It has been shown that in some robotic applications, where the execution times cannot be assumed to be independent and identically distributed, a Markov Chain with discrete emission distributions can be an appropriate model. In this paper we investigate whether execution times can be modeled as a Markov Chain with continuous Gaussian emission distributions. The main advantage of this approach is that the concept of distance is naturally incorporated. We propose a framework based on Hidden Markov Model (HMM) methods that 1) identifies the number of states in the Markov Model from observations and fits the Markov Model to observations, and 2) validates the proposed model with respect to observations. Specifically, we apply a tree-based cross-validation approach to automatically find a suitable number of states in the Markov model. The estimated models are validated against observations, using a data consistency approach based on log likelihood distributions under the proposed model. The framework is evaluated using two test cases executed on a Raspberry Pi Model 3B+ single-board computer running Arch Linux ARM patched with PREEMPT_RT. The first is a simple test program where execution times intentionally vary according to a Markov model, and the second is a video decompression using the ffmpeg program. The results show that in these cases the framework identifies Markov Chains with Gaussian emission distributions that are valid models with respect to the observations.

In the recent works that analyzed execution-time variation of real-time tasks, it was shown that such variation may conform to regular behavior. This regularity may arise from multiple sources, e.g., due to periodic changes in hardware or program state, program structure, inter-task dependence or inter-task interference. Such complexity can be better captured by a Markov Model, compared to the common approach of assuming independent and identically distributed random variables. However, despite the regularity that may be described with a Markov model, over time, the execution times may change, due to irregular changes in input, hardware state, or program state. In this paper, we propose a Bayesian approach to adapt the emission distributions of the Markov Model at runtime, in order to account for such irregular variation. A preprocessing step determines the number of states and the transition matrix of the Markov Model from a portion of the execution time sequence. In the preprocessing step, segments of the execution time trace with similar properties are identified and combined into clusters. At runtime, the proposed method switches between these clusters based on a Generalized Likelihood Ratio (GLR). Using a Bayesian approach, clusters are updated and emission distributions estimated. New clusters can be identified and clusters can be merged at runtime. The time complexity of the online step is $O(N^{2}+ NC)$ where N is the number of states in the Hidden Markov Model (HMM) that is fixed after the preprocessing step, and C is the number of clusters. 

In real-time systems analysis, probabilistic models, particularly Markov chains, have proven effective for tasks with dependent executions. This paper improves upon an approach utilizing Gaussian emission distributions within a Markov task execution model that analyzes bounds on deadline miss probabilities for tasks in a reservation-based server. Our method distinctly addresses the issue of runtime complexity, prevalent in existing methods, by employing a state merging technique. This not only maintains computational efficiency but also retains the accuracy of the deadline-miss probability estimations to a significant degree. The efficacy of this approach is demonstrated through the timing behavior analysis of a Kalman filter controlling a Furuta pendulum, comparing the derived deadline miss probability bounds against various benchmarks, including real-time Linux server metrics. Our results confirm that the proposed method effectively upper-bounds the actual deadline miss probabilities, showcasing a significant improvement in computational efficiency without significantly sacrificing accuracy.

Computationally demanding tasks with highly variable execution times may require parallel processing. Scheduling such tasks with low deadline miss rates but without significant overprovisioning is challenging. This issue arises in applications like nonlinear optimization for Model Predictive Control (MPC). The Constant Bandwidth Server (CBS) provides timing isolation, supporting both hard and soft real-time tasks. However, scheduling parallel, time-varying jobs across multiple CBS instances requires static job-to-server assignments, which can lead to resource underutilization due to queued jobs awaiting specific servers. This paper introduces the Job Acceptance Multi-Server JAMS, a mechanism in which multiple CBS instances share a common job queue, enabling flexible job dispatching for parallel workloads. JAMS incorporates a job dismissal mechanism to address overloads, ensuring that only jobs with guaranteed resource availability are accepted. Each CBS instance checks if it can complete a job by its deadline, given probabilistic knowledge on its execution times, dismissing unfeasible jobs to avoid excessive tardiness across queued tasks. Implemented in Linux, JAMS  is evaluated with computation times drawn from an MPC task and synthetic datasets. The extensive experimental results we provide, demonstrate that JAMS effectively controls the deadline miss rate, maintaining it below a specified design threshold.

In scheduling real-time tasks, we face the challenge of meeting hard deadlines while optimizing for some other objective, such as minimizing energy consumption. Formulating the optimization as a Multi-Armed Bandit (MAB) problem allows us to use MAB strategies to balance the exploitation of good choices based on observed data with the exploration of potentially better options.In this paper, we integrate hard real-time constraints with MAB strategies for resource management of a Stochastic Parallel Synchronous Task. On a platform with M cores available for the task, m<=M cores are initially assigned. Prior work has shown how to compute a virtual deadline such that assigning all M cores to the task if it has not completed by this virtual deadline guarantees that the deadline will be met. An MAB strategy is used to select the value of m. A Dynamic Power Management (DPM) energy model considering CPU sockets and sleep states is described. Experimental evaluation shows that MAB strategies learn consistently suitable m, and perform well compared to binary exponential search and greedy methods.