; U ) -I (X; U |Y ) • H(Y |X) I (X; U |Y ) -I (X; U ) , (14)
where

I (X; Y ) = H (X) -H (X | Y ) = 1 -H b (P e ) , (15)
I (X; U |Y ) = H (U | Y ) -H (U | X, Y ) = H (U | Y ) -H (U | X) = H b (D X * P e ) -H b (D X ) , (16)
I (X; U ) = H (X) -H (X | U ) = 1 -H b (D X ) , (17)
and

H (Y | X) = H b (P e ) . (18)

C. Outage Probability in Two-Phase Orthogonal Transmission Case

Outage is defined as the event where the rate pair does not fall into the admissible rate regions. Hence, the outage probability of one-source-one-helper WZ systems can be expressed by twofold integrals with respect to the pdf of the instantaneous SNRs of each link. In the case of the orthogonal transmission, the intersection of the WZ and MAC regions does not have to be taken into account. Hence, the outage probability can also be derived from the exact rate region expressed in [START_REF] Zhou | Outage probability of correlated binary source transmission over fading multiple access channels[END_REF]. In this subsection, we first calculate the outage probability based on both (9) and the approximation [START_REF] Gamal | Network information theory[END_REF], and then the next subsection compares the results derived from ( 9) and [START_REF] Gamal | Network information theory[END_REF].

Since the SD and HD links are both point-to-point, the ratedistortion pair (R i , D i ), i = 1, 2, is achievable, if Shannon's source-channel separation theorem

R i (D i )R c,i ≤ C(γ i ) (19)
is satisfied, where C (γ i ) = log 2 (1 + γ i ) is the channel capacity of each link. R c,i denotes the normalized spectrum efficiency, of which the modulation multiplicity and the channel coding rate are taken into account [START_REF] Frias | Near-Shannon/Slepian-Wolf performance for unknown correlated sources over AWGN channels[END_REF]. R i (D i ) is the rate distortion function for lossy source coding. With the uniformly distributed binary source assumption,

R i = 1 -H b (D i ).
For the information transmitted by the helper,

R 2 = 1 -H b (p ). Hence p = H -1 b {1 -R 2 } = H -1 b 1 -C(γ2) Rc,2
. Since the inverse of the binary entropy function H -1 b {•} can not be expressed explicitly, we use the expression obtained by a curve fitting technique shown in [START_REF] Zhou | Exact and approximated outage probability analyses for decode-and-forward relaying system allowing intra-link errors[END_REF], as

H -1 b (x) ≈ 2 ax b -2 -cx d e , (20)
where a = 0.6794, b = 0.7244, c = 0.1357, d = 21.8026 and e = 1.9920.

The outage probability derived from the exact WZ region in the orthogonal transmission can be expressed by P out,exact = P out,exact,1 + P out,exact,2 , with

P out,exact,1 = Pr {0 ≤ R 1 ≤ I (X; U |Y ) ; R 2 ≥ 0} = Pr {Φ (0) ≤ γ 1 ≤ Φ [I (X; U |Y )] ; γ 2 ≥ Φ (0)} = Φ[I(X;U |Y )] Φ(0) +∞ Φ(0) p (γ 1 ) p (γ 2 ) dγ 2 dγ 1 = 2 R c,1[ H b (D X * Pe )-H b (D X )] -1 0 +∞ 0 p (γ 1 ) p (γ 2 ) dγ 2 dγ 1 , (21)
and

P out,exact,2 = Pr {I (X; U |Y ) ≤ R 1 ≤ I (X; U |V ) ; 0 ≤ R 2 ≤ H(Y )} = Pr Φ [I (X; U |Y )] ≤ γ 1 ≥ Φ [I (X; U |V )] ; Φ (0) ≤ γ 2 ≤ Φ [H(Y )] = Φ[H(Y )] Φ(0) Φ[I(X;U |V )] Φ[I(X;U |Y )] p (γ 1 ) p (γ 2 ) dγ 1 dγ 2 = 2 R c,1 -1 0 2 R c,2[ H b (D X * Pe * p )-H b (D X )] -1 2 R c,2[ H b (D X * Pe )-H b (D X )] -1 p (γ 1 ) p (γ 2 ) dγ 1 dγ 2 , ( 22
)
where the relationship between the rate and the instantaneous SNR is given by

γ i = Φ (R i ) = 2 Rc,iRi -1, (23)
and

R i = Φ -1 (γ i ) = 1 R c,i log 2 (1 + γ i ) . ( 24
)
The outage probability with the approximated WZ region in orthogonal transmission is expressed by P out,approx = P out,approx,1 + P out,approx,2 , with

P out,approx,1 = Pr {0 ≤ R 1 ≤ I (X; U |Y ) ; R 2 ≥ 0} = Pr {Φ (0) ≤ γ 1 ≤ Φ [I (X; U |Y )] ; γ 2 ≥ Φ (0)} = Φ[I(X;U |Y )] Φ(0) +∞ Φ(0) p (γ 1 ) p (γ 2 ) dγ 2 dγ 1 = 2 R c,1[ H b (D X * Pe )-H b (D X )] -1 0 +∞ 0 p (γ 1 ) p (γ 2 ) dγ 2 dγ 1 , (25)
Fig. 6: Outage probabilities calculated from exact and approximated WZ rate regions in two-phase orthogonal transmission. and

P out,approx,2 = Pr {I (X; U |Y ) ≤ R 1 ≤ I (X; U ) ; 0 ≤ R 2 ≤ αR 1 -β} = Pr Φ [I (X; U |Y )] ≤ γ 1 ≤ Φ [I (X; U )] ; Φ (0) ≤ γ 2 ≤ Φ (αR 1 -β) = Φ[I(X;U )] Φ[I(X;U |Y )] Φ(αR1-β) Φ(0) p (γ 1 ) p (γ 2 ) dγ 2 dγ 1 = 2 R c,1[ 1-H b (D X )] -1 2 R c,1[ H b (D X * Pe )-H b (D X )] -1 2 -R c,2 β (1+γ1) R c,2 R c,1 α -1 0 p (γ 1 ) p (γ 2 ) dγ 2 dγ 1 . (26)

D. Comparison of Outage Probabilities in Orthogonal Transmission

According to the expressions provided in Subsection III-C, this subsection compares the outage probabilities derived from the exact and approximated WZ rate regions in the orthogonal transmission. Without loss of generality, R c,1 = R c,2 = 0.5 is assumed throughout this paper. As shown in Fig. 6, the solid curves represent the outage probability obtained by the exact WZ rate region, while the dotted curves denote that obtained by the approximated WZ rate region. The results indicate that the difference is negligibly small. It is found that the smaller the gap, the larger the average SNR as well as the smaller the D X . It is also found that even though the gap is small, the outage probability derived from the approximated WZ region is slightly larger than that derived from the exact region. This is because the probability that the rate pair falls into the area indicated by A in Fig. 5 is ignored in the approximation. It can be concluded that the approximation is very accurate. Hence, we utilize the approximated WZ rate region in WZ-MAC outage probability analysis.

IV. OUTAGE PROBABILITY ANALYSIS IN ONE-PHASE MAC

A. MAC Rate Region

Since the information from the source and the helper is assumed to be transmitted over the MAC, the transmission rates have to satisfy MAC rate region for successful transmissions, as shown in Fig. 7. The achievable MAC rate region can be expressed by

   R 1 ≤ C (γ 1 ) , R 2 ≤ C (γ 2 ) , R 1 + R 2 ≤ C (γ 1 + γ 2 ) . (27)
In the WZ systems, MAC transmission is successful when the WZ and MAC rate regions intersect, with which sourcechannel separation holds, since the rate pair is included in the both regions. Replacing R i in (27) with R i R c,i , the MAC region with the source-channel separation can be further represented by

   R 1 R c,1 ≤ C (γ 1 ) , R 2 R c,2 ≤ C (γ 2 ) , R 1 R c,1 + R 2 R c,2 ≤ C (γ 1 + γ 2 ) . (28)

B. Outage Probability Expressions in MAC Case

The outage probability can be calculated by twofold integrals, of which the boundaries are determined by the admissible rate regions. For the ease of twofold integral calculations, we use the approximated WZ rate region. Two scenarios with which outage happens are shown in Fig. 8. The outage probability can be expressed by P out,M AC = P out,M AC,1 + P out,M AC,2 . In Fig. 8a, since 1 Rc,1 C (γ 1 ) ≤ I (X; U |Y ), the WZ and MAC regions do not intersect. Hence,

P out,M AC,1 = Pr 0 ≤ 1 R c,1 C (γ 1 ) ≤ I (X; U |Y ) ; 1 R c,2 C (γ 2 ) ≥ 0 = Pr {Φ (0) ≤ γ 1 ≤ Φ [I (X; U |Y )] ; γ 2 ≥ Φ (0)} = Φ[I(X;U |Y )] Φ(0) +∞ Φ(0) p (γ 1 ) p (γ 2 ) dγ 2 dγ 1 = 2 R c,1[ H b (D X * Pe)-H b (D X )] -1 0 +∞ 0 p (γ 1 ) p (γ 2 ) dγ 2 dγ 1 . (29)
In Fig. 8b, with I (X; U |Y ) ≤ 1 Rc,1 C (γ 1 ) ≤ I (X; U ), the regions do not intersect when 0 ≤ -Rc,1 Rc,2 R 1 + Fig. 9: Outage probabilities of WZ systems in two-phase orthogonal and one-phase MAC transmissions for P e = 0.1.

1 Rc,2 C (γ 1 + γ 2 ) ≤ αR 1 -β. Hence, P out,M AC,2 = Pr I (X; U |Y ) ≤ 1 Rc,1 C (γ 1 ) ≤ I (X; U ) ; 0 ≤ - Rc,1 Rc,2 R 1 + 1 Rc,2 C (γ 1 + γ 2 ) ≤ αR 1 -β = Pr Φ [I (X; U |Y )] ≤ γ 1 ≤ Φ [I (X; U )] ; Φ (0) ≤ γ 2 ≤ M = Φ[I(X;U )] Φ[I(X;U |Y )] M Φ(0) p (γ 1 ) p (γ 2 ) dγ 2 dγ 1 = 2 R c,1[ 1-H b (D X )] -1 2 R c,1[ H b (D X * Pe )-H b (D X )] -1 M 0 p (γ 1 ) p (γ 2 ) dγ 2 dγ 1 , (30)
with

M = 2 -βRc,2 (1 + γ 1 ) αR c,2 R c,1 +1 -1 -γ 1 .

C. Numerical Results of Outage Probability in One-Phase MAC Transmission

According to the expressions provided in Subsection IV-B, numerical results of the outage probabilities are presented in this subsection. We first compare the outage probabilities of two sources transmitted using one transmission phase with MAC and using two orthogonal transmission phases. This comparison is fair because their transmit powers are the same. Fig. 9 shows the outage probabilities of WZ systems in the orthogonal and MAC transmissions, with the distortion D X as a parameter. The results show that the difference is negligibly small. However, the transmission efficiency in the MAC transmission is twice as large as that in the orthogonal transmission, i.e., two phases are needed to transmit the source and helper information independently by using orthogonal transmission, while only one phase by MAC.

We also take into account the effect of the helper locations on the outage probability, where d hd = θ • d sd with θ ∈ (0, 1]. Since d sd is normalized to the unity, the geometric gain makes change to the average SNR of the HD link, and is expressed as G 2 = θ -3.52 [START_REF] Youssef | Distributed serially concatenated codes for multi-source cooperative relay networks[END_REF]. Fig. 10 shows the outage probability with θ as a parameter. Obviously, it is found that the outage probability decreases with smaller θ since the distance between the helper and the destination decreases.

We also investigate the relationship between the outage probability and the distortion. It is found in Fig. 11 that the larger the acceptable distortion, the smaller the outage probability. When the acceptable distortion is larger than 0.35, the outage probability decrease more rapidly, of which the tendency is consistent to [START_REF] Lin | Lossy-forward relaying for lossy communications: rate-distortion and outage probability analyses[END_REF]. Also, lower outage probability can be achieved when the distance between the helper and the destination decreases. Fig. 11 suggests that the trade-off between the distortion and the outage probability can be well controlled, and the required average SNR to satisfy the outage probability requirement can be identified.

V. WZ SYSTEMS WITH HELPER SELECTION

A. WZ Systems with Helper Selection

We investigate a one-source-multi-helper system in this section, where a helper-selection technique is introduced. As shown in Fig. 12, since one source and L helpers observe the same target, the information obtained by the source and helpers is correlated. Instead of sending information from all the helpers H i , i = 1, 2, ..., L, the best helper H î is assumed to be selected with the help of higher layer protocols which is, however, not detailed at this stage of the paper. Let P e,i represents the crossover probability of the information between the source and the helper H i , and p i denote the crossover probability between the original and decoded information via the i-th HD link. The best helper H î has to satisfy [START_REF] Hu | Performance analysis of decode-and-forward relaying with selection combining[END_REF] î = arg min i {P e,i * p i },

where P e,i is set as a fixed parameter while p i is a random variable depending on the instantaneous SNR of the HD link.

Even though finding the best helper following (31) may be possible with the help of higher layer protocol, it is not practical in the systems requiring low latency.

Even though in practice P e,i is a random variable, because of the justification provided in Appendix A, we assume P e,i = P e,max , i = 1, 2, ..., L2 , even though they are random variable. With this assumption, the best helper H î to be selected should be the one having the largest instantaneous SNR, as 3î = arg max

i {γ i } . (32)
The source and the selected helper then transmit their observations over MAC. One-source-multiple-helper with helper selection can be seen as a simple solution to one-sourcemulti-helper problems. The pdf of the instantaneous SNR in Rayleigh fading with helper selection is equivalent to the selection combining diversity, expressed as [START_REF] Goldsmith | Wireless communications[END_REF] p

s (γ 2 ) = L Γ 2 exp - γ 2 Γ 2 1 -exp - γ 2 Γ 2 L-1 . (33)

B. Outage Probability Calculation Results

The outage probability of the helper-selection WZ systems in the orthogonal and MAC transmissions can be obtained by repacing the pdf p (γ 2 ) by p s (γ 2 ) in ( 25), (26), (29) and (30). In Fig. 13, outage probabilities of the helper-selection WZ systems with orthogonal transmission in both lossless and lossy cases are presented. The result indicates that with the lossy transmission, lower outage probability can be achieved. It is found that the outage probability decreases, of which decay corresponds to the (L+1)-order diversity with L helpers in a small value range of the average SNR, while it converges into L as the average SNR becomes large. This is because the source and helper information is not fully correlated (P e,i = 0).

According to the numerical results, we make a comparison of the outage probability of the helper-selection WZ systems in the orthogonal and MAC transmissions. As shown in Fig. 14, the difference in outage probability between the orthogonal and MAC transmissions is negligibly small. Hence, the transmission efficiency with MAC can be improved without sacrificing outage probability. In both cases, the diversity order with helper selection is L + 1 when the average SNR is small and approaches L when the average SNR becomes large. It should be noted that Fig. 14 does not take account the overhead due to the help of the higher layer protocol.

VI. CONCLUSION

In this paper, the outage probabilities of WZ systems both in the orthogonal and MAC transmissions have been evaluated. For MAC transmission, we defined successful transmissions as the case the WZ and MAC rate regions intersect. Conversely, outage happens when they have no intersection. In order to make the outage probability calculations tractable, an approximation of WZ rate region has been proposed. The results have demonstrated that the difference in outage probabilities between the exact and approximated WZ regions is negligibly small. It has been found that compared to lossless transmissions, lower outage probability can be achieved in lossy transmissions, and the larger the acceptable distortion, the lower the outage probability. In addition, when the helper is located closer to the destination, outage probability decreases. Furthermore, we have investigated the WZ systems with helper selection, of which the outage probabilities both in the orthogonal and MAC transmissions are theoretically calculated. The results show that the outage probability decreases, of which decay corresponds to the (L+1)-order diversity with L helpers in a small value range of the average SNR, while it converges into L as the average SNR becomes large. It has been verified that without sacrificing outage probability, the transmission efficiency in the MAC transmission is twice as large as that in the orthogonal transmission in one-source-one-helper and helper-selection WZ systems.

APPENDIX

In practice, since the relative positions of the source and the helpers are not the same, P e,i is a random variable. It may be reasonable to assume P e,i follows the uniform distribution, where the maximum value is P e,max . To evaluate the effect of the correlation between the source and the helper, two extreme cases are considered: Case (1): the first helper has the lowest correlation to the source, and the other helper are fully correlated to the source, as P e,i = P e,max , for i = 1, 0, for i = 2, 3, ..., L.

which is the assumption used in Section V. Numerical results of the outage probabilities for Case (1) and [START_REF] Parvez | A survey on low latency towards 5G: RAN, core network and caching solutions[END_REF] with P e,max = 0.2 and D X = 0.1 are shown in Fig. 15. It is found that the outage probability curves with the assumption of (35) are above that with (34) for all the considered L values. Hence, the assumption of (35) yields the upper bound of the outage probability. It is important to see that the gap between the two cases becomes smaller with larger L. This observation inherently justifies the statements described in footnote 1, and hence, the assumption of P e,i = P e,max , i = 1, 2, ..., L in the outage probability analysis is indirectly proven to be reasonable.

Fig. 1 :

 1 Fig. 1: Example of autonomous vehicle driving with lossy observation.

Fig. 4 :

 4 Fig. 4: Admissible rate regions of one-source-one-helper in lossless and lossy cases.

Fig. 5 :

 5 Fig. 5: Exact and approximated Wyner-Ziv rate regions; solid and doted curves represent exact and approximated rate regions, respectively.

  where the operator * indicates the binary convolution a * b = a(1 -b) + b(1 -a).

Fig. 7 :

 7 Fig. 7: MAC rate region.

2 Fig. 8 :

 28 Fig. 8: Two scenarios that WZ and MAC rate regions do not intersect.

Fig. 10 :

 10 Fig. 10: Outage probability with different helper locations d hd = θ • d sd , θ ∈ {0.25, 0.5, 0.75, 1.0}.

Fig. 11 :

 11 Fig.11: Relationship between outage probability and distortion for θ ∈ {0.25, 0.5, 0.75, 1.0} with the average SNR of the SD link is 5 dB.

Fig. 12 :

 12 Fig. 12: Helper-selection WZ system.

Fig. 13 :

 13 Fig. 13: Outage probabilities of WZ with helper selection in both lossless and lossy two-phase orthogonal transmissions for P e,i = 0.01.

Fig. 14 :

 14 Fig. 14: Outage probability of helper-selection WZ systems in two-phase orthogonal and one-phase MAC transmissions for P e,i = 0.01 and D X = 0.1.

Fig. 15 :

 15 Fig. 15: Outage probability of helper-selection one-phase MAC WZ system in two extreme cases for P e,max = 0.2 and D X = 0.1.

This only describes the meaning of the inequality set in[START_REF] Zhou | Outage probability of correlated binary source transmission over fading multiple access channels[END_REF] and doesn't specify the decoding order in practice. As mentioned in[START_REF] Anwar | Accumulator-assisted distributed turbo codes for relay systems exploiting source-relay correlation[END_REF], joint iterative (turbo) decoding can be used to obtain U and V . Since the channel conditions vary block-by-block, fixing decoding order is not meaningful. Since in this paper we only focus on the theoretical calculation, rather than practical implementation, the details of decoders are not investigated in this paper.

This assumption is reasonable because the P e,i value determines the average SNR value where the diversity order changes from L + 1 to L. Obviously, the decrease in the diversity order changes the decay of the curve of outage probability versus the average SNR. However, since the more the helpers, the sharper the decay, the average SNR corresponding to the decay change can not be clearly identified. Under this assumption, the outage probabilities shown in Fig.13and Fig.14are upper bounds, according to Appendix A which investigates the effect of the correlation between the source and the helpers in helper-selection WZ systems in two extreme cases.

For example, a short packet is transmitted to the destination sequentially from all the helpers during the 'monitoring phase' in the protocol. The destination identifies which helper's received instantaneous SNR is the largest, and requests the best helper for the transmission. This 'monitoring phase' does not reduce the overall throughput significantly.

This work was in part conducted when he was with JAIST.