Booking limits

How it works

Background

Cheap fares sell first and expensive ones sell last, so a flight left to itself fills up with the cheapest passengers. Revenue management sets a cap on each fare class and keeps the rest for later, higher-paying bookings. The field started there: in 1972 Ken Littlewood, at BOAC, showed that you should keep protecting one more seat as long as the chance that high-fare demand reaches it is greater than the low fare divided by the high fare. The second step applies that rule down a ladder of six classes with EMSR-a and EMSR-b and tests the caps on thousands of simulated flights; the third decides how many extra bookings to accept for no-shows.

Two fares: Littlewood's rule

One flight leg with C seats and two fares, f_H > f_L. Low-fare customers book first and are plentiful. High-fare demand D_H is uncertain: a normal draw with average μ and spread σ, rounded down to whole passengers (a negative draw counts as zero). Protecting y seats means the low fare is sold until C − y seats are gone (its booking limit), and the last y are kept for the high fare.

Revenue. With y seats protected, average revenue is R(y) = f_L (C − y) + f_H E[min(D_H, y)], and for whole-number demand E[min(D_H, y)] = Σ_k=1…y P(D_H ≥ k).

Littlewood's rule. Protecting the y-th seat changes revenue by R(y) − R(y − 1) = f_H P(D_H ≥ y) − f_L. That gain shrinks as y grows, so the best protection level y* is the largest y with P(D_H ≥ y) ≥ f_L / f_H, capped at C. Because demand is a normal draw rounded down, P(D_H ≥ y) = 1 − Φ((y − μ) / σ) exactly for whole y ≥ 1, which is what the bars show, and y* is the whole part of μ + σ · Φ⁻¹(1 − f_L / f_H). With the default numbers, the ones in RM 101's worked example: 40 + 10 × 0.253 = 42.5, so protect 42 seats.

Fill the flight. "Too few" and "too many" protect one spread (σ, rounded, at least one seat) below and above y*. Each flight draws one high-fare demand from the forecast; all three cabins see the same customers. "New flight" draws another. The random generator is seeded, so the page always opens on the same flight. Which seat a passenger takes is only for the picture: airlines protect a number of seats, not particular seats.

From two fares to many. EMSR-b protection for classes 1 to j against class j+1 uses the composite fare f̄ = Σ f_iμ_i / Σ μ_i and demand N(Σ μ_i, Σ σ_i²): y_j = μ̄ + σ̄ · Φ⁻¹(1 − f_j+1 / f̄), rounded to the nearest seat as in the many-fares tab, whose default flight it uses. Rounding to the nearest seat rather than down can give one seat more than the two-fare rule above.

Many fares: seat allocation

Classes are ranked from the highest fare f₁ to the lowest f_n. Demand for class i is independent and normal, D_i ~ N(μ_i, σ_i²). Everything rests on Littlewood's rule: keep one more seat for a higher fare f_H as long as f_H · P(D_H ≥ y) ≥ f_L, the lower fare you would turn away. The left-hand side is the expected marginal seat revenue (EMSR) of the y-th seat, and the charts plot it seat by seat.

EMSR-b (Belobaba, 1992) protects classes 1 to j together against class j+1 by merging them into one composite class, with fare f̄ = Σ f_iμ_i / Σ μ_i and demand N(Σ μ_i, Σ σ_i²). The protection level is y_j = μ̄ + σ̄ · Φ⁻¹(1 − f_j+1 / f̄). EMSR-a (Belobaba, 1987) applies Littlewood's rule to each higher class on its own against class j+1 and adds up the results. Protection levels are rounded to the nearest seat, kept between 0 and the capacity, and never allowed to fall as you go down the classes. The booking limit of class j+1 is the capacity minus y_j; the top class can sell the whole cabin.

Monte Carlo: 2,000 departures, each with a fresh demand draw per class (rounded, and floored at zero). Bookings arrive strictly from the cheapest class to the most expensive, which is the textbook assumption behind EMSR and the case where control matters most. All policies see the same draws. The random generator is seeded, so the page opens on the same results every time; "Run 2,000 new departures" changes the seed.

Overbooking

With A bookings accepted and a show-up rate p, the number of passengers at the gate is binomial, S ~ Bin(A, p). The expected cost is c_DB · E[(S − C)⁺] + c_sp · E[(C − S)⁺], where C is the capacity. One more booking is worth accepting while the chance that everyone else leaves a seat free, P(S_A ≤ C − 1), is above the critical fractile c_DB / (c_DB + c_sp). The page also computes the expected cost for every level and checks that the minimum is where the rule says. With separate business and leisure rates, bookings are split by the business share and show-ups are the sum of two binomials; the page then takes the level with the lowest expected cost directly.

Sources

• Littlewood, K. (1972), "Forecasting and control of passenger bookings", AGIFORS Symposium; reprinted in the Journal of Revenue and Pricing Management 4, 2005.
• Belobaba, P. (1987), Air travel demand and airline seat inventory management, PhD thesis, MIT (EMSR-a); Belobaba, P. (1989), "Application of a probabilistic decision model to airline seat inventory control", Operations Research 37(2); Belobaba, P. (1992), "Optimal vs. heuristic methods for nested seat allocation", AGIFORS Reservations and Yield Management Study Group (EMSR-b).
• Talluri, K. and van Ryzin, G. (2004), The Theory and Practice of Revenue Management, Springer: chapter 2 (single-resource capacity control) and chapter 4 (overbooking).
• Denied boarding costs, for scale: in the US, involuntary denied boarding compensation is 200% or 400% of the one-way fare, capped at $1,075 and $2,150 (14 CFR 250.5, as amended in October 2024, checked September 2026). In Canada the minimum for large airlines is CA$900, CA$1,800 or CA$2,400 depending on the delay (Canadian Transportation Agency guide, updated February 2026). Most passengers who give up a seat are volunteers paid less than that, so the $900 default is an assumption, not a figure for any airline. US denied boarding counts are published in the DOT's Passengers Denied Confirmed Space report.
• Fares, for scale: the US average domestic itinerary fare was $387 in 2025 (BTS). The six example fares are invented around that level.

Limitations

• In the two-fare tab, low fares all book before high fares. When they arrive mixed, a fixed protection level is no longer exactly optimal; real systems re-run the calculation many times before departure (RM 101, chapter 04).
• The two-fare tab assumes low-fare demand is large enough to buy every seat on sale to it. That doesn't change the rule: on a weak low-fare day the booking limit simply isn't reached and the spare seats stay open to high-fare customers. It does make the revenue figures there, and the gain over protecting nothing, too high for such days.
• Demand in each class is independent of what else is open. Real passengers buy down to a cheaper class when it's available, which Littlewood and EMSR ignore; see the buy-down simulator and RM 101, chapter 05.
• One flight leg, one pass. Real systems re-optimise many times before departure, forecast from unconstrained history, and control whole itineraries rather than single legs (chapter 06).
• Strict low-before-high arrivals flatter EMSR against first come, first served. With mixed arrival orders the gap is smaller.
• No cancellations, group bookings or upgrades, and overbooking is modelled separately from seat allocation. In practice the two are combined by setting booking limits on a virtual capacity above the physical one.
• The overbooking model treats every denied passenger as costing the same and every no-show as independent. Families and groups tend to no-show together, which widens the spread and argues for a lower authorisation level.

This is a personal project and isn't affiliated with any airline.