A computationally efficient Bayesian Monte Carlo for Monotonic (BMCM) models for reliability based design of engineering systems is described in this paper. The model employs Gaussian distribution and monotonicity principles that have been implemented in the Dynamic Bounds (DB) method (Rajabalinejad 2009) integrated with a Bayesian Monte Carlo (BMC) technique. Signficant improvements in the computational speed of coupled DB and BMC methods are realized by incorporating a weighted logical dependence between neighboring points of the Limit-State Equation (LSE) as prior information and global uncertaintiy concept for quantifying variations of the controlling input variables. The outcomes of preceding simulations are factored in subsequent calculations to accelerate computing efficiency of the Monte Carlo method. The theory and numerical algorithms of the BMCM are described in this paper, and extension of the BMCM to multi-dimensional problems is provided.