Repair cycles can have memory even when their averages match

In an alternating renewal model, a cycle consists of an up period U followed by a down period D before a comparable new cycle begins. Assume independent identically distributed cycles, positive finite expected cycle length, and the regularity needed for the long-run limit. Counting up time as reward gives long-run availability E[U]/E[U+D]. This ratio of expectations is not generally the expectation of each cycle's up fraction. Long cycles occupy more observed time, so averaging short and long cycles with equal weight can answer a different question.

The model does not require U and D to be exponential. Dependence within a cycle can be allowed when the regenerative-cycle assumptions remain valid, but dependence across successive cycles or a changing environment needs a richer model. A successful repair that returns a service to a known reference state is a candidate renewal boundary. A repair that leaves accumulated wear, degraded redundancy, or a backlog of related incidents may not start an identical new cycle. State what is reset rather than assuming every incident closure makes the process as good as new.

An exponential down duration has a memoryless remaining lifetime: waiting ten minutes does not change its conditional distribution of additional waiting. A deterministic repair duration has the opposite structure: after most of the required work has elapsed, little time remains. Weibull and lognormal families can represent other shapes, subject to evidence. A single mean cannot distinguish these conditional behaviors. If a runbook gives an expected remaining recovery time based on elapsed incident age, it is implicitly making a distributional claim that should be checked against completed and still-open incidents.

Failure-rate models for repairable systems also differ from nonrepairable lifetime models. An as-good-as-new replacement can support renewal lifetimes. Minimal repair may restore operation while leaving the age-dependent failure intensity largely unchanged. NIST distinguishes homogeneous and nonhomogeneous Poisson repair-rate models from lifetime distributions; a power-law process is not simply a sequence of iid Weibull lifetimes. Confusing these models can make an improving or deteriorating system appear stationary. Select the model for the actual maintenance effect and the observation timeline rather than matching a distribution name to a familiar curve.

Collect up and down durations with consistent start and end definitions, including observation gaps and open incidents. Distinguish detection, mitigation, full repair, and restoration of redundancy; they can be different endpoints. A service can resume user traffic before the system is ready to tolerate another fault. Model the endpoint relevant to the objective and avoid silently mixing phases into one MTTR field. Validate cycle independence across maintenance campaigns, software versions, and shared events. A stationary estimate can be useful for a chosen period while failing as a forecast after a major operational change.

The Markov lab is a memoryless discrete baseline, not a general renewal simulator. Reset, set BOTH p=0.2 and q=0.25, predict stationary down probability 0.2/0.45≈0.4444, then Play or Step. Reset and re-set BOTH p=0.2 and q=0.5 for a baseline of about 0.2857. n=floor(2t), and edits retain t/recompute from initial up state. The sliders specify per-step transitions, not deterministic or Weibull repair durations. Explain which elapsed-age information this two-state chain omits before using the cycle formula and repair evidence to choose a different model.