How MRT Passenger Capacity Is Calculated Using Mathematics: Why a Full Train Is Not a Full Railway

A crowded MRT train looks full.

But that does not necessarily mean the railway is using all of its capacity.

Another train may be approaching only two minutes behind it. Some carriages may be packed while others have space. The busiest section of the line may last for only three stations. The opposite direction may be lightly loaded. A platform doorway, terminal crossover, junction or irregular train gap may be limiting the service more than the physical size of the train.

This is why the question:

“How many people fit inside an MRT train?”

is only the beginning of a capacity calculation.

The deeper question is:

How many passenger journeys can an entire MRT line deliver safely, regularly and repeatedly through its busiest place and busiest time?

That question takes us into arithmetic, algebra, rates, graph theory, queueing theory, statistics, probability, optimisation and systems engineering.

Singapore provides a useful public example. LTA states that a Circle Line train has a maximum capacity of up to 931 passengers. Since the opening of Circle Line Stage 6 on 12 July 2026, LTA has described peak frequencies of about two minutes on the main loop, with about four minutes on the Dhoby Ghaut–Promenade–Bayfront–Marina Bay section.

Those two public numbers allow us to perform a simple theoretical calculation:

2-minute headway = 30 trains/hour

30 × 931
= 27,930 passenger places/hour/direction

But 27,930 is not automatically the line’s actual passenger throughput, guaranteed usable capacity or comfortable operating capacity.

It is a theoretical product of a published maximum train capacity and an approximate published frequency.

To understand real MRT capacity, we must ask what happens to that number after passengers, stations, direction, reliability and bottlenecks enter the system.

This article continues the eduKateSG MRT mathematics series. Begin with How MRT Timing Works Using Mathematics, then explore braking, headway, station dwell, delay recovery, timetable construction, and energy optimisation.

The RFE — Why Does MRT Capacity Mathematics Exist?

The weakest possible definition of capacity is:

put the maximum possible number of bodies into every train

That is not the purpose of public transport.

A railway exists to move people, not to manufacture crowding.

The capacity calculation must therefore preserve the whole human job:

move enough people
+
move them safely
+
move them regularly
+
let them board and alight
+
serve passengers with different mobility needs
+
recover when the timetable is disturbed
+
repeat the service throughout the operating period

The Reason for Existence is not maximum compression.

It is useful, deliverable and resilient movement.

A good capacity number should therefore answer:

How much passenger demand can this system carry
through its critical point,
in the required direction,
during the required time,
without making the service unstable or inaccessible?

Capacity is not the largest number a railway can touch once. It is the amount of public service it can sustain.

Prompt 1 — What Does “MRT Capacity” Actually Mean?

The word capacity is used for several different quantities.

Confusing them creates bad reasoning.

1. Vehicle capacity

This is how many passengers one train can carry under a stated loading definition.

Ptrain = passengers/train

LTA’s published maximum of 931 passengers for a Circle Line train is a vehicle-capacity figure.

2. Train frequency

This is how many trains pass a point during a period.

q = trains/hour

If headway H is measured in seconds:

q = 3600/H

3. Theoretical line capacity

Multiply train capacity by trains per hour:

Ctheoretical = Ptrain × q
             = Ptrain × 3600/H

This is commonly expressed as passengers per hour per direction.

The abbreviation is often:

pphpd

The phrase matters because capacity is not one undirected daily total.

It is measured:

per hour
through a point or section
in one direction

LTA uses exactly this kind of directional hourly capacity for the Johor Bahru–Singapore RTS Link, which is targeted to begin passenger service at the end of 2026 and is designed to serve up to 10,000 commuters per hour in each direction during peak periods.

4. Scheduled capacity

The timetable may schedule fewer trains than the technical maximum.

So:

Cscheduled = Pplanning × qscheduled

where P_planning may be a practical planning load rather than the absolute published maximum.

5. Delivered capacity

Scheduled trains must actually operate and arrive with useful spacing.

A simplified delivered-capacity model is:

Cdelivered
=
Cscheduled
× ηservice
× ηregularity

where the efficiency factors conceptually represent service delivery and the loss caused by irregular spacing.

This is an educational model, not an LTA formula.

6. Used capacity

This is the passenger demand actually carried.

Cused ≤ Cdelivered ≤ Cscheduled ≤ Ctheoretical

Not every inequality must be strict, but the chain helps us distinguish the concepts.

A line can possess high theoretical capacity and low used capacity during off-peak hours.

It can also have strong vehicle capacity but weak delivered capacity during a disruption.

“How much could fit?”, “how much was scheduled?”, “how much arrived?” and “how many travelled?” are four different questions.

Prompt 2 — How Is the Capacity of One MRT Train Calculated?

A train contains seats, standing space, designated spaces, doorways, gangways, equipment and areas that passengers cannot occupy.

A simple conceptual calculation is:

Ptrain
=
S
+ ρAusable
+ Pdesignated

where:

• S is seated passenger capacity,
• A_usable is usable standing area,
• ρ is the assumed standing density,
• P_designated represents capacity allocated through designated spaces and the planning treatment applied to them.

The exact engineering definition depends on the rolling stock, standards and loading condition being quoted.

The important lesson is that capacity does not come from floor area alone.

It depends on the assumptions placed on that floor area.

Standing density changes the number

Suppose a train has:

S = 300 seats
Ausable = 150 m²

At a planning density of four standing passengers per square metre:

P = 300 + 4(150)
  = 900 passengers

At six standing passengers per square metre:

P = 300 + 6(150)
  = 1,200 passengers

The train did not become physically larger.

The assumed density changed.

This is why maximum capacity, planning capacity and comfortable capacity should not be treated as synonyms.

A capacity number is partly a geometric fact and partly a definition.

Longer trains

If each car provides effective capacity P_car and there are n similar cars:

Ptrain ≈ nPcar

Doubling the number of cars can approximately double train capacity.

But longer trains require:

• longer platforms,
• compatible platform screen doors,
• longer station footprints,
• more traction power,
• larger depots and maintenance facilities,
• and suitable signalling and terminal infrastructure.

Vehicle capacity cannot be increased independently of the railway that receives the vehicle.

Doors create usable capacity

A train may contain physical standing space that passengers cannot reach quickly enough during the available dwell.

If passengers cluster near doors while the centre remains less crowded, the train has geometric capacity but poor usable capacity.

Let η_distribution represent how effectively passenger space is used:

Pusable = ηdistribution Pmaximum

0 < ηdistribution ≤ 1

This is a teaching abstraction.

It captures a real systems idea:

Space that exists but cannot be reached during the boarding process does not behave like fully usable capacity.

LTA’s public guidance encouraging passengers to move further into trains is therefore not merely etiquette. Better distribution can release capacity that already exists physically.

Prompt 3 — How Does Headway Turn One Train into Line Capacity?

A railway reuses the same track corridor repeatedly.

Headway determines how often.

If H is the number of seconds between trains:

q = 3600/H

Theoretical line capacity is:

Ctheoretical
=
Ptrain × 3600/H

Suppose one hypothetical train holds 1,000 passengers.

At 180-second headway:

q = 20 trains/hour
C = 20,000 pphpd

At 120 seconds:

q = 30 trains/hour
C = 30,000 pphpd

At 100 seconds:

q = 36 trains/hour
C = 36,000 pphpd

The train did not become larger.

Capacity increased because the track was reused more often.

This is why high-frequency urban rail can move extremely large passenger flows without every individual train being enormous.

A Circle Line public-data illustration

LTA states:

• maximum Circle Line train capacity: up to 931 passengers,
• main-loop peak frequency after CCL6 opening: about two minutes.

The simple theoretical product is:

q = 3600/120
  = 30 trains/hour

Cmaximum,theoretical
= 30 × 931
= 27,930 passenger places/hour/direction

This should not be read as a claim that 27,930 passengers are actually carried through every Circle Line section every peak hour.

The calculation assumes:

• every train arrives at exactly that frequency,
• every train reaches the stated maximum capacity,
• the chosen section receives the full train flow,
• stations and terminals sustain the pattern,
• and no operational margin reduces usable capacity.

It is therefore a useful ceiling calculation, not a complete operating-capacity statement.

Planning load factor

Let α represent a planning load factor relative to stated maximum train capacity:

Pplanning = αPmaximum

For illustration only, if:

α = 0.80

then a 931-passenger maximum becomes:

Pplanning = 0.80 × 931
          ≈ 745 passengers

At 30 trains per hour:

Cplanning ≈ 745 × 30
          ≈ 22,350 pphpd

The 80 per cent factor is purely educational and is not an LTA planning standard.

It shows how a practical capacity estimate can differ from an absolute maximum.

Vehicle capacity supplies the size of each packet. Headway decides how many packets the railway can send.

Prompt 4 — Why Is the Busiest Section More Important Than the Average Train Load?

A train does not carry the same number of passengers for its entire journey.

Passengers board and alight at every station.

Let L_s be the train load after leaving station s.

If B_s passengers board and A_s alight:

Ls = Ls−1 − As + Bs

This is a conservation equation.

Passengers after the station equal passengers before, minus those who leave, plus those who enter.

A six-station example

Suppose train capacity is 1,000 passengers.

Station A:
board 500
load after A = 500

Station B:
alight 50, board 300
load after B = 500 − 50 + 300 = 750

Station C:
alight 100, board 250
load after C = 750 − 100 + 250 = 900

Station D:
alight 350, board 100
load after D = 900 − 350 + 100 = 650

Station E:
alight 300, board 50
load after E = 650 − 300 + 50 = 400

Station F:
remaining passengers alight

The train’s average load across the sections is:

(500 + 750 + 900 + 650 + 400)/5
= 640 passengers

Average load factor is:

640/1000 = 64%

The line may sound comfortably below capacity.

But the C→D section carries 900 passengers:

900/1000 = 90%

That section is the bottleneck.

Unused capacity after Station E cannot be sent backwards in time to help passengers trying to board at Station C.

A railway can look half-empty on average and still be capacity-constrained at its critical link.

The bottleneck load factor

For train capacity P:

φs = Ls/P

The critical load factor is:

φcritical = maxs(φs)

Capacity planning should therefore study the maximum section load, not only the average load across the full route.

Origin–destination mathematics

A more complete model begins with an origin–destination matrix.

Let:

Oij(t)

represent passengers travelling from origin station i to destination station j during time interval t.

The load crossing section s is:

Ls(t)
=
Σ Oij(t)
for all journeys with i ≤ s < j

In words:

count every passenger whose journey crosses that section.

LTA DataMall lists passenger-volume datasets for individual train stations and origin–destination station pairs. It also lists real-time and forecast station crowd-density datasets. These forms of data allow passenger demand to be studied by place, direction and time instead of as one undifferentiated daily total.

This is how tap-in and tap-out activity can eventually become a load profile along the railway.

fare transactions
→ origin–destination demand
→ passenger assignment
→ section loads
→ bottleneck identification
→ service planning

Direction matters

Suppose the eastbound peak carries 30,000 passengers per hour while westbound demand is 12,000.

Total bidirectional demand is 42,000.

But a railway with 25,000 passenger places per hour in each direction cannot average the two directions and declare success.

The eastbound direction is overloaded even though the combined total is below 50,000.

capacity must satisfy demand
by direction
and by critical time interval

An empty return-direction train cannot directly create space on a full train moving the other way.

Prompt 5 — How Can Stations and Doors Limit Line Capacity?

A train may have enough internal space.

Passengers still need to reach and use it during the available dwell time.

This turns station capacity into part of line capacity.

A useful systems equation is:

Csystem
=
min(
Cvehicle,
Csignalling,
Cstation,
Cjunction,
Cterminal,
Cpower,
Cfleet,
Creliability
)

The minimum matters.

The lowest effective capacity in the chain can govern the whole railway.

Boardable space

Suppose a train arrives with L passengers.

A passengers alight.

If maximum train capacity is P, the space available for boarding is:

V = P − (L − A)
  = P − L + A

If Q passengers are waiting, the number able to board is:

B = min(Q, V)

New onboard load becomes:

Lnew = L − A + B

This tiny group of equations explains why alighting creates boarding capacity.

It also explains why the same train can accept many passengers at one station and very few at another.

Doorway capacity

Let μ_j be the effective flow rate through door j and T the usable exchange time.

Idealised doorway throughput is:

Kj = μjT

If n doors operate in parallel:

Kdoors ≈ Σ μjT

But passenger exchange does not finish until the busiest relevant doorway clears.

If N_j passengers use each door:

Texchange ≈ maxj(Nj/μj)

This is why uneven passenger distribution can reduce usable capacity.

Several doors may be lightly loaded while one door determines when the entire train may leave.

Platform circulation

The platform is also a flow system.

Its population changes according to:

Pnext
=
Pnow
+ entrants
+ alighting passengers
− boarding passengers
− passengers leaving the platform

In rate form:

dP/dt
=
λentry
+ λalight
− λboard
− λexit

If people enter or alight faster than stairs, escalators, lifts, passageways and fare gates can remove them, platform density grows.

Then station flow can become the bottleneck even when trains still have theoretical capacity.

A Singapore infrastructure example

LTA notes that Jurong East MRT station was retrofitted with two additional platforms in 2012 to increase capacity, allowing two trains to enter and depart the station at the same time.

This is a beautiful example of physical infrastructure changing the capacity equation.

The trains themselves did not need to become larger.

The station’s ability to process train movements increased.

A line can be limited by how quickly passengers pass through a door or how quickly trains pass through a station.

Prompt 6 — How Do Queues and Denied Boarding Change Capacity?

Suppose passengers arrive at a platform at rate λ passengers per second.

With headway H, approximate arrivals between trains are:

Narrival ≈ λH

Let Q_k be the waiting queue after train k departs.

Let B_k+1 be the number who can board the next train.

The queue evolves as:

Qk+1
=
max(0,
Qk + λHk − Bk+1)

If each train can board fewer passengers than arrive during the gap, the queue grows across successive trains.

A simplified stability requirement is:

E[B] > λE[H]

Average boardable capacity must exceed average arrivals over the relevant period.

But averages can hide short bursts.

During a surge:

λpeak H > B

and a temporary queue develops even if the longer-term average is manageable.

Denied boarding

When a waiting passenger cannot enter the arriving train, they remain for another headway.

Their platform wait becomes:

W
=
initial wait
+ one or more additional headways

If probability of being left behind is p and additional headway is H, an elementary expected extra-wait term is:

E[Wextra] ≈ pH

If passengers can be denied repeatedly, a geometric-style model can be used.

If each arriving train independently offers boarding probability r, the expected number of train opportunities needed is:

E[Ntrains] = 1/r

Expected extra train intervals are:

E[Ntrains − 1] = (1 − r)/r

so an illustrative extra-wait estimate is:

E[Wextra]
≈ H(1 − r)/r

Real boarding events are not independent and r changes with crowding, train load and passenger position, but the model reveals why capacity failure becomes nonlinear.

Once a passenger misses one train, they add to the queue confronting the next.

insufficient capacity
→ passengers left behind
→ larger next queue
→ slower next boarding
→ longer dwell
→ distorted headway
→ still larger passenger batch

The railway may enter positive feedback.

Why 100 per cent loading is not automatically optimal

A train filled to its stated maximum may appear perfectly efficient.

But very high crowding can:

• slow boarding and alighting,
• make internal space harder to reach,
• increase dwell variability,
• reduce the probability that passengers can board at downstream stations,
• and make the service less accessible and comfortable.

So effective throughput need not increase indefinitely as crowding approaches its physical maximum.

A conceptual throughput curve can rise, flatten and even deteriorate when crowding increases dwell enough to reduce train flow.

passengers/train rises
but
trains/hour may fall
if dwell becomes unstable

Since:

passenger throughput
=
passengers/train × trains/hour

maximising the first factor while damaging the second may not maximise the product.

A full train can reduce railway capacity if the fullness prevents trains from remaining regular.

Accessibility changes the usable capacity definition

Passengers do not all occupy or use space in identical ways.

Wheelchair users need designated spaces and accessible paths. Families may travel with strollers. Elderly or less mobile passengers may need more time and safer movement near doors. LTA provides priority queue zones at MRT and LRT stations for vulnerable commuters.

A capacity model built only around maximum standing density can therefore miss the railway’s social purpose.

The system should preserve:

physical capacity
+
boarding feasibility
+
accessible space
+
sufficient dwell
+
safe circulation

Accessibility is not outside capacity engineering.

It defines what useful capacity must be able to do.

Prompt 7 — Why Can Reliability Change Capacity Even When the Number of Trains Does Not?

Suppose a timetable schedules 30 trains in one hour.

Each train has an effective planning capacity of 900 passengers.

Scheduled capacity is:

30 × 900
= 27,000 passenger places/hour

Now suppose one train is not delivered during that hour.

Capacity becomes:

29 × 900
= 26,100 passenger places/hour

Loss:

900 places
= 3.33% of scheduled capacity

But the passenger impact can be larger than 3.33 per cent because the missing train creates one unusually long gap.

If normal headway is 120 seconds, removing one trip can create a gap approaching 240 seconds at some point in the sequence.

That larger gap collects more passengers and loads the following train more heavily.

So service loss changes both:

number of passenger places
and
distribution of those places through time

Regularity penalty

Consider two sequences with the same average headway.

Regular:

120, 120, 120, 120 seconds

Irregular:

60, 180, 60, 180 seconds

Both average 120 seconds.

But the 180-second gaps accumulate three times as many passengers as the 60-second gaps if arrival rate is constant.

That imbalance can overload some trains while others carry spare capacity.

Total geometric capacity has not changed.

Usable capacity has.

A simple regularity factor can be imagined:

ηregularity
=
usefully distributed capacity
/
geometric scheduled capacity

When trains bunch, η_regularity falls.

This is why automatic train regulation and delay recovery matter to capacity, not only punctuality.

Capacity needs reserve

A railway operated permanently at every absolute limit may have no room for:

• a longer dwell,
• a slow boarding passenger,
• a small running-time variation,
• a temporary train fault,
• or a brief signalling constraint.

So resilient capacity may be defined below absolute capacity.

Cresilient
=
Cmaximum − Creserve

The reserve is not necessarily wasted.

It is what allows the railway to continue operating when reality deviates from the average.

Capacity without reliability is a number the railway cannot consistently deliver.

Prompt 8 — How Can an MRT System Increase Passenger Capacity?

There is no single capacity lever.

A railway can act on several parts of the equation.

1. Increase train capacity

Possible methods include longer trains, more usable interior space, wider gangways, more effective layouts or additional doors.

But stations, platforms, depots, power and signalling must be compatible.

2. Increase train frequency

Reduce headway:

C = P × 3600/H

But signalling, dwell, junctions, terminals, fleet and power all have to sustain the smaller H.

3. Improve station throughput

Reduce doorway conflicts, distribute passengers more evenly, improve stairs and escalators, add platforms or redesign circulation.

The Jurong East additional-platform project is a Singapore example of increasing the station’s ability to process trains.

4. Improve terminal and junction capacity

Faster and more flexible turnbacks can permit more frequent train cycles.

The Pasir Ris rail turnback completed in 2024 is one Singapore example of infrastructure supporting faster crossover and shorter East-West Line frequencies when required.

5. Improve reliability and regulation

Deliver the scheduled trains and preserve regular headways.

This releases usable capacity without necessarily changing train size or nominal frequency.

6. Redistribute demand through the network

New lines and interchanges can give passengers alternative paths.

LTA states that the fully completed Thomson-East Coast Line is expected to relieve crowding on the East-West and North-South Lines while serving about one million commuters daily.

This is network capacity.

The old line may not carry more trains, but its critical passenger load can fall because journeys are reassigned to another route.

new route option
→ passenger redistribution
→ lower bottleneck load
→ more resilience on existing lines

7. Match frequency to time-varying demand

LTA describes typical network frequencies of about two to three minutes during the morning peak and about five to seven minutes off-peak.

Running peak frequency all day would provide more capacity than necessary during quieter periods and increase operating and energy costs.

Running off-peak frequency during the peak would under-supply demand.

The timetable therefore tries to make:

scheduled supply(t)
≈
passenger demand(t)
+ resilience margin

This is capacity allocation through time.

The optimisation problem

A conceptual objective might be:

minimise J
=
w1(passengers left behind)
+ w2(waiting time)
+ w3(crowding)
+ w4(fleet cost)
+ w5(energy)
+ w6(delay risk)
+ w7(accessibility failure)
+ w8(capital cost)

subject to:

train capacity limits
minimum safe headway
station dwell constraints
platform capacity
junction and terminal constraints
available rolling stock
power-system limits
required service reliability

The objective is not to maximise one number.

It is to balance the entire transport job.

Capacity engineering is the art of finding which constraint should be improved next.

A Complete Worked MRT Capacity Model

Let us build a fictional line from first principles.

This is an educational example, not an actual Singapore MRT operating model.

Step 1 — Define train capacity

Pmaximum = 1,000 passengers/train

Use a practical planning factor:

α = 0.85

Pplanning = 0.85 × 1,000
          = 850 passengers/train

Step 2 — Define scheduled headway

H = 120 s

Train flow:

q = 3600/120
  = 30 trains/hour

Step 3 — Calculate scheduled practical capacity

Cscheduled
= 850 × 30
= 25,500 pphpd

Step 4 — Apply service delivery

Suppose 98 per cent of the scheduled train capacity is delivered during the study period:

ηservice = 0.98

C1 = 25,500 × 0.98
   = 24,990 pphpd

Step 5 — Apply regularity

Suppose irregular spacing means only 96 per cent of geometric capacity is usefully distributed through time:

ηregularity = 0.96

Cdelivered
= 24,990 × 0.96
≈ 23,990 pphpd

The factors are hypothetical.

They show how an apparently 30,000-pphpd maximum can become a smaller usable number after practical loading and service performance are considered.

Step 6 — Compare with demand

Suppose peak demand on the critical section is:

Dcritical = 23,000 passengers/hour

Capacity margin is:

M = Cdelivered − Dcritical
  = 23,990 − 23,000
  = 990 passengers/hour

Utilisation is:

u = Dcritical/Cdelivered
  ≈ 23,000/23,990
  ≈ 95.9%

The line is operating close to its illustrative delivered capacity.

Step 7 — Test a passenger surge

Suppose demand rises by 8 per cent:

Dsurge = 23,000 × 1.08
       = 24,840 passengers/hour

Now:

Dsurge − Cdelivered
= 24,840 − 23,990
= 850 passengers/hour

A queue can accumulate unless the railway temporarily delivers more capacity or demand redistributes.

Step 8 — Test a shorter headway

Suppose infrastructure and operations can safely reduce headway to 110 seconds.

q = 3600/110
  ≈ 32.73 trains/hour

In an actual timetable, trains occur as discrete events, so the realised hourly count depends on the exact period boundaries. For a rate calculation:

Cscheduled rate
≈ 850 × 32.73
≈ 27,818 pphpd

Apply the same illustrative factors:

Cdelivered
≈ 27,818 × 0.98 × 0.96
≈ 26,174 pphpd

The demand surge can now be absorbed in the numerical example.

But only if the shorter headway does not worsen dwell, regularity, terminal performance or reliability.

The final condition is therefore:

shorter headway is useful
only when the full system can sustain it

The Capacity Cube: Space × Time × Reliability

One elegant way to summarise MRT capacity is as three dimensions.

Space

How many passengers can each train and station physically accommodate?

Time

How often can trains, doors, platforms, junctions and terminals repeat their operations?

Reliability

How consistently can those spatial and temporal capacities actually be delivered?

Conceptually:

Useful capacity
≈
space per train
× repetitions per hour
× delivery quality

Remove any one dimension and the number collapses.

A giant train that arrives once an hour has low hourly capacity.

Tiny trains arriving frequently may still carry too few people.

Large, frequent trains that arrive in bunches waste capacity through irregularity.

MRT capacity is not merely space. It is space delivered through time with sufficient reliability.

The Capacity Deletion Tests

We can understand the system by removing one component.

Remove train capacity

Frequent trains arrive, but each carries too few passengers.

Remove frequency

Large trains exist, but the track corridor is reused too slowly.

Remove station throughput

Passengers cannot enter, leave or clear the platform quickly enough to sustain train flow.

Remove directional analysis

Empty trains in one direction disguise overload in the other.

Remove section-load analysis

A comfortable route average disguises a critical bottleneck between two stations.

Remove reliability

Capacity exists on paper but is not delivered consistently.

Remove accessibility

The system may claim a larger physical number while failing part of the population it exists to serve.

The deletion tests reveal that capacity is a system property.

No single train owns it.

The Capacity Paradoxes

Paradox 1 — A full train can indicate too little capacity

High occupancy may mean demand exceeds the service being supplied.

Paradox 2 — A full train can also indicate good utilisation

If passengers can still board reliably and dwell remains stable, high utilisation may be efficient.

The load alone does not tell us which case we are seeing.

Paradox 3 — More standing space may reduce capacity elsewhere

Changing interiors can affect seats, accessibility, doorway circulation and passenger comfort.

Paradox 4 — A shorter headway can deliver less useful capacity

If stations or terminals cannot sustain it, irregularity and bunching may worsen.

Paradox 5 — A new line can increase capacity on an old line without adding trains to it

Passenger demand is redistributed through the network.

Paradox 6 — Spare capacity may exist but be unusable

It may be on the wrong carriage, wrong section, wrong direction or wrong time of day.

Why Passenger Capacity Is a Graph Problem

An MRT network can be represented as a graph:

G = (V,E)

where stations are nodes V and rail connections are edges E.

Each edge can have:

• passenger demand,
• train capacity,
• frequency,
• travel time,
• crowding cost,
• reliability,
• and alternative routes.

Passenger assignment asks which path travellers choose.

When one edge becomes crowded, some passengers may choose another route if the network provides one.

This means demand on a line is not always fixed.

It can respond to:

• travel time,
• waiting time,
• number of transfers,
• crowding,
• disruption,
• and new connections.

Academic transit-assignment models therefore include capacity constraints and crowding because a passenger’s route choice can depend on whether a train is likely to have usable space.

Capacity planning is no longer:

one line carries one fixed demand

It becomes:

network design
→ route choice
→ passenger distribution
→ section loading
→ crowding
→ new route choice

Again, the output returns as a new input.

Why Capacity Should Be a Probability, Not Only a Fixed Number

Passenger demand varies.

Train delivery varies.

Dwell varies.

So a robust capacity plan should ask:

What is the probability demand exceeds usable capacity?

Let random demand be D and delivered capacity be C.

Overload probability is:

P(D > C)

A planning constraint might conceptually require:

P(D ≤ C) ≥ 1 − ε

where ε is a chosen small probability of demand exceeding planned capacity during the defined condition.

This is not an LTA formula.

It illustrates the difference between:

capacity that matches average demand
and
capacity that remains adequate under normal variation

Suppose average peak demand is 20,000 pphpd.

A line with exactly 20,000 pphpd of usable capacity will be overloaded roughly whenever demand rises above its mean.

A resilient system needs margin related to the distribution, not merely the average.

The average passenger is not the busiest morning.

The World Return — When Passenger Data Answers the Capacity Model

The planner predicts demand.

The timetable supplies trains.

The railway operates.

Then reality returns evidence:

• tap-in and tap-out volumes,
• origin–destination journeys,
• train loading,
• platform crowd density,
• dwell times,
• passengers left behind,
• headway regularity,
• and actual service delivery.

Let predicted section load be L̂_s,t and observed load be L_s,t.

Model error is:

es,t = Ls,t − L̂s,t

If one station is repeatedly more crowded than predicted, the mismatch may reveal:

• a new travel pattern,
• an incorrect origin–destination assumption,
• a transfer surge,
• uneven passenger distribution,
• or under-delivered train capacity.

The planning loop becomes:

forecast demand
→ schedule capacity
→ operate
→ observe passenger flow
→ locate bottleneck
→ compare prediction with reality
→ revise service or infrastructure

LTA DataMall’s station-volume, origin–destination and crowd-density datasets show the public-data side of this logic: transport demand can be observed in increasingly spatial and time-specific ways.

The model should not defend itself against the measurement.

It should learn from it.

RFE Return — What Is a Good MRT Capacity?

A good capacity is not:

the largest possible number of passengers per square metre

It is not:

the smallest possible number of seconds between trains

And it is not:

the maximum theoretical pphpd written in a design document

A good capacity performs the entire public job:

enough space
+
enough trains
+
regular spacing
+
workable station flow
+
accessible boarding
+
manageable crowding
+
reliable delivery
+
recovery margin
+
capacity where and when demand actually occurs

That is why a half-empty train is not automatically wasteful.

It may be necessary to position the train for a later peak, serve a quieter section, preserve frequency or carry passengers in the reverse direction.

And a packed train is not automatically efficient.

It may be a symptom of irregular headway, insufficient service, poor passenger distribution or a bottleneck elsewhere.

The correct capacity is not the fullest railway. It is the railway that moves the required people with the least avoidable waiting, crowding and instability.

Conclusion — A Full Train Is Only One Frame of the Film

Stand on an MRT platform during the peak.

A train arrives crowded.

It is tempting to conclude that the line is full.

But capacity exists across space and time.

We need to know:

• how large the train is,
• how frequently trains arrive,
• how regularly they arrive,
• where passengers board and alight,
• which section is busiest,
• whether passengers can reach available space,
• whether doors and platforms can process the flow,
• whether terminals and junctions sustain the service,
• and whether the timetable can recover from variation.

Only then can we understand the line.

The mathematics travels through a chain:

train geometry
→ passengers per train
→ headway
→ trains per hour
→ passenger places per hour
→ origin–destination demand
→ section load
→ bottleneck
→ platform queue
→ dwell
→ regularity
→ delivered capacity
→ passenger experience

A train can be full while the next train has space.

A line can be unconstrained on average while one section is overloaded.

A network can gain capacity by adding another route rather than enlarging an existing train.

A timetable can lose usable capacity without losing a single physical carriage if its headways become irregular.

And a railway can deliberately preserve empty space because that space is doing another job: allowing downstream boarding, accessibility, recovery and resilience.

The passenger sees a crowded carriage.

The vehicle engineer sees floor area and doors.

The timetable planner sees trains per hour.

The data scientist sees an origin–destination matrix.

The station designer sees flow rates.

The operations researcher sees a bottleneck.

The city feels whether all of them agreed.

An MRT line is not full when one train is full. It is full when the next unit of passenger demand cannot be carried through the critical part of the system without unacceptable waiting, crowding or instability.

That is how MRT passenger capacity is calculated using mathematics.

Key Equations

Ptrain = S + ρAusable + Pdesignated
Conceptual train capacity

Pusable = ηdistribution Pmaximum
Capacity adjusted for space utilisation

q = 3600/H
Trains per hour from headway

Ctheoretical = Ptrain × 3600/H
Theoretical line capacity

Cscheduled = Pplanning × qscheduled
Scheduled practical capacity

Cdelivered = Cscheduled × ηservice × ηregularity
Illustrative delivered capacity

Ls = Ls−1 − As + Bs
Train-load conservation at a station

φs = Ls/P
Section load factor

φcritical = maxs(φs)
Bottleneck load factor

Ls(t) = ΣOij(t), for i ≤ s < j
Section load from origin–destination demand

V = P − L + A
Boardable space after alighting

B = min(Q,V)
Passengers able to board

Qk+1 = max(0,Qk + λHk − Bk+1)
Platform queue evolution

E[B] > λE[H]
Simplified queue-stability condition

E[Wextra] ≈ H(1−r)/r
Illustrative extra wait under repeated denied boarding

Csystem = min(Cvehicle,Csignalling,Cstation,
              Cjunction,Cterminal,Cpower,Cfleet,Creliability)
Whole-system bottleneck capacity

M = Cdelivered − Dcritical
Capacity margin

u = Dcritical/Cdelivered
Capacity utilisation

P(D ≤ C) ≥ 1 − ε
Illustrative probabilistic capacity requirement

es,t = Ls,t − L̂s,t
Observed minus predicted section load

Reader-safety note: The formulas, planning factors and fictional examples in this article are educational abstractions. They do not reproduce proprietary Singapore MRT loading standards, internal capacity models, restricted operating data, signalling rules or control procedures. Actual capacity planning uses line-specific, train-specific and safety-validated engineering assumptions.

Continue the MRT Mathematics Series

• How MRT Timing Works Using Mathematics — the system-wide pillar.
• How MRT Braking Works Using Mathematics — stopping distance and regenerative braking.
• How MRT Headway Works Using Mathematics — how train spacing becomes throughput.
• How MRT Station Dwell Time Works Using Mathematics — queues, doors and platform flow.
• How MRT Delays Propagate and Recover Using Mathematics — instability, regulation and resilience.
• How an MRT Timetable Is Built Using Mathematics — demand, train cycles and robustness.
• How MRT Energy Use Is Optimised Using Mathematics — traction, coasting and regenerative energy.

The next natural leg is How MRT Routes and Transfers Are Optimised Using Mathematics: graph theory, shortest paths, generalized travel cost, interchange penalties, crowding, resilience and why the geographically shortest route is not always the best route.

Sources and Further Reading

• Land Transport Authority — Circle Line train maximum capacity
• Land Transport Authority — Circle Line Stage 6 opening and current operating frequencies
• Land Transport Authority — Rail network operating hours and typical peak/off-peak frequencies
• LTA DataMall — train-station volume, origin–destination and crowd-density datasets
• Land Transport Authority — Jurong East additional platforms and station capacity
• Land Transport Authority — Thomson-East Coast Line and relief of crowding on existing lines
• Land Transport Authority — RTS Link capacity of up to 10,000 commuters per hour in each direction
• Land Transport Authority — inclusive transport and priority queue zones
• Transportation Research Part C — train-following performance and bottleneck capacity sections
• Urban Rail Transit — integrated train and pedestrian capacity analysis at passenger rail hubs
• Soft Computing — integrated urban-rail timetabling, passenger demand and rolling-stock capacity
• Journal of Rail Transport Planning & Management — dynamic passenger crowding and capacity bottlenecks
• Transportation Research Part C — capacity-constrained passenger assignment in transit networks
• Urban Rail Transit — international analysis of passenger boarding and alighting rates

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