This paper presents a solution for resource allocation of Industry 4.0 micro-service applications across serverless fog federation. The research addresses the challenge of fulfilling the fault-tolerant and real-time constraints of Industry 4.0 applications on resource-limited fog systems in remote industrial sites. The proposed approach dynamically and seamlessly federates nearby fog systems to enable cloud-like elasticity. It develops serverless resource allocation solutions that are aware of the applications' software architecture, their latency requirements, and the distributed nature of the underlying infrastructure. The paper proposes methods to seamlessly and optimally partition micro-service-based applications across the federated fog. Experimental evaluation shows that the proposed method can serve around 20% more tasks on-time than existing methods. The paper explores the stochastic behaviors and in-depth structure of Industry 4.0 applications and addresses the challenges of modern computing platforms. It discusses the problem of resource allocation in Industry 4.0, the use of fog computing for Industry 4.0 use cases, ML applications and serverless platforms, and software-hardware system stacks. The paper presents a system model for Industry 4.0 applications, which are often in the form of microservice workflows deployed via containers. The system model comprises a gateway in each fog system that receives arriving application requests and transparently allocates them across the federation. The paper also presents a partitioning method for micro-service application workflows and a resource allocation method across serverless fog federation. The partitioning method, called Probabilistic Partitioning (ProPart), is used to partition a micro-service workflow in a way that the application can meet its deadline. The resource allocation method is based on the notion of relevance, which is defined as a fog system that maximizes the likelihood of meeting the deadline for a given request. The paper evaluates the proposed partitioning and resource allocation methods against baseline solutions and shows that the proposed method can serve around 20% more tasks on-time than existing methods.This paper presents a solution for resource allocation of Industry 4.0 micro-service applications across serverless fog federation. The research addresses the challenge of fulfilling the fault-tolerant and real-time constraints of Industry 4.0 applications on resource-limited fog systems in remote industrial sites. The proposed approach dynamically and seamlessly federates nearby fog systems to enable cloud-like elasticity. It develops serverless resource allocation solutions that are aware of the applications' software architecture, their latency requirements, and the distributed nature of the underlying infrastructure. The paper proposes methods to seamlessly and optimally partition micro-service-based applications across the federated fog. Experimental evaluation shows that the proposed method can serve around 20% more tasks on-time than existing methods. The paper explores the stochastic behaviors and in-depth structure of Industry 4.0 applications and addresses the challenges of modern computing platforms. It discusses the problem of resource allocation in Industry 4.0, the use of fog computing for Industry 4.0 use cases, ML applications and serverless platforms, and software-hardware system stacks. The paper presents a system model for Industry 4.0 applications, which are often in the form of microservice workflows deployed via containers. The system model comprises a gateway in each fog system that receives arriving application requests and transparently allocates them across the federation. The paper also presents a partitioning method for micro-service application workflows and a resource allocation method across serverless fog federation. The partitioning method, called Probabilistic Partitioning (ProPart), is used to partition a micro-service workflow in a way that the application can meet its deadline. The resource allocation method is based on the notion of relevance, which is defined as a fog system that maximizes the likelihood of meeting the deadline for a given request. The paper evaluates the proposed partitioning and resource allocation methods against baseline solutions and shows that the proposed method can serve around 20% more tasks on-time than existing methods.