A generalized framework for the generation of Periodic Breathing (PB), caused by delay variation, hypocapnia and sleep along with its management with oxygen therapy is presented. For this, a minimal model of respiratory regulation with cardiovascular component and two delays is proposed. This model is linearized and analyzed for stability by the proposed algorithms using Lyapunov-Krasovskii functional. Oscillation in this model is produced by the increase of delays, an increase of chemoreceptor gains (hypocapnia) and a decrease in minute ventilation (sleep induced PB). For delay variation, it is established that both the delays are responsible for oscillation in the system. However, maximum tolerable delay limit for the peripheral chemoreceptors is lower (0.3  min) compared to the central chemoreceptors (5.2  min). Stability analysis shows that application of additional oxygen is capable of suppressing the oscillation in the system by increasing the tolerable delay limit. Hypocapnia caused by hyperventilation is modeled by increased chemoreceptor gain. 50% increase in chemoreceptor gain along with 46% decrease in lung carbon dioxide storage makes the system oscillatory, which increases average minute ventilation by 19.42%. Application of additional oxygen makes the system stable. For sleep induced PB, it is shown that lowering minute ventilation causes oscillation in the system. A parameter is introduced to limit the minute ventilation in sleep, and its upper limit is calculated (8.7% drop in minute ventilation) for producing oscillation in the system. Application of higher oxygen makes the system stable by compensating for the reduction. Finally, two simulation studies are presented, showing the delay limits in hyperventilation and sleep condition. In these conditions, as the gains increase or minute ventilation decreases, tolerable delay limits become smaller.