Availability rates and step probabilities are different model inputs

Let an up service fail at constant rate λ and a down service recover at constant rate μ. These are nonnegative inverse-time quantities, not probabilities. The continuous-time Markov model assumes exponential state holding times with no memory of elapsed time in the state. With positive rates, mean up duration is 1/λ and mean down duration 1/μ. A repair process with staged diagnosis, a scheduled human handoff, or an aging hazard may not fit that assumption. Adding states or using a non-Markov renewal model can be more appropriate than assigning one convenient average rate.

The stationary up probability is μ/(λ+μ), while transient availability also depends on the starting state. Starting up gives a decaying approach from one to the stationary value. The stationary result is a long-run model property, not a promise about every finite interval or every customer. If both rates are zero there is no unique stationary distribution and the initial state remains fixed. If exactly one is zero, an absorbing state changes the interpretation. Keep these cases visible rather than dividing by zero or announcing that every two-state model necessarily mixes.

A discrete-time chain instead assigns a probability p of moving from up to down in one step and q of moving back. Its stationary up probability q/(p+q), when defined, resembles the continuous formula but uses different inputs. For an exact sampled two-state continuous process at interval Δ, the transition probabilities include the possibility of several transitions inside the interval. Simply setting p=λΔ and q=μΔ is only a small-step approximation under appropriate limits and can even produce invalid probabilities for large Δ.

Discrete chains can also behave in ways that no finite-rate two-state continuous process sampled at a positive finite interval does. With p=q=1, the chain alternates forever from its specified initial state rather than converging to its stationary distribution. With p=q=0 it never moves. A displayed stationary vector alone does not establish convergence or a physical failure rate. State the observation interval, initial state, and whether the model is a sampled process, an approximation, or a discrete operational rule before interpreting its animation as time spent unavailable.

Measure actual up and down intervals, transition counts, repair phases, and missing observations. Monitoring that polls once a minute can miss short outages and repairs, so observed state changes are not automatically the underlying transition events. A stable average can hide correlated incidents or changing repair policy. Fit and validate the model against the question being asked: instantaneous availability, finite-window downtime, and mission survival are different targets. Report parameter uncertainty and test whether exponential holding times and a constant environment are plausible for the chosen data.

The Markov lab is explicitly discrete. Reset, set BOTH p=0.1 and q=0.4, predict down probabilities 0.1 and 0.15 after one and two transitions, then Step twice. The stationary down value is 0.2. Reset and re-set BOTH p=q=1 to inspect alternation. Each Step advances one transition because n=floor(2t); animation seconds are not physical repair hours. Edits retain t and recompute from initial up state. No continuous rates or exponential repair durations are simulated, so do not label the sliders λ and μ.