MAST30001 Stochastic Modelling
The Complete Exam Bible · Semester 2 2026MAST30001 Overview
- University of Melbourne
- Semester 2 2026
- 8 chapters
- Statistics
MAST30001 Stochastic Modelling develops mathematical models for systems whose random variation affects prediction or decision making. The subject begins by asking what a model should retain from a real system, then builds a set of connected process families.
- Assessed by Assignment 1 10% · Assignment 2 10% · Written examination 80%
- Key terms Stochastic Process, Markov Property, Transition Matrix, Stationary Distribution
- How to prepare Treat the subject as a decision tree.
- Most asked What should I prioritise for the final exam?
How MAST30001 is assessed
| Component | Weight | Format |
|---|---|---|
| Assignment 1 | 10% | Written problems posted in Week 6 and submitted by 5pm Friday of Week 7 |
| Assignment 2 | 10% | Written problems posted in Week 11 and submitted by 5pm Friday of Week 12 |
| Written examination | 80% | Three hours during the examination period |
What MAST30001 covers
Move from probability foundations to Markov chains, arrival processes, queues, renewal cycles, Monte Carlo methods and Brownian motion.
Probability Foundations for Stochastic Models
State choice, conditioning, expectation, transforms and simulation foundationsA stochastic model begins with a system, a state description and an explicit account of randomness.
- Why physical, mathematical and computational models simplify real systems
- Choosing a state that retains the information needed for prediction
- Events, conditioning and the law of total probability
- Expectation, variance, independence and random sums
- Exponential waiting times and memorylessness
- Inverse-transform simulation from uniform random values
Discrete-Time Markov Chains
Transition matrices, communicating classes, stationarity and first-step analysisDiscrete-time Markov chains model systems that move among states at integer steps. This chapter develops transition matrices and multi-step probabilities, then uses the directed graph to find communicating classes, recurrence and period.
- The Markov property and homogeneous one-step transitions
- Transition matrices, initial distributions and matrix powers
- Directed graphs and communicating classes
- Recurrence, transience, period and positive recurrence
- Stationary and limiting distributions
- First-step equations for hitting probabilities and times
Poisson Processes
Counts, arrival times, thinning, superposition and compound processesPoisson processes describe random event streams with independent increments and a constant rate. The same process can be viewed through counts N(t), arrival epochs T(k), or exponential interarrival times.
- Definition through independent increments and Poisson counts
- Count, arrival-time and interarrival descriptions
- Exponential waiting times and gamma arrival epochs
- Superposition of independent streams
- Independent thinning and marked arrivals
- Compound Poisson sums and PASTA
Continuous-Time Markov Chains
Generators, transition equations, stationary flow and birth-death chainsContinuous-time Markov chains move among states at random times. Their transition family P(t) obeys Chapman-Kolmogorov, while the generator Q stores instantaneous jump rates.
- Continuous-time Markov property and homogeneity
- Chapman-Kolmogorov for the transition family P(t)
- Generator rows, holding times and embedded jumps
- Forward and backward transition equations
- Stationary equations and long-run flow
- Birth-death chains and adjacent balance
Queueing Systems
Stability, stationary queue lengths, waiting times, PASTA and Little's lawQueueing models connect random arrivals, service capacity and congestion. The M/M/1 system uses a Poisson arrival stream, exponential service and one server, making its queue length a birth-death process.
- Arrival process, service mechanism, servers, capacity and discipline
- M/M/1 queue as a birth-death chain
- Traffic intensity and the stability condition
- Stationary queue-length distribution
- Waiting-time distributions and PASTA
- Little's law for system and queue measures
Renewal Theory
Cycle counts, limit theorems, rewards, age, overshoot and size biasRenewal theory studies repeated independent cycles with an arbitrary common lifetime distribution. It preserves the relationship between event counts and cumulative arrival times without requiring exponential gaps.
- Renewal process definition and cumulative renewal times
- Count-time event identities
- Long-run renewal rate from the mean cycle
- Renewal central limit approximation
- Renewal-reward rates and long-run proportions
- Age, overshoot and size-biased intervals
Markov Chain Monte Carlo
Target stationarity, detailed balance, proposal moves, acceptance and mixingMarkov chain Monte Carlo constructs an ergodic chain whose stationary distribution is a target that may be difficult to sample directly.
- Sampling a target through long-run Markov-chain visits
- Unnormalised target weights and proposal probabilities
- Metropolis-Hastings acceptance ratio
- Detailed balance and stationary distributions
- Rejected proposals and self-transition probability
- Mixing, autocorrelation and proposal tuning
Brownian Motion
Normal increments, covariance, hitting times, maxima and random-walk limitsBrownian motion is a continuous process with independent increments and normally distributed changes whose variance equals elapsed time.
- Continuous paths and independent normal increments
- Time scaling and marginal normal distributions
- Joint Gaussian distributions and covariance min(s,t)
- Conditional distributions from fresh increments
- First-hitting times and the reflection principle
- Path maxima and random-walk limits
Discrete-time Markov chains describe stepwise state changes through transition matrices, communicating classes, recurrence, period, stationarity and first-step hitting analysis.
Poisson processes connect event counts, arrival times and exponential gaps, with thinning, superposition and compound sums extending the basic model.
Continuous-time Markov chains replace one-step probabilities with jump rates stored in a generator. Their birth-death structure leads directly to queueing systems, where stability, stationary queue lengths, PASTA and Little's law connect congestion to waiting time.
Renewal theory removes the exponential assumption and studies repeated independent cycles, long-run renewal rates, renewal rewards, age and overshoot. Markov chain Monte Carlo turns stationarity into a computational tool by constructing a chain whose visit frequencies match a target distribution.
Brownian motion closes the map with continuous paths, independent normal increments, covariance, hitting times and maxima.
The published assessment consists of two written assignments worth 10% each and a three-hour written examination worth 80%. Because the exam carries most of the subject mark, revision should focus on model recognition as well as calculation.
Every worked problem in this guide uses original settings and numbers. The method is consistent across chapters: define the random object, state the assumption that licenses the model, write the governing relation, compute with units, and interpret the result in the original system.
Worked example · free
Stability and delay in a single-server queue
- 1Traffic intensity is 6/10 = 0.60, below one, so stationary formulas apply.
- 1Mean number is 0.60/(1-0.60) = 1.50 requests.
- 1Mean total time is 1/(10-6) hour = 0.25 hour = 15 minutes.
- 1Little's law verifies 6(0.25) = 1.50.
Key terms
- Stochastic Process
- A collection of random variables indexed by time or another ordered parameter, used to model a system that evolves randomly.
- Markov Property
- The conditional law of the future depends on the past only through the current state under the chosen model.
- Transition Matrix
- A row-stochastic matrix whose entry from row i to column j is the one-step probability of moving from state i to state j.
- Stationary Distribution
- A probability distribution that remains unchanged after the process evolves for one transition or any elapsed time.
- Poisson Process
- A counting process with independent increments whose count over length t is Poisson with mean equal to rate times t.
- Generator Matrix
- A matrix of continuous-time jump rates with nonnegative off-diagonal entries and rows that sum to zero.
- Traffic Intensity
- The arrival rate divided by service capacity, used first to test whether a queue can have a stationary regime.
- Little's Law
- The long-run relation equating mean number in a system to throughput rate multiplied by mean time in the system.
- Renewal Process
- A counting process whose successive cycle lengths are independent and identically distributed nonnegative random variables.
- Detailed Balance
- Pairwise stationary flow equations that provide a sufficient reversible route to a stationary distribution.
- Brownian Motion
- A continuous process with independent normal increments and variance equal to elapsed time in the standard model.
- Hitting Time
- The first time a stochastic process enters a specified state or reaches a specified level.
MAST30001 FAQ
How is the subject assessed?
The current subject page lists two written assignments worth 10% each and a written examination worth 80%. The first assignment is posted in Week 6 and due in Week 7; the second is posted in Week 11 and due in Week 12. The examination lasts three hours during the examination period.
What should I prioritise for the final exam?
Practise identifying the process before applying a formula. The review outline spans discrete-time chains, Poisson processes, continuous-time chains, queues, renewal theory and Brownian motion, while current materials also include Markov chain Monte Carlo. Rehearse definitions, licensing conditions, complete numerical working and one interpretation sentence for each result.
How much probability background is needed?
The opening material reviews probability spaces, random variables, conditional reasoning, expectation and simulation before the main process topics. Tutorial 1 is explicitly positioned as a diagnostic review. If conditioning, exponential waiting times or expectations feel slow, repair those foundations early because every later chapter uses them.
When can a Markov chain use a stationary distribution?
A stationary vector must satisfy the relevant balance equation and normalise to one. Interpreting it as a limiting distribution also requires structural conditions. For an ergodic discrete-time chain, irreducibility, aperiodicity and positive recurrence matter. For queueing and birth-death models, first verify that the stationary weights are summable and the stability condition holds.
Which queueing formulas are most important?
For M/M/1, begin with traffic intensity and require it to be below one. Then connect the geometric stationary distribution to mean system size, total time and waiting time. Little's law provides the strongest cross-check: use L = lambda times total time and the queue-only version with matching populations and units.
What materials are permitted in the examination?
The captured subject assessment block confirms a three-hour written examination but does not publish a permitted-materials rule. It also does not state a fixed section split or question count. Check the current subject LMS and the instructions issued for the actual examination sitting rather than assuming an open-book or closed-book format.
Questions students ask in each MAST30001 chapter
- Why must the state be defined before calculating?
- The state determines whether the future can be predicted from current information alone and fixes the units and sample paths. Omitting relevant history can destroy the Markov property, while including unnecessary history makes the model harder without improving its predictive law. Chapter 01 · Probability Foundations for Stochastic Models
- How do I orient a transition matrix?
- Under the course convention, row i describes the current state and column j the next state, so p_ij means from i to j. Each row must be nonnegative and sum to one, and a row-vector distribution evolves by multiplying on the right. Chapter 02 · Discrete-Time Markov Chains
- When should I use a count rather than a waiting time?
- Use a count when the event asks how many arrivals occur by a fixed time. Use an arrival time when it asks when a numbered event occurs. The identity T(k) at most t exactly matches N(t) at least k, so choose whichever distribution is easier. Chapter 03 · Poisson Processes
- How is a generator different from a transition matrix?
- A transition matrix contains probabilities and has rows summing to one. A generator contains instantaneous rates, has nonnegative off-diagonal entries and rows summing to zero, with negative diagonal entries equal to minus the total departure rate from each state. Chapter 04 · Continuous-Time Markov Chains
- Why must traffic intensity be below one?
- The server must complete work faster on average than work arrives. If arrival rate meets or exceeds service rate, the M/M/1 stationary weights cannot be normalised and the queue grows without a stationary regime under the unchanged assumptions. Chapter 05 · Queueing Systems
- How is a renewal process different from a Poisson process?
- A renewal process allows any independent identically distributed nonnegative gap law. A Poisson process is the special exponential-gap case, which adds memorylessness and a Markov count process. General renewal counts usually need age information for a Markov state. Chapter 06 · Renewal Theory
- Why can the target be unnormalised?
- The Metropolis-Hastings ratio divides one target weight by another. Any common unknown normalising constant appears in both numerator and denominator and cancels, so relative positive weights are enough to construct the accepted transitions. Chapter 07 · Markov Chain Monte Carlo
- Are Brownian values at different times independent?
- No. Values share the earlier path history, giving covariance equal to the earlier time. What is independent are increments taken over disjoint intervals. Decomposing the later value into the earlier value plus a fresh increment makes this distinction clear. Chapter 08 · Brownian Motion
How to study for the exam
Treat the subject as a decision tree. First ask whether time is discrete or continuous and whether the object is a state, count, waiting time, queue length, renewal cycle, simulated target or continuous path.
Second, write the property that licenses the method: Markov dependence through the state, independent Poisson increments, exponential memorylessness, queue stability, regenerative cycles, detailed balance or independent Brownian increments. Third, solve one original problem without notes and check it independently.
Transition rows must sum to one, generators to zero, probabilities stay between zero and one, thinned rates recombine, queue measures satisfy Little's law, renewal rates have reciprocal-time units, MCMC transition rows include rejected mass, and Brownian variances equal elapsed time. Keep an error log organised by model choice, set-up, algebra, units and interpretation.
In the final week, mix chapters so the process is not announced by the page heading. Because the published examination is three hours and carries 80%, use the paper's own marks and instructions to set the sitting-specific time budget.
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