A shared device is either available or busy at each inspection. A deterministic average occupancy does not describe the actual sequence of switches. Instead, define probabilities for the next state conditional on the current state. The time between inspections is part of the model; a one-minute transition matrix cannot simply be relabeled as a one-hour matrix.
For this synthetic example, an available device stays available with probability 4/5, while a busy device becomes available with probability 2/5.
Predict
If half the devices are available now, predict the next available proportion. Do not average 4/5 and 2/5 without checking the weights: they are conditional probabilities associated with the current state distribution.
Experiment
Write the row-stochastic transition matrix with state order available, busy.
Current state
Next available
Next busy
Available
4/5
1/5
Busy
2/5
3/5
If p is the current available probability, next p is 4p/5+2(1−p)/5=2/5+2p/5. From p=1/2 this gives 3/5. Solving p=2/5+2p/5 gives stationary probability 2/3. Calculate another step from 3/5 and compare its distance from 2/3 with the previous distance.
Understand
Every transition entry must be nonnegative and every row must sum to one. With row probability vectors, evolution is π next=πP. The row convention determines the multiplication order; a column convention is equally valid if used consistently. A stationary distribution satisfies πP=π, together with nonnegativity and total probability one.
Stationarity means the distribution stays unchanged, not that each device remains in its current state. Individual paths can keep switching while the ensemble proportion remains stable. Likewise, the expected state is not necessarily an admissible individual state: a device cannot literally be two-thirds available.
The Markov assumption says that, conditional on the current state, the next-state distribution does not depend on earlier history. Time homogeneity additionally says the transition law does not change with inspection time. These are distinct assumptions. A device whose chance of becoming free depends on how long it has been busy may require a duration state or a non-Markov model. Rush-hour behavior may require time-varying probabilities even if the state description is otherwise adequate.
Observed transition counts can estimate rows, but small counts leave uncertainty. A row with no observed departures has no empirical conditional distribution; assigning zeros would violate the row-sum requirement. State a prior, collect more observations or report the row as unestimated. A synthetic assumed row should remain labeled as such.
Apply
Build a three-state process with clearly observable categories. Check every row, calculate one step from a chosen distribution and identify a pattern of duration or time dependence that would challenge your assumptions. Distinguish an exact calculation under the proposed matrix from evidence that the matrix describes a real process.
Continue
Networks next model how resources move through connections. Their static flow constraints can certify throughput, but they do not automatically account for the random timing and queues hidden by an aggregate model.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
State A persists with probability 3/4; B moves to A with probability 1/2. If P(A)=2/5, find next P(A).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For that two-state chain, find stationary P(A).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
State and verify the requirements for a row-stochastic transition matrix.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Give an observation that would challenge a two-state time-homogeneous Markov assumption.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Why does a stationary distribution not say an individual trajectory stays constant?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Construct a three-state model for a bounded operational process.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
