Reverse Poisson Counting Process with Random Observations

This paper introduces the Reverse Poisson Counting Process (RPCP), a stochastic model derived from the Poisson counting process with limited and random capacity, characterized by counting backward from a defined maximum level. Explicit analytical formulas are developed for key stochastic properties, including the probability mass function and mean, specifically for scenarios involving random capacity. The model is shown to represent extreme cases of established stochastic processes, such as the M/M/1 queue with instant service completion and the death-only process. Practical applications are discussed, focusing on enhancing streaming service availability and modeling transaction validation in decentralized networks with dynamic node populations on lightweight devices. Analysis under Binomial random capacity demonstrates the impact of node accountability on MSER and network reliability, highlighting the value of incorporating stochastic variations for optimizing system performance. The explicit formulas facilitate straightforward analysis and optimization of system functionals.