Two-Tier Cache-Aided Full-Duplex Hybrid Satellite-Terrestrial Communication Systems

Enabling global Internet access is challenging for cellular-based Internet of Things (IoT) due to the limited range of terrestrial network services. One viable solution is to deploy IoT over satellite systems for coverage extension. However, operating a hybrid satellite-terrestrial network (STN) might incur high satellite bandwidth consumption and excessive service latency. Aiming to reduce the content delivery latency from the Internet-connected gateway to the users, this work proposes a wireless two-tier cache-enabled model with full-duplex transmissions where content caches are deployed at the satellite and ground station. A closed-form solution for the successful delivery probability (SDP) of the files is derived considering the requested content distributions and channel statistics. Then, the SDP performance under common caching policies can be conveniently evaluated. The results are also used to optimize cache placement under caching capacity constraints. Numerical results demonstrate the performance improvements of the proposed system over those of single-tier cache-aided and half-duplex transmission systems.


I. INTRODUCTION
Although being around for many years, cellular-based technologies for the Internet of Things (IoT) have not fulfilled the demand of globally connecting everyone and everything. To offer service across all geographic regions, integrating satellites into IoT networks has been touted.
With better service reliability and coverage, satellite communications can provide more effective solutions for IoT than cellular networks. For instance, satellites can provide more than 99.9% availability with proper constellation arrangements [1] and can cover an enormous area of the Earth that is suitable for bringing connectivity to rural or remote areas. Reflecting the utilization of satellite systems as a promising solution complimenting terrestrial networks, many projects have been developed such as the new radio supporting integrated satellite-terrestrial networks by 3GPP [2], the Satellite and Terrestrial Network for 5G (SaT5G) [3], SpaceX's Starlink satellite project [4].
Operating a hybrid satellite-terrestrial IoT system, however, poses challenges in service delays and satellite bandwidth costs. A content delivered to end-users from the Internet-connected gateway is relayed through satellite(s) and ground station(s), which extends the serving time in addition to very pricey and often limited satellite bandwidth. Excessive content delivery delay can hinder potential use of satellite IoT systems to support delay-sensitive applications. To address these challenges, caching has been proposed as a mechanism to relocate content storage to edge devices that are closer to users [5]- [12]. In general wireless networks, edge caching has been shown to shorten delivery latency and reduce network congestion [13]. In addition, caching contents in satellites or terrestrial stations helps avoid re-transmissions of the same contents and significantly reducing in-network traffic [14].
In this paper, we study a two-tier cache-aided satellite IoT model considering end-to-end communications (i.e., from Internet-connected gateway via satellite and ground station to endusers). Our work focuses on the content delivery time analysis in terms of the successful delivery probability (SDP) considering full-duplex (FD) transmissions at the satellite and ground station, albeit with imperfect self-interference cancellation. Deploying FD communications will potentially shorten the service delivery time compared to the half-duplex (HD) mode providing sufficiently effective self-interference cancellation, and hence increasing the SDP. The main contributions of this paper are summarized as follows.
1) A closed-form SDP expression for the two-tier cache-enabled satellite IoT systems is derived under the FD mode taking into account the requested content characteristics, satellite/terrestrial realistic channel statistics, and caching configurations. Consequently, three common caching policies, namely uniform caching, ground station most popular caching, and satellite most popular caching, are considered; 2) Using the derived results, the SDP performance under different transmission modes (FD with and without FastForward relaying or HD mode) and caching configurations is thoroughly investigated. The network capacity in terms of maximum supportable users given a minimum SDP requirement is also studied; 3) A SDP maximization-based cache placement is designed subject to the caching capacity constraints of the two cache tiers. While the optimal solution is hard to compute, a lowcomplexity algorithm is developed achieving superior performance as compared with that of common caching configurations. 4) Various simulation scenarios are conducted to illustrate the superior performance of the proposed system over its single-tier cache-aided counterparts as well as HD transmission systems. Results show that the greater self interference cancellation parameter is, the worse performance FD transmission achieves, and the critical point where the FD mode performs worse than the HD mode is also determined. In addition, numerical results show that the proposed system outperforms the single-tier cache systems in [8]- [10] with at least 25% higher average SDP.
Other parameters such as the file popularity, file size, the number of supportable ground users and caching capacity of the two layers that affect the system performance are also investigated.
The remainder of this paper is organized as follows. Section II introduces the related work on edge caching in satellite-terrestrial networks. Section III describes the system model and the transmission scheme for full-duplex communications. Section IV analyzes the successful delivery probability of the general caching and special caching policies. Full-duplex without FastForward relaying protocol, half-duplex transmission scheme and single-tier cache system are employed as baselines for the proposed two-tier cache full-duplex system. Cache placement design and numerical results are presented in Sections V and VI, respectively. Finally, conclusions are made in Section VII.

II. RELATED WORK
In this section, the state-of-the-art edge caching techniques in satellite-terrestrial networks (STNs) are reviewed.
Two types of satellites that are commonly used in STNs are the low earth orbit (LEO) and geosynchronous equatorial orbit (GEO) satellites. Employing LEO satellites in STNs can achieve lower transmission latency than using GEO satellites. However, using LEO satellites poses a very challenging problem on dynamic network topology since they are only in contact with stations on the ground for a brief period due to their traveling speeds.
Taking into account the relative motion of LEO satellites and terrestrial users, [5] proposes a cross-time-slot graph for the time-varying topology file distribution in combination with the back-tracing partition directed on-path caching distribution mechanism (BPDM) to reduce the caching overhead. The proposed BPDM focuses on cache location decision using a modified Dijkstra routing algorithm and cache content placement using the popularity-aware multiple regions cooperative cache strategy. Proposing a time-evolving covering set in-network caching mechanism, [6] exploits the local connectivity of different regions in choosing the intermediate nodes with better connectivity that relay the objective file. The proposed mechanism includes a novel event updating graph that captures the time-varying topology of the network, a minimum time-evolving covering set algorithm for cache location decision, and a query-based minimum delay distribution path algorithm for optimal path finding during file access procedure. Leveraging the broadcast characteristic of satellite to efficiently provide traffic offloading for terrestrial networks, [7] proposes an integrated LEO satellite/terrestrial cooperative caching scheme, in which the terrestrial access point is able to share its cached content with the satellite.
Employing GEO satellites in STNs is suited to latency-non-sensitive services due to the satellite high altitude. STNs with GEO satellites have the advantage of the static network topology and non-interrupting transmission between the satellites and devices on the ground. Incorporating wireless caching into STNs, [8] proposes an amplify-and-forward relaying protocols, where cache is enabled at relays on the ground. The proposal in [8] considers the most popular and uniform content-based cache placement schemes, which shows a substantial improvement in outage probability over the traditional approach without caching. To offload the backhaul of terrestrial networks, [9] proposes using hybrid STN in combination with an off-line edge caching algorithm. Caches are placed at base stations, in which each cache is divided into two parts.
The first part is filled with the most local popular files whilst the other part is filled with the most global popular files. On the performance investigation of 5G edge caching over satellite, [10] uses off-line caching approaches to offload terrestrial backhaul networks. The proposal in [10] focuses on the multimode satellite backhaul in two use cases, i.e., dense urban areas and sparsely populated regions. The effectiveness of the system is studied by virtue of the cache hit ratio and cost per bit for satellite transmission.
It should be emphasized that the system models in [5]- [10] are single-tier cache-enabled systems where caching is used for terrestrial devices only. For STNs employing GEO satellites, [8]- [10] consider dual-hop downlink satellite-relay(s)-users transmission. The satellite uplink communications from the Internet-connected gateway to the satellite is not considered, which is usually the bottleneck in satellite IoT systems due to large bandwidth consumption.
Enabling cache capacity at satellite, [11], [12] propose two-tier caching models, where the firstand second-tier caches are placed in the terrestrial base stations and the satellite, respectively. The two-tier cache system will potentially: (i) reduce service latency since more contents are stored closer to the end-users; (ii) efficiently utilize satellite bandwidth as the contents are prefetched at the satellite during off-peak hours; Furthermore, when same contents are requested by multiple users, having the contents cached at satellite and broadcasting them at once can save satellite bandwidth. The work in [11] focuses on analyzing the satellite bandwidth consumption with an off-line caching scheme where popular contents are cached at each ground station for local services, while the satellite's cache is used for the most popular contents within its coverage (containing multiple ground stations) to take advantage of the satellite's broadcast nature to the ground stations. Non-cached contents can be retrieved from the gateway if needed. On the other hand, our current work focus on investigating the file delivery time through the successful delivery probability (SDP) metric considering the realistic channel propagation models. SDP analysis is typically relevant in delay-sensitive applications [15]. In addition, while [11] considers a scenario where a file is fully cached, our work considers more general caching policies allowing partial caching of files. Hence, caching capability can be exploited better for improved performance.
Employing LEO satellite into a two-tier caching model, [12] designs a terrestrial link between the core network and base stations to maintain services when quality satellite connection is absent.
The design, however, does not consider the handover problem of the link between a satellite and base stations, and the dynamic of the satellite link.

III. SYSTEM MODEL
The satellite-assisted IoT system is composed of a satellite S, an Internet-connected satellite gateway G, a ground station G s , and a set of K ground end-users U i , i = 1, . . . , K as depicted in Fig. 1. In this system, S is a GEO satellite and G s is a low power base station equipped with a satellite receiver. Both S and G s are cache enabled. Assuming there is no direct link from user U i to satellite S and from the ground station to the gateway due to weather factors, long distance, and heavy shadowing.

A. Two-tier Caching Model
The caching model consists of two tiers, namely the first tier at the ground station with storage capacity of C 1 (bits) and the second tier at the satellite with storage capacity C 2 (bits).
The gateway is connected to the internet and hosts N files W 1 , . . . , W N , which are assumed to be of equal size of F (bits). This assumption is for the scenario of heterogeneous IoT applications.
For other IoT applications, the analysis can be easily extended to the case of unequal file size.
Without loss of generality, assume C i /F = N i , i = 1, 2 being integer numbers with N 1 +N 2 < N as it is not possible to cache all the file contents. The generic caching policy considered in this paper is represented by are the portions of W n fetched at the first-tier cache and the second-tier cache, respectively. Assuming that the cached portions of the same file do not have overlapping contents. Other conditions of the caching policy are It is always optimal to utilize all caching capacities.

B. File Popularity Model
The request probability q n for file W n follows the Zipf distribution, i.e., where 0 < α < 1 denotes the Zipf skewness factor [16]. A large value of α means the requests on high-popularity files, whereas a small value of α is related to the requests with heavy-tailed popularity. Without loss of generality, it is assumed that files W 1 , . . . , W N have decreasing popularity.

C. Channel Model
We consider block-based communications, where a transmission section is accomplished within a coherence time T (seconds). The satellite-terrestrial downlink and gateway-satellite uplink channels are similarly modeled; Hence, only the satellite downlink channel model is described.
The satellite-terrestrial links have multipath fading and shadow fading. The shadow fading is composed of the line-of-sight (LOS) shadow fading and multiplicative shadow fading; The multipath fading composes of one LOS and many weak scatter components [17]. Therefore, the channel coefficienth S (t) for the satellite-terrestrial link can be modeled as where j 2 = −1, A(t) is the independent stationary random process representing the amplitude of the scatter components and following the Rayleigh distribution; Z(t) represents the LOS component and follows the Nakagami distribution; ϑ(t) is the stationary random phase process with a uniform distribution over [0, 2π); ζ 0 is the deterministic phase of the LOS component.
The normalized channel power gain h S (t) = |h S (t)| 2 /σ 2 , where σ 2 is the additive white Gaussian noise power, has the probability distribution function (PDF) f h S (x) given by [17], [18] with coefficients represents the average power of the scatter components ; represents the average power of the LOS component; m 1 = Ω 2 1 /Var[|Z(t)| 2 ] is the Nakagami parameter; 1 F 1 (., ., .) is the confluent hypergeometric function of the first kind [19, eq.(9.210.1)]: Note that we omit the time-dependent index due to the stationarity assumption.
can be derived using elementary functions and finite summations Using the following result on integration with exponential function: we can derive (4).
To validate (4) As can be observed from Fig. 2, as x becomes large, F h S (x) approaches one.
The terrestrial link between the ground station and user U i is modeled as a Rayleigh fading channel with the (normalized) channel power gain h i (t) governed by the following distribution withh i is the average channel power gain taking into the effects of path-loss, small-scale fading, etc.

D. Full-duplex Transmission
When caching IoT data, it is essential to maintain data freshness [20]. This model assumes that the users will be served in a time-division multiple access (TDMA) manner which allows each user to be consecutively active in T /K (seconds). The time-slot T /K will be large enough to ensure data freshness in practical IoT applications. Under TDMA, inter-user interference does not exist. Both the satellite and ground station operate in the FD mode. When the channel state information is known at the transmitter, rate adaption is employed and the achievable rates (bps) on the links are where B G , B S and B Gs are the bandwidths (Hz) of the satellite uplink, downlink, and the terrestrial downlink respectively; P S , P G , and P Gs denote the transmit powers of the satellite, gateway, and ground station respectively; βP S and βP Gs represent the normalized residual selfinterference (SI) powers at the satellite and ground station respectively, which are assumed to be proportional to the transmit powers with coefficient β ≥ 0 being the SI cancellation quality parameter [21]. For notational simplicity, denote: P S P S /(βP Gs + 1),P G P G /(βP S + 1).
When a user requests file W n , the gateway needs to transfer the non-cached (1 − µ n 1 − µ n 2 ) portion to the satellite, which also needs to send accumulated (1 − µ n 1 ) portion to the ground station. By employing the FastForward protocol [22] with the FD mode at the satellite, Similarly, by deploying the FastForward protocol with the FD mode at the ground station, the (end-to-end) delivery time for transferring the entire file to the user is determined by the time for sending the (1 − µ n 1 ) portion over the satellite downlink to the ground station and the time for sending the whole file over the terrestrial link to the user, i.e., We will further consider FD transmissions without the FastForward protocol in the ensuing section.

A. SDP Analysis
The successful delivery probability (SDP) of serving the requested file W n is the probability that a user receives the file within the user's active time-slot. The following analysis is for file W n and user U i , and is true for all users. The SDP of file W n is ψ n,i (µ n 1 , µ n 2 ) = Pr (t n,i ≤ T /K) .
The probability operator is taken with respect to the channel power gain variables. A closed-form expression for ψ n,i is given in (9).
Proof. Substituting (7) into (8) gives rise to where we have and τ F/(T /K) is the average file serving rate.
We have the following special cases: • If W n is cached at ground station, we have ψ n,i (1, 0) = exp (2 τn/B Gs − 1)/(P Gshi ) ; • If W n is cached at the satellite, the SDP is ψ n,i (0, 1); • If W n is fully retrieved from the gateway, the SDP is ψ n,i (0, 0).
The SDP of file W n under the effect of µ n 1 and µ n 2 is illustrated in Fig. 3. It shows that the SDP can be improved by deploying the second-tier caching at the satellite, and the larger the caching capacity is, the more SDP improvement is. The SDP is the largest or smallest when the file is cached at the ground station or fully retrieved from the gateway, respectively. The average SDP of the system is defined as a weighted sum of the users' SDPs: with weighting coefficients ω i ∈ (0, 1) and i ω i = 1. In the following, without loss of generality, we assume homogeneous users with ω i = 1/K, and average channel gainsh i =h k , ∀i, k. Thus, ψ = ψ i .

B. SDP Under Different Caching Policies
In this section, we analyze the average SDP under three different caching policies, i.e., uniform caching, ground station most popular caching, and satellite most popular caching. This caching policy tends to be better if only a number of files are most popular, i.e., the skewness factor α is large. In this case, by caching these files at the ground station, we can increase the SDP.

3) Satellite Most Popular Caching:
The satellite caches N 2 most popular files, and the ground station caches N 1 most popular files of the remaining files. The average SDP ψ sat is This caching policy is generally more suitable for a two-tier cache-enabled system with multiple ground stations. When most popular files are requested by users associated with different ground stations, the files, when cached at the satellite, are broadcast from the satellite to the ground station only once. In this case, each ground station does not need to separately cache these popular files. For the system under consideration, we have the following result.
It is true that ψ grnd > ψ sat .
Proof. We have q n (ψ n,i (1, 0) − ψ n,i (0, 1)) > 0 C. Baselines 1) Single-tier Caching: Our derived results can be used to evaluate the SDP of single-tier caching policies. In the single-tier scenario, only cache C 1 is enabled at the ground station. There are two cases: • When the same portion of N files are cached at the ground station, we have: • When N 1 most popular files are cached at the ground station, we have: q n ψ n,i (0, 0).

2) Full-duplex Without FastForward Relaying:
The delivery time of file W n is completed in three slots. In the first slot, the ground station transmits µ n 1 F to the user while receiving µ n 2 F from the satellite; the satellite transmits µ n 2 F to the ground station while receiving (1 − µ n 1 − µ n 2 )F from the gateway. In the second slot, the ground station transmits µ n 2 F to the user while receiving (1 − µ n 1 − µ n 2 )F from the satellite. Finally, in the third slot, the ground station transmits (1 − µ n 1 − µ n 2 )F to the user. The delivery time is given by: The SDP of file W n isψ n,i = Pr t n,i ≤ T /K . We will then evaluate the system SDPψ = n q nψn,i using Monté Carlo simulation in Section VI.

3) Half-duplex Transmission:
In this transmission scheme, both the satellite and ground station adopt the HD mode. The achievable rates on the links are The timeline for delivering file W n is divided into three slots as shown in Fig. 4. The delivery time for file W n is The SDP of file W n for the HD mode case is ψ HD n,i = Pr t HD n,i ≤ T /K . While ψ HD = n q n ψ HD n,i is too complex to be found in closed form, it is evaluated using Monté Carlo simulation in Section VI.

V. CACHE PLACEMENT DESIGN
Cached contents are placed into caches C 1 and C 2 via satellite uplink and downlink transmissions during off-peak hours. In this section, the cached portions µ n 1 and µ n 2 are optimized with the objective of maximizing the average SDP. Under the generic caching policy, the optimization problem for the prefetched content in the two cache tiers subject to caching capacity constraints is formulated as follows: The problem is a non-linear non-convex optimization one, and hence it is very challenging to find the optimal solution. We resort to sub-optimal caching placement solutions by confining the original 2N -dimensional search space into two N -dimensional search spaces. Specifically, we consider that the cached files are prefetched with the same portions, i.e., µ n i =μ i , i = 1, 2 if file W n is cached at the respective tier, and µ n i = 0 otherwise. We consider the following two policies.
It is required thatN 1 +N 2 ≤ N . Similar to the generalized uniform caching policy, this policy only choosesN 1 andN 2 most popular files with higher portion compared to that of the ground station most popular caching policy (Section IV-B2) to cache at the ground station and satellite, respectively. This caching policy degrades to the ground station most popular caching policy Then the resulting problem can be formulated as We can conduct an exhaustive search to find solutions to (12) with a computational complexity of O(n 2 ). First, we search forN 1 andN 2 constraining the policy being one of the generalized caching policies. Then, we determine the caching solution with higher SDP.

Remark 1.
The caching solution to problem (12) always achieves a higher SDP than the ground station most popular and uniform caching strategies.
With the focus on caching the likely requested files, the generalized caching policies allow more popular contents to be cached closer to users. Therefore, the serving time under generalized caching policies will be shorter than that of the generic uniform and ground station most popular caching policies.

VI. NUMERICAL RESULTS
The values for key parameters [23] used in this section are presented in Table I. Because it requires more than 100 dB of self-interference cancellation for the full-duplex system to achieve the same signal-to-noise-ratio-plus-interference as that of the half-duplex system [24], we choose the self-interference cancellation quality parameter β = 0.0001 for most  We first investigate the average SDP when β varies. Consider C 1 = 50 Gb and C 2 = 10 Gb, and K = 20 users under uniform caching policy.
With the HD scheme as a reference, the effect of β on the FD scheme SDP is displayed in Fig. 5. As β increases (i.e., worse self-interference cancellation in the FD system), the SDP decreases and starts getting worse than that of HD transmission. The FD scheme achieves the same SDP as that of HD with β = 0.02. The self-interference cancellation in the FD system in this case is 97 dB to 100 dB which is almost the recommended value in [24]. The SDP of FD is 20% worse than that of HD with β = 0.1 where self-interference cancellation is 85 dB to 90 dB. We also present numerical result for hybrid FD/HD transmission. In this hybrid scheme, the channel realizations are generated, then the delivery time of each file is the minimum delivery time between the FD and HD modes. In practical applications, once knowing the channel condition, devices in the system can dynamically switch between the FD and HD modes for better performance.

B. Two-tier versus Single-tier Caching
The single-tier cache-enabled system in this section is adopted from STN system models in [8]- [10], where content caches are placed at ground stations only. The caching policies for the single-tier cache system correspond to those at the ground station in the two-tier cache system.
To compare the performance between two-tier and single-tier caching, we set up both systems with the ground station cache capacity of 80 Gb serving 20 users. The uniform and ground station most popular caching policies are applied to study the average SDP.
Under the uniform caching policy, N 1 /N varies and N 2 /N is set to be 0.2 in the two-tier caching (meaning the satellite cache capacity is 20 Gb). The result is shown in Fig. 6 with the two-tier scheme outperforming its single-tier counterpart. At best performance with N 1 /N = 0.7, the achieved SDP in the two-tier scheme is 25% higher than that of the single-tier scheme. Under the ground station most popular caching policy, the number of most popular file cached N 1 varies; N 2 is fixed at 200 files in the two-tier scheme. The two-tier scheme gives approximately 40% SDP higher than that of the single-tier scheme in best performance scenarios with the ground station caching 600 most popular files.

C. FD: FastForward versus Non-FastForward Relaying Protocols
To compare the performance of the FD protocols, the system is set up with varying values for

D. Network Capacity
To examine the serving capability, the system is set up with the two layers' caches C 1 of 50 Gb and C 2 of 20 Gb. The number of users K varies while other parameters are listed in Table I. Fig. 8 shows the number of supportable users by the two-tier system under two caching policies.
A smaller average SDP under uniform caching is achieved when supporting a small number of users; however, it surpasses the SDP of the ground station most popular caching scheme with a larger number of users. The acceptable SDP value depends on practical applications. In this case, with SDP of 50%, the results follow a common understanding that the uniform caching scheme is not as effective as the ground station most popular caching scheme.

E. Caching Capacity
In this section, we analyze the two-tier caching capacity when serving 20 users. The system is configured using the parameters specified in Table I. The cache capacities of the two layers are shown in Fig. 9. As expected, the ground station most popular caching policy outperforms uniform caching. The more cache capacity is allocated to the ground station, the higher the SDPs are attainable; and the more content is cached at the satellite, the more the SDPs improve. Under uniform caching, the highest SDP is achieved at 70%, while it is 80% under ground station most popular caching.

F. Effect of File Popularity
In this subsection, to examine the effect of the file popularity on the SDP under the ground station and satellite most popular caching policies, the system is configured with two tier caches C 1 and C 2 of 50 Gb and 20 Gb, respectively, and serves 20 ground users. Through varying Zipf skewness factor α, the resulting SDP is plotted in Fig. 10. Higher SDPs are achieved for both most popular caching schemes with an increase in the Zipf skewness factor. The results also match Proposition 1. This is because having one ground station in the system, the ground station most popular caching policy always gives better SDP results than under satellite most popular caching.

G. Effect of the File Size
To investigate the effect of the file size on SDP, we vary the sizes of N files at the gateway following a uniform distribution with a mean of 100 Mb and a standard deviation of 50 Mb .
Two caching schemes are set up as follows: (i) C 1 is prefetched with the largest files and then C 2 is filled with the largest files among the remained files; (ii) C 2 is prefetched with the largest files and then C 1 is filled with the largest files among the remained files. The results are shown in Fig. 11. The average SDP is better when the largest files are cached at the ground station, resulting in a gain of approximately 10% to 20% in SDP.

H. Generalized Caching Solutions
To investigate the performances of the generalized caching policies, we find the solutions to the problem in (12) when the caching capacity at satellite is fixed at 20 Gb and at ground station varies up to 80 Gb. The performances are illustrated in Fig. 12. As can be seen from the figure, generalized caching policies yield higher SDPs than the naive uniform and ground station most popular caching policies. The solutions for each case of C 1 is detailed in Table II.

VII. CONCLUSION
In this paper, we studied the SDP of a two-tier cache enabled satellite IoT system with full-duplex communications. A thorough analysis for the SDP under the FD mode with the FastForward relaying protocol was carried out under different caching policies. Cache placement was also designed with the goal of maximizing the average SDP. Using the FD mode without the FastForward protocol, HD mode and single-tier cache system as benchmarks, numerical results were presented when investigating the system performance in terms of SDP, maximum number of supportable users, caching capacity in each tier, and the effect of file popularity on the system performance under different caching policies. For a better understanding on parameters that affect the proposed system performance, a special case with a variety of file sizes was also investigated.