Haidong Zhao

Zakaria Benomar

Tobias Pfandzelter

Nikolaos Georgantas

Supporting Multi-Cloud in Serverless Computing

Keywords: serverless, multi-cloud, performance and cost, vendor lock-in

Serverless computing is a widely adopted cloud execution model composed of Function-as-a-Service (FaaS) and Backend-as-a-Service (BaaS) offerings. The increased level of abstraction makes vendor lock-in inherent to serverless computing, raising more concerns than previous cloud paradigms. Multicloud serverless is a promising emerging approach against vendor lock-in, yet multiple challenges must be overcome to tap its potential. First, we need to be aware of both the performance and cost of each FaaS provider. Second, a multi-cloud architecture needs to be proposed before deploying a multi-cloud workflow. Domain-specific serverless offerings must then be integrated into the multi-cloud architecture to improve performance and/or save costs. Finally, we require workload portability support for serverless multi-cloud.

In this paper, we present a multi-cloud library for crossserverless offerings. We develop an analysis system to support comparison among public FaaS providers in terms of performance and cost. Moreover, we present how to alleviate data gravity with domain-specific serverless offerings. Finally, we deploy workloads on these architectures to evaluate several public FaaS offerings.

I. INTRODUCTION

Serverless computing is a promising cloud execution model that supports event-driven resource provisioning while eliminating the need for resource management and allocation. This paradigm enables a fine-grained pay-as-you-go billing model [START_REF] Jonas | Cloud programming simplified: A berkeley view on serverless computing[END_REF]. Given its widespread adoption, cloud vendors improve and differentiate their serverless offerings, aiming at locking in their clients, as has already been the case with previous cloud computing paradigms [START_REF] Petcu | Portability and interoperability between clouds: challenges and case study[END_REF].

Vendor lock-in is exacerbated in serverless computing as public providers only allow customers to use Function-as-a-Service (FaaS) and Backend-as-a-Service (BaaS) offerings [START_REF] Jonas | Cloud programming simplified: A berkeley view on serverless computing[END_REF]. More importantly, a public BaaS offering can only natively trigger a FaaS offering from the same provider. Customers are locked into a single provider due to this tight coupling.

Migrating even a simple workload may encounter tricky issues or possibly a dead-end [START_REF] Yussupov | Facing the unplanned migration of serverless applications: A study on portability problems, solutions, and dead ends[END_REF]- [START_REF] Taibi | Serverless computing-where are we now, and where are we heading?[END_REF]. One solution to mitigate the effect of vendor lock-in is to deploy multi-cloud [START_REF] Castro | Hybrid serverless computing: Opportunities and challenges[END_REF]. Multicloud is not only a technical problem but also an economic and social one. Several factors may propel us to adopt and deploy it. First, it can increase availability. Second, we cannot make sure that the currently used FaaS provider is suitable for every workload, as FaaS providers typically have different computeto-memory ratios, resource management policies [START_REF] Wang | Peeking behind the curtains of serverless platforms[END_REF], and pricing schemes. Third, FaaS providers may offer heavy discounts to a company that holds a contract with them. Fourth, FaaS is still in its infancy and lacks geo-distributed deployment. To meet regulatory constraints and improve performance, a company may have to adopt a new vendor that enables local FaaS deployment. As a result, we employ an analysis system that takes into account both performance and cost factors.

A real workload in serverless computing is typically supported by both FaaS and BaaS. To extend a workload across clouds, a well-designed multi-cloud architecture to manage these serverless offerings should be used. Besides, domainspecific offerings from small providers may need to be integrated into multi-cloud architectures to make up for what dominant public providers cannot support, helping solve some long-lasting issues such as data gravity [START_REF] Faction | What is Data Gravity? How it Can Influence Your Cloud Strategy[END_REF]. As more and more data accumulates, it becomes increasingly difficult for data portability since cloud providers charge prohibitively expensive fees for data egress. Data gravity also worsens the vendor lock-in in serverless computing. Consider that using a hardware-accelerated FaaS to retrieve data from another provider's storage service can incur a expensive data egress fee. On the other hand, it is typically free to move data from a storage service to the same provider's FaaS. Thus, this locks customers to a single provider and may limit serverless applications to simple workloads.

The Serverless Framework [START_REF]Serverless Framework[END_REF] can help in deploying serverless multi-cloud applications. An early effort [START_REF]Multicloud solutions with the Serverless Framework[END_REF] with it struggled to orchestrate a well-designed multi-cloud architecture, including synchronizing data from multiple storage providers and paying a data egress fee at least twice due to its limited application templates. Hence, a multi-cloud library would be more practical for designing a multi-cloud architecture with flexible vendor choices and facilitating workload portability.

In summary, this paper 1 makes the following contributions: 1) We discuss the motivation of multi-cloud deployments as well as the necessity of redesigning a multi-cloud library for serverless computing (Section II). 2) We propose a serverless multi-cloud library that supports workload deployment and portability among clouds with ease (Section III-B). 3) We implement the End Analysis System (EAS) to support the performance evaluation and fine-grained cost estimation of public FaaS providers (Section III-C). 4) We discuss the design of multi-cloud architectures and present proof-of-concept architectures that include domain-specific BaaS offerings. In addition, we report the benchmarking results of public FaaS providers in these architectures (Section IV).

II. BACKGROUND AND MOTIVATION

A. Multi-Cloud Classification

We investigated multi-cloud deployment in both industry and academia [START_REF] Chasins | The sky above the clouds[END_REF], [START_REF] Iorio | Multicloud trends[END_REF], and classify the multi-cloud in terms of workload distribution:

Each workload only runs on a single cloud. One classical case is hybrid clouds, which migrate some workloads to public clouds. According to a survey [START_REF] Adler | Cloud Computing Trends: Flexera 2022 State of the Cloud Report[END_REF], the hybrid cloud approach dominates the multi-cloud market. Another representation of this classification is running segmented solutions in different clouds. For example, using Google Docs for collaborative writing and Outlook's email service.

A workload runs across multiple clouds. This refers to the workload migration enabled. This deployment requires us to choose shared services among providers. Deploying a workload across multiple clouds can improve availability and help in finding the best FaaS provider for each workload.

A workload is divided among multiple clouds. A pipeline task or a workload can be separated into several stages. At each stage, we can choose an optimal cloud offering instead of all stages in a single cloud. Researchers from Berkeley proposed a transparent cloud targeting this objective [START_REF] Chasins | The sky above the clouds[END_REF], [START_REF] Stoica | From cloud computing to sky computing[END_REF]. A FaaS offering that collaborates with another provider's BaaS offering is also regarded as this multi-cloud category.

B. Benefits of Multi-Cloud Deployment

Mitigate the risks of data gravity and being locked into a FaaS provider. Ingress data is typically free of charge by cloud providers. However, when data flees from cloud providers to the Internet, providers usually charge a data egress fee, as shown in Table I. Consider if we want to retrieve 10 TB of data from a storage site in London and the storage service is provided by Alibaba, we need to pay $143.36 to save a month and $716.8 to move out, 5 times the price difference. With Amazon S3 [START_REF] Amazon | [END_REF], $239.76 for saving a month and $891 for moving out. Even if moving out data with high bandwidth just takes minute-level time, it is much more costly than saving the same amount of data for a month. This is data gravity. Customers tend to store their data with one provider for performance considerations. Therefore, with data growing, it is more unlikely to move data out with the prohibitively expensive data egress fee. Moreover, FaaS is a data-shipping architecture [START_REF] Hellerstein | Serverless computing: One step forward, two steps back[END_REF]. Data is unidirectionally moved to the FaaS. Outbound data from an object storage site to the same provider's FaaS is normally free, but to the Internet is costly, especially for large-scale data. This worsens the lockin issue and precludes customers from using other domainspecific FaaS, such as FaaS with hardware acceleration. In light of this, we may need to employ some low or free egress fee storage services to the multi-cloud architecture.

Increase availability, but what about performance enhancement and cost savings. Compared with traditional compute instances, deploying multi-cloud with FaaS offerings to increase availability does not suffer from being double charged because of its pay-as-you-go billing model [?], [START_REF] Altwaijiry | Cloud computing present limitations and future trends[END_REF]. Public FaaS providers have different memory-to-compute, resource management policies [START_REF] Wang | Peeking behind the curtains of serverless platforms[END_REF], pricing schemes, etc. Therefore, one serverless provider may perform well for one workload but may be clumsy in another. Thus, it is necessary to evaluate which FaaS provider is most suitable for each workload. Moreover, public FaaS providers usually have deployments in metropolises but with sporadic deployments in other regions, especially developing countries. To adopt serverless computing, a company in this greenfield may use a new provider's FaaS for its local deployment. A FaaS cooperating with another provider's BaaS offering is also a kind of multicloud, as described in Section II-A.

Mitigate the risk of an application being locked to a serverless provider. Due to the lack of a standard and proprietary serverless offerings among providers, we may find it challenging or impossible to migrate an application [START_REF] Yussupov | Facing the unplanned migration of serverless applications: A study on portability problems, solutions, and dead ends[END_REF], [START_REF] Hartauer | Cloud function lifecycle considerations for portability in function as a service[END_REF]. Multi-cloud deployment suggests choosing common services in order to preclude being locked into a vendor. However, a company is also able to reserve some proprietary services after risk assessment.

C. Redesign Multi-Cloud Library

Traditional multi-cloud libraries help in accessing general services with a common interface. In serverless, a multi-cloud library should only focus on FaaS and BaaS offerings. Besides, to tap the full potential of multi-cloud, the library should meet the following requirements:

Versatile and Scalable. In addition to supporting BaaS offerings from well-known public clouds, the serverless multicloud library should take into account some nimble providers who may offer domain-specific services.

Conversion-enabled and one-for-all. One is able to use the multi-cloud library to access any BaaS provider's services with ease. Besides, we also account for a situation where a company and its engineers are integrated and trained by a cloud provider, respectively. If this company plans to port some workloads to other providers, the multi-cloud library should also provide the conversion from one provider's software development kits (SDKs) to that of another one. This allows engineers to access different providers' serverless offerings with their accustomed SDKs.

Lightweight. Using too many third-party libraries that the function instance runtime does not provide initially may run out of memory and worsen the "cold start" issue since an instance should spend more time preparing for the runtime (i.e., downloading code and third-party libraries from persistent storage).

III. DESIGN AND IMPLEMENTATION

Fig. 1 depicts the overview of the system design, which is composed of three parts: the workload deployed in the multicloud architecture; the multi-cloud library, which enables the utilization of FaaS and BaaS offerings from different vendors with ease; and the End Analysis System, which evaluates the performance and cost of the FaaS providers.

M u l t i -c l o u d L i b r a r y BaaS FaaS BaaS FaaS M u l t i -c l o u d L i b r a r y

BaaS FaaS

Query from clouds Fig. 1: System design overview

A. System Design Overview

As described in Section II-B, a workload may be divided into several stages. Each workload/task stage may work on the most optimized cloud offering. Consider that if a workload stage is to store infrequently used data, it may opt for a storage service with the lowest fee for long-term data storage. On the other hand, if this workload stage is for computing, hardware acceleration FaaS may be employed on it. As a result, a workload may be served by more than one cloud provider.

we design and implement the End Analysis System (EAS) and a serverless multi-cloud library, respectively. To achieve a more accurate workload performance measurement, we may need to log the request that will trigger a serverless offering before sending it. EAS can also be used to evaluate the FaaS providers' performance and cost in a real-world deployment workload, albeit some performance metrics will not be supported. The Multi-cloud library enables us to use common interfaces or familiar SDKs to manage serverless offerings from different providers.

B. The Design of the Multi-Cloud Library

One obstacle to enabling workload portability across multiple clouds exists at the API level, where developers may need to learn new SDKs or APIs of new FaaS and/or BaaS offerings from several cloud providers. Thus, we suppose that the multicloud library could be the best approach to help developers get familiar with new services.

First, we designed a common interface, acting as an abstraction layer on top of the proprietary APIs or SDKs of cloud providers. In doing so, we make multi-cloud workload deployment easier by allowing developers to use a common interface to access the cloud services that the current multi-cloud library supports. However, considering that if we are currently using a cloud service and want to extend our workloads to multi-cloud deployment, at this moment, we would hope not to learn too many new things, i.e., new providers' SDKs. In view of this, we consider that it is necessary to implement SDK-conversion. To allow this, we further developed provider-like SDKs on top of the proprietary SDKs and APIs for direct conversion, e.g., a mocked AWS S3-like library module that can access other providers' object storage services while maintaining the semantics and syntax of the proprietary SDKs of AWS S3. By doing this, we can access the services of other providers using our familiar SDKs.

C. End Analysis System (EAS)

We developed the EAS to benchmark which FaaS provider performs best for each workload in terms of performance and cost. The EAS is composed of the following components:

Authentication. To gain access to FaaS and BaaS offerings of different providers, the keys or passwords are saved in configuration files. As a request needs to be authenticated in order to access public cloud services, the configuration files are integrated with the multi-cloud library.

Adapter. One objective of multi-cloud is to select the best FaaS provider for each workload. FaaS triggered requests are diverse, as a function instance can be triggered via direct HTTP invocation or indirect operations in BaaS offerings. The adapter helps in managing these trigger requests and switching between benchmarked scenarios with ease. Cloud's Logging Query. Serverless providers allow developers to debug their programs by reading logs that contain metrics and state information about serverless functions. Cloud logging is typically asynchronous. We query cloud logs from FaaS providers and save them to EAS for further analysis.

Local Logging. Local logging is important. First, the cloud logs only tell us when a function instance starts and finishes. Second, we may need to evaluate the response time of the request. Moreover, we may need to log the status of running function instances.

Cost Model. Table I summarizes the pricing schemes of some public serverless providers we use for evaluation in this paper. The duration fee, request fee, and network traffic fee are the three main components of the cost of FaaS offerings. This cost table of FaaS providers is summarized at the time of writing. How to update it as time goes by needs to be discussed. We propose to publish this data on the shared database and maintain the updates on price changes with the help of the community and beneficiaries.

Analysis. We perform analysis based on the cloud and local logs on two dimensions: performance and cost. For the performance, we focus on the billing duration and latency (request serving latency, response time, etc.). For the cost, we support fine-grained estimation, including the duration fee, the invocation fee, the data egress fee, and perhaps the disk fee.

IV. EVALUATION

A. Proof-of-Concept Multi-Cloud Architectures

As discussed in Section II-B, we may also benefit the domain-specific services from some small providers. We propose two proof-of-concept multi-cloud architectures, in which we employ Cloudflare R2, a zero data egress fee storage service.

1) Pipeline-Enabled Multi-Cloud Architecture: Fig. 2 depicts a two-stage workload architecture. This architecture enables pipeline processing, as we let a FaaS offering be triggered by a BaaS offering from the same provider.

Optimization 1: Avoid data distribution by saving data at one storage site. The Serverless Framework with limited application templates does not support flexible vendor-choice multi-cloud architectures and may lead to at least double the data egress fee as the data needs to be synchronized across storage providers [START_REF]Multicloud solutions with the Serverless Framework[END_REF]. What we need to do is to put processed data from FaaS providers into one storage service.

Optimization 2: Leverage the zero egress fee storage service. In Section II-B, we have discussed the prohibitively expensive data egress fee and how the data gravity issue has limited serverless applications. In light of this, we employ Cloudflare R2 to store data in order to save money on network traffic fees and avoid data gravity.

2) Pipeline-Disabled Multi-Cloud Architecture: Fig. 3 depicts a multi-stage workload architecture. In this architecture, we further break the boundary between FaaS and BaaS offerings and do not require that they come from the same provider. Therefore, this architecture does not support pipeline processing anymore, as a public provider does not allow its BaaS offering to trigger a FaaS offering from another provider. This multi-cloud architecture may appear in several scenarios. For example, a company would like to port some workloads to FaaS but may find that its cooperating cloud vendor does not support local FaaS deployment. As a result, in order to satisfy its performance and regulatory requirements, this company may need to look for a new cloud provider. Most public FaaS providers impose time-out limits and rigid compute-to-memory ratios on function instances. Besides, they only support general-purpose hardware. Nimble FaaS providers, which want to attract customers and differentiate their products, may offer solutions in light of this situation. This would attract more customers to migrate more diverse workloads to it.

3) Data Portability: As discussed in Section II-B, data gravity is the real threat, especially in serverless computing, hindering data portability. Cloud providers do not impede data movement but charge a prohibitively expensive data egress fee. As a result, cloud customers may find that they are progressively trapped in data lock-in. This motivates us to employ a zero egress fee storage service in the architecture as shown above. How to migrate data from traditional storage services to this zero egress fee storage service needs to be discussed nevertheless. In an optimal situation, new processed data can be stored in a zero-egress fee storage service first, and then customers can retrieve it from it. This avoids double billing. Otherwise, it should plan how to move redundant data out based on the company's own situation and needs.

B. Benchmark

Serverless benchmark workloads can be classified into micro-benchmark or application-oriented benchmarks [START_REF] Kim | Functionbench: A suite of workloads for serverless cloud function service[END_REF]- [START_REF] Copik | Sebs: A serverless benchmark suite for function-as-a-service computing[END_REF]. To obtain a more practical evaluation of FaaS providers, we focus on application benchmarks.

1) Pipeline-Enabled Multi-Cloud Architecture Benchmark: In this benchmarking, we stressed the FaaS platforms to test the auto-scaling capability of their providers. We sent burst requests to the FaaS platforms, reaching their maximum currency limit. We set each function instance's timeout to 20s and memory allocation to 1 GB.

Image Processing. Specifically, we chose the image resizing application. The function instance downloads the picture from a bucket to disk space before image processing. We chose a data size of 5MB to ensure that the ephemeral storage can handle 50 pictures in a batch (maximum concurrency limit). We use PILLOW [START_REF] Pillow | [END_REF] library to process images.

2) Pipeline-Disabled Multi-Cloud Architecture Benchmark: This architecture may employ the low or zero data egress fee storage service to support domain-specific FaaS, thus supporting data-intensive or computer-intensive serverless workloads.

Machine Learning Training. A bulk of papers have discussed how to use FaaS for machine learning jobs [START_REF] Jiang | Towards demystifying serverless machine learning training[END_REF]- [START_REF] Sánchez-Artigas | Experience paper: Towards enhancing cost efficiency in serverless machine learning training[END_REF]. There are some positive results show that using FaaS to train some machine tasks performs better, but with more cost for the moment. We trained a machine learning model based on a 100 MB truncated dataset collected from Amazon Fine Foods reviews. We applied TF-IDF (term frequency-inverse document frequency) to evaluate the importance a word has to a document among a set of documents and logistic regression model for classification ,using the machine learning library scikit-learn [START_REF]scikit-learn[END_REF]. After cleaning the dataset, we removed the 100 most infrequently used words in text feature extraction and limited the maximum number of iterations to 1000 in the logistic regression model.

C. Experimental Setup

We deployed the End Analysis System on Azure Standard D8s v3, with 8 vCPUs, 32 GiB of memory, and hosted in Paris. We deployed all our FaaS and BaaS offerings in London.

D. Pipeline-Enabled Multi-Cloud Architecture Evaluation

Image Processing Workload Analysis. Fig. 4 show the cumulative distribution of evaluated FaaS providers' billing duration and the end-to-end delay, i.e., the elapsed time when a request is handled by a function instance.

Performance comparison: As Fig. 4 and Table II present, AWS Lambda outperforms its counterparts in terms of the lowest average billing duration and 99 th percentile latency. Although we may find that Alibaba's most requests have the lowest latency for being served. However, this serving latency is not as stable as AWS Lambda as a certain proportion of requests are served late. As the pressure comes, AWS Lambda may struggle to handle requests as the tail latency of AWS Lambda exceeds 80s. According to the cloud log, the requests that lead to tail latency experience timeout and appear in two scenarios: after the first time-out, the request experiences a II shows that the overall fee for this workload is dominated by the network fee and the duration fee. AWS Lambda and Alibaba's Function Compute are both economical choices.

In conclusion, AWS Lambda is the best FaaS provider for this workload since it offers the best performance and (one of) the lowest overall costs. Besides, we cannot consider the duration fee as the only main expense for FaaS because the results show that the data egress fee may account for a significant portion of the overall cost.

E. Pipeline-Disabled Multi-Cloud Architecture Evaluation

ML Training Workload Analysis. Table III shows each FaaS provider's compute power when the memory allocation is 2 GB. Even if we chose a small dataset (100 MB) for training traditional ML algorithms, it would take more than 2 minutes to converge. The time-out duration is short in FaaS, which means deploying deep learning algorithms on it will be ineffective for a large training dataset as continued training is needed after a timeout. Besides, we notice the imbalance between memory allocation and usage. We need to allocate more memory to the FaaS in order to gain powerful computing capabilities. However, the computing power increases with the allocated memory for most public FaaS providers. In particular, when training a model with a relatively small dataset, many memory resources will be wasted. We suppose that there would be an opportunity for the FaaS market to support more flexible configurations: GPU enabled (which AFC has enabled), flexible compute-to-memory ratio, etc.

V. RELATED WORK

We are motivated to deploy serverless multi-cloud to mitigate the effects of vendor lock-in as migrating an even simplified workload may encounter tricky issues or dead-ends [START_REF] Yussupov | Facing the unplanned migration of serverless applications: A study on portability problems, solutions, and dead ends[END_REF], [START_REF] Hartauer | Cloud function lifecycle considerations for portability in function as a service[END_REF]. Lithops is a multi-cloud serverless computing framework for big data analytics and parallel jobs, supporting running the same code across multiple clouds [START_REF] Sampe | Toward multicloud access transparency in serverless computing[END_REF]. Baarzi et al. proposed developing a latency-aware virtual service provider to schedule requests to public FaaS and proved that multi-cloud provides more stable performance when failure comes [START_REF] Baarzi | On merits and viability of multi-cloud serverless[END_REF]. We focus on selecting best FaaS provider for general cases. Some works extended FaaS to on-premises environments and designed scheduling algorithms in the edge-cloud continuum [START_REF] Aske | Supporting multi-provider serverless computing on the edge[END_REF]. Jindal et al. designed schedule algorithms in a heterogeneous FaaS deployment environment [START_REF] Jindal | Courier: delivering serverless functions within heterogeneous faas deployments[END_REF]. Real-world workloads are typically supported by both FaaS and BaaS offerings. This paper also considers the performance of FaaS when it is triggered by BaaS offerings. We are also inspired by Sky Computing, which envisioned a transparent cloud on top of cloud providers to provide best-of-breed service in terms of performance and cost for each workload [START_REF] Chasins | The sky above the clouds[END_REF], [START_REF] Stoica | From cloud computing to sky computing[END_REF].

VI. CONCLUSION

In this paper, we discuss how to benefit from multi-cloud deployment in addition to mitigating the effect of vendor lock-in. We present two proof-of-concept multi-cloud architectures that aim to avoid some common problems such as data distribution and data gravity. Besides, we developed the End Analysis System to quantitatively evaluate the performance and cost of FaaS providers, assisting customers in selecting the best FaaS provider for their workloads. To facilitate multi-cloud deployment, we designed the first version of a multi-cloud library that allows developers to access resources provisioned by different clouds via a common interface or their familiar SDKs.

Fig. 2 :

 2 Fig. 2: Pipeline-enabled multi-cloud architecture

Fig. 3 :

 3 Fig. 3: Pipeline-disabled multi-cloud architecture

  Fig. 4: Image Processing

TABLE I :

 I Pricing scheme of public FaaS providersProvider Duration Fee (GB-second) Invocation Fee (million) Data Egress Fee (GB / Month) Ephemeral Storage Fee (GB-second)

				Free quotas		Beyond	Free quotas	Beyond	Free quotas	Data Egress	Free quotas	Beyond
		AWS Lambda a [15]	400,000 $1.67 × 10 -5	1	$0.20	100 GB	$0.09	512 MB	$3.58 × 10 -8
	Google Cloud Functions b [16]	400,000	$2.5 × 10 -6	2	$0.40	5 GB	$0.12	Integrated with memory
	Alibaba Function Compute [17] 400,000 $1.67 × 10 -5	1	$0.20	$0.07 in Frankfurt and London	512 MB	NAS File System
	a x86 pricing scheme with tiered data egress pricing: $0.09 in the first 10 TB, $0.085 in the next 40 TB, and $0.07 in the following 100 TB.
	b Tier 1 location pricing. GCF also charges for CPU time
		Workload stage 1	Workload stage [2, N-1]	Workload stage N
		BaaS	FaaS
	M u l t i -c l o u d L i b r a r y	BaaS BaaS	FaaS FaaS
					End Analysis System (EAS)
		Workload	Adaptor	Cloud Logging Query	Cost Model
		Request logging	Authentication	Local Logging		Analysis

TABLE II :

 II The cost table of experiments

			Image Processing
		AWS	Google	Alibaba
	avg. Billing Duration	1937 ms 2421.5 ms 2380.7 ms
	avg. Egress Data Size	482 KB	482 KB	482 KB
	99 th percentile Latency	5388 ms 12254 ms	9496 ms
	1M req. Request Fee	$ 0.2	$ 0.4	$ 0.2
	1M req. Network Fee	$ 43.38	$ 57.84	$ 33.74
	1M req. Duration Fee a	$ 32.33	$ 39.95	$ 39.70
	1M req. Cost	$ 75.91	$ 98.19	$ 73.64
	1M req. Cost (Free-tier) $ 60.05	$ 94.19	$ 66.77
	a We merge GCF's CPU charges into duration fee.

TABLE III :

 III The FaaS configuration and cost table

		Machine Learning Training
		AWS	Google	Alibaba
	Data Size	100 MB	100 MB	100 MB
	vCPU	1.16	1	1.33
	Memory Allocation	2 GB	2 GB	2 GB
	Memory Usuage	≈700 MB ≈675 MB ≈720 MB
	avg. Billing Duration	133 s	159 s	138 s
	1K req. Duration Fee * $ 4.43	$ 4.64	$ 4.63

long period to be served again; or the first serving time for a request is very late. These scenarios occur when the request is throttled due to a lack of instances to handle.

Cost comparison: Table

Our software is in progress and we make it available as open-source at https://github.com/hd-zhao/serverless multicloud

ACKNOWLEDGEMENT

We thank our anonymous reviewers for their valuable feedback that improved this work.


Jordi Orrit-Prat

Laurent Boudou

Christina Villeneuve

Gilbert Teyssedre

Samuel Behar

Laurence Ressier

Régis Diaz

Effect of film thickness on the dielectric properties and charge storage in PMMA thin films

Keywords: Polymethylmethacrylate, thin films, charge writing by atomic force microscopy, Kelvin force microscopy, conductivity I

In this work, we demonstrate the excellent charge retention capability of spin-coated Polymethylmethacrylate (PMMA) 200 nm thin films previously charged by atomic force microscopy. In order to elucidate the mechanisms involved in the charge transport and charge storage in PMMA films, the conductivity of 200nm thin PMMA films was analyzed on Metal-Insulation-Metal (MIM) structures. A wide range of electric fields was applied on these structures and the charge and discharge currents were recorded. The thermal dependence of the conduction mechanisms was determined for temperatures below the PMMA glass transition temperature. These results were compared to those obtained for 50µm thick PMMA films in which the conductivity appears much smaller. The effects of PMMA thickness on charge transport are discussed.

INTRODUCTION

Polymethylmethacrylate (PMMA) is a polymeric material widely used in electrical and optical applications. Recently, it has been introduced in the Organic Thin Film Transistors (OTFT) [START_REF] Park | Studies on poly(methyl methacrylate) dielectric layer for field effect transistor: Influence of polymer tacticity[END_REF], showing excellent properties as gate dielectric or buffer layer. This has motivated some research on the electrical properties of PMMA thin films [START_REF] Miyairi | AC electrical breakdown and conduction in PMMA thin films and the influence of LiClO4 as an ionic impurity[END_REF]. In parallel, other innovative fields incorporate the use of PMMA thin films. It is the case of nanoxerography [START_REF] Palleau | Coulomb Force Directed Single and Binary Assembly of Nanoparticles from Aqueous Dispersions by AFM Nanoxerography[END_REF], which consists in the directed assembly of colloidal nano-objects onto electrostatically patterned dielectric surfaces. This technique can lead to the fabrication of nanodevices in various domains such as electronics, optics or sensor applications [START_REF] Shipway | Nanoparticle arrays on surfaces for electronic, optical, and sensor applications[END_REF]. In fact, PMMA thin films can be efficiently electrostatically patterned by different charging methods, like Atomic Force Microscopy (AFM) charge writing [START_REF] Palleau | Coulomb Force Directed Single and Binary Assembly of Nanoparticles from Aqueous Dispersions by AFM Nanoxerography[END_REF], electrical microcontact printing [START_REF] Jacobs | Submicrometer patterning of charge in thin-film electrets[END_REF] or electrical nanoimprint [START_REF] Ressier | Electrical nano-imprint lithography[END_REF].

Nonetheless, some well-known characteristics of PMMA are no longer valid for nanometric-thick films. Film thickness strongly affects the molecular dynamics and the glass transition temperature (Tg) [START_REF] Hartmann | Molecular dynamics in thin films of isotactic poly(methyl methacrylate)[END_REF]. With regard to the specific dielectric characteristics, the breakdown strength depends on the physical dimensions [START_REF] Miyairi | AC electrical breakdown and conduction in PMMA thin films and the influence of LiClO4 as an ionic impurity[END_REF]. Charge transport mechanisms and charge retention capability would also be influenced by the film thickness, which can be an important issue for nanoxerography applications. Better understanding of the electrical conduction in PMMA thin films would give more information about the breakdown mechanism and polymer characteristics at the nanometric scale, which are related to several promising applications.

In this study, after charge writing into PMMA films of two different thicknesses (200nm and 50µm) by AFM, charge retention was followed by Kelvin force microscopy (KFM). Conduction characteristics of these PMMA films were also analyzed from I(V) measurements. Finally, the thickness impact on dielectric properties, charge transport and charge storage of PMMA is discussed.

II. EXPERIMENTAL

A. Material and samples

50µm thick commercial films were obtained from Goodfellow (GF). Two kinds of 200nm thin PMMA films were elaborated by spin coating on n-doped (10 15 cm -3 ) silicon substrates and glass substrates. The first ones (named SA thin films) were elaborated from PMMA powder provided by Sigma Aldrich (SA) dissolved into methyl isobutyl ketone (MIBK). The second ones (named GF thin films) were prepared from above-mentioned 50µm thick commercial films.

B. AFM charge writing and surface potential measurements by KFM

AFM charge writing was performed on 200nm SA and GF thin PMMA films spin coated on silicon substrates and on 50μm thick PMMA films. 20 µm x10 µm positively (resp. negatively) charged rectangles were written into each PMMA film by applying voltage pulses to a conductive AFM tip with an external generator. The z-feedback was adjusted to control the tip-film separation during charge writing. The pulse length and frequency were fixed at 1 ms and 50 Hz, respectively, while the tip velocity was fixed at 10µm/s. After AFM charge writing, amplitude modulation KFM was used to measure the surface potentials of charged patterns (lift height = 50nm). Charge retention in PMMA films was measured by recording successively 12 KFM images.

C. Electrical conduction characteristics in PMMA films

For 50µm thick films, 30nm thick silver electrodes with a diameter of 5cm were deposited by sputtering on each face. For both SA and GF thin films, two 150nm thick parallel silver stripes were deposited on glass substrates before spin coating 200nm thin PMMA films. Two columns of five silver stripes perpendicular to the previous ones were finally deposited on top of the PMMA films. Thus, 10 Metal-Insulator-Metal (MIM) regions of 12 mm 2 were available on each sample.

The conduction measurements on PMMA films were performed in a chamber with a temperature-controlled helium environment. The samples were polarized at constant temperatures for 1000 seconds and then short-circuited for the same duration. The polarization and depolarization currents were recorded by a Keithley 6512 electrometer. Several electric fields conditions were applied for each temperature.

III. RESULTS

A. KFM measurements

Fig. 1 shows a typical KFM image just after charge writing in a 200nm SA thin film spin-coated on a silicon substrate. The upper negatively charged rectangle written with -75V voltage pulses present a surface potential of -10.0V whereas the lower positively charged rectangle written with +75V voltage pulses has a surface potential of +9.6V. Charge writing on 200nm GF thin films gave similar results. The same experiments performed for 50µm thick films revealed that charge injection is far less efficient than charge injection in 200nm thin films. A stronger dissymmetry of charge injection between negative and positive charges was also observed: the rectangle written with -75V (resp. +75V) voltage pulses presented a surface potential of -6.5 V (resp. +4.8V).

The temporal evolution of the surface potential of the negatively and positively charged rectangles written in 200nm SA thin films and 50µm thick films are plotted in Fig. 2. Measurements on 200nm GF thin films (data not shown) revealed no significant difference with those obtained on 200nm SA thin films. Fig. 2 reveals a strong decrease of the surface potential of charged rectangles during the 30 first minutes after charge writing for thin and thick films. The charge loss is weaker after, especially for positive charges into 200 nm SA thin films (less than 25% of charge loss during the following 2 hours). The charge retention of thin films appears to be much better than that of thick films, especially for positive charges. About 50% of the positive charges written in 200nm PMMA thin films are still present after 3 hours, making PMMA an excellent electret.

B. Conductivity measurements 1) 200nm thin films

Charge and discharge currents were measured at different temperatures (25-90°C) below Tg (around 100°C) and for several fields conditions (in the 0.5-250kV/mm range). Fig. 3 presents a plot of the conduction current density as a function of the applied electric field, for various temperatures in the above-mentioned ranges, in case of the 200nm GF thin films. A linear dependence between the conduction current density and the applied electric field is observed in all cases. This implies that the dominant charge transport mechanism is the ohmic conduction, related to the following expression:

 F J    
where J is the current density, σ the conductivity and F the applied electric field. The σ values obtained from the linear fits made on the experimental data range from 4.13×10 -15 S/m at 25°C up to 1.09×10 -12 S/m at 90°C. On the other hand, the conductivity dependence on the temperature, presented in Fig. 4 for the same PMMA films, can be addressed by the Arrhenius law, given below:

 kT E a e   0    
where σ0 is a pre-exponential factor, Ea the activation energy, k the Boltzmann constant and T the temperature. From the exponential fit, a value for the activation energy of 0.79 was obtained.

In the case of 200nm SA thin films, a very different trend was observed. At room temperature the current dependence on the electric field is no longer linear, but it can be approximated by hyperbolic sinus (Fig. 5). This trend is characteristic of

        kT eFd J 2 sinh  
where d is the distance between traps. By fitting the theoretical function to the experimental data, a value of d=1.4nm was extracted. At low fields (5-25kV/mm) the curve J-F can be approximated by a straight line, then the conduction can be considered as ohmic. Thereby, the conductivity can be estimated at 1.09×10 -15 S/m. At higher temperatures, a steady current for fields higher than 17.5 kV/mm could not be obtained, and in some cases, even the breakdown took place. Fig. 6 shows the Arrhenius diagram for 200nm SA thin films. The point corresponding to conductivity at 90°C lies out of the straight line fitting the other points. This can be explained by the closeness of the glass transition, despite in GF thin films no change in the trend was observed at such temperature. From the Arrhenius fit, the activation energy was found to be of 0.10eV. It is worth noting, for comparison purposes, that a trap depth of 0.15eV and an average distance between traps of 1.5nm have been estimated for polyethylene [START_REF] Meunier | Molecular modeling of electron trapping in polymer insulators[END_REF].

2) 50µm thick films

In the case of 50µm thick PMMA films, electrical fields as high as those used for thin films could not be applied. At low temperatures, the current needed too much time to stabilize and the discharge current was of the same order of magnitude as the charge current. So, the stationary conduction current could not be accurately obtained. However, by taking the polarization current along with the depolarization one, and assuming that the measurements are within the ohmic regime, the conductivity could be properly calculated [START_REF] Diego | Annealing effect on the conductivity of XLPE insulation in power cable[END_REF]. The hypothesis of considering the electrical conduction as ohmic was verified by performing I(V) characteristics at temperatures over 50°C (Fig. 7). The calculated conductivity values increases from 3.53×10 -17 S/m at 25°C, up to 1.08×10 -14 S/m at 90°C. The estimated activation energy is 0.79eV (Fig. 8 these differences between GF and SA thin films are consistent with the presence of a little amount of elastomer in GF PMMA (according to the supplier), whereas SA PMMA powder does not include any kind of additive. Furthermore, ohmic conductivity is found to be higher for thin films compared to thick films. We assume an effect of PMMA thickness on its conduction properties. As film thickness becomes closer to the polymer chain length, the possibility for charge carriers to cross the structure without changing the polymer chain increases. Also, the existence of large defects, like voids, becomes less probable. Such physical defects can hinder the charge carriers motion. Thus, the effective mobility should increase if the large defects amount is reduced. A similar reasoning could be applied to the dielectric strength. Since breakdown can be originated by defects like protrusions or voids [START_REF] Fothergill | Electrical, microstructural, physical and chemical characterization of HV XLPE cable peelings for an electrical aging diagnostic data base[END_REF], the breakdown strength would raise when the physical dimensions of the sample are lowered to the nanometric scale. Even so, a dielectric strength increase with a thickness decrease could be also explained by the theory of thermal breakdown caused by Joule heating [START_REF] Miyairi | AC electrical breakdown and conduction in PMMA thin films and the influence of LiClO4 as an ionic impurity[END_REF].

However, charge retention measurements show that nanometric thin PMMA films present better charge retention capabilities than micrometric ones. This seems to be in contradiction with the fact that ohmic conductivity is apparently higher in the former. Nonetheless, conductivity has been determined from the current flowing through metallic coated films, while an electric field was applied. On the other hand, PMMA films analyzed by KFM were previously charged by an AFM tip, without any conductive coating on film surface. Thus, injected charges could be dissipated by surface conduction along with bulk charge transport. In addition, the traps responsible for the long-run charge retention are not necessary involved in the dominant charge transport process. For instance, while shallow traps can enhance the charge transport, the deep localized states originate the steady trapped charge. In this sense, spin coating process, used for preparation of both GF and SA thin films, could lead to the incorporation of impurities that would act as deep traps. Further work on charge trapping, detrapping and recombination in PMMA films charged by AFM is needed to clear up the mechanisms involved in such processes.

  Cite as: J. Orrit-Prat, L. Boudou, C. Villeneuve, Gilbert Teyssedre, S. Behar, L. Ressier, R. Diaz, "Effect of film thickness on the dielectric properties and charge storage in PMMA thin films", IEEE Intern. Conf. on Solid Dielectrics (ICSD), Bologna, Italy, 30 Jun.-4 Jul. 2013. Proc. ICSD-2013, pp. 350-353, 2013. DOI: 10.1109/ICSD.2013.6619883

Fig. 5 Fig. 2

 52 Fig.5 Current density vs. electric field in 200nm SA PMMA thin films at T=25°C. (■, experimental points; -, calculated fit) 0 40 80 120 160 200

  Cite as: J. Orrit-Prat, L. Boudou, C. Villeneuve, Gilbert Teyssedre, S. Behar, L. Ressier, R. Diaz, "Effect of film thickness on the dielectric properties and charge storage in PMMA thin films", IEEE Intern. Conf. on Solid Dielectrics (ICSD), Bologna, Italy, 30 Jun.-4 Jul. 2013. Proc. ICSD-2013, pp. 350-353, 2013. DOI: 10.1109/ICSD.2013.6619883

Fig. 7 Fig. 6 Fig. 4 Fig. 3 Fig. 5

 76435 Fig.7 Current density vs. electric field in 50µm GF thick PMMA films, at: ■, 60°C.; •, 70°C; ▲, 80°C; ▼, 90°C. (calculated fits are represented by straight lines)

  ). IV. DISCUSSION A summary of conduction mechanisms and extracted parameters is shown in Table I. 50µm GF thick films and 200nm GF thin films present the same conduction mechanism and similar activation energies. It consists in a charge transport by extended states thermally assisted by the detrapping of charge carriers at deep localized states (around 0.80eV of depth). On the other hand, 200nm SA thin films show conduction by hopping mechanism between shallow localized states. Knowing that band structure and localized states are determined by chemical structure and polymer morphology, Cite as: J. Orrit-Prat, L. Boudou, C. Villeneuve, Gilbert Teyssedre, S. Behar, L. Ressier, R. Diaz, "Effect of film thickness on the dielectric properties and charge storage in PMMA thin films", IEEE Intern. Conf. on Solid Dielectrics (ICSD), Bologna, Italy, 30 Jun.-4 Jul. 2013. Proc. ICSD-2013, pp. 350-353, 2013. DOI: 10.1109/ICSD.2013.6619883

Fig. 8

 8 Fig.8 Conduction vs. reciprocal of temperature (Arrhenius diagram) in 50µm GF thick films, for an electric field of 20kV/mm. The calculated activation energy is displayed in the figure.

TABLE I

 I

		.	SUMMARY RESULTS
	Sample	Conductivity at Room Temperature (S/m)	Activation Energy (eV)	Electrical Conduction
	GF 50 µm	3.53×10 -17		0.79	Ohmic
	GF 200 nm	4.13×10 -15		0.79	Ohmic
	SA 200 nm	1.09×10 -15 (low fields)	0.10	Hopping
	Fig. 7 Current density vs. electric field in 50µm GF thick PMMA films,
	at: ■, 60°C.; •, 70°C; ▲, 80°C; ▼, 90°C. (calculated fits are represented
	by straight lines)

ACKNOWLEDGMENT

This work was supported by the French National Agency (ANR) in the frame of its program 'Emergence' (Elec-NanoPrint project no ANR-2011-EMMA-015-01).


H Chapuis
email: herve.chapuis@inrae.fr

M Lague

H Gilbert

Genetic parameters of feeding behaviour traits in parental lines of a mule duck

Automatic feeders were used to characterize the feeding behaviour traits of two populations of ducks (White Pekin and Muscovy). In addition to feed conversion ratio, six phenotypes were derived at the scale of the day and the meal and genetic parameters were estimated. Feeding rate in the Pekin population was twice as large as in the Muscovy population (19 g/min vs. 9 g/min). Feeding traits exhibited moderate (0.38) to high (0.67) heritabilities. A similar pattern of correlations could be observed between these traits in the two populations. Feed conversion ratio did not show any remarkable correlation with feeding traits in the Pekin line, except with daily feed intake, but was negatively correlated with the number of meals and meal feed intake in the Muscovy population (0.68). Further studies will be needed to decipher how these traits can be incorporated in breeding programs without impairing ducks' abilities to produce fatty liver.

Introduction

The mule duck contributes to more than 90% of the production of French fatty liver. It is a sterile hybrid obtained by crossing common duck females (Anas platyrynchos) and Muscovy drakes (Cairina moschata). Selection to improve fatty liver production is carried out on its parental populations, and only limited knowledge is available about the relationships between parental and crossbred traits. In addition, as recording individual feed intake in both parental populations is still a challenge, feed efficiency remains poor. We set up a design of purebred ducks to investigate detailed feeding traits to improve feed efficiency. We computed feeding behaviour traits using automated feeders (Bley and Bessei, 2008) and estimated genetic parameters for these traits in the parental populations, and their relationships with feed efficiency.

Materials & Methods

The present study was conducted in agreement with the 2010/63/EU regulation for use of animals for research purposes. Animals were bred at the INRAE Duck farm (UEPFG, Benquet, France) which has been approved for animal experimentation (C40-037-1).

Birds and housing.

Three successive generations of animals were produced in a dam White Pekin (PEK) common duck line and in a sire Muscovy (MUS) line. Electronically tagged animals were reared up to the age of 7 (PEK) or 8 (MUS) weeks in a 96 m² pen equipped with 6 automatic feeders similar to those used by Basso et al. (2014), including animal weighing at each visit. To facilitate duck management, the area was divided in two parts with a barrier, so each bird had effective ad libitum access to 3 feeders. In the PEK line, we reared 273 males and 467 females, while only males (n=460) were studied in the MUS line.

Data acquisition.

The date and time at the beginning and end of each visit to the feeder was recorded. Animal weight and feed consumption were recorded for each visit. Feeding behaviour traits were then computed. Daily Feed Intake (DFI) and Daily Feeding Time (DFT) definitions are straightforward. Feeding Rate (FR) was computed as the ratio of feed intake over feeding time for each visit and averaged over the whole period. Following Howie et al.'s (2009) method 3 to compute the so-called "meal criterion", visits were aggregated into meals in order to allow for reliable comparisons of feeding behaviours between the two populations. For each duck, visits were defined as part of a single meal if the interval between visits was lower than the meal criterion (1616 s in PEK, 2113 s in MUS). We therefore recorded the number of meals per day (NM), Meal Duration (MD) and Meal Feed Intake (MFI). Feed efficiency was addressed through computation of feed conversion ratio (FCR), as the ratio of total feed intake over the observed weight gain. Average Daily Gain was also computed. We planned to collect and analyse early feed and growth data over a 5-week span (i.e. 35 days). In order to limit the occurrence of misidentified visits, which were found numerous at early age in the Muscovy population [START_REF] Cobo | Proc. of « Douzièmes Journées de la Recherche Avicole et Palmipèdes à Foie Gras[END_REF], the 35-day control period spanned from Day22 to Day56 in the MUS line, while it started at Day15 and ended at Day49 in the PEK line.

Statistical analysis.

The phenotypic comparison of performances between lines was obtained through a linear model accounting for the effects of batch and sex in the PEK line. Then, genetic parameters were estimated using ASRemL [START_REF] Gilmour | ASReml User Guide Release 4.1 Functional Specification[END_REF], with the model:

y = µ + βbatch (+ γsex) + u + e
(1) where y is the phenotype, µ the general mean, βbatch and γsex are the fixed effects of batch and sex (PEK line only), respectively, u the additive genetic effects, and e the residual effects. In order to normalize the data, DFT and MD were log-transformed prior to analyses. Values more than three standard deviations from the mean were considered as outliers and removed from the analysis. First, univariate analyses were run to compute heritabilities, and next multivariate models were applied with the same effects to obtain correlations.

Results

The least squares means of the linear model applied for phenotypic line comparisons are given in Table 1. MUS ducks spent daily almost twice as much time feeding as PEK ducks, while the amount of feed they consumed was 12% lower. This resulted in a feeding rate more than twice as great for the PEK ducks, as reported in [START_REF] Cobo | Proc. of « Douzièmes Journées de la Recherche Avicole et Palmipèdes à Foie Gras[END_REF] where only male data were considered. Compared to the MUS line, the number of meals per day was greater and meal duration was roughly 2.5 times shorter in the PEK line ( Table 1). FCR was better (i.e. lower) in the Muscovy line. The average daily gain was greater in the MUS line. Yet, it should be remembered that the cut-off points occurred one week earlier in the PEK population. The genetic parameters are displayed in Table 2. In both lines, feeding traits were highly heritable; most heritabilities were higher than 0.5, such as NM and MFI in both populations.

DFT and FR exhibited remarkably high heritability values (~0.6) in the PEK line and around 0.4 in the MUS line.

In PEK, DFI was moderately correlated with MFI and FCR (~0.35), while these correlations were much higher in MUS (~0.70). DFI and MD were positively but not significantly correlated in PEK, while the correlation was positive in MUS. In both lines, DFT and FR were strongly and negatively correlated.

Discussion

The lower FCR in the MUS line compared to the PEK line was expected, as Muscovy duck is known to produce a leaner meat than the White Pekin and is popular in France for this very reason. Here we studied only males in the MUS population, which is known to exhibit a larger sexual dimorphism for body weight than the PEK line [START_REF] Tai | [END_REF]. This may contribute to the difference in FCR observed between lines. We computed a meal criterion within each batch using Howie et al.'s (2009) method 3. This method is based on analyses of the probability for an animal to start feeding given the time since the last visit. This requires finding the nadir of a potentially flat curve, which may be not precisely estimated. For the PEK line, the meal criterion was close to the value of 1,725 seconds found by Howie et al. ( 2009) in another Pekin duck population.

Our genetic parameters for feeding behaviour traits were in good agreement with those obtained on Pekin broilers by Le [START_REF] Mignon | Proc of X th European Symposium on Poultry Genetics[END_REF] and [START_REF] Thiele | available at Feed Efficiency and Feeding Behaviour in Pekin Ducks -Lohmann Breeders[END_REF]. With a heritability of 0.33±0.11 for feed intake from Day15 to Day42, [START_REF] Zhang | Asian[END_REF] obtained a slightly lower value than ours. Some common features can be noted between our study and that of Le [START_REF] Mignon | Proc of X th European Symposium on Poultry Genetics[END_REF]: (1) Strong negative correlations between the number of meals and the meal size, reflecting that an increase in the meal number does not lead to increased feed intake. If the animal has reached satiety, an increase in the number of meals does not respond to an urge to eat more feed but simply illustrates a variability in feeding behaviour;

(2) Moderately negative correlations between number of meals and feeding rate. Birds exhibiting a large FR may swiftly reach satiety and not feel the need to start another feeding sequence;

(3) A large and positive, but significantly different from unity, correlation between meal duration and total time spent feeding, emphasizing an exploitable variability of feeding behaviours. As expected, the latter is also strongly negatively correlated with FR. These three points are also common to our two populations, which might suggest some generality about ducks' feeding behaviours.

For feed efficiency, we focused on FCR. Feed efficiency could also be profitably addressed through the fraction of feed intake that is not explained by maintenance and production requirements, aka Residual Feed Intake (RFI). Yet, in the case of fatty duck production, attention should be paid to an adequate derivation of RFI, i.e. accounting for body lipid composition, as in [START_REF] Drouilhet | [END_REF]. With estimates of 0.24 and 0.31, FCR was less heritable in the two Pekin lines studied by Le [START_REF] Mignon | Proc of X th European Symposium on Poultry Genetics[END_REF] than in our study. In the PEK line, our estimates of the correlation between FCR and feeding behaviour traits are in accordance with those obtained by [START_REF] Thiele | available at Feed Efficiency and Feeding Behaviour in Pekin Ducks -Lohmann Breeders[END_REF] and Le [START_REF] Mignon | Proc of X th European Symposium on Poultry Genetics[END_REF]: they are of low magnitude (<0.20), except for DFI. Conversely, in the MUS line, FCR exhibited strong correlations with DFI and MFI, which may suggest a larger part of feed allocated to growth compared with the PEK line. In addition, in MUS, the negative correlation between NM and FCR, though not accurately estimated, could indicate that selecting for increased number of meals would improve FCR, which contradicts the literature in which efficient animals generally show less feeding activity.

These first genetic analyses of a large design focusing on feeding characteristics of the parental lines of the mule duck show that feeding traits are heritable in these lines, and some are correlated to feed efficiency. Further studies will be conducted to decipher how these traits can be incorporated in breeding programs without impairing the ducks' abilities to produce fatty liver.

This study was funded by the French interprofessional association for fatty liver production (CIFOG), and INRAE animal genetic division. The authors thank UEPFG technical staff for animal care and data collection.

Table 1 . Mean values (Least squares means 1 ± standard error) of feeding behaviour traits.

 1

	Trait	PEKIN dam line	MUSCOVY sire line P line
	Daily Feed Intake (g/d)	225.00 ± 0.72	200.00 ± 1.16	<0.0001
	Daily Feeding Time (s/d)	811.0 ± 12.1	1,595.0 ± 19.5	<0.0001
	Feeding Rate (g/min)	18.84 ± 0.15	8.86 ± 0.24	<0.0001
	Number of daily meals/d	4.46 ± 0.03	3.54 ± 0.05	<0.0001
	Meal Feed Intake (g)	53.2 ± 0.5	59.1 ± 0.8	<0.0001
	Meal Duration (s)	191.00 ± 3.98	470.00 ± 6.41	<0.0001
	Feed Conversion Ratio (kg/kg) 3.25 ± 0.01	2.56 ± 0.01	<0.0001
	Average Daily Gain (g/d)	69.1± 0.24	79.7± 0.38	<0.0001

1 Least squares means from a linear model including the fixed effect of batch (PEK and MUS) and sex (MUS).

Table 2 . Genetic parameters of feeding traits 1 in two duck lines.

 2

		DFI	DFT	FR	NM	MFI	MD	FCR
	PEKIN
	DFI 0.49±0.08 0.13±0.13 0.12±0.13 -0.15±0.13 0.36±0.12 0.24±0.14 0.33±0.12
	DFT		0.57±0.07 -0.95±0.01 0.44±0.10 -0.41±0.11 0.61±0.08 -0.05±0.13
	FR			0.61±0.07 -0.43±0.10 0.48±0.09 -0.57±0.08 0.06±0.13
	NM				0.57±0.07 -0.95±0.08 -0.42±0.08 -0.06±0.08
	MFI					0.63±0.07 0.44±0.08 0.10±0.08
	MD						0.48±0.07 0.01±0.08
	FCR							0.48±0.07
	MUSCOVY
	DFI	0.54±0.12 0.23±0.20 0.18±0.20 -0.20±0.18 0.69±0.10 0.50±0.19 0.73±0.15
	DFT		0.43±0.11 -0.88±0.05 0.52±0.17 -0.10±0.20 0.68±0.12 0.32±0.24
	FR			0.39±0.11 -0.49±0.18 0.41±0.18 -0.60±0.14 -0.06±0.26
	NM				0.54±0.11 -0.90±0.04 -0.31±0.18 -0.44±0.21
	MFI					0.67±0.11 0.44±0.16 0.68±0.16
	MD						0.38±0.11 0.01±0.08
	FCR							0.25±0.09

1 DFI = Daily Feed Intake; DFT = Daily Feeding Time; FR = Feeding Rate; NM = Number of daily meals; MFI = Meal Feed Intake; MD = Meal Duration; FCR = Feed Conversion Ratio.