How MRT Station Dwell Time Works Using Mathematics: Why Doors Can Limit an Entire Railway

An MRT train can travel quickly through a tunnel and still lose the railway at the platform.

That sounds strange until we notice where a high-frequency metro spends a surprising amount of its time: standing still with its doors open.

In November 2025, Singapore’s Ministry of Transport stated that MRT and LRT dwell times generally range from about 35 to 60 seconds at interchange stations and about 30 to 50 seconds at non-interchange stations. The exact dwell depends on operating conditions and passenger loading, and the time must be sufficient for passengers — including elderly commuters and people using wheelchairs or personal mobility aids — to board and alight safely.

Thirty seconds. Forty seconds. Sixty seconds.

Those numbers can be comparable to a significant fraction of the running time between neighbouring stations.

This creates one of the most important mathematical facts about an MRT:

The train may be controlled by machines, but the timetable must also be controlled by human movement.

This article continues our Singapore MRT mathematics series. The main pillar is How MRT Timing Works Using Mathematics. We have also examined MRT braking mathematics and MRT headway mathematics.

Now we stop the train and look at the platform.

The RFE — Why Does Dwell-Time Mathematics Exist?

The easiest answer would be:

open doors
→ let people out
→ let people in
→ close doors

But a railway cannot simply make this process as short as physically possible.

It has to give enough time for real people to move safely. At the same time, every extra second the train remains at the platform occupies track and platform capacity that the following train may soon need.

So dwell time exists inside a tension:

too short
→ unsafe or incomplete passenger exchange

too long
→ capacity loss and delay propagation

The mathematical purpose is therefore not:

minimise dwell time

It is:

provide enough dwell
for safe and inclusive passenger exchange
while keeping the railway stable,
regular and high-capacity

That is the Reason for Existence.

Prompt 1 — What Exactly Is MRT Dwell Time?

Dwell time is the period during which a train remains at a station as part of the passenger exchange and departure process.

It is tempting to imagine dwell as one block:

Tdwell = 40 seconds

But internally it can be understood as several pieces.

Tdwell
≈ Topen
+ Texchange
+ Tclear
+ Tclose
+ Tdeparture

where:

• Topen covers the door-opening process,
• Texchange is the time passengers need to alight and board,
• Tclear represents the time needed for the doorway and platform edge to be safely clear,
• Tclose covers the door-closing process,
• Tdeparture represents any remaining departure process before the train moves.

Not every railway defines every component in exactly this way. The decomposition is educational. The important idea is that dwell is not one indivisible number.

Fixed time and variable time

Some parts of dwell are relatively mechanical and repeatable.

Passenger exchange is not.

So we can split dwell conceptually into:

Tdwell = Tfixed + Tpassenger

The first part is comparatively deterministic.

The second is affected by:

• how many passengers alight,
• how many board,
• how crowded the train already is,
• where passengers are standing on the platform,
• how evenly they are distributed across doors,
• whether boarding passengers give way to those alighting,
• door geometry and interior layout,
• mobility differences among passengers,
• and ordinary randomness in human movement.

This gives us the central idea of the article:

The train is highly controlled. The passenger exchange is stochastic.

Prompt 2 — Why Is Passenger Exchange a Queueing Problem?

Passengers do not appear at the platform all at once.

They arrive over time.

Let:

λ = average passenger arrival rate
H = train headway

A simple estimate of passengers accumulating between trains is:

N ≈ λH

Suppose:

λ = 2.5 passengers/s
H = 120 s

Then approximately:

N = 2.5 × 120
  = 300 passengers

arrive during the gap.

If headway rises to 180 seconds:

N = 2.5 × 180
  = 450 passengers

A delayed train can therefore meet a larger passenger batch simply because more time has passed.

This connects headway directly to dwell.

larger headway
→ more passenger arrivals
→ larger boarding demand
→ potentially longer passenger exchange
→ potentially longer dwell

That is the beginning of delay propagation.

The platform queue

Let Q_k be the number of passengers left waiting after train k departs.

Let C_k+1 be the effective number the next train can take from that queue.

A deliberately simple queue equation is:

Qₖ₊₁ = max(0, Qₖ + λHₖ − Cₖ₊₁)

This tells us something important.

If passenger arrivals exceed the number that can be taken away, the queue survives the departure and continues growing.

A station then moves from:

each train clears the waiting demand

towards:

passenger backlog persists across trains

That is a different operating state.

The platform has become a queueing system under load.

Prompt 3 — Why Is Dwell Time Not Simply Proportional to Passenger Count?

Suppose 20 people need to leave a carriage.

Now suppose 40 people need to leave.

We might guess that the second case takes exactly twice as long.

Human flow does not always behave that cleanly.

Research on metro boarding and alighting repeatedly finds nonlinear effects from crowding, passenger interaction, onboard density, door use and boarding/alighting conflict. A 2025 study of crowded metro dwell times, for example, treats dwell-time delay and passenger load probabilistically because high-volume station dwell is variable and difficult to represent with a single fixed estimate.

A conceptual dwell model might begin:

Td = T0 + αA + βB

where:

• A = number alighting,
• B = number boarding,
• α and β = average service-time effects.

But once crowding and interference become important, a better educational model might include a nonlinear term:

Td = T0 + αA + βB + γI(A,B,C)

where C describes crowding and I represents interaction.

For example, purely as a mathematical illustration:

I = AB/(1 + W)

could represent the idea that simultaneous opposing passenger flows create more interference when boarding and alighting volumes are both high, while greater usable doorway width W reduces the penalty.

This is not an LTA or SMRT formula. It is a teaching model.

The point is structural:

Two passenger streams do not simply add. They can interfere with one another.

Why “alight first” is mathematics, not merely etiquette

SMRT asks passengers to remain within queue lines and allow passengers to alight before boarding. LTA’s Thoughtful Bunch campaign similarly encourages commuters to queue to one side of the doors and let passengers alight first.

This behaviour is courteous.

It is also flow management.

If boarding and alighting happen directly against each other, two opposite pedestrian streams attempt to use the same narrow doorway at once.

If boarding waits briefly:

outflow first
→ carriage space increases
→ doorway conflict decreases
→ inflow begins with more internal space

A 2026 simulation study of metro boarding and alighting specifically found that non-compliance with an “alight before board” rule prolonged operation times in its tested scenarios. The exact numbers belong to that study and its layout, not automatically to Singapore, but the mechanism is easy to understand.

Courtesy reduces interference.

Interference consumes seconds.

Seconds consume capacity.

Prompt 4 — Why Can One Door Control the Whole Train?

This is one of the most useful mathematical insights in station operations.

A train has many doors.

Passenger exchange therefore happens in parallel.

If door j needs service time S_j, then the train cannot normally complete passenger exchange until the slowest relevant doorway is finished.

Texchange ≈ max(S1, S2, S3, ... , Sn)

Notice the word max.

The average door may not determine dwell.

The busiest door can.

Suppose six abstract door groups require:

18 s, 19 s, 20 s, 21 s, 22 s, 39 s

The mean is:

23.2 s

But the passenger exchange cannot finish in 23.2 seconds if one doorway still needs 39 seconds.

The bottleneck door dominates:

Texchange ≈ 39 s

This explains why passenger distribution matters.

If commuters cluster near one escalator or one convenient platform position, several doors can be lightly used while another becomes overloaded.

The train has plenty of theoretical doorway capacity.

But it is poorly distributed.

In a parallel system, imbalance can waste capacity that physically exists.

Door service rate

Let μ_j be the effective passenger service rate at door j, measured in passengers per second.

If N_j passengers need to use that doorway:

Sj ≈ Nj / μj

So passenger exchange can be thought of as:

Texchange ≈ max(Nj/μj)

Now both distribution and service rate matter.

A door can become slow because:

• N_j is unusually large,
• μ_j falls because the carriage is crowded,
• boarding and alighting streams interfere,
• passengers pause or reorganise near the doorway,
• or movement around that door is constrained.

This gives us a much better mental model than “there are many doors, so boarding should always be fast.”

Prompt 5 — How Does Train and Station Design Change the Mathematics?

Geometry changes passenger flow.

Door count, doorway width, carriage interior layout, platform circulation and the position of stairs and escalators all affect how passengers reach the doors and move through them.

Singapore provides an easy example. LTA has highlighted that Thomson-East Coast Line trains are longer than Circle and Downtown Line trains and use five doors on each side rather than four, specifically noting that the arrangement helps boarding and alighting.

Why should another doorway matter mathematically?

If n doors operate in parallel and each could ideally serve μ passengers per second, an idealised total service rate is:

μtotal = nμ

Increase n and the ideal capacity rises.

But real passenger distribution is imperfect, so we can introduce an efficiency factor η:

μeffective = η n μ

0 < η ≤ 1

If passengers spread perfectly and every door is equally usable, η approaches 1.

If they crowd around a few doors, η falls.

This explains an important design truth:

Adding physical capacity is useful, but using that capacity evenly is another problem.

The station behind the platform

The passenger does not disappear after leaving the train.

They join stairs, escalators, lifts, passageways and fare gates.

If these downstream elements cannot remove people from the platform quickly enough, crowd density rises before the next train arrives.

So platform state follows another conservation-style equation:

Platform population next
=
platform population now
+ arrivals from entrances
+ passengers alighting
− passengers boarding
− passengers leaving platform

In symbols:

Pₖ₊₁ = Pₖ + Eₖ + Aₖ − Bₖ − Xₖ

where E represents entrants and X represents people exiting the platform area.

A station therefore has its own flow balance.

The train timetable and station architecture meet inside that equation.

Prompt 6 — Why Is Accessibility Part of the Mathematical Constraint?

One of the easiest mistakes in optimisation is to optimise for the average person.

Suppose most passengers can complete an exchange within 30 seconds.

Does that mean the doors should always close after 30 seconds?

No.

The Ministry of Transport explicitly says dwell times must be long enough for all passengers to board and alight safely, including elderly commuters and those using wheelchairs and personal mobility aids.

LTA also provides priority queue zones at MRT/LRT stations for elderly passengers, less mobile commuters and families with young children.

Mathematically, this means the system should not be designed only around:

E[Tpassenger]

the average passenger-exchange time.

A robust timetable may instead care about a high percentile of the distribution.

For example, conceptually:

P(Tpassenger ≤ Tdwell) ≥ 1 − ε

where ε is a small acceptable probability of the planned dwell being insufficient under the model.

This is not an LTA scheduling formula.

It is a powerful way to understand inclusive design.

The timetable is not merely asking:

How quickly does the average person move?

It must ask:

How much time makes this process reliably usable
for the population the railway is meant to serve?

Accessibility converts ethics into an engineering constraint.

This is one of the most important reasons the mathematically minimum dwell time and the socially correct dwell time are not always the same number.

Prompt 7 — How Does a Long Dwell Become a Railway Delay?

Suppose the timetable expects a train to dwell for:

40 seconds

but actual passenger exchange takes:

55 seconds

The excess dwell is:

ΔTd = 15 seconds

If nothing else changes, the train leaves 15 seconds later.

That changes two headways at once.

The gap from the train ahead grows.

The gap to the following train shrinks.

Suppose desired headway is 120 seconds.

Before delay:
A → 120 s → B → 120 s → C

After Train B loses 15 seconds:

A → 135 s → B → 105 s → C

Now passenger arrivals respond.

The larger 135-second gap before B brings B more passengers.

The smaller 105-second gap before C brings C fewer.

If B therefore needs another long dwell while C needs a shorter one, the spacing can become even more unequal.

long gap
→ more waiting passengers
→ longer exchange
→ later departure
→ even longer gap

short gap
→ fewer waiting passengers
→ shorter exchange
→ earlier relative departure
→ even shorter gap

This is the mechanism behind bunching.

A dwell-delay recurrence

Let δ_k be train delay after station k.

A conceptual recurrence is:

δₖ₊₁
=
δₖ
+ ΔTdwell,k
+ ΔTrun,k
− Rₖ

where R_k is time recovered elsewhere.

If the timetable and running profile contain enough recovery margin, the delay may shrink.

If new dwell delays exceed recovery, it can grow.

A platform event has now become a line event.

Prompt 8 — What Is the Mathematically Best Dwell Time?

There is no universal answer.

The best dwell depends on station demand, interchange activity, accessibility needs, train load, platform layout, door arrangement, headway, recovery margin and service objectives.

We can imagine an optimisation problem:

minimise J(Td)

where:

J
=
w1(passengers left behind)
+ w2(delay)
+ w3(headway irregularity)
+ w4(safety risk)
+ w5(accessibility failure)
+ w6(capacity cost)

subject to:

Td ≥ mechanical minimum
Td ≥ safe passenger-exchange requirement
headway remains feasible
platform conditions remain acceptable

Notice that this objective is not simply:

min Td

because an aggressively short dwell can create a different cost: passengers left behind, unsafe rushing, accessibility failure, repeated door interruptions or an unstable passenger queue.

Likewise, an unnecessarily long dwell wastes line capacity.

The optimal dwell is not the shortest stop. It is the shortest stop that still performs the whole human job reliably.

A Worked Dwell-Time Model

Let us create an educational model of a busy station.

This is not an actual Singapore MRT operating formula or dataset.

Suppose the fixed mechanical and clearance portion is:

Tfixed = 12 s

Suppose four abstract doorway groups have exchange times:

S1 = 19 s
S2 = 22 s
S3 = 24 s
S4 = 31 s

The train needs the slowest active doorway to finish:

Texchange = max(19,22,24,31)
          = 31 s

Total simplified dwell is:

Tdwell = 12 + 31
       = 43 s

Now suppose passenger distribution improves so that the busiest doorway falls from 31 seconds to 25 seconds while the others remain similar.

Tdwell = 12 + 25
       = 37 s

Six seconds have been saved without making any passenger walk through a doorway faster.

The gain came from distribution.

Now repeat that across 20 station stops:

20 × 6 = 120 s

Two minutes of cumulative timetable effect have appeared from changing how parallel door capacity is used.

Real railways would not simply subtract every saved second linearly — recovery, schedules and other constraints intervene — but the example shows why doorway balancing matters.

The Crowding Threshold Problem

At low passenger density, adding another passenger may add only a small amount of exchange time.

At high density, the same extra passenger can have a larger effect because movement itself becomes constrained.

We can represent effective door service rate as a function of crowding C:

μ = μ(C)

A simple illustrative form is:

μ(C) = μ0 / (1 + αCᵖ)

where μ₀ is an uncongested service rate and α and p determine how sharply efficiency falls with crowding.

Again, this is a teaching function, not a Singapore operating formula.

The important idea is that:

more crowding
→ lower movement efficiency
→ longer exchange time
→ greater dwell variability

Recent metro research continues to find threshold-like deterioration in boarding/alighting efficiency under dense conditions, reinforcing why crowded-station dwell is better treated as a variable distribution than as one perfectly predictable number.

Why Dwell Time Should Be a Distribution, Not Just an Average

Suppose a station averages 40 seconds of dwell.

Compare two possible distributions.

Station A:

38, 39, 40, 41, 42 seconds

Station B:

25, 28, 32, 45, 70 seconds

The average could be similar.

The timetable risk is not.

So planners may care about:

mean dwell
variance
95th percentile
probability of exceeding scheduled dwell
correlation with passenger load

A 2025 study on crowded metro stations specifically proposed a bivariate probability framework relating passenger volume and dwell-time delay, precisely because point estimates become less reliable under heavy crowding.

This is a beautiful example of why statistics exists.

The average tells us where the centre is.

The distribution tells us how often the railway gets hurt.

The Human Bottleneck Paradox

Suppose signalling is improved.

Trains can safely run closer together.

Suppose rolling stock accelerates efficiently.

Suppose the terminal can turn trains around quickly.

But a busy interchange repeatedly needs 60 seconds to exchange passengers.

Then the platform can become the effective bottleneck.

This leads to a general capacity equation:

Hoperating ≥ max(
Hsignal,
Hstation,
Hjunction,
Hturnback,
Hpower,
Hrecovery
)

Even if:

Hsignal = 100 s

the whole line cannot automatically sustain a 100-second service if another component requires more space in the operating cycle.

This is why a railway upgrade is rarely one machine.

It is a system.

The fastest signalling system in the world cannot make passengers teleport through the doors.

Why Queue Lines Matter More Than They Look

The lines painted on a platform seem like a tiny piece of infrastructure.

But they are attempting to alter boundary conditions for pedestrian flow.

Without organisation:

boarding flow ↔ alighting flow
compete for the same opening

With passengers waiting to the sides:

central outflow first
→ reduced conflict
→ side inflow begins

The painted line is therefore a low-cost control system.

It does not move passengers mechanically.

It changes the probability of people choosing more efficient trajectories.

LTA and SMRT’s public guidance to let passengers alight before boarding is therefore not separate from railway operations.

Passenger behaviour is part of railway operations.

A Door Is a Tiny Capacity Machine

Suppose one doorway can effectively process μ passengers per second under a given crowd condition.

Over exchange time T:

Kdoor ≈ μT

With n equally productive doors:

Kideal ≈ nμT

But if only a fraction η of that parallel capacity is effectively used:

Kactual ≈ ηnμT

This one equation gives us three different ways to improve passenger exchange:

• increase n — provide more usable door capacity,
• increase μ — improve the flow through each door,
• increase η — distribute passengers more evenly so capacity is not wasted.

And of course we can increase T — keep the doors open longer — but that consumes timetable capacity.

This makes dwell-time engineering a resource allocation problem.

The Mathematics of “Move In”

LTA’s public campaigns also encourage passengers to move further into the train instead of clustering around doors.

This too has a mathematical interpretation.

Imagine a carriage with unused standing space in the centre but dense crowding near the door.

Total carriage density may appear acceptable.

Local doorway density is not.

Let average carriage density be:

Cavg = passengers / carriage area

But doorway flow depends more directly on a local density:

Cdoor = passengers near doorway / doorway-zone area

It is possible to have:

Cavg moderate
while
Cdoor very high

Then boarding becomes difficult even though empty space still exists elsewhere.

Moving inward redistributes density.

It does not increase the geometric size of the train.

It increases the usable capacity.

Good distribution can create effective capacity without creating new physical space.

The Platform as a Control Volume

Engineers often analyse physical systems using a control volume: choose a boundary and count what enters, leaves and accumulates inside.

We can do the same with a platform.

Over a time interval Δt:

ΔP
=
passengers entering platform
+ passengers alighting
− passengers boarding
− passengers leaving platform

In rate form:

dP/dt = λin + λalight − λboard − λexit

If:

dP/dt > 0

the platform population is growing.

If that persists, density rises.

Eventually flow rates themselves may deteriorate because crowding makes movement less efficient.

That produces another feedback loop:

platform accumulation
→ higher density
→ slower movement
→ lower exit/boarding flow
→ more accumulation

A crowded station is therefore not merely “many people”.

It can become a nonlinear flow system approaching a capacity boundary.

Why Interchanges Need More Time

MOT’s published range is higher for interchange stations — around 35 to 60 seconds — than for non-interchanges, around 30 to 50 seconds.

That makes intuitive mathematical sense.

An interchange can create:

• larger alighting batches,
• larger boarding batches,
• directional passenger surges connected to another line,
• more simultaneous movement through stairs and escalators,
• and greater variance between trains depending on transfer arrivals.

Passenger arrivals may also be less like a smooth Poisson-style trickle and more like batches.

If another train arrives on a connecting line, it can release a pulse of passengers.

So instead of a constant λ:

λ(t) = constant

an interchange may experience:

λ(t) = low → spike → low → spike

This makes dwell and crowding more time-dependent.

A Singapore MRT Dwell-Time Thought Experiment

Consider a hypothetical interchange with a scheduled dwell of 45 seconds.

Suppose fixed door and clearance processes use 12 seconds, leaving:

45 − 12 = 33 s

for the passenger exchange in this simplified model.

Now suppose one heavily loaded doorway needs 38 seconds because passengers are clustered there.

Actual dwell becomes:

12 + 38 = 50 s

Excess dwell:

ΔTd = 5 s

Five seconds sounds trivial.

But if the line is trying to maintain a 120-second headway, those five seconds represent:

5/120 ≈ 4.17%

of the separation budget.

If similar excess dwell appears at several consecutive busy stations, the cumulative pressure becomes substantial.

Now suppose passengers are distributed more evenly and the maximum doorway exchange falls to 31 seconds.

Dwell becomes:

12 + 31 = 43 s

The train now has two seconds of margin relative to the 45-second schedule.

No train motor was upgraded.

No top speed changed.

The improvement happened entirely at the human-machine interface.

Dwell Time and the Mathematics of Reliability

A strong timetable needs more than a good expected dwell.

Suppose:

E[Tdwell] = 40 s

but standard deviation is:

σ = 12 s

That station behaves very differently from another station with the same mean but:

σ = 2 s

So reliability engineering asks questions such as:

P(Tdwell > 45 s) = ?
P(Tdwell > 50 s) = ?
P(Tdwell > scheduled allowance) = ?

This is a more useful question than:

What is the average?

because the railway is often damaged by the tail of the distribution.

Average dwell tells us the centre of the station. Extreme dwell tells us how delay enters the railway.

RFE Return — Why Not Just Close the Doors Earlier?

Because the railway is not transporting equations.

It is transporting people.

The mathematically fastest possible door closure may leave someone behind who reasonably needed more time.

The mathematically longest possible dwell may destroy capacity for everyone else.

So the correct problem is not an extreme.

It is a balance:

Safety
+ Accessibility
+ Passenger completion
+ Punctuality
+ Headway regularity
+ Capacity
+ Recoverability

all competing for the same seconds.

This is why MOT’s explanation matters. Dwell is deliberately long enough to support safe boarding and alighting for different kinds of passengers.

Those seconds are not necessarily waste.

They are doing a job.

Conclusion — The Railway Bottleneck Can Be a Doorway

An MRT station looks still.

Mathematically, it is full of flow.

Passengers enter the platform.

Passengers accumulate.

A train arrives.

Passengers alight.

Passengers board.

Doorways process several flows in parallel.

The busiest doorway determines when exchange is complete.

The train departs.

The next headway begins.

And immediately:

passengers begin accumulating again

So the platform sits inside a feedback loop:

headway
→ arrivals
→ queue
→ doorway distribution
→ boarding/alighting
→ dwell
→ departure
→ new headway

That is why station timing cannot be separated from railway timing.

And it leads to one of the most counterintuitive lessons in transport mathematics:

The capacity of a multi-billion-dollar railway can be influenced by how efficiently human beings pass one another through a doorway.

The passenger sees doors.

The mathematician sees queues.

The station designer sees flow capacity.

The operator sees dwell variance.

The timetable sees seconds.

And the city feels whether all of them worked together.

Key Mathematical Ideas

Tdwell ≈ Tfixed + Tpassenger

N ≈ λH
Passenger arrivals between trains

Qₖ₊₁ = max(0, Qₖ + λHₖ − Cₖ₊₁)
Simplified platform queue

Td = T0 + αA + βB + γI(A,B,C)
Illustrative nonlinear dwell model

Sj ≈ Nj/μj
Doorway service time

Texchange ≈ max(S1,...,Sn)
Slowest-door bottleneck

μeffective = ηnμ
Parallel door capacity with utilisation factor

Pₖ₊₁ = Pₖ + Eₖ + Aₖ − Bₖ − Xₖ
Platform population balance

dP/dt = λin + λalight − λboard − λexit
Continuous flow balance

P(Tpassenger ≤ Tdwell) ≥ 1 − ε
Illustrative reliability/accessibility constraint

δₖ₊₁ = δₖ + ΔTdwell,k + ΔTrun,k − Rₖ
Delay propagation model

Important: These equations are educational models used to explain the structure of the problem. Actual Singapore MRT dwell rules, passenger-flow models and control systems use line-specific, train-specific and safety-validated engineering parameters that are not reproduced here.

Continue the MRT Mathematics Series

• How MRT Timing Works Using Mathematics — the system-wide pillar.
• How MRT Braking Works Using Mathematics — stopping distance, braking curves and energy recovery.
• How MRT Headway Works Using Mathematics — how seconds between trains become capacity.

The next leg follows what happens after dwell goes wrong: how a small MRT delay propagates through a line, why trains bunch, and how mathematics can bring the railway back towards its intended rhythm.

Sources and Further Reading

• Singapore Ministry of Transport — Dwell Times for Trains at Interchange and Non-Interchange Stations
• SMRT — Platform safety, queue lines and allowing passengers to alight before boarding
• Land Transport Authority — Thoughtful Bunch and passenger-flow behaviour
• Land Transport Authority — Inclusive Public Transport and Priority Queue Zones
• Land Transport Authority — Thomson-East Coast Line train and boarding/alighting design
• Journal of Rail Transport Planning & Management — Dwell-time models for crowded metro stations
• Physica A — Modelling and simulation of metro passenger boarding and alighting
• Physica A — Crowd dynamics during metro boarding and alighting
• Transportation Research Part A — Passenger boarding and alighting simulation at metro platforms

Return to the complete MRT Mathematics hub: How MRT Works | It’s Mathematics.