Simulation and Stochastic Experimentation

Introduction and context

Operations managers frequently make decisions in systems where demand, processing times, failures, arrivals, transport times, and customer behaviour are uncertain. Analytical models are powerful when assumptions are simple enough to produce tractable equations. However, real systems often contain multiple interacting queues, priorities, resource constraints, time-dependent demand, disruptions, and nonlinear policies. In such settings, simulation provides a laboratory for experimentation: a model of the system is executed repeatedly under controlled conditions, and the resulting observations are used to estimate performance and compare alternatives.

Simulation does not eliminate uncertainty. It represents uncertainty explicitly and estimates the consequences of that uncertainty. The quality of a simulation study therefore depends not only on programming skill but also on conceptual modelling, data analysis, statistical design, verification, validation, and managerial interpretation.

This chapter focuses on Monte Carlo simulation and discrete-event simulation (DES), with emphasis on operations-management applications such as inventory, logistics, healthcare, manufacturing, service operations, and supply chains.

Learning objectives

By the end of this chapter, the learner should be able to:

1. Distinguish Monte Carlo, discrete-event, continuous, and hybrid simulation.
2. Formulate a stochastic operational system as entities, resources, events, states, and performance measures.
3. Fit and assess probability distributions for simulation inputs.
4. Generate random observations using pseudorandom numbers and appropriate transformations.
5. Build event-scheduling and process-interaction models.
6. Verify computer logic and validate model representation and outputs.
7. Design warm-up, replication, run-length, and variance-reduction strategies.
8. Analyse simulation outputs using confidence intervals and paired comparisons.
9. Compare operational alternatives while accounting for statistical and managerial significance.
10. Recognise limitations, ethical risks, and governance requirements in simulation studies.

1. Simulation as an operations-research method

A simulation model is a simplified representation of a real or proposed system. It specifies entities, activities, resources, rules, state variables, random inputs, and outputs. A simulation experiment changes one or more controllable assumptions and observes the resulting system behaviour.

A Monte Carlo simulation repeatedly samples random quantities and evaluates a mathematical or logical model. It is appropriate when the principal problem is uncertainty propagation rather than time-dependent system dynamics. For example, a firm may simulate annual project cost by sampling uncertain demand, price, exchange rate, and delay.

A discrete-event simulation represents a system whose state changes at identifiable points in time. Examples include an arrival, service completion, machine failure, replenishment receipt, or truck departure. Between events, the state is usually constant. DES is especially useful for queues, production lines, hospitals, warehouses, call centres, and transport networks.

A continuous simulation models state variables that change continuously, often through differential equations. A hybrid model combines continuous and discrete-event components, such as a chemical process with discrete maintenance decisions.

Simulation is most valuable when:

• interactions among components are important;
• the system is dynamic and path-dependent;
• policies involve complicated rules;
• analytical assumptions such as exponential service times are unrealistic;
• the proposed system has not yet been built;
• performance must be evaluated under alternative scenarios.

Simulation is not automatically superior to analytical methods. If an exact or well-established queueing, inventory, or optimization model is available and its assumptions are acceptable, the analytical method may be faster, more transparent, and more accurate.

2. The conceptual structure of a stochastic simulation

A useful simulation model separates the following elements.

The model boundary must be explicit. For example, a warehouse model may include supplier lead time as an input rather than modelling the supplier’s internal production process. Boundary decisions affect both credibility and computational cost.

A stochastic model can be represented as a state-transition system. Let [[M:S(t)]] denote the system state at time [[M:t]]. If event [[M:e]] occurs at time [[M:t_e]], the state transition may be written as:

Here, [[M:S(t_e^-)]] is the state immediately before the event, [[M:S(t_e^+)]] is the state immediately after it, [[M:T]] is the transition rule, and [[M:X_e]] represents any random quantities associated with the event. The notation is general: it does not imply that every model has a closed-form transition equation.

Typical output measures include average waiting time, maximum queue length, throughput, utilisation, service level, stockout probability, total cost, and cycle time. Measures must be defined carefully. For example, resource utilisation over an observation interval is:

where [[M:B]] is the total time the resource is busy and [[M:T]] is the length of the observation interval. Both quantities must use the same time unit, and the interval must be defined consistently, particularly when a warm-up period is excluded.

3. Probability models for simulation inputs

3.1 Sources of input data

Input distributions may be obtained from:

1. historical observations;
2. time studies and process measurements;
3. expert judgement;
4. engineering specifications;
5. contractual or policy assumptions;
6. theoretical distributions justified by system mechanisms.

The analyst should distinguish aleatory uncertainty, arising from inherent variability, from epistemic uncertainty, arising from incomplete knowledge. More data may reduce epistemic uncertainty; it does not necessarily reduce genuine process variability.

3.2 Empirical and theoretical distributions

An empirical distribution uses observed values directly, possibly with interpolation or grouping. It is attractive when data are abundant and no theoretical form is credible. A theoretical distribution uses a parameterised family such as normal, lognormal, exponential, gamma, Weibull, triangular, or discrete categorical distributions.

The choice must be based on process knowledge, data quality, tail behaviour, and goodness of fit—not merely on the distribution with the smallest numerical test statistic.

Important examples include:

• Bernoulli: a binary outcome with probability [[M:p]] of success.
• Binomial: number of successes in [[M:n]] independent Bernoulli trials.
• Poisson: count of events in a fixed interval when events occur independently at a constant average rate [[M:λ]].
• Exponential: nonnegative time with rate [[M:λ]], often used for memoryless interarrival or failure times.
• Normal: symmetric continuous variation, though it is unsuitable for inherently nonnegative skewed times unless truncation or another model is justified.
• Lognormal: positive, right-skewed quantities whose logarithms are normally distributed.
• Gamma and Weibull: flexible positive distributions used for service, repair, reliability, and lead-time modelling.
• Triangular: minimum, most likely, and maximum values; useful when data are scarce but expert estimates are available.

For an exponential random variable [[M:X]] with rate [[M:λ]] measured in events per unit time:

and its mean is [[M:E[X] = 1/λ]]. The exponential assumption implies the memoryless property and should not be used simply because it is convenient.

3.3 Parameter estimation and goodness of fit

Parameters may be estimated by methods such as maximum likelihood, method of moments, or least squares. For independent observations [[M:x_1, x_2, …, x_n]], the sample mean is:

and the unbiased sample variance is:

The sample standard deviation is [[M:s = √s²]]. These statistics describe the sample; they do not prove that a theoretical distribution is correct.

Useful diagnostic tools include histograms, empirical cumulative distribution functions, Q–Q plots, box plots, and time-series plots. Formal tests include the chi-square test, Kolmogorov–Smirnov test, and Anderson–Darling test. Formal tests can be overly sensitive with large samples and weak with small samples. Operational plausibility and tail behaviour must therefore accompany statistical testing.

Data should also be checked for dependence, seasonality, trends, batching, censoring, missingness, and changes in operating policy. Fitting a single stationary distribution to data collected across different regimes can produce a misleading model.

3.4 Dependence and correlation

Many simulation inputs are correlated. Demand may be related to delivery time; workload may influence service time; failures may be clustered; and successive observations may be autocorrelated. Treating correlated inputs as independent can materially distort risk and capacity estimates.

Possible approaches include joint distributions, conditional distributions, copulas, time-series models, scenario-based sampling, and resampling blocks of observations. A simple model that represents major dependence is usually preferable to a sophisticated model whose assumptions cannot be defended.

4. Random-number generation and random variate generation

Computers generally use deterministic algorithms to produce pseudorandom numbers. A generator produces a sequence intended to behave like independent observations from the uniform distribution on [0,1]. Let [[M:U]] denote such a number. Reproducible seeds are essential for debugging and controlled comparisons, while independent or carefully separated streams are useful for replications and input factors.

A common linear congruential generator has the form:

where [[M:m]] is the modulus, [[M:a]] the multiplier, [[M:c]] the increment, and [[M:Z_0]] the seed. Modern libraries generally use better generators; the formula is presented to clarify the structure, not as a recommendation for production use without testing.

For inverse-transform generation, if a continuous random variable has cumulative distribution function [[M:F]], then:

has distribution [[M:F]] when [[M:U]] is uniform on [0,1]. For an exponential variable with rate [[M:λ]]:

Because [[M:1−U]] is also uniform, [[M:X = −ln(U)/λ]] is commonly used when [[M:U]] is strictly between 0 and 1. Units are consistent: [[M:λ]] has units of inverse time, so [[M:1/λ]] and [[M:X]] have units of time.

Normal variates may be generated using specialised library algorithms or transformations such as the Box–Muller method. Discrete distributions can be generated by cumulative-probability intervals. For example, if categories have probabilities 0.2, 0.5, and 0.3, a uniform draw in [0,0.2) selects category 1, [0.2,0.7) category 2, and [0.7,1) category 3.

Random streams should be documented. A simulation report should identify the generator, seed policy, software version, distributions, parameter values, and treatment of random-number streams.

5. Discrete-event simulation design

5.1 Event scheduling

In an event-scheduling approach, future events are stored in an event list ordered by event time. The simulation clock advances to the earliest event, the event is processed, the state is updated, and newly scheduled events are inserted.

A generic event loop is:

1. Initialise clock, state variables, resources, statistics, and event list.
2. Remove the event with the smallest scheduled time.
3. Advance the clock to that time.
4. Update time-persistent statistics using the elapsed time.
5. Execute the event logic.
6. Schedule any future events generated by the logic.
7. Continue until the stopping condition is reached.
8. Report observations and statistical estimates.

Time-persistent quantities require time weighting. If queue length is [[M:Q(t)]], the time-average queue length over [0,T] is:

In a DES, this integral is usually calculated by summing rectangles between events. If queue length remains [[M:Q_j]] for duration [[M:∆t_j]], then:

Counting statistics and time-persistent statistics must not be confused. The average number of customers served is a count-based measure; average number in queue is generally time weighted.

5.2 Process interaction

In process-oriented modelling, an entity follows a sequence such as:

Arrive → Join queue → Seize resource → Receive service → Release resource → Depart

This representation is often easier for managers to read, while the software translates it into event logic. Priority rules, balking, reneging, batching, preemption, setup times, routing, and failures must be stated explicitly.

5.3 Inventory and supply-chain simulation

An inventory simulation may update inventory position after demand, receipt, and review events. A common policy is an order-up-to policy: when inventory position falls to or below a reorder point, an order is placed to raise the position toward a target level. Inventory position typically includes on-hand inventory plus on-order inventory minus backorders, although exact definitions vary by policy.

A simulation should distinguish customer demand, fulfilled demand, lost sales, backorders, replenishment lead time, and holding or shortage costs. Otherwise, a high service level may be reported while substantial hidden backorders are ignored.

6. Verification and validation

Verification asks: “Was the model implemented correctly?” Validation asks: “Is the model an adequate representation for the intended decision?” The distinction is essential. A perfectly coded model can still represent the wrong system.

6.1 Verification techniques

Useful verification practices include:

• modular development and unit testing;
• tracing a small, hand-calculated scenario;
• conservation checks, such as arrivals = departures + entities remaining;
• extreme-condition tests, such as zero demand or unlimited capacity;
• event-log inspection;
• animation used cautiously as a diagnostic rather than proof;
• independent review of logic and code;
• checking units, queue disciplines, and resource-release rules;
• testing random streams and seed reproducibility.

6.2 Validation techniques

Validation may involve subject-matter expert review, comparison with historical system performance, face validation, comparison with analytical special cases, and sensitivity analysis. If a queueing model reduces to a known simple case, the simulation should approximately reproduce the known result under equivalent assumptions.

Validation is purpose-dependent. A model adequate for comparing staffing policies may be inadequate for predicting individual patient outcomes. The intended use, decision horizon, and acceptable error must be stated.

No model is universally valid. Validation supports a bounded claim such as: “The model is suitable for comparing the proposed staffing alternatives under the demand and operating conditions represented.”

7. Experimental design and output analysis

7.1 Initialisation bias and warm-up

A terminating simulation has a natural start and end, such as a one-day project. A non-terminating or steady-state simulation, such as a continuously operating warehouse, may begin empty even though the real system is normally active. Early observations then reflect artificial initial conditions.

A warm-up period is discarded before collecting steady-state statistics. The warm-up length can be assessed through time-series plots, moving averages, replication analysis, or change-point diagnostics. Discarding data is not a substitute for sound initialisation; if a realistic initial state is known, it should be used.

7.2 Replications and run length

A replication is an independent simulation run using a different random-number stream or seed. Multiple replications provide observations for estimating sampling variability. Increasing run length within one replication and increasing the number of replications address variability differently. Independent replications are often convenient for confidence intervals and alternative comparisons.

For independent replication outputs [[M:Y_1, …, Y_n]], the estimated mean is:

The estimated standard error is:

where [[M:s_Y]] is the sample standard deviation of the replication outputs. A large-sample approximate 95% confidence interval is:

The [[M:t_{0.975,n−1}]] value is the 97.5th percentile of the Student’s t distribution with [[M:n−1]] degrees of freedom. The interval assumes independent, approximately representative replication outputs. For strongly autocorrelated observations collected within one long run, ordinary independent-observation formulas are inappropriate unless methods such as batch means are used.

7.3 Comparing alternatives

Suppose two alternatives are simulated using common random numbers, producing paired outputs [[M:Y_{Ar}]] and [[M:Y_{Br}]] for replication [[M:r]]. Define:

The mean difference is estimated from the [[M:D_r]] values, and a confidence interval for the difference is constructed using their sample standard deviation. Common random numbers can reduce comparison noise because both alternatives experience comparable random conditions. They are helpful only when streams are synchronised appropriately; careless implementation can increase variance or introduce unintended dependence.

A statistically significant difference is not necessarily operationally important. Managers should consider effect size, confidence interval width, cost, risk, implementation feasibility, resilience, and strategic fit.

7.4 Variance reduction

Variance-reduction methods seek more precise estimates without proportionally increasing computation. Major methods include:

• Common random numbers: use matched random conditions for alternatives.
• Antithetic variates: use negatively related random inputs, such as [[M:U]] and [[M:1−U]], when this reduces covariance.
• Control variates: adjust an output using a correlated quantity with a known expected value.
• Importance sampling: sample rare but important events more frequently and reweight observations.
• Stratified sampling: divide the input space into strata and sample each systematically.

These methods require statistical reasoning and can be counterproductive if their assumptions are not met.

8. Worked example: staffing a service centre

Consider an illustrative service centre operating for an eight-hour day. Customers arrive according to an average rate of 18 per hour. Two staffing alternatives are under consideration: two servers or three servers. Service times are assumed independent and exponentially distributed with mean 5 minutes. The objective is to estimate waiting time and server utilisation.

The service rate of one server is:

The offered workload is 18 customers per hour. With two servers, nominal capacity is 24 customers per hour; with three servers, it is 36 customers per hour. The approximate capacity utilisation, if all arrivals are eventually served, is:

where [[M:λ = 18]] customers/hour, [[M:µ = 12]] customers/hour/server, and [[M:c]] is the number of servers. Thus, for two servers, [[M:ρ = 18/(2×12) = 0.75]]; for three servers, [[M:ρ = 18/(3×12) = 0.50]]. These are workload ratios, not simulation estimates of time-average utilisation under every possible policy.

A DES would generate exponential interarrival and service times, route each customer to an available server, place arrivals in a first-come-first-served queue when all servers are busy, and record arrival time, service-start time, departure time, and server state. Customer waiting time is:

For each alternative, the analyst should run many independent daily replications, report the mean and confidence interval for average waiting time, estimate the probability that waiting exceeds a service target, and calculate staffing cost. The decision should not be based on a single random run.

If the three-server alternative reduces average waiting by 4 minutes but costs an additional 8 labour-hours per day, management must compare the value of reduced waiting, abandonment, and congestion with that cost. The simulation informs this trade-off; it does not determine the value judgment.

9. Case study: illustrative distribution-centre redesign

Background

A regional e-commerce distribution centre experiences evening congestion. Orders arrive throughout the day, but demand is concentrated between 16:00 and 21:00. Each order requires picking, packing, and dispatch. Management is considering either adding a packing station or introducing an order-release rule that smooths the workload.

Problem

The existing spreadsheet uses average processing times and assumes that all activities occur independently. It predicts adequate capacity, yet observed dispatch delays and overtime are increasing. The organisation needs to compare alternatives while representing variable arrivals, correlated workload, priority orders, equipment failures, and shift boundaries.

Analysis and approach

The project team first defined system boundaries from order release to carrier departure. Historical data were cleaned to remove cancelled orders and to identify changes in operating policy. Arrival rates were estimated by time block rather than using one daily average. Picking and packing times were fitted separately by order type. Failure records were analysed for downtime distributions, and supervisors confirmed that large orders require both more work and more packing material.

A DES was constructed with order entities, picker and packer resources, queues, order attributes, shift calendars, failure events, and a carrier cut-off. The model included the existing release policy, an added packing station, a smoothed release rule, and a combined alternative. Each scenario used common random numbers across paired replications. A warm-up was unnecessary for each finite daily run because the operating day had a defined start, but several days were simulated within each replication to capture repeated operating conditions.

Verification included event tracing for a small test day, conservation checks, and extreme tests with zero arrivals and unlimited packing capacity. Validation compared simulated throughput, queue length, overtime, and late-dispatch percentage with historical ranges. Differences were investigated rather than hidden by changing parameters arbitrarily.

Outcome

The simulation showed that the added packing station reduced peak packing queues but did not fully resolve late dispatch because order release created a downstream burst. The smoothed release rule reduced peak queue length and overtime, but slightly increased average order completion time for low-priority orders. The combined alternative produced the best late-dispatch performance but required training and a revised priority policy.

Management selected a pilot of the combined alternative, with explicit monitoring of late dispatch, worker workload, injury indicators, priority-order treatment, and customer outcomes. The model was retained as a decision-support tool, not as an automatic scheduling authority.

Learning points

1. Average capacity can conceal time-dependent congestion.
2. Correlation between order size and processing time matters.
3. A local bottleneck intervention may shift congestion elsewhere.
4. Validation must compare several outputs, not only throughput.
5. Operational feasibility and human consequences are part of the decision.

10. Implementation, technology, and governance

Simulation software ranges from spreadsheets and general-purpose programming languages to specialised DES platforms. The choice should reflect model complexity, required transparency, integration needs, computational scale, and staff capability. A transparent, reproducible script may be preferable to a visually impressive model that cannot be audited.

Cloud computing and parallel execution make large experimental designs feasible. Automated experiment managers can run scenarios, collect outputs, and generate confidence intervals. Digital twins connect simulation models to current operational data, but real-time connectivity does not guarantee model validity. Data latency, sensor error, changing process logic, and cyber risk must be addressed.

AI can support distribution fitting, anomaly detection, surrogate modelling, and experiment prioritisation. It should not replace subject-matter validation. Black-box predictive models may reproduce historical correlations without representing causal responses to a new policy. Any AI-assisted model should document data provenance, training period, performance limits, human review, and failure modes.

Governance requirements include version control, access control, audit trails, documented assumptions, data privacy, and approval of sensitive scenarios. In healthcare, workforce, credit, or public-service applications, simulation outputs may influence allocation decisions affecting individuals. Analysts should test for disparate impacts, avoid using protected or proxy variables without justification, and communicate uncertainty rather than presenting point estimates as facts.

11. Challenges and limitations

Simulation studies commonly fail because of:

• poorly defined objectives and system boundaries;
• insufficient or biased input data;
• inappropriate independence or stationarity assumptions;
• omission of rare but consequential events;
• coding errors and incorrect event ordering;
• inadequate warm-up or too few replications;
• reporting point estimates without confidence intervals;
• comparing alternatives with inconsistent random conditions;
• excessive model detail that obscures key mechanisms;
• validation against only one historical aggregate;
• confusing model precision with real-world accuracy.

Simulation is computationally expensive and can create false confidence. A narrow confidence interval around a biased model output is still misleading. Scenario analysis should include sensitivity tests for uncertain parameters, alternative demand regimes, disruption conditions, and policy compliance.

Best-practice workflow

A disciplined simulation study normally follows these steps:

1. Define the decision, performance measures, alternatives, horizon, and required precision.
2. Map the real process and establish the model boundary.
3. Collect, clean, and document input data.
4. Identify dependence, nonstationarity, batching, and exceptional conditions.
5. Select and justify input distributions.
6. Build a simple conceptual model before coding.
7. Implement modularly with reproducible random streams.
8. Verify logic using tests, traces, conservation laws, and extreme cases.
9. Validate behaviour against experts, data, and analytical benchmarks.
10. Design warm-up, run length, replications, and variance reduction.
11. Conduct scenario experiments and calculate confidence intervals.
12. Interpret statistical and practical significance together.
13. Document assumptions, limitations, data provenance, and governance controls.
14. Pilot important recommendations and update the model as evidence changes.

Chapter Summary

Simulation is a structured method for analysing stochastic, dynamic, and complex operational systems. Monte Carlo simulation propagates uncertainty through repeated random sampling, while discrete-event simulation represents state changes at events such as arrivals, service completions, failures, and replenishments. Effective models define entities, resources, events, state variables, inputs, outputs, and decision policies.

Reliable simulation depends on defensible input distributions, appropriate random-number generation, explicit treatment of dependence, and careful event logic. Verification establishes that the model was implemented correctly; validation establishes that it is adequate for its intended purpose. Output analysis requires attention to initialisation bias, warm-up, independent replications, confidence intervals, autocorrelation, and fair comparison of alternatives. Variance-reduction techniques can improve precision when properly designed.

Simulation results are estimates, not certainties. Managers must evaluate statistical uncertainty, practical effect size, cost, implementation feasibility, resilience, ethics, and governance. The strongest simulation studies combine data analysis, process expertise, software discipline, statistical experimentation, and transparent communication.

Key Terms

Simulation: An executable representation of a real or proposed system used for experimentation.

Monte Carlo simulation: Repeated random sampling used to estimate outcomes of a stochastic model.

Discrete-event simulation: Simulation in which system state changes at discrete event times.

Entity: An object that flows through a simulated system.

Resource: A capacity-constrained facility, worker, machine, or service channel.

Event: An instantaneous occurrence that changes system state.

State variable: A variable describing the system’s condition at a particular time.

Input distribution: A probability model used to generate uncertain model inputs.

Pseudorandom number: A deterministic computer-generated number designed to emulate randomness.

Random stream: A reproducible sequence of pseudorandom numbers assigned to a simulation component.

Verification: Establishing that the implemented model follows its intended logic.

Validation: Establishing that the model is adequate for its intended use.

Warm-up period: Initial simulation time excluded to reduce bias from artificial starting conditions.

Replication: An independent simulation run used to estimate sampling variability.

Confidence interval: An interval estimate expressing uncertainty around a simulation performance estimate.

Common random numbers: A variance-reduction method using matched random inputs for alternatives.

Time-persistent statistic: A measure accumulated over time, such as average queue length.

Terminating simulation: A simulation with a natural, meaningful start and end.

Steady-state simulation: A simulation intended to represent long-run operation of a continuing system.

Digital twin: A data-connected model intended to represent an operating physical or organisational system.

Review Questions

5 Short Answer Questions

1. What is the difference between Monte Carlo simulation and discrete-event simulation?
2. Define verification and validation in the context of simulation.
3. Why may a warm-up period be required in a steady-state simulation?
4. What is the purpose of using multiple independent replications?
5. Why can treating correlated inputs as independent produce misleading results?

5 Long Answer Questions

1. Explain the principal components of a discrete-event simulation model and illustrate each with a service-operation example.
2. Discuss how an analyst should select and justify probability distributions for arrival, service, and failure data.
3. Explain event scheduling, process interaction, and the calculation of time-average queue length.
4. Describe a complete verification and validation programme for a warehouse simulation.
5. Evaluate the managerial and ethical risks of using AI-assisted simulation for workforce or public-service decisions.

3 Analytical / Application Questions

1. A call centre receives 30 calls per hour, and each agent handles an average of 10 calls per hour. Compare the nominal workload utilisation for three and four agents. Explain why a simulation may still be needed before changing staffing.
2. Design a simulation experiment to compare two inventory policies when demand is seasonal, lead times are correlated with order size, and stockouts are treated as lost sales. Specify inputs, outputs, replications, and validation checks.
3. A simulation reports that Alternative A has a mean cost of $105,000 with a 95% confidence interval of [$103,000, $107,000], while Alternative B has a mean cost of $108,000 with an interval of [$104,000, $112,000]. Explain what can and cannot be concluded, and identify additional managerial information required for a decision.

Multiple Choice Questions with Answers

Question 1

Which statement best distinguishes verification from validation?

A. Verification checks whether the model was implemented according to its logic; validation checks whether it is adequate for its intended use.

B. Verification estimates confidence intervals; validation generates random numbers.

C. Verification concerns historical data only; validation concerns computer code only.

D. Verification and validation are identical terms.

Correct Answer: A. Verification checks whether the model was implemented according to its logic; validation checks whether it is adequate for its intended use.

Question 2

If one server’s mean service time is 5 minutes, its mean service rate is:

A. 5 customers per hour

B. 10 customers per hour

C. 12 customers per hour

D. 60 customers per hour

Correct Answer: C. 12 customers per hour

Question 3

Which quantity is generally calculated as a time-weighted statistic in a discrete-event queueing model?

A. Number of arrivals during the run

B. Average queue length

C. Number of completed services

D. Number of replications

Correct Answer: B. Average queue length

Question 4

For an exponential random variable with rate [[M:λ]], the inverse-transform observation is commonly generated as:

A. [[M:X = λU]]

B. [[M:X = −ln(U)/λ]]

C. [[M:X = U/λ^2]]

D. [[M:X = λ/U]]

Correct Answer: B. [[M:X = −ln(U)/λ]]

Question 5

What is the main purpose of independent replications?

A. To eliminate all model uncertainty

B. To provide observations for estimating sampling variability

C. To guarantee that the input distribution is correct

D. To remove the need for validation

Correct Answer: B. To provide observations for estimating sampling variability

Question 6

Which issue is most directly associated with starting a steady-state simulation empty?

A. Parameter identifiability

B. Initialisation bias

C. Unit inconsistency

D. Overfitting a regression model

Correct Answer: B. Initialisation bias

Question 7

Common random numbers are primarily used to:

A. Increase the number of entities in the system

B. Reduce comparison noise by exposing alternatives to matched random conditions

C. Convert a discrete-event model into a continuous model

D. Replace input-data collection

Correct Answer: B. Reduce comparison noise by exposing alternatives to matched random conditions

Question 8

If arrivals occur at 18 customers per hour and there are two servers, each serving 12 customers per hour, the nominal workload utilisation is:

A. 0.50

B. 0.67

C. 0.75

D. 1.50

Correct Answer: C. 0.75

Question 9

Which practice is most appropriate when input observations show strong dependence?

A. Assume independence without testing

B. Use a single normal distribution automatically

C. Consider joint, conditional, time-series, or block-resampling approaches

D. Delete all extreme observations

Correct Answer: C. Consider joint, conditional, time-series, or block-resampling approaches

Question 10

A narrow confidence interval around a simulation output proves that:

A. The model is unbiased and valid

B. The real system will achieve exactly the reported value

C. Sampling uncertainty is relatively small under the specified experiment

D. No further managerial judgement is required

Correct Answer: C. Sampling uncertainty is relatively small under the specified experiment