Lorentzian-geometry-based analysis of airplane boarding policies highlights “slow passengers first” as better

We study airplane boarding in the limit of a large number of passengers using geometric optics in a Lorentzian metric. The airplane boarding problem is naturally embedded in a ($1+1$)-dimensional space-time with a flat Lorentzian metric. The duration of the boarding process can be calculated based on a representation of the one-dimensional queue of passengers attempting to reach their seats in a two-dimensional space-time diagram. The ability of a passenger to delay other passengers depends on their queue positions and row designations. This is equivalent to the causal relationship between two events in space-time, whereas two passengers are timelike separated if one is blocking the other and spacelike if both can be seated simultaneously. Geodesics in this geometry can be utilized to compute the asymptotic boarding time, since space-time geometry is the many-particle (passengers) limit of airplane boarding. Our approach naturally leads to the introduction of an effective refractive index that enables an analytical calculation of the average boarding time for groups of passengers with different aisle-clearing time distribution. In the past, airline companies attempted to shorten the boarding times by trying boarding policies that allow either slow or fast passengers to board first. Our analytical calculations, backed by discrete-event simulations, support the counterintuitive result that the total boarding time is shorter with the slow passengers boarding before the fast passengers. This is a universal result, valid for any combination of the parameters that characterize the problem: the percentage of slow passengers, the ratio between aisle-clearing times of the fast and the slow group, and the density of passengers along the aisle. We find an improvement of up to 28% compared with the fast-first boarding policy. Our approach opens up the possibility to unify numerous simulation-based case studies under one framework.

Figure 1

(a) Illustration of the stepwise advance of the queue during the boarding process, with $N=8$ passengers, four rows and two seats per row. There is space for two passengers per row along the aisle. Each passenger is marked as a circle with designated row number. At each time step, the queue moves forward, and passengers that have reached their designated rows sit down simultaneously. Red arrows indicate passengers that take their seat at that time step, and each group of passengers that sit down simultaneously is color coded. (b) Each passenger is marked as a point in a $qr$ diagram: The initial queue position of the passenger is on the horizontal axis and the designated row number is on the vertical axis. The points of passengers that sit down simultaneously are joined by line segments.

Figure 2

Comparison of four different boarding policies. We used a realistic congestion parameter $k=4$, $20\%$ slow passengers ($p=0.2$), and assumed that the slow passengers clear the aisle on average five times slower than the fast passengers ($C=0.2$). We assumed there is a single aisle and 6 seats per row, and a total of $N=240$ passengers. The percentage of seated passengers is plotted as a function of time. Remarkably, on average, the fast-first policy (leftmost, red solid line) is leading all the way to around $\sim 98\%$. However, the slow-first policy eventually seats all passengers in a shorter time, relative to all the other policies. The fast-first policy is second (FF, $+7\%$), random boarding comes third (R, $+23\%$) and the back-to-front policy turns out to be the worst (BTF, $+40\%$). That the slow-first policy is superior can be intuitively explained by that it is the most parallel among all the policies, i.e., it better exploits the possibility to seat passengers simultaneously, while the other policies are more serial in structure. The graph is an average of $10000$ discrete-event runs.

Figure 3

The $qr$ diagrams for four different boarding policies, with each of the $N=240$ passengers marked as a point, $h=6$ seats per row, and congestion $k=1$. (a) Random boarding policy: The passengers are uniformly distributed over the diagram. (b) Back-to-front policy with two equal-sized groups: The first part of the queue is heading for the rows in the back of the airplane. (c) Slow-first policy with two equal-sized groups: The slow passengers are in the first part of the queue (blue bullets). (d) Fast-first policy with two equal-sized groups: The fast passengers are in the first part of the queue (red diamonds). For all policies the boarding time is the sum of the aisle-clearing times for passengers that belong to the longest chain (dashed lines). The preceding passenger in a chain must take his seat before the next in the chain can sit down. In all four diagrams, the longest chain that determines the boarding time follows the asymptotic limit (solid line, the geodesic), up to statistical fluctuations that are diminishing as the number of passengers increases (see Sec. 6 for further details).

Figure 4

Relative difference in average boarding time $D=(\langle T_{\text{FF}}\rangle -\langle T_{\text{SF}}\rangle )/\langle T_{\text{SF}}\rangle$ between the fast-first and the slow-first policies when $k=4$. (a) Number of passengers $N\to \infty$. The slow-first policy is superior for all $(p,C)$ combinations, and the maximum relative difference is $20\%$ for small $p$ and $C$. For the parameter choice in Fig. 2, the relative difference is $11\%$ (black circle). (b) Simulation results for finite numbers of passenger $N$ confirm that slow-first is superior to fast-first, i.e., $D=(\langle T_{\text{FF}}\rangle -\langle T_{\text{SF}}\rangle )/\langle T_{\text{SF}}\rangle >0$ for increasing $N$ for all combinations of parameter values $p\in \{0.1,0.5,0.9\}$ and $C\in \{0.2,0.5,0.8\}$. The rightmost points are asymptotic values taken from the indicated positions in the inset contour plot from (a). There are six seats per row, $k=4$, and the accuracy is $\pm 0.0002$ (as a result of ${10}^6$ runs for each finite-$N$ data point).

Figure 5

With $N=4000$ passengers, there is closer correspondence between the longest chain and the longest curve than in Fig. 3 (where $N=240$). Passengers (points) seat simultaneously in consecutive wave fronts (black thin lines). For clarity, only every 20th wave front is shown. (a) Random boarding policy: $k=4$, with the same aisle-clearing time for all passengers. (b) Slow-first policy: $(k,p,C)=(1.5,0.3,0.33)$. The longest curve is refracted when the aisle-clearing time changes value on the border between the slow (blue bullets) and the fast (red diamonds) passengers. The aisle-clearing time plays the role of a refractive index.

Figure 6

Example of blocking relations for the same case as in Fig. 1. The distance $d$ between consecutive rows is twice the distance $w$ occupied by each passenger in the aisle. Passenger $A$ blocks both passengers $B_1$ and $B_2$. (a) Passenger $B_1$ is blocked by $A$ since $B_1$ is heading for row 4, which is beyond row 2 where $A$ is taking a seat. Passenger $B_2$ is blocked by $A$ by displacement as in Eq. (1): The space $3w$ occupied in the aisle by passenger $A$ and the two passengers between $A$ and $B_2$ is larger than the distance $d=2w$ between the designated rows of $A$ and $B_2$. (b) The blocking relations are indicated by arrows. There are several other blocking relations that are not shown in the figure.

Figure 7

(a) Coordinate transformation from space-time to queue row when $k=0$. The $qr$ diagram is in the future light cone of the origin. Passenger $B$ can only be blocked by $A$ if $B$ is within the light cone of $A$. (b) Blocking by displacement for $k>0$ according to Eq. (1). Since passenger $B$'s designated row is in front of $A$'s designated row, $A$ can only block $B$ by displacement. The value of $k$ and the number of passengers in between $A$ and $B$ in the queue determine the extent of the displacement. Most of the passengers in between are heading for the rows behind $A$'s designated row (shaded area). (c) Passengers nearby $A_1$ and $A_2$ must be within their respective future light cones in order to be blocked. The cones of each blocking passenger have a wider angle when the designated row is near the front of the airplane, reflecting a larger potential for blocking fellow passengers (here $k=4$). Passengers can never block passengers who are in front of them in the queue, and therefore the line emanating upward is always part of the light-cone boundary.

Figure 8

The optimal path (green thick curves) depends on the metric and is orthogonal to the contour lines (black thin curves) which defines the equidistant points from the starting point (origin). (a) Under the Euclidean metric, the contour lines are circle shaped. The shortest paths from the starting point to any other point (red squares) are straight lines. (b) An appropriate Lorentzian metric is used in airplane boarding, and the starting point is $(0,0)$ in the $qr$ diagram (here $k=0.6$). The contour lines coincide with passenger wave fronts when $N\to \infty$. The longest curves (geodesics) from the starting point to any other point are everywhere orthogonal to the contour lines (wave fronts).

Figure 9

The shape of the longest curve for random boarding can be either ordinary or piecewise. (a) Ordinary-type curve ($O_R$) when $0<k\le ln\left(2\right)$. (b) Piecewise curve ($P_R$) consisting of a constant function ($C_R$) and an upward-going ordinary-type curve ($U_R$) when $k>ln\left(2\right)$.

Figure 10

For fixed crossing height $\delta$, the shape of the longest curve for slow-first can be either piecewise ($P_S,P_F$) or ordinary ($O_S,O_F$) in both the slow and the fast regions, respectively. Hence, the total curve can take four different shape types. The subfunction that defines the weight $W_{\text{SF}}\left(\delta \right)$ in Eq. (12) depends on the values of ${\delta }_S(k,p)$ and ${\delta }_F(k,p)$ (expressions are given in the text): (a) $W_{\text{SF}1}$, $\delta <min\{{\delta }_S,{\delta }_F\}$; (b) $W_{\text{SF}2}$, ${\delta }_F⩽\delta <{\delta }_S$; (c) $W_{\text{SF}3}$, ${\delta }_S⩽\delta <{\delta }_F$; and (d) $W_{\text{SF}4}$, $max\{{\delta }_S,{\delta }_F\}⩽\delta$. The $\delta$ that maximizes $W_{\text{SF}}\left(\delta \right)$ also depends on the relative aisle-clearing time $C={\tau }_F/{\tau }_S$.

Figure 11

The top row shows the subdomains of the $(p,C)$ unit square where the slow-first boarding time is represented by the different subfunctions in Eq. (13). The bottom row shows the corresponding subdomains for the fast-first policy in Eq. (A3).

Figure 12

Average boarding time estimates for the (a) slow-first and (b) fast-first policies for different $(k,p,C)$ parameter settings. Simulation results for an increasing number of passengers are compared to the asymptotic results for all combinations of parameter values $p\in \{0.1,0.5,0.9\}$ and $C\in \{0.2,0.5,0.8\}$. Here $k=4$ and the accuracy is $\pm 0.002$ (as a result of ${10}^6$ runs for each finite-$N$ data point). The rightmost points are asymptotic values.

Figure 13

(a) Initial position of the optimal queue ordering, with $N=8$ passengers, 4 rows, and 2 seats per row. (b) The symmetry of the optimal queue ordering enables four passengers to sit down in each wave, which gives a minimal boarding time of $T=2$. The time step $t=1$ is not explicitly shown.