A passenger sees an MRT timetable as a promise.
The first train will come around a certain time. During the morning peak, another train should follow a few minutes later. The journey planner estimates when we will arrive. The last train determines whether we can still get home.
Inside the railway, however, a timetable is not merely a list of clock times.
It is a mathematical construction made from passenger demand, running time, dwell time, braking, headway, junction conflicts, terminal turnaround, fleet size, energy, maintenance access and recovery margin.
Every arrival must be connected to a previous departure. Every departure must leave enough safe space behind the train ahead. Every train that leaves one terminal eventually has to return, reverse direction, enter another trip or leave service. Every minute given to passengers is also a minute that cannot be used by another train, a maintenance team or a different operating movement.
An MRT timetable is a city converted into events, intervals and constraints.
Singapore’s Land Transport Authority states that the rail network generally operates from about 5.30am until around midnight, with typical frequencies of about two to three minutes during the 7am–9am peak and five to seven minutes during off-peak periods. Those passenger-facing numbers are the visible edge of a much deeper scheduling problem.
This article continues the eduKateSG MRT mathematics series. Begin with How MRT Timing Works Using Mathematics, then explore braking, headway, station dwell, and delay propagation and recovery.
Now we ask the planning question that comes before all of them:
How do we build the sequence of times that tells an entire MRT line when to move?
The RFE — Why Does an MRT Timetable Exist?
A weak answer is:
so trains know when to leave
A stronger answer begins with people.
The timetable exists to convert changing passenger demand into a safe, repeatable and recoverable transport service using finite infrastructure.
people wanting to travel + tracks, stations, trains and power + safety and operating constraints + limited time = a feasible public service
Its Reason for Existence is therefore not to make a beautiful spreadsheet.
It is to answer a real city problem:
Who needs to move? When do they need to move? How many trains are required? Where must those trains be? How closely can they operate? How much uncertainty can the plan absorb? When must the railway stop carrying passengers so that it can be maintained?
A good timetable is therefore:
useful to passengers + feasible for trains + safe for the railway + economical in resources + robust when reality changes
That is the RFE.
Prompt 1 — How Does Passenger Demand Become a Train Frequency?
The timetable should not begin with trains.
It should begin with the transport job.
At 6am, passenger demand may be modest. At 8am, it can be much larger. During the middle of the day, it may fall. In the evening, direction and intensity can change again.
So passenger arrival demand is better represented as a function of station, direction and time:
λs,d(t)
where:
- s identifies the station,
- d identifies direction,
- t identifies time,
- λ represents passenger arrival rate.
A timetable that ignores this time pattern may send too many trains when they are not needed and too few when demand is strongest.
From demand to required capacity
Suppose a section of railway must carry D passengers per hour in one direction.
If one train provides an effective usable capacity of P passengers under the assumptions of the planning model, the minimum number of trains required per hour is approximately:
qrequired ≥ D/P
If q trains are required per hour, corresponding headway is:
H = 3600/q
For an illustrative demand of 24,000 passengers per hour and an effective planning load of 1,000 passengers per train:
qrequired = 24,000/1,000
= 24 trains/hour
H = 3600/24
= 150 seconds
This does not yet prove that a 150-second timetable is feasible. It only describes the demand side.
The railway must still ask whether signalling, stations, terminals, fleet and power can sustain it.
Demand is not evenly distributed
Even within one hour, demand can surge.
A simple hourly average may hide a ten-minute peak.
Suppose 12,000 passengers arrive during one hour. That is an average of 200 passengers per minute.
But if 4,000 of them arrive during one especially intense ten-minute interval:
4,000/10 = 400 passengers/minute
The short-period demand is twice the hourly average rate.
This is why timetable design uses time-dependent demand rather than one daily total.
Research on demand-driven metro scheduling explicitly treats passenger demand as varying across time and space, and optimises headway, train service and capacity around that pattern.
Waiting-time mathematics
If trains arrive perfectly regularly every H seconds and passengers arrive randomly relative to the service, expected waiting time is approximately:
E[W] = H/2
At a 180-second headway:
E[W] = 90 s
At 120 seconds:
E[W] = 60 s
At 100 seconds:
E[W] = 50 s
Shorter headway reduces waiting and also reduces the batch of passengers accumulating before each train.
But shorter headway usually requires more trains and leaves less operating room.
Demand asks for frequency. Infrastructure decides whether that frequency is possible.
A current Singapore example of differentiated frequency
A timetable does not have to use one frequency everywhere.
When Circle Line Stage 6 opened in 2026, LTA stated that from 12 July trains would arrive at about two-minute frequencies on the main loop during peak hours, about four minutes on the Dhoby Ghaut–Promenade–Bayfront–Marina Bay section, and about five to six minutes during off-peak periods and weekends.
This is an excellent illustration of timetable geometry.
Different segments can receive different train flows because routes, branches, demand and operating patterns differ.
The timetable is therefore not simply:
one line = one frequency
It can be:
route pattern + segment demand + branch interaction + terminal availability = different effective frequencies
Prompt 2 — How Is a Timetable Represented as Events and Activities?
A timetable can be modelled as a network of events.
An event might be:
- Train 7 arrives at Station C,
- Train 7 departs Station C,
- Train 7 reaches the terminal,
- Train 7 begins its return trip.
An activity connects two events.
Examples include:
- running between stations,
- dwelling at a platform,
- turning back at a terminal,
- waiting for a transfer,
- maintaining headway from another train.
This creates an event-activity network.
arrival event → dwell activity → departure event → running activity → next arrival event
Let:
Aᵢ,ₖ = arrival time of train i at station k Dᵢ,ₖ = departure time of train i at station k rᵢ,ₖ = running time from station k to k+1 dᵢ,ₖ = dwell time at station k
Then two fundamental relationships are:
Dᵢ,ₖ = Aᵢ,ₖ + dᵢ,ₖ Aᵢ,ₖ₊₁ = Dᵢ,ₖ + rᵢ,ₖ
These equations form the spine of the timetable.
Every departure comes from an arrival plus dwell.
Every next arrival comes from the previous departure plus running time.
Minimum and maximum activity times
An activity usually cannot take any arbitrary length of time.
For running activity a:
la ≤ ta ≤ ua
where:
- la is the lower bound,
- ua is the upper bound,
- ta is the scheduled activity time.
The lower bound may represent the shortest safe and feasible time.
The upper bound prevents unnecessary slowness or may reflect service-quality requirements.
For a dwell activity:
dmin ≤ dᵢ,ₖ ≤ dmax
The minimum must still permit safe passenger exchange.
The timetable therefore becomes a large set of connected inequalities.
Headway constraints
Two successive trains cannot be scheduled arbitrarily close.
At a reference point:
Dᵢ,ₖ − Dᵢ₋₁,ₖ ≥ Hmin,ₖ
This represents the minimum feasible separation under the relevant assumptions.
LTA states that the upgraded CBTC signalling on the North-South and East-West Lines has a technical capability of 100-second headways, compared with 120 seconds before 2018. That is a signalling capability, not a requirement that every passenger service always operate at precisely 100 seconds.
The operating timetable must satisfy all constraints, not only the signalling one.
Periodic timetables
Many public transport timetables repeat a pattern.
In a metro, the pattern may be a regular headway rather than a passenger memorising every exact departure.
If the period is T, events repeat modulo T.
time of event in next cycle = time of event now + T
Periodic Event Scheduling Problem models use this event-and-activity structure to construct feasible repeating timetables.
A timetable is a graph whose edges are made of time.
Prompt 3 — Where Do the Running Times Come From?
The timetable cannot simply invent a two-minute interstation time.
Running time must be physically achievable.
A train begins at one state:
position = Station A velocity = 0
and must reach another:
position = Station B velocity = 0
while respecting:
- traction limits,
- braking limits,
- speed limits,
- gradients and curves,
- passenger-comfort limits,
- signalling movement authority,
- and energy objectives.
Velocity is a function of time:
v = v(t)
Distance is the integral of velocity:
L = ∫ v(t)dt
For a section of length L, running time may also be written conceptually as:
r = ∫₀ᴸ dx/v(x)
subject to the train’s speed remaining inside the permitted envelope.
Minimum time is not automatically scheduled time
Suppose the physically achievable running time under a permitted high-performance trajectory is:
rmin = 76 s
The timetable may schedule:
rscheduled = 82 s
The difference:
srun = 82 − 76 = 6 s
is a simplified running-time supplement.
It can provide:
- energy-efficient coasting,
- recovery capability,
- allowance for ordinary variability,
- and smoother interaction with neighbouring trains.
SMRT’s Green CBTC programme offers a real Singapore example of why a timetable and speed profile belong together. SMRT states that Automatic Train Operation controls acceleration, braking and coasting along optimised curves, seeking energy savings while maintaining punctuality.
The scheduled running time therefore helps define which trajectories are available.
If the timetable is extremely tight, the train may need to use a high-energy trajectory more often and retain less recovery room.
If it is unnecessarily loose, passenger journeys become longer and line capacity can be wasted.
Running time is not only a prediction of physics. It is also a budget that the timetable gives to the control system.
Prompt 4 — How Are Dwell Times, Transfers and Stations Put into the Timetable?
A train can be fast between stations and still operate a poor timetable if it cannot exchange passengers predictably.
For train i at station k:
Dᵢ,ₖ = Aᵢ,ₖ + dᵢ,ₖ
Dwell di,k must cover the necessary station process.
Singapore’s Ministry of Transport stated in November 2025 that typical MRT/LRT dwell times are approximately 35–60 seconds at interchange stations and 30–50 seconds at non-interchange stations, depending on operating conditions and passenger loading.
Those ranges immediately tell us that station time is not trivial.
If an interstation run takes 80 seconds and dwell takes 40 seconds, the train spends one-third of that departure-to-departure cycle stationary:
40/(80 + 40) = 1/3
A timetable that models running time precisely but treats dwell casually is incomplete.
Flow-dependent dwell
A simple educational dwell model is:
d = d0 + αA + βB + γI
where:
- A is the number alighting,
- B is the number boarding,
- I represents crowding or interaction,
- d0 is the minimum process time.
This is not an LTA formula. It shows why scheduled dwell should respond to passenger conditions.
Academic timetable research explicitly treats dwell as flow-dependent because passenger volumes influence the time needed at stations.
Transfers
At an interchange, one timetable can affect another.
Suppose a passenger needs at least τ minutes to walk from the arrival platform of Line A to the departure platform of Line B.
A planned connection requires:
DB − AA ≥ τ
where AA is the arrival on Line A and DB the departure on Line B.
In a very frequent metro, passengers may not need a formally protected connection because another train follows soon.
But transfer waves still matter.
An arriving train can release a batch of passengers into another platform. That can alter boarding demand and dwell on the connecting line.
Line A arrival → transfer walking time → Line B platform pulse → Line B boarding demand → Line B dwell
The timetable has become a network rather than a single line.
Prompt 5 — How Are Headways, Junctions, Branches and Terminals Fitted Together?
Once running and dwell times are known, the scheduler still has to prevent trains from demanding the same railway resource at incompatible times.
Following trains
Successive departures must satisfy minimum headway:
Dᵢ,ₖ − Dᵢ₋₁,ₖ ≥ Hmin
But desired headway may be larger than the technical minimum:
Hscheduled = Htechnical + Hmargin
The margin helps the timetable remain operable when dwell and running times vary.
Junction conflicts
At a junction, two trains may require conflicting routes.
If Train A occupies the junction during interval:
[tAstart, tAend]
and Train B requires a conflicting route, their occupation intervals cannot overlap after required safety separation is considered.
A simplified either-or condition is:
B after A: tBstart ≥ tAend + s or A after B: tAstart ≥ tBend + s
This is a disjunctive scheduling constraint.
Similar mathematics appears in runway scheduling and manufacturing machines: one scarce resource cannot process two incompatible jobs simultaneously.
Branches and merges
Suppose two branches each send one train every four minutes into a common trunk.
If they are perfectly offset by two minutes:
Branch A: 00, 04, 08, 12 Branch B: 02, 06, 10, 14 Common trunk: 00, 02, 04, 06, 08, 10...
The trunk receives a two-minute service.
If the offset is poor:
Branch A: 00, 04, 08, 12 Branch B: 01, 05, 09, 13 Common trunk gaps: 1 min, 3 min, 1 min, 3 min...
The average frequency is still one train every two minutes.
The passenger experience is worse because headway is irregular.
Branch timetabling is therefore an offset problem.
Terminal turnaround
At the end of a line, a train must become a train in the opposite direction or leave passenger service.
If arrival at the terminal is Ai,T and minimum turnback time is τturn:
Dreturn,i ≥ Aᵢ,T + τturn
The terminal also has finite crossover and platform capacity.
If turnback cannot process trains at the desired headway, it becomes the bottleneck for the whole line.
LTA’s completion of the Pasir Ris rail turnback in April 2024 is a concrete Singapore example of physical infrastructure enabling faster crossover movements and supporting shorter East-West Line frequencies when required.
A timetable is limited by the slowest repeated operation in its cycle.
Prompt 6 — How Does the Timetable Determine How Many Trains Are Needed?
A headway cannot exist without enough trains to fill it.
Suppose a complete cycle includes:
- one-way running time,
- one-way dwell time,
- terminal turnaround,
- return running and dwell,
- turnaround at the original end.
Then:
Tcycle = Toutbound + Tturn,far + Tinbound + Tturn,home
For a regular headway H, the minimum number of trainsets simultaneously needed in the repeating cycle is approximately:
Ncycle = ceil(Tcycle/H)
The ceiling function matters because a railway cannot operate 33.4 trains.
It needs 34.
Worked hypothetical line
Consider a deliberately fictional 15-station metro line.
There are 14 interstation sections.
Suppose one-way running times sum to:
Σr = 1,330 s
Suppose dwell at the 13 intermediate stations sums to:
Σd = 455 s
One-way time is:
Tout = 1,330 + 455
= 1,785 s
= 29 min 45 s
Assume the return direction is similar and each terminal receives a four-minute turnaround:
Tcycle = 1,785 + 240 + 1,785 + 240 = 4,050 s = 67 min 30 s
At a 120-second headway:
Ncycle = ceil(4,050/120)
= ceil(33.75)
= 34 trains
At a 100-second headway:
Ncycle = ceil(4,050/100)
= ceil(40.5)
= 41 trains
Reducing headway by twenty seconds requires seven additional trainsets in this fictional cycle before any allowance for spare trains, maintenance or operational contingencies.
This is not an estimate of any actual Singapore MRT line.
It demonstrates the connection:
running time + dwell + turnback = cycle time cycle time ÷ headway = active fleet requirement
Why one saved minute can remove a whole train from the cycle
At a two-minute headway, reducing cycle time by two minutes can reduce the theoretical active requirement by approximately one trainset.
ΔN ≈ ΔTcycle/H
But the ceiling function creates thresholds.
Reducing cycle time from 4,050 seconds to 3,960 seconds at 120-second headway gives:
ceil(3,960/120) = ceil(33) = 33 trains
A 90-second cycle improvement crossed an integer boundary and removed one train from the minimum repeating cycle.
This is why small timetable improvements can have discrete fleet effects.
Fleet size is continuous mathematics forced to buy whole trains.
Prompt 7 — How Is Robustness Built into a Timetable?
A timetable that works only when every event occurs at its minimum time is not a strong timetable.
It is a demonstration run.
Real operations need margin.
Suppose minimum activity time is la and scheduled time is ta.
Activity slack is:
sa = ta − la
Slack can be placed in:
- running time,
- dwell time,
- headway buffers,
- terminal turnaround,
- or scheduled layover.
The difficult question is not whether to include margin.
It is where margin produces the most resilience for the least passenger cost.
Uniform slack versus strategic slack
Suppose a line has 20 activities and ten minutes of total margin.
One approach distributes 30 seconds everywhere.
Another concentrates more margin after busy interchanges, difficult junctions or sections with greater variability.
The second may be more useful if disturbances are not equally likely everywhere.
Let σa represent variability of activity a.
A simple planning idea is:
allocate more slack where σa is larger or where propagation cost is greater
The timetable is then protecting its most vulnerable connections.
Probability rather than certainty
Running and dwell times vary.
So a robust timetable may use chance-style requirements such as:
P(actual activity time ≤ scheduled activity time) ≥ 1 − ε
This is an educational expression, not an LTA operating formula.
It captures the idea that a timetable should work with high probability, not merely under an average case.
The capacity–robustness trade-off
More margin improves recovery.
But margin consumes time.
If every train receives longer running, dwell and turnback allowances, cycle time grows and more trains may be needed to preserve the same headway.
more margin → greater robustness → longer cycle time → potentially more fleet or less frequency
The optimum lies between two poor extremes:
too little margin → fragile timetable too much margin → slow and resource-heavy timetable
A timetable is robust when it carries enough unused time to survive reality without carrying so much that it wastes the railway.
The maintenance timetable
Passenger service is only one user of railway time.
Tracks, signalling, power and stations must be inspected, tested, repaired and renewed.
LTA’s passenger information states that normal rail operation is generally from about 5.30am to around midnight. The hours outside passenger service are not empty.
They are engineering access.
Several LTA service adjustments during 2025 and 2026 shortened or altered Circle, Thomson-East Coast and Downtown Line operating hours to create longer blocks of engineering time for integration, testing, commissioning or tunnel works.
This demonstrates a deeper timetable equation:
24 hours = passenger service + start-up and closure processes + engineering access + contingency margin
Extending passenger service reduces maintenance access unless something else changes.
A railway timetable is therefore also an asset-care timetable.
Prompt 8 — How Is the Whole Timetable Optimised and Then Operated?
By now, the timetable has many decision variables:
- first and last departure times,
- peak and off-peak headways,
- arrival and departure times at every station,
- running times,
- dwell times,
- branch offsets,
- junction order,
- terminal turnbacks,
- train insertion and withdrawal,
- fleet circulation,
- and recovery margins.
It also has multiple objectives.
A simplified multi-objective cost function might be:
minimise J = w1(passenger waiting) + w2(crowding) + w3(fleet requirement) + w4(energy consumption) + w5(transfer inconvenience) + w6(headway irregularity) + w7(delay propagation risk) + w8(maintenance-access cost)
subject to all safety and feasibility constraints.
This is often formulated using mathematical programming, including mixed-integer models where some choices are continuous times and others are discrete decisions.
Why integers appear
Some decisions are yes/no:
Does Train A go before Train B at the junction? Does this trip short-turn? Is this train inserted from the depot? Is another train required in the cycle?
These can be represented by binary variables:
x ∈ {0,1}
Other decisions are continuous:
departure time = 08:01:37 running allowance = 5.5 s
Putting both types together creates a mixed-integer optimisation problem.
The timetable changes across the day
A full operating day may contain phases:
start-up → early morning → morning peak → inter-peak → evening peak → late evening → close-down → engineering hours
Headway may change between phases.
Trains may enter or leave service.
Branch patterns may change.
The timetable must transition between frequencies without creating badly spaced trains.
For example, changing from 180-second to 120-second headway is not achieved by merely changing a label.
Additional trains must be inserted at appropriate times and places so that the new sequence forms smoothly.
First and last trains
The first train at one station is not necessarily the first train leaving a terminal.
Trains can begin from different stabling or depot positions and enter service at different stations.
Likewise, the last train must be planned with:
- end-of-service passenger demand,
- connections,
- terminal arrival,
- depot or stabling movement,
- and the beginning of engineering access.
The timetable therefore has boundary conditions at the start and end of the operating day.
The plan meets reality
Once service begins, the timetable is no longer alone.
Actual running and dwell times return from the world.
planned arrival vs actual arrival planned headway vs actual headway planned passenger load vs observed passenger load
The operating system then regulates trains within its permitted rules.
It may use running-time flexibility, speed-profile adjustment, dwell management or other operational measures to return the service towards a useful rhythm.
The timetable is therefore not the controller.
It is the reference trajectory the controller tries to realise.
plan → operate → observe → compare → regulate → recover → learn for the next plan
A timetable is finished on paper only when it begins being tested by the world.
A Complete Worked Timetable Construction
Let us build a small fictional metro timetable from the ground up.
This example is purely educational and is not an actual Singapore line.
Step 1 — Define the line
Five stations:
A — B — C — D — E
Running times:
A→B = 90 s B→C = 80 s C→D = 100 s D→E = 90 s
Dwell:
B = 35 s C = 45 s D = 35 s
Terminal turnaround:
A = 180 s E = 180 s
Step 2 — Calculate one-way time
Running = 90 + 80 + 100 + 90
= 360 s
Dwell = 35 + 45 + 35
= 115 s
One-way = 360 + 115
= 475 s
= 7 min 55 s
Step 3 — Calculate cycle time
Assume the return journey is identical:
Tcycle = 475 + 180 + 475 + 180 = 1,310 s = 21 min 50 s
Step 4 — Choose target headway
Suppose passenger demand requires a 130-second headway and all infrastructure constraints permit it.
H = 130 s
Step 5 — Calculate active train requirement
N = ceil(1,310/130) = ceil(10.077) = 11 trains
Ten trains are insufficient because they would cover only:
10 × 130 = 1,300 s
which is ten seconds shorter than the full cycle.
Step 6 — Build departure sequence
From Terminal A:
08:00:00 08:02:10 08:04:20 08:06:30 08:08:40 ...
Each departure is 130 seconds after the previous one.
Step 7 — Propagate arrivals and departures
For the 08:00 train:
Depart A 08:00:00 Arrive B 08:01:30 Depart B 08:02:05 Arrive C 08:03:25 Depart C 08:04:10 Arrive D 08:05:50 Depart D 08:06:25 Arrive E 08:07:55
The next train follows the same sequence shifted by 130 seconds.
Step 8 — Check terminal feasibility
The first train reaches E at 08:07:55.
With a 180-second turnback:
earliest return departure = 08:10:55
The reverse-direction sequence must fit that availability and all other terminal movements.
Step 9 — Add robustness
Suppose the timetable contains two seconds of running supplement on A→B and five seconds of dwell margin at C.
Those seconds can absorb ordinary variation.
But they also increase cycle time unless offset elsewhere.
Step 10 — Test a disturbance
Suppose one train needs 15 extra seconds at C.
Five seconds are absorbed by dwell margin.
Ten seconds remain as delay.
The scheduler or simulator then checks whether:
- the train can recover time later,
- the following headway remains safe,
- the terminal slot is still feasible,
- and passenger accumulation amplifies the delay.
A timetable is not accepted merely because it fits once.
It must be tested against variation.
The Timetable Deletion Tests
One useful way to understand a system is to remove one component and observe what breaks.
Remove passenger demand
The timetable may still be mathematically feasible, but there is no reason for its frequency pattern.
Remove headway constraints
Trains can be scheduled into conflicting or unsafe separations.
Remove dwell
The schedule stops serving passengers and becomes a non-stop train exercise.
Remove terminal turnaround
Trains arrive at the end of the line and never become available for another trip.
Remove fleet circulation
The timetable demands trains that may not physically exist at the required place and time.
Remove margin
The timetable may look maximally efficient but becomes brittle.
Remove engineering hours
The railway can carry passengers for longer today while reducing the time available to keep carrying them safely tomorrow.
These tests reveal the whole job.
A timetable is not a departure table. It is the agreement among every process that must happen before the next departure is possible.
The Counterexample: Why the Fastest Timetable Can Be Slower for Passengers
Suppose Timetable A uses minimal running and dwell times.
Scheduled journey time is 30 minutes.
But it has almost no recovery margin, so ordinary disturbances frequently create irregular headways and long platform waits.
Timetable B schedules 31 minutes.
It contains well-placed margin and maintains regular service more reliably.
Which is faster?
On paper, A.
For the passenger over many journeys, B may produce lower total travel time because waiting and disruption are smaller.
Passenger journey time is:
Tpassenger = access time + platform waiting + onboard time + transfer waiting + disruption cost
Optimising only onboard running time can worsen the full journey.
The fastest train schedule is not necessarily the fastest transport system.
How the Mathematics Grows from Primary School to University
Primary level
- time intervals,
- speed, distance and time,
- tables and patterns,
- whole-number capacity.
Secondary level
- linear equations,
- inequalities,
- graphs,
- rates,
- quadratic braking relationships,
- statistics and variance.
Junior College
- calculus,
- kinematics,
- probability distributions,
- optimisation,
- differential equations.
University and operations research
- graph theory,
- mixed-integer programming,
- stochastic optimisation,
- periodic event scheduling,
- model predictive control,
- simulation,
- robust and resilient timetable design.
The MRT is therefore not one mathematics chapter.
It is a place where the chapters are forced to cooperate.
The World Return — How the Timetable Learns
A timetable is built from models.
Models predict:
- passenger demand,
- running time,
- dwell time,
- energy use,
- delay propagation,
- and required fleet.
Then the world returns observations.
predicted dwell vs observed dwell predicted demand vs observed demand planned headway vs actual headway predicted energy vs measured energy predicted recovery vs actual recovery
The difference becomes evidence.
Let prediction be ŷ and observation be y.
Model error is:
e = y − ŷ
Repeated error suggests the model or timetable assumptions need revision.
This creates the planning loop:
observe city → estimate demand → build timetable → simulate → operate → measure → diagnose mismatch → rebuild
The timetable is therefore not a permanent truth.
It is a carefully tested hypothesis about how the railway and the city will interact.
RFE Return — What Is a Good MRT Timetable?
A good MRT timetable is not the one with the smallest numbers everywhere.
It is not automatically the one with:
- the shortest running times,
- the shortest dwell times,
- the shortest headways,
- the smallest fleet,
- or the longest passenger operating hours.
Each extreme can damage another part of the system.
A good timetable performs the whole job:
enough trains when people need them + feasible running and dwell + safe and regular separation + workable junctions and terminals + enough trainsets + energy-aware trajectories + recovery margin + engineering access + useful first and last trains
It makes the railway feel ordinary.
That ordinary feeling is the achievement.
A timetable succeeds when millions of constrained calculations disappear into one simple passenger experience: the next train comes.
Conclusion — The City Becomes a Clock
Before the first train moves, the timetable has already imagined the day.
It has imagined where passengers will be.
How quickly trains can travel.
How long doors must remain open.
How closely trains may follow.
How branches merge.
How terminals turn trains around.
How many trainsets must be circulating.
Where recovery time should be stored.
And when passenger service must end so that the railway can be prepared for the next day.
Then reality begins.
Passengers arrive differently from the forecast.
Dwell varies.
One train loses seconds.
The operating system tries to preserve the pattern.
And the timetable reveals what it really is:
not a prediction that nothing will go wrong, but a structure designed so that the railway can keep working when small things do.
The passenger sees 8.02am.
The planner sees an event.
The engineer sees a feasible trajectory.
The operations researcher sees a constrained optimisation.
The controller sees a reference state.
The maintenance team sees the boundary of engineering access.
The city sees a train arrive.
Demand → frequency → headway → event network → running time → dwell → junctions → terminals → cycle time → fleet → robustness → daily operation → World Return
That is how an MRT timetable is built using mathematics.
A city is translated into seconds.
Then, every morning, the railway tries to turn those seconds back into movement.
Key Equations
qrequired ≥ D/P Required train flow from passenger demand H = 3600/q Headway-frequency relationship E[W] = H/2 Expected wait for regular service and random arrivals Dᵢ,ₖ = Aᵢ,ₖ + dᵢ,ₖ Departure = arrival + dwell Aᵢ,ₖ₊₁ = Dᵢ,ₖ + rᵢ,ₖ Next arrival = departure + running time la ≤ ta ≤ ua Activity-time bounds Dᵢ,ₖ − Dᵢ₋₁,ₖ ≥ Hmin Minimum headway constraint L = ∫v(t)dt Distance from the velocity trajectory r = ∫dx/v(x) Conceptual running-time integral srun = rscheduled − rmin Running-time supplement Dreturn ≥ Aterminal + τturn Terminal turnback constraint Tcycle = Tout + Tturn,far + Tin + Tturn,home Train-cycle time Ncycle = ceil(Tcycle/H) Simplified active train requirement sa = ta − la Activity slack P(actual ≤ scheduled) ≥ 1 − ε Illustrative robustness condition J = Σ weighted passenger, fleet, energy, regularity, reliability and maintenance costs Multi-objective timetable optimisation e = y − ŷ Observed minus predicted model error
Reader-safety note: The equations and examples in this article are educational abstractions. They do not reproduce proprietary Singapore MRT timetables, internal control logic, restricted operating data or safety procedures. Real timetables use line-specific, train-specific and safety-validated engineering parameters.
Continue the MRT Mathematics Series
- How MRT Timing Works Using Mathematics — the full system pillar.
- How MRT Braking Works Using Mathematics — stopping distance, braking curves and energy.
- How MRT Headway Works Using Mathematics — how train spacing becomes capacity.
- How MRT Station Dwell Time Works Using Mathematics — passenger queues and doorway bottlenecks.
- How MRT Delays Propagate and Recover Using Mathematics — bunching, margins, control and resilience.
The next natural leg is How MRT Energy Use Is Optimised Using Mathematics: traction, coasting, regenerative braking, power peaks and the timing relationship between trains.
Sources and Further Reading
- Land Transport Authority — Singapore Rail Network, operating hours and typical frequencies
- Land Transport Authority — NSEWL CBTC and 100-second headway capability
- Land Transport Authority — Circle Line Stage 6 operating frequencies from July 2026
- Ministry of Transport — MRT and LRT dwell-time ranges
- Land Transport Authority Annual Report 2023/24 — Pasir Ris rail turnback
- Land Transport Authority — Engineering hours for Circle Line testing and commissioning
- Land Transport Authority — TEL and DTL service adjustments for engineering works
- SMRT — Green CBTC, optimised running curves and punctuality
- Transportation Research Part B — Dynamic passenger-demand-oriented metro train scheduling
- Transportation Research Part C — Demand-driven timetable design for metro services
- Public Transport — Dwell times consistent with timetable design and passenger flows
- Transportation Research Part B — Cycle-time optimisation and stable periodic railway timetables
- Periodic Event Scheduling Problem — modelling periodic public transport timetables
- Integrated optimisation of urban rail line planning, timetables and rolling-stock schedules
Return to the complete MRT Mathematics hub: How MRT Works | It’s Mathematics.