Airport Lounge Queuing Problem

Martin Ganco, HyeJin Youn

1. Description

One hundred travelers are waiting in the passenger lounge of the airport. When the airline agent at the gate announces that passengers may begin to board the aircraft, people start forming the line at the gate. Our model focuses on the line formation mechanism, its impact on the distribution of individual waiting times and evolution of line preferences.

The key question that we ask is why, when we assume that the objective of all passengers is to get inside the plane as soon as possible, we observe that not all people rush to the line immediately?

When we look at the dynamics in the waiting area more carefully, there are usually some people who keep their seats until the queue gets shorter. Some passengers stay seated while others tend to join the line soon. Our model assigns a “line tolerance” parameter to every passenger at the beginning of each boarding. Hence, a passenger's action - whether in the queue or not - is determined by f(line tolerance, number of people on the queue).

However, the overall waiting time is determined by the ordering of agents as they enter the plane. The progression of line is slowed down whenever passengers enter the plane in a “wrong” order – when a passenger blocks seating of a passenger immediately behind. There is no explicit coordination built into the model. The model was implemented in Matlab and Python.

2. Assumptions

Agents:

• Tol (line tolerance): uniformly distributed between [0:1], controls tendency to stay seated vs. join the line (we tested the robustness of this assumption using Normal distribution with no substantial impact on the shape of the curves and no impact on our conclusions).

• ID (assigned seat number): [1:100] represents a location of a seat from head to tail of the cabin
• Action (decision to join the line): bool, On - if agent decides to join the line, Off otherwise.

Environment:

• Nq: number of passengers in the queue
• The cabin is assumed to be a long row of seats with numbers assigned in ascending order
• The seat number has no effect on the tendency of passengers to join the line

Procedure:

• T_delay: if agent ‘j’ has a lower ID than agent 'i' who is next in line, the agent 'i' has to wait for 'T_delay'.
• Fq: frequency of updating the queue by allowing the first person to enter the plane

3. The Model

Initialization:

• Tol is randomly given to every agent drawn from uniform distribution
• ID is given to every agent from the interval [1:100]

Process:

• Agent i is drawn randomly to determine whether it will join the queue or not according to f(Tol, Nq). (The fundamental results are robust to different matching methods, e.g. when the entire waiting area is searched for passengers satisfying the rule in one time period, etc.)
• bool f(Tol, Nq) = 1(Tol>Nq), 0 (otherwise)
• if f is on, join the queue.

Note: the decision rule has the simple form: if g(line length) < line tolerance of agent i => join the line, otherwise stay seated, where in our case g(x)=x but g(.) can be any increasing function.

4. Results

4-1. No penalty for wrong order

• Hump shaped line length over time
• Higher mean line tolerance increases mean line length
• Uniformly distributed individual waiting times (reflecting uniform preferences)
• Distribution of line tolerance has no effect on the average waiting time (the mean waiting time is the same due to exogenous line update frequency and no penalty for wrong order)

Mean waiting time = 150

4-2. Penalty for wrong order

• Mean waiting time increases with penalty
• The line length decreases slowly with time
• Distribution is no longer uniform - people with low line tolerance penalized disproportionately more
• The distribution seems bimodal (although, we did not perform significance tests)

Mean waiting time = 450

5. Adding Evolution

As a simple extension, we ask the question whether the passengers can evolve a line tolerance that would match their seating (if we assume this is constant across runs). We add a simple hill climbing rule - each period (airline boarding), each agent draws a new tolerance value. If it leads to shorter time, it keeps it otherwise returns to the old tolerance value.

So, could the group evolve socially optimal line tolerance that would "match" their seat assignment and optimize overall waiting time?

The result shows that they cannot. After 100 runs the mean waiting time and distribution did not change. The cause of the slowdown related to the “wrong” order is a pure negative externality. This result justifies an outside intervention.

6. Applications and Extensions

• In a general sense, the model applies to situations where the objective is possible to achieve only when an action is taken that yields temporary disutility (standing in line) and status quo yields temporary benefits (keep sitting). The queue formation model could be thus modified to model participation in negotiations, perhaps social unrest, etc. The particular form of the model applies to all similar line formation problems.
• As possible extensions, one might consider agents traveling in groups like families or possible impact of outside coordination by seat number groups or even individual line up by IDs. However, in the current simple version of our model, the result of such extensions seems obvious – decrease of mean waiting time and convergence of distribution toward the uniform distribution.

7. Conclusion

The purpose of our model was to investigate the mechanism of line formation and its impact of the distribution of individual waiting times. Our results suggest that delays cause a bimodal distribution of waiting times. The model also shows that due to negative externality the decentralized agents do not have the capability to optimize the overall boarding time by matching their line tolerance with seat assignment. This indicates that outside intervention is justified and necessary.