Airplane boarding meets express line queues

Highlights

• First analysis of group boarding with different aisle clearing times.

• Relation between airplane boarding and express line queues.

• An index of refraction for distributions in the context of airplane boarding.

• Algorithm for finding good queue/row domains for boarding groups.

Abstract

Airlines have been implementing several policies which partition passengers into groups of fast and slow passengers and placing them in different queue/row combinations. For example, the policy which provides boarding precedence to passengers needing assistance and families traveling with small children, or a policy which provides precedence to passengers with no overhead bin luggage. The idea of separating customers into groups with different service times also appears in the very different setting of express line queues, for example, in the supermarket or in server farms. Such queues have been extensively explored in the last 20 years in the queueing theoretic literature.

We show that the two systems, airplane boarding with slow and fast passenger groups and express line queues are intimately related in the asymptotic regime where the ratio between the slowest and fastest customer becomes large. We produce good algorithms for placing different groups of passengers in the airplane boarding setting with competitive guarantees compared to optimal placements. We then show how we can analyze the airplane boarding setting using results from the express line setting. This relation provides a novel bridge between the theory of project management and the critical path method to which airplane boarding belongs and queueing theory to which express line queues belong.

The analysis uses very basic notions and arguments from geometry, but in the setting of space-time (Lorentzian) geometry, a geometry which was invented to study relativity theory.

Introduction

The problem of analyzing airplane boarding strategies has been considered both analytically and using computer simulations in many recent studies, see Jaehn and Neumann (2015) for a recent comprehensive survey.

In this paper we consider policies which partition passengers into several distinct groups which differ in the amount of time it takes them to clear the aisle. The aisle clearing time is the time between the moment that a passenger arrives at his/her designated row and the moment that the passenger is seated and the aisle is clear. One common example is to allow passengers with small children or passengers who need special assistance (slow passengers) to board before others (fast passengers). Recently, a few airlines implemented a policy which allowed passengers with no overhead bin luggage (fast passengers) to board before other passengers (slow passengers) (Reed, 2013, Seatguru). Such policies, which separate passengers into groups which relate to their aisle blocking time and which differentiate their boarding placements, have been studied less extensively, (Audenaert, Verbeeck, & Berghe, 2009) using simulations. They have also been considered (again via simulations) more recently in Milne and Kelly (2014), Milne and Salari (2016) and Notomista et al. (2016), however, in these works it is assumed that airlines have very tight control over each and every passenger during the boarding procedure. This level of control is much stronger than practiced by airlines.

We will consider a general family of boarding policies which separate groups with different aisle clearing time distributions. We assume that some criterion defines a partition of all passengers into m groups which we denote by G_i, $i=1,\dots ,m$. Each group G_i has a relative portion p_i of all passengers. For example, suppose that G₁ is comprised of all passengers with no overhead bin luggage and suppose this group consists of 37% of all passengers, then we will have $p_1=0.37$. Thus, we can think of the p_i as giving the probability that a uniformly randomly chosen passenger is in group i. We let X_i be the distribution of aisle clearing times of passengers in group G_i, and X denotes the overall distribution of aisle clearing times of all passengers. Clearly we have that for any time t, $Pr(X<t)={\sum }_i{p}_{i}Pr(X_i<t)$ where Pr denotes the probability of an event. In this sense we have the relation

$$X=\sum _{j=1}^m{p}_i{X}_i.$$

which states that the overall aisle clearing time distribution is linear in the distributions of the different groups.

A partition into groups and the associated partition given in Eq. (1) still does not fully determine the boarding process. We still need to assign the different groups to various placements in the queue and in the airplane. As examples we can consider the slow first and fast first policies noted above. While such simple to implement policies are practiced, we can explore more general possibilities with the assignment depending also on the location in the airplane and not only on queue location. For example, we can decide to place slow passengers (assuming we know them to be so) in the front rows of the airplane or at the back rows. We can even think of placement policies involving a mix of queue locations and row locations,

To express complex policies mathematically and more generally to analyze boarding policies, we represent passengers as points (q, r) in the unit square. The q coordinate may be thought of as representing the relative queue location of the passenger (passenger 37 out of 115 in the queue has $q=37/115$) and similarly, r represents the relative row location of the passenger. To cover any conceivable scenario, regardless of the difficulty of implementing it, we consider the case in which each group G_i is assigned a general domain D_i in the plane with coordinates representing queue location and row location in the airplane. For example, if G₁ is the group of slow passengers, then the slow first policy assigns it the domain D₁ given by q ≤ p₁, while the fast first policy will assign to G₁ the domain D₁ given by $q\ge 1-p_1$. The slow in the front policy would assign the domain D₁ given by r ≤ p₁. An example of a mixed policy would be to consider two groups of equal size, i.e., $p_1=1/2$and let D₁ be the union of two sub-domains, the first given by q ≤ 1/2 and r ≥ 1/2, and the second given by q ≥ 1/2 and r ≤ 1/2. This policy first boards fast passenger who are seated in the front half of the airplane together with slow passengers who are seated in the back half of the airplane. It then boards the remaining passengers. More complicated domains/assignments will be considered later.

The distribution X is not in our control, rather it is a property of the passenger population. However, we can hope to control to a certain extent, the partition of the passengers into groups G_i, and consequently control the partition variables p_i, X_i and we can also control to a certain extent the assignment of domains D_i to each group (representing a boarding policy for the different groups). The choice of the control variables p_i, X_i and the auxiliary control variables D_i defines a boarding process. We wish to study how different choices of group partitions and domain assignments affect the boarding process and in particular the boarding time, from the entry of the first passenger into the airplane until everyone is seated.

Even with the choice of the control variables and consequently the boarding process, the boarding time still depends on parameters which are related to the layout of the airplane and the occupancy level of the flight. As it turns out these can be combined to form a parameter called congestion which we denote by k. The parameter k may be defined as the number of lengths of the airplane needed to fit all passengers standing one after the other comfortably (with reasonable personal space). In other words, we measure the length of a queue which holds all passengers, we measure the total aisle length (all aisles) of the airplane and we divide the first length by the second to obtain k.

Some boarding methods are very sensitive to k. For example, back-to-front boarding (passengers from row 40 and above, followed by row thirty and above,...) is very effective for low values of k, but very detrimental for large values of k, Bachmat and Elkin (2008). Given a specific partition p_i, X_i and looking at various choices of queue/row domains D_i for the different groups, the merit of using some specific domains can vary greatly depending on the value of k. Domains which are good for small values of k will not be good for large values and vice versa.

There is a final parameter, n, the number of passengers, but in this paper, we will always let n go to infinity and examine the asymptotic behavior of a normalized version of boarding time in that limit. It turns out that (in our model) this asymptotically scales boarding time by a factor of $\sqrt{n},$ so we will consider a normalized version of average boarding time, where we divide by $\sqrt{n}$. The parameter k, the control variables p_i, X_i, and the auxiliary control variables D_i (subject to the restriction given by Eq. (1), together determine the (normalized) boarding time.

The challenge is:

Given a distribution X, to find good (possibly parameter k dependent) control variables p_i, X_i such that coupled with a good (possibly k dependent) auxiliary control variables D_i, the boarding time will be minimized.

We break this up into two sub-challenges:

- Given the partition control variables p_i, X_i, find good (k dependent) domains D_i(p_i, X_i, k). We provide a solution to this problem which is differs in performance from the optimal choice, by at most a constant factor independent of p_i, X_i, k. We show that the slow first and fast first placement policies do not have this property, their assignments can be worse than the optimal assignment by an unboundedly large factor. The good domains are indeed k dependent, and the corresponding policies much more complicated than slow first or fast first. It turns out that with this choice of domains, the dependence of the boarding time on the parameter k can be essentially factored out, i.e., it affects boarding time in much the same way regardless of p_i, X_i.

- Find good partition control variables p_i, X_i , that coupled with the good (k dependent) domains from the first sub-challenge, will nearly minimize the boarding time. We solve this problem by showing a surprisingly strong connection to a completely different scenario, express line queues, where the problem of finding good partitions p_i, X_i to a distribution X has been the subject of much study.

In express line queues, as in the supermarket and elsewhere, customers with less than a certain number of items (10 in many cases) have a reserved queue. Sometimes there are a couple of such queues, say one for 1–10 items and another for 11–20 items. More generally, we may consider queueing systems, with m identical servers, with a Poisson arrival process, where server i is dedicated to a group of customers G_i which take up a portion p_i of all customers and whose service time is sampled i.i.d. from a service time distribution X_i.

The challenge is:

Given a general service time distribution X, to find good partitions satisfying (1) that minimize average waiting time.

This problem has been studied extensively over the past 15 years (Bachmat, Sarfati, 2010, Feng, Misra, Rubenstein, 2005, Harchol-Balter, 2002, Harchol-Balter, 2013, Harchol-Balter, Crovella, Murta, 1999). In this case, there is no analogue of the choice of queue/row placements for the various groups, just the partition into different servers and there is no congestion parameter k. On the other hand there is a major parameter that does not exist in airplane boarding, but plays a major role express line queues and that is utilization ρ.

To state the relation, we need to control parameters outside p_i, X_i which appear in just one of the scenarios. The utilization is the ratio of the arrival rate of customers and the service rate of a single server. For a stable system (no exploding queue lengths) with m servers, the utilization ρ is in the range (0, m), with a utilization ρ < 1 meaning that we can service all requests using a single server. The integer part of the utilization [ρ] has a major qualitative influence on the performance of such systems. For example, a choice of partition p_i, X_i which is good for utilizations in the range [1/4, 3/4], will not necessarily be good for utilizations in the range of [5/4, 7/4] and good partitions for either of these utilization ranges will not be good in the range [9/4, 11/4] and so on. A system can be considered to be truly with m servers only when ρ < 1, because in that case there are no stability conditions that obstruct the choice of some partitions. Under these circumstances, we can consider the auxiliary parameter ρ in a range, say, [1/4, 3/4], i.e., bounded away from 0 and 1. The results for any other range within these bounds will be similar. We will simplify further by fixing the utilization to be 1/2. Any other choice in (0, 1) will yield similar results. With this choice, we let W(p_i, X_i) be the average waiting time of an express line system with such a service time partition.

The main result that will be stated formally later on can be summarized as saying that, roughly speaking, the effect of a partition p_i, X_i on normalized average boarding time is the square root of its effect on normalized waiting time in the express line queues setting, in the asymptotic regime where the support of X is large. In particular, good partitions for minimizing average boarding time are also good partitions for minimizing average waiting time in express line queues and vice versa.

The relation between airplane boarding with group partitions and express line queues is not trivial in the sense that it depends on the two dimensionality of the description of passengers in terms of queue and row locations and on the choice of good domains.

The proof is composed of three parts. The first defines and then estimates the effect of an aisle clearing time distribution X on the boarding time of a non partitioned population of passengers. We attach to each distribution a number C_X which we can think of as an index of refraction. It determines how slow is a group of passengers with aisle clearing time distribution X. The value C_X was first defined in Bachmat, Berend, Sapir, Skiena, and Stolyarov (2009), and we provide the first estimates for it. Via an elementary argument it is shown that C_X, is given by $\sqrt{E\left(X^2\right)},$ up to a multiplicative error of size $O\left(\sqrt{\ln \left(T\right)}\right),$ where T is the ratio between the longest aisle clearing time to the shortest one (the aspect ratio of the distribution X).

The second part is a geometric algorithm for choosing good domains with bounded competitive guarantees. The constructed domains also satisfy a property of being balanced, a property which is important in some cases for making the model realistic. The construction of the good domains involves very elementary geometric arguments, but in the less familiar setting of space-time geometry. We provide all the needed background.

We complement the algorithmic construction with matching lower bounds which show that the boarding time estimate for the algorithmic choice of domains is multiplicatively bounded from the boarding time estimate with an optimal choice of domains. For any k, p_i, X_i, the lower bounds show the construction yields domains which are (multiplicatively) at most 12 times worse than an optimal choice. One bound is very straight forward but is particular to airplane boarding and balanced domains. The second requires a bit more geometry but applies in a very general setting and so can be used in other applications.

The third part is to show that tying together the first two parts leads to the desired result. This part is purely formal.

Once we obtain the theorem, we can use it to transfer much of the well developed theory of express line queues to provide a fairly detailed asymptotic analysis of partition boarding policies. We will provide several examples of results that can be carried over to the new setting. We also provide a much tighter analysis in the case of asymptotic analysis of bounded Pareto distributions. this setting was explored in great detail in the express line queue literature with very precise results. We show that in that case, we can construct domains which are at most a few percent worse than optimal.

The constructions and notions presented in this paper, have their roots in space-time geometry, also known in the mathematics literature as Lorentzian geometry. This geometry was invented by Minkowski in 1908 with the sole purpose of mathematically modeling relativity theory. At the time, only special relativity existed and Minkowski constructed the flat version, known as Minkowski space. It is sufficient for the purposes of special relativity theory. Minkowski space is the analogue of Euclidean space. Euclidean geometry generalizes to curved spaces such as a sphere or a saddle and the more general geometry is known as Riemannian geometry. Likewise, Minkowski space can be generalized to include curvature and the general geometry is known as Lorentzian geometry. The more general Lorentzian geometry is the right structure to mathematically model general relativity theory which is the theory that describes gravity, the major force in the history of the universe in terms of shaping stars and galaxies. As it turns out, the same geometric theory (Lorentzian geometry) models airplane boarding! In short, we can use the same type of geometry to model the universe or airplane boarding and not much in between (with a few interesting exceptions). This peculiar feature makes airplane boarding into one of the most interesting processes studied in operations research. Understanding the geometry is central to analyzing, understanding and gaining intuition about airplane boarding. In the following (rather long) section, we will provide a completely qualitative (no equations) explanation of the analogy between airplane boarding and relativity theory and how the same type of geometry models both. The main point is that the central construction of the paper is grounded in very basic geometry, but not of the type we are used to.

To place this paper in the broader context of operations research, we can think of airplane boarding as a project consisting of tasks, one for each passenger. The task of the passenger is to sit down. Each task has a weight which is the aisle blocking time. The aisle blocking of passengers by other passengers generates the precedence relations which can be described by a weighted directed acyclic graph (WDAG), which by transitive closure also corresponds to a partially ordered set with weights. The boarding time is given by the weight of the heaviest path in the WDAG, i.e., the critical path. We can apply the standard CPM technique to figure out the boarding time for each specific boarding scenario.

The boarding process is affected by three basic ingredients:

(1) An airline policy.

(2) The congestion parameter which is determined by the seat layout and the occupancy level (actual number of passengers divided by number of seats).

(3) The aisle blocking time distributions of passengers in groups (either a single group or several).

In previous research, (Bachmat, Berend, Sapir, Skiena, Stolyarov, 2009, Bachmat, Elkin, 2008), we analyzed policies which affect the distribution of passengers in the q, r unit square. For example, if we board first a third of the passengers from the back of the airplane (now boarding rows 31–45, and afterwards the remaining passengers) then there will only be passengers in the square 0 ≤ q ≤ 1/3, 2/3 ≤ r ≤ 1 and the square 1/3 ≤ q ≤ 1, 0 ≤ r ≤ 2/3, with densities, 3 and 1.5, respectively. However, we assumed that the two groups of passengers have the same aisle blocking time distribution, i.e., the policy only affects the density of passengers on the q, r unit square.

In this work we consider policies with a uniform distribution of passengers in the unit square, as in the no policy case, but do specify groups and domains with different aisle clearing time distributions. The analysis in this case, has a different nature.

When an airline fixes some boarding policy, which holds for many flights, the situation becomes stochastic in two different ways. The ordering of passengers in the queue with respect to their seat location changes between flights. This ordering determines the partial order relation of the WDAG (the DAG part), which is in turn stochastic due to fluctuations in the boarding order between flights. Thus, the policy can be viewed as determining a random variable with DAG (partial order) values. As explained above, all the policies considered in this paper have the same DAG valued random variable which coincides with that of the uniformly random (no) policy.

The aisle blocking time (weight) of each of the passengers in line also changes from flight to flight and so we have random variables for the weight (W of the WDAG). In standard project management PERT is used for handling the stochastic nature of the weight, but the partial order is fixed (deterministic), here we have a doubly stochastic situation where both the ordering and the weights are stochastic. Consequently, even assessing a single policy may be a bit challenging.

What comes to the rescue is that for a given policy and a given congestion parameter, the DAGs which are generated typically by the policy can be viewed as discretizations of a well studied continuous geometric structure on the unit square. Moving from a discrete model to a continuous model as the number of tasks becomes large is standard procedure in queueing theory and optimization and is referred to as a fluid model. It is shown in such cases that the fluid limit describes with ever increasing accuracy (as the number of tasks grows) the limiting behavior of the discrete system. The fluid model is deterministic and describes the law of large numbers for the discrete systems. Typical examples can be found in Kurtz (1970), Kurtz (1981) and Mitzenmacher, Richa, and Sitaraman (2001). The state of the system is described by some differential equations. The fluid model for airplane boarding has been constructed and the limiting behavior has been shown to match the fluid model in Bachmat, Berend, Sapir, Skiena, and Stolyarov (2006), Bachmat et al. (2009) and, in full detail, in Bachmat (2014). The fluid limit object is the unit square equipped with a space-time geometry (a Lorentzian metric). This is the same type of geometric structure that captures the mathematics of relativity theory. In relativity theory, the mathematical model is continuous to begin with. The points of the geometric object correspond to events (space and time coordinates). There is a natural partial order between events, which is the past-future (or causality) relation. Event A is smaller than (in the past of) an event B if it is possible to move from A to B under or at the speed of light (information coming from A can reach B). The fact that particles carry information means that they travel along trajectories (curves) which are chains in the partial order (the past of a particle can influence its future). A possible trajectory of a particle (any curve which traces motion at or below the speed of light) is called a causal curve precisely because its points are in a causal relation. The Lorentzian metric which defines the space-time structure is a gadget for defining the length of a causal curve from its initial event to its final event. The physical interpretation of the length in relativity theory is the time that passes on a clock attached to a particle that follows the trajectory. This length is called proper time and is an intrinsic measure of time which does not depend on observers. In addition to measuring lengths of causal curves, the metric defines a natural notion of volume. We can use the volume to naturally sample points from a domain in space-time.

According to general relativity theory, among all possible paths between two events, a free falling particle (a particle only under the influence of gravity) will follow a trajectory that maximizes (locally, between any two nearby points on the trajectory) the time passing on a clock attached to the particle. Such free fall trajectories are known as geodesics.

Given a domain, say the unit square and a space-time structure on it, one can consider curves whose length is maximal among all curves. Such curves are known to exist, see Penrose (1972), and they are called maximal curves in the space-time geometry literature. Given an airplane boarding policy, the aisle blocking time distribution and the occupancy of the airplane we can construct a Lorentzian metric (space-time structure) on the q, r unit square which serves as a fluid model. The main property of the fluid model is that as the number of passengers, N, becomes large, with high probability, the boarding time will be roughly the length of a maximal curve in the fluid model times $\sqrt{N}$. Moreover, if we look at the sequence of passengers which are on a critical path, their q, r coordinates roughly follow a maximal curve. In other words, maximal curves are the fluid limits of critical paths. A maximal curve is composed of boundary components and geodesics when not constrained by the boundary. Geodesics satisfy a differential equation, which by a miracle, in the cases of interest for airplane boarding can be solved explicitly. It is also known that in a maximal curve, the boundary component and the geodesic should have the same tangent at their meeting points. These facts are sufficient to identify the maximal curve in many cases of interest and to perform explicit calculations. More generally, we can use geometry to gain insights into airplane boarding and to find good or optimal policies among large classes of policies. Geometry points in the right direction among the very large (infinite) set of possibilities.

The case of policies which partition into groups with different aisle blocking time distributions is similar to a resource allocation problem. We can think of the fast/slow passengers as tasks which were given more/less resources. Since the policy leads only to a stochastic effect on the ordering of passengers, we are limited in our ability to allocate the resources. We can find good policies (choice of domains) by using some basic geometric intuition and that is the source of the construction.

The results of this paper can be viewed as the beginning of a geometric theory of projects. It may also be viewed as a beginning for a geometric theory of queues, Whereas classical queueing theory studies the one dimensional case, or equivalently linear orderings, in the geometric version, we replace the linear order by partial orders which are naturally defined in geometric terms, i.e., the elements of the partial order are points in some geometric space and the order relation is also naturally defined in geometric terms. Most of the results in the paper can be generalized to projects (families of projects) which have a geometric setting in any number of dimensions.