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Breakdowns and Buffers by the Numbers

Markov models for repairable machines: availability, stoppage counts, series lines, buffers, repair crews, spares, redundancy and the cost of downtime

Model breakdowns, repairs, buffers, maintenance crews, spares and standby machines as Markov processes, and use the results to choose improvements that raise a line's output.

10 chapters · About 100 minutes · Free to read · Intermediate production, maintenance and reliability engineering

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KEVOS® LibraryOPERATIONS↑→↓←VALUEBreakdowns andBuffers by theNumbersKEVOS® ORIGINAL BOOK10 CHAPTERS

About this book

Machines stop. A packing line, a press, a compressor or a pump runs for a while, fails or jams, waits for someone to fix it, and runs again. Over a shift the pattern looks like noise. Over a month it has a shape that probability describes well: stops arrive at a fairly steady average rate, repairs take a fairly steady average time, and the machine spends a predictable share of its life in each state. That description is a Markov model, and once a machine is written down that way, questions that look like guesswork become calculations. What output will this line make? What is a bigger buffer worth? Is one fitter enough for four machines? How many spare drives should be on the shelf? How often will both compressors be down at once?

This book builds the models one step at a time. It starts with a single machine that is either running or stopped, tests whether real stoppage data are random enough for the model, and adds states for minor and major stops. It then joins machines in series, with and without a buffer between them, and turns to the people and parts that repairs need: the fitter, the maintenance crew and the spares shelf. It finishes with standby machines and the time until both are down, and with the decision that all of this serves: which improvement buys the most output for the money.

It is for production, maintenance and reliability engineers, operations managers, planners and students who keep or analyse stoppage records. It assumes algebra, averages and the idea of probability; the exponential function is explained where it appears. Each chapter works the running case with real numbers, includes one plausible mistake and how it was found, links the KEVOS® tool that performs the chapter’s calculation where one exists, and ends with an exercise and its worked answer.

The running case

Every chapter follows one fictional line. This is an illustration. A snack-food manufacturer runs four identical packing lines. Each line has a bagger, a vertical form-fill-seal machine making 60 bags a minute, feeding a case packer rated at the equivalent of 66 bags a minute through an accumulation conveyor that holds five minutes of the bagger’s output, 300 bags. The plant works two 8-hour shifts a day, 250 days a year: 500 shifts.

  • The bagger: stops on average every 2.0 running hours and takes 15 minutes on average to restart. Most stops are minor film and product jams cleared by the operator in a few minutes; about one in five is a breakdown that needs a fitter for about an hour.
  • The case packer: stops on average every 4.0 running hours and takes 12 minutes on average to restart, with most stops a short carton jam and a few long ones.
  • Maintenance: the four lines share one fitter per shift for breakdowns; the site’s wider crew looks after 40 machines.
  • Parts and services: 24 servo drives across the four lines, repaired by the maker overseas, and two air compressors, one needed, serving the packing hall.
  • Money: each bag contributes A$0.10 towards fixed costs and profit, excluding GST.

The line, its stoppage data, the costs and the contribution are invented. None describes KEVOS® work or a real business.

How to use the book

Read Chapters 1 to 3 in order: they set out the two-state model, the tests of randomness and the balance equations that every later chapter uses. Chapters 4 and 5 deal with lines of machines and should be read together. Chapters 6 to 9 each take one supporting resource (the fitter, the crew, the spares and the standby machine) and can be read in any order. Chapter 10 brings the results together into a decision.

The models are solved with closed-form results where they exist and with a simulation, checked against those results, where they do not. Values are given to more figures than the data could justify, so that a reader can follow each step.

Source limits and professional responsibility

Markov models describe machines whose stops arrive at random and whose repairs are independent of the past. Real machines also wear out, follow shift patterns and depend on the people who run them, and the book shows how to test for those effects and when to use other methods instead. Its numbers are only as good as the stoppage records behind them.

Clearing jams and repairing machines are among the most hazardous tasks in a factory. Work health and safety law requires plant to be isolated before such work, and Safe Work Australia’s model code of practice on managing the risks of plant sets out how. Nothing in this book’s arithmetic about shortening repair times justifies skipping isolation. The book teaches how to model and improve the output of repairable machines. It is educational and is not a maintenance plan for any site.

Who this book is for

Production, maintenance and reliability engineers, operations managers, planners and students who need to predict and improve the output of machines that stop and get repaired.

What you will learn

  • Build two-state and multi-state Markov models of repairable machines from a stoppage log and solve their balance equations.
  • Test whether stoppages are random enough for a Markov model, with Poisson counts, control charts and simple checks on intervals.
  • Predict the output of machines in series, with and without a buffer, and judge what a buffer is worth.
  • Size a repair crew and a spares holding with birth–death models, and know when an infinite-source queue overstates the waits.
  • Calculate the time to first failure of a redundant pair with repair, including common-cause failures, and rank improvements by the output and contribution they add.

Inside this book

Chapters

  1. 01Up and down: the two-state modelA repairable machine as a two-state Markov process: failure and repair rates from a stoppage log, steady-state availability, settling after a restart, and why stop definitions matter.
  2. 02Are the stops random? Poisson counts and memoryless clocksTesting a stoppage log for the randomness a Markov model assumes: Poisson counts per shift, a c chart for special causes, exponential intervals, repair time spread and wear-out parts.
  3. 03More states and the balance equationsSplitting stops into minor jams and breakdowns: a three-state Markov model, its balance equations and detailed balance, slower settling, long-stop risk, and availability inside OEE.
  4. 04Machines in series without a bufferTwo machines coupled directly: the Markov chain of a series line, operation-dependent and time-dependent failures, why downtime ratios add, and what long rigid lines lose.
  5. 05Buffers between machinesHow an accumulation buffer decouples two machines, why a speed margin is needed to use it, output against buffer size from a validated model, and why the mix of stop lengths sets a buffer's worth.
  6. 06One fitter, four machines: the machine repair modelA birth–death model for machines that share one repairer: how often breakdowns queue, the waiting hidden in logged repair times, fitter utilisation, and what a second fitter would buy.
  7. 07Sizing the maintenance crewBreakdown calls from many machines as a queue: the M/M/c model and Erlang C, utilisation and waiting, Little's law, when an infinite-source queue overstates waits, and crew size.
  8. 08Spares and the repair pipelineSpares for random failures: Poisson demand over the lead time, Palm's theorem for units out for repair, service level against the cost of a stock-out, and the hours that set the failure rate.
  9. 09Redundancy and the time to first failureTwo compressors, one needed: mean first passage time to losing both, active and standby redundancy, why repair makes redundancy work, failures to start, and common causes.
  10. 10From model to decision: what to fix firstPricing a minute of downtime, ranking improvements by the output they add on a buffered line, wear-out parts on a schedule, checking predictions against results, and the Pareto chart that misled.

Apply This in Practice

Use these KEVOS® tools to try the method from this reading with your own numbers.

References and source notes

The manuscript and worked examples are original KEVOS® writing. References inform particular concepts; their authors do not endorse this book. Web references were consulted on 30 October 2026.

  • L. E. Reichl, John Wiley & Sons. A Modern Course in Statistical Physics, 2nd edition (1998). In copyright. Used only as a map of the probability behind the book: the Poisson distribution, Markov chains, the master equation, detailed balance, mean first passage times and birth–death processes. No text, structure, figures, tables or problems are reproduced.
  • National Institute of Standards and Technology. NIST/SEMATECH e-Handbook of Statistical Methods, Chapter 8: Assessing Product Reliability (2012). Open reference for lifetime distributions, repairable-system models and the treatment of failure data used in Chapters 1, 2 and 10.
  • R. G. Gallager, MIT OpenCourseWare. 6.262 Discrete Stochastic Processes (2011). Open university course used as the checkable reference for Poisson processes, finite-state Markov chains, birth–death processes and queues.
  • L. Kleinrock, John Wiley & Sons. Queueing Systems, Volume 1: Theory (1975). Named in Chapters 6 and 7 as a standard text for the M/M/c queue, the Erlang C formula and the finite-source (machine repair) model. No text is reproduced.
  • J. D. C. Little, Operations Research 9(3), 383–387. A proof for the queuing formula L = λW (1961). Named in Chapters 5 and 7 as the source of Little's law.
  • J. A. Buzacott and J. G. Shanthikumar, Prentice Hall. Stochastic Models of Manufacturing Systems (1993). Named in Chapters 4 and 5 as a standard text on transfer lines, operation-dependent failures and buffers between machines. The book's line results come from its own calculations and simulation.
  • C. C. Sherbrooke, Kluwer Academic Publishers. Optimal Inventory Modeling of Systems: Multi-Echelon Techniques, 2nd edition (2004). Named in Chapter 8 as a standard account of Palm's theorem and spares for repairable items.
  • International Electrotechnical Commission. IEC 61165 Application of Markov techniques (current edition). Named by number and scope as the international standard on applying Markov models in dependability analysis. Paywalled; no clauses or values are quoted.
  • Safe Work Australia. Model Code of Practice: Managing risks of plant in the workplace (current edition). Named in Chapters 3 and 6 for isolation before clearing jams and repairing machines. Readers check whether the code applies in their state or territory.

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