Most existing research in probabilistic real-time scheduling analysis has primarily focused on systems with only stochastic execution times, neglecting the stochastic nature of task release patterns in many real-world applications. Current approaches for handling stochastic release times rely on computationally expensive convolution-based methods, which has poor scalability, especially when both execution and release times are stochastic. This paper presents novel techniques to apply the Chernoff Bound approach to the analysis of systems with both stochastic execution and release times. The key challenge lies in adapting the Chernoff Bound, which traditionally operates on a fixed number of random variables, to handle the stochastic job counts resulting from stochastic release patterns. Our main contribution is a new technique for bounding convolutions involving random numbers of random variables using Chernoff principles. Through comprehensive evaluation, we demonstrate that our approach achieves several orders of magnitude speedup compared to state-of-the-art convolution-based methods while simultaneously improving analysis precision.