A MAP/PH/1/K Queueing Model with N-Policy for Optimal Regeneration of a Diesel Particulate Filter

This paper analyzes a MAP/PH/1/K queue with N-policy, setup, interruptions, reset, and a random environment. Arrivals are the MAP; service, setup, interruption, and reset times are PH-distributed. Under the N-policy, the server idles until the queue length is equal to N, and then performs setup. Interruptions return the system to idle and re-enable the N-policy. At capacity K, a reset empties the system. The random environment modulates parameters for different regimes. Motivated by Diesel Particulate Filter (DPF) regeneration, soot accumulation is mapped to arrivals, burning to service, regeneration triggers to N-policy, heating to setup, engine changes to interruptions, and cleaning to reset. Environmental states represent driving patterns. Regeneration succeeds if either the system empties via service or an interruption occurs with remaining soot less than or equal to level L. We derive the block-structured generator, obtain stationary probabilities via matrix-analytic methods, and optimize the threshold N via average cost. Numerical results quantify how correlation and driving conditions affect performance and costs, offering tools to balance fuel consumption, engine performance, and filter longevity.