Optimum warmness levels in general standby systems

This study explores general standby systems featuring multiple primaries and backups. All primaries operate at a uniform warmness level of one. Each backup operates at an adjustable warmness level, which ranges from zero (cold standby) to one (hot standby) with intermediate values representing warm standbys. The objective is to identify the optimum warmness level for each backup to minimize the average total cost per unit time. This total cost comprises the operating cost of each component and the opportunity loss incurred when waiting for a backup with a warmness level below one to become available as a primary when needed. Both operating and opportunity costs vary depending on component warmness levels, with expected values contingent upon system states and transitions. The state and transition probabilities calculations employ the Markov model, assuming all components have lifetimes that follow independent and not necessarily identical exponential distributions. The Markov model is represented in a cost-probability transition diagram. The model is utilized to investigate characteristics of optimum warmness levels, both with and without energy constraints. Its applications for series-standby systems are demonstrated.