Bounds on the performance of back-to-front airplane boarding policies

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1 Bounds on the pefomance of bac-to-font aiplane boading policies Eitan Bachmat Michael Elin Abstact We povide bounds on the pefomance of bac-to-font aiplane boading policies. In paticula, we show that no bac-to-font policy can be moe than 0% bette than the policy which boads passenges andomly. Intoduction The pocess of aiplane boading is expeienced daily by millions of passenges woldwide. Reductions in gate delays would yield significant economic benefits fom moe efficient use of aicaft and aipot infastuctue and would also impove passenge expeience. See Van Landeghem and Beuselinc, [7], Maelli et al., [4] and Van den Biel et al., [5, 6] fo an extensive discussion. Aiplane boading has been studied using detailed compute simulations by Van Landeghem and Beuselinc, [7], Maelli et al., [4], Van den Biel et al. [5, 6] and Feai and Nagel, [3]. Bachmat et al., [, ], have intoduced an analytical model which was shown to be in nealy complete ageement with the esults of the afoementioned simulation studies. Ailines have adopted a vaiety of boading stategies in the hope of shotening the boading pocess fo aiplanes. Many ailines pactice bac-to-font boading policies, namely, the ailine boads passenges fom the bac of the aiplane fist. These stategies ae paametized by the choice of which goups of ows ae allowed to join the boading queue at any given time. Seveal policies of this type have been studied both via simulations and analytically, [-3,5-7], and the esults showed that these policies povide no impovement, and may even be detimental. In this lette we ty to explain this phenomenon by poving bounds on the effectivness of bac-tofont boading policies in the setting of the analytical model of [, ]. The analytical methods have identified a congestion paamete which plays a cucial ole in assessing the effectivness of boading policies. The paamete depends on the design of the aiplane, namely, on the distance between successive ows (leg oom) and the numbe of passenges pe ow. We show that fo an aiplane whose design leads to a congestion facto bac-to-font policies can educe boading time in compaison to andom boading by at most a facto of. () + ln Depatment of Compute Science, Ben-Guion Univesity, patially suppoted by an IBM faculty awad, ebachmat@cs.bgu.ac.il Depatment of Compute Science, Ben-Guion Univesity, elinm@cs.bgu.ac.il

2 As was agued in [], in eality the congestion facto is aound 4. Fo this value of the congestion facto, ou esult shows that no bac-to-font policy can impove upon andom boading by moe than 0%. Moeove, since the expession () tends to as gows to infinity, ou lowe bound is, in fact, asymptotically optimal. Modeling the aiplane boading pocess In this section we explain how to estimate analytically the boading time of a given bac-to-font policy, using the mathematical model of []. We epesent a bac-to-font policy by a monotone deceasing sequence of numbes = ( 0,,..., m ), = 0 > >... > m > m = 0. The sequence = (,..., m ) is efeed to as a patition of size m. Assume that the aiplane has n ows and n passenges. We will assume that the aiplane is full, and so n = Θ(n ). The set of passenges who ae seated between ows i n and i n is called the ith goup of passenges. The bac-to-font policy coesponding to the patition allows the passenges fom the fist goup of ows to join the queue fist, followed by passenges fom the second goup and so on. We epesent passenges by points (q, ) in the unit squae [0, ]. The ow coodinate epesents the ow of the passenge divided by n. The queue coodinate q epesents the position of the passenge in the boading queue divided by n. The boading policy detemines a joint density function p(q, ), which descibes the pobability that a passenge sitting in ow will have queue position q. We note that in a bac-to-font policy with paametes, passenges in the ith goup occupy positions ( i )n to ( i )n in the queue, theefoe, the coodinates of passenges (q, ) in the ith goup satisfy and i i () i q i. (3) We denote the squae given by these inequalities by S i. The set of squaes S i, i =,..., m, contains the anti-diagonal segment given by q + =, 0 q,. Fo each i =,,..., m, let B i be the bottom edge of the squae S i. See Figue. Since a passenge in a ow is equally liely to have any of the allowable queue positions, the pobability density function p is defined by p(q, ) = /( i i ) if (q, ) S i, i =,..., m, and p(q, ) = 0 othewise (outside the squaes S i ). In addition to the density function p the model also uses a congestion paamete. The congestion paamete is a cetain function of the numbe of passenges pe ow, the aveage aisle length occupied by a single passenge, and the aisle distance between a pai of successive ows (the leg-oom ). Substituting ealistic values of these paametes, one obtains a value of oughly equal to fou ( = 4) []. Given the pobability density function p = p which is detemined by the boading policy at hand, and the congestion paamete of the aiplane, the model defines the boading time of the policy as follows. Set α(q, ) = p(q, z)dz. The boading time T (, ) is now given by the solution to the following vaiational poblem. Conside the set Φ of all piecewise diffeentiable functions ϕ(q) defined on an inteval [q, q ], 0 q < q, with values in the unit inteval [0, ], and which satisfy ϕ (q) + α(q, ϕ(q)) 0. (4)

3 S B S B 0 0 S4 Figue : A gaphic illustation of a patition into five boading goups of diffeent sizes. Each squae coesponds to one goup. The bottom edges B, B,..., B 5 of the squaes S, S,..., S 5, espectively, ae depicted by a solid thic line. This is a patition of size 5. B 4 q 3

4 Let whee L(ϕ) = T (, ) = T (p, ) = max ϕ Φ q L(ϕ), (5) q p(q, ϕ(q))(ϕ (q) + α(q, ϕ(q))))dq. (6) This estimate T fo the boading time was validated in [] against detailed tace diven simulations, paticulaly those of van Landeghem and Beuselin, [7]. We note that this vaiational poblem has a natual intepetation in tems of spacetime (Loentzian) geomety. L(ϕ) is the length (pope-time) of the gaph of ϕ with espect to the Loentzian metic ds = dq(dq + αd). The class of functions ove which the maximum is taen consists of the time-lie cuves with espect to the metic, hence T is the pope time of the maximal cuve in the model. See [] fo futhe details. The value T = T (p, ) is given by the maximum of the functional L(ϕ) ove a lage class of functions ϕ(q). Ou stategy fo obtaining lowe bounds on T is to pesent a paticula cuve ϕ Φ with a lage value L(ϕ). Let b(q) denote the piecewise linea function defined by the union of the bottom edges of S i, namely, b(q) = i fo i q < i, i =,,..., m (see Figue ). Lemma. Fo all points (q, ), 0 q, 0 b(q), it holds that α(q, b(q)) =. Poof: By definition, α(q, ) = p(q, z)dz. Let i = i(q) be the index such that i q i. Then by definition of the density function p(q, ), p(q, z)dz = i i /( i i )dz =. Definition. Given a patition = ( = 0,,..., m, m = 0), and an index j {,,..., m}, we define a piecewise linea continuous function ϕ (,j) (q) = ϕ j (q) as follows. The vaiable q is in the ange [0, j ]. The gaph of the function ϕ j is composed of h = h(j, ) linea segments, ψ,..., ψ h whee h is an intege, h j. The segments ae of two types. A segment ψ of the fist type is a hoizontal segment (that is, a segment with slope 0), and it is necessaily a subsegment of some bottom edge B i fo an index i between and j. Moeove, the segment ψ contains the left endpoint ( i, i ) of the segment B i. Finally, the ight-most segment ψ h is of the fist type and consists of the entie bottom edge B j of the squae S j. A segment ψ of the second type is a segment with slope ( ) that ends in a point ( i, i ) fo some index i, i j. Moeove, fo all values of q fo which ψ(q) is defined, the inequality ψ(q) b(q) holds. Fix an index j, j m. The cuve ϕ j (q) is the unique piecewise linea continuous cuve in which segments of the fist and second types altenate. The sequence (ψ, ψ,..., ψ h ) is called the segment decomposition of the cuve ϕ j. See Figues and 3 fo an illustation. 4

5 ψ ψ ψ 3 ψ 4 ψ 6 ψ 5 Figue : The cuve ϕ m with m = 5 and h = 6. The segments of the second type ae depicted by diagonal lines, and the segments of the fist type ae depicted by hoizontal lines. These segments altenate. The next obsevation follows fom Definition. by a basic geometic agument. See Figue 4 fo an illustation. Obsevation.3 Conside an index j, j < j m. If ( q, ) is a point of ϕ j which belongs to the bottom edge B j of the squae S j then the cuves ϕ j and ϕ j coincide in the ange [0, q]. Fo a fixed patition, let Ω = {ϕ, ϕ,..., ϕ m } be the family of m cuves as above. Fo cuves ϕ Ω thee is a combinatoial desciption of the functional L(ϕ). Specifically, conside a cuve ϕ Ω, and let ψ, ψ,..., ψ h, be its decomposition into linea segments. By definition, L(ϕ) = l L(ψ l). Since p(q, ) = 0 fo all points (q, ) with 0 q and < b(q), it follows that fo any segment ψ of the second type, we have L(ψ) = 0. Conside now a segment ψ of the fist type defined on an inteval [q (ψ), q (ψ)], with q = q (ψ) = i. The segment ψ is contained in the bottom edge B i of the squae S i, fo some index i between and m, and so q i. By (6) we obtain L(ψ) = (q q ). (7) i i Hence L(ϕ j ) is the sum of the contibutions of the hoizontal segments. We ecall that T is defined as the maximum of the functional L(ϕ) ove all piecewise diffeentiable functions which satisfy condition (4). Obviously, segments of the fist type satisfy the condition (4). By Lemma., segments of the second type also satisfy the condition (4). Consequently, the cuves ϕ (,j) satisfy condition (4). We will show that fo evey patition, the cost L(ϕ m ) of the cuve ϕ m is at least, and conclude that T. 5

6 ψ ψ ψ 3 ψ 4 Figue 3: The cuve ϕ 4 fo the same patition. Note that h = h(, 4) = 4. The linea segments of ϕ 4 ae denoted by ψ, ψ, ψ 3, and ψ 4. Compaing this cuve with the cuve ϕ 5 (see Figue ) we see that ψ = ψ, ψ = ψ, ψ3 = ψ 3, but ψ 4 ψ 4. Specifically, ψ4 is the entie bottom edge B 4 of the squae S 4, while ψ 4 is a (pope) subsegment of B 4. ( q, ) B 4 ϕ 5 ( q, ) B 4 ϕ 5 q q Figue 4: The cuve ϕ 4 (espectively, ϕ 5 )) is depicted on the left-hand (esp., ight-hand) figue. In tems of the notation in the text, j = 4, j = 5. The condition q, ) B 4 ϕ 5 implies that the cuves ϕ 4 and ϕ 5 coincide fo q [0, q]. 6

7 To illustate ou appoach we conside the case m =. In this case thee is the unique patition = ( 0 =, = 0). Fo this patition, S is the entie unit squae, and the density function p(q, ) is given by p(q, ) = fo all points (q, ) in the squae S. This patition coesponds to the policy of allowing passenges to boad the aiplane in andom ode, in othe wods, the ailine does not employ a boading policy. We will compae all othe policies with this one. It has been shown in [] that fo this patition T = + ln(). In this case the family Φ = {ϕ } of cuves contains just one single cuve ϕ (q) = 0 fo all q, 0 q, and by equation (7), L(ϕ ) =. Theoem.4 If > then fo any patition = ( 0 =,,..., m, m = 0), we have L(ϕ (,m) ). Poof: The poof is by induction on m. The induction base m = was established above. Let F m = min{l(ϕ (,m )) m m, is a patition of size m }. Let = ( 0 =,,.., m, m+ = 0) be a patition of size m +, and conside ϕ (,m+). Let (ψ, ψ,..., ψ h ) be the segment decomposition of the cuve ϕ (,m+). We split the agument into two cases, depending on the value of m. Fist suppose that m. By definition, the last linea segment ψ h of ϕ m+ has the fom ψ h (q) = 0, fo m q, and so L(ϕ m+ ) L(ψ h ) = m. (8) We theefoe assume that m <. (9) Conside the line l given by the equation (q) = q + ( m ) which passes though the point E = (q E, E ) = ( m, 0) and has slope. Let j = j( ) m be the lagest index such that the line l intesects the bottom edge B j of the squae S j. Let D = (q D, D ) = (( m ) j /, j ) be the intesection point of the line l with B j. By definition of the cuve ϕ m+, the next to last segment ψ h coincides with the segment of the line l connecting the points D and E. (See Figue 5.) Let C = (q C, C ) = ( m, m) be the intesection point of l with the anti-diagonal. Note that since the squaes S i cove the anti-diagonal, the q coodinate of the point D is no smalle than that of C, i.e., q C = m q D m. Moeove, the coodinate of D, D, is no lage than the coodinate of C, C, i.e., m j = D C = m. (0) Let ϕ m+ = ϕ m+ ( ) be the pat of ϕ m+ consisting of ψ,... ψ h, i.e., the cuve ϕ m+ esticted to the ange [0, q D ]. By (9) and (0), this ange is not empty. Since ψ h is a segment of the second type, L(ψ h ) = 0. Consequently, L(ϕ m+ ) = L( ϕ m+ ) + L(ψ h ) + L(ψ h ) = L( ϕ m+ ) + L(ψ h ). By (8), L(ψ h ) = m. Next, we estimate L( ϕ m+ ). The index j = j( ) detemines the cuve ϕ j (see Definition.). Since the point D lies on the bottom edge B j, by Obsevation.3, the cuve ϕ m+ is also the 7

8 l C B j j D ψ E q D = ( m ) j / Figue 5: The line l contains the segment ψ h, and intesects the bottom edge B j of the squae S j. The squaes of the patition, the anti-diagonal and the line l ae all depicted by solid lines, and the dotted line is used to connect the point D with its pojection on the axis q. q 8

9 G H I D F q D γ q q F = j Figue 6: The piecewise linea cuve GHID is ϕ m+, and the cuve GHIF is ϕ j. The segment DF is γ. 9

10 estiction of ϕ j to the ange [0, q D ]. The cuve ϕ j is defined in the domain [0, j ]. Let γ be the estiction of ϕ j to the complementay domain [q D, j ] = [ m j /, j ]. See Figue 6 fo an illustation. By definition of the functional L, L(ϕ j ) = L( ϕ m+ ) + L(γ). By (7), L(γ) = ( j q D ) = ( m j j j j j). The segment γ is contained in the bottom edge B j of the squae S j. The length of γ is m j, and the length of B j is j j. It follows that j j m j, and so ( L(γ) m ) j. () We conclude that ( L( ϕ m+ ) L(ϕ j ) m ) j. To estimate L(ϕ j ), conside the affine map U : IR IR given by U(q, ) = (( j )q, ( j ) + j ). This map contacts the plane by a facto of j aound the fixed point (0, ). Conside the patition = ( = 0 j, j,..., j j, j j = 0) of size j < m +. This patition detemines the cuve ˆϕ = ϕ (,j). By the definition of the functional L (see (6) and (7)), L(U(ϕ)) = j L(ϕ) fo any cuve ϕ. Since j < m +, by the induction hypothesis, L( ˆϕ). Theefoe, L(ϕ j ) ( )( j ). By (), Consequently, L( ϕ m+ ) = L(ϕ j ) L(γ) L(ϕ m+ ) = L( ϕ m+ ) + L(ψ h ) Let a = j / m be the atio between j and m. By (0), ( )( j ) ( m j). ( ) ( )( j ) ( m j) + m. a. () 0

11 It follows that L(ϕ m+ ) a m ( ) a m. Let g( m, a) denote the ight-hand side. Next, we pove that fo all a and m, a 0 m, g( m, a). (3) and Obviously, this will complete the poof. Diffeentiating the function g( m, a) with espect to the vaiable m we get g ( m ) ( m, a) = ( a) ( ) a m a. m The equality g ( ( m) m, a) = 0 holds when ( ) a ( a m ) = a m. Since m < and a, both sides ae non-negative, and thus squaing both sides esults in the following equivalent equation. ( ) a ( a m ) = a ( ) m. (4) Fix a and conside (4) as an equation in the single vaiable m. This is clealy a linea equation. The fee coefficient of this equation is positive, and thus this equation has at most one solution. Since g(0, a) = g(, a) = fo all values of a, by the mean value theoem this equation has exactly one solution. Hence the function g a ( m ) = g( m, a) has a unique extemum in the inteval 0 m. Moeove, since lim g m 0 ( m) (0, a) = it follows that this extemum is a maximum. Consequently, fo all values of a, a, and m <, it holds that g( m, a) g(0, a) =. Refeences [] Bachmat, E., Beend, D., Sapi, L., Siena, S., N. Stolyaov Analysis of aiplane boading, spacetime geomety and andom matix theoy. Jounal of physics A: Mathematical and geneal, 39, pages L453-L459. [] Bachmat, E., Beend, D., Sapi, L., Siena, S., N. Stolyaov Analysis of aiplane boading times, submitted. Available at [3] Feai, P., K. Nagel Robustness of efficient passenge boading in aiplanes, Tanspotation Reseach Boad Annual Meeting, pape numbe , Washington D.C., Available at

12 [4] Maelli, S., G. Mattocs, R. Mey The ole of compute simulation in educing aiplane tun time, Boeing Aeo Magazine, Issue. [5] Van den Biel, M., J. Villalobos, G. Hogg The Aicaft Boading Poblem. Poc. of the th Industial Eng. Res. Conf., IERC, CD ROM only, aticle numbe 53. [6] Van den Biel, M., J. Villalobos, G. Hogg, T. Lindemann, A.V. Mule Ameica West develops efficient boading stategies, Intefaces, 35, 9-0. [7] Van Landeghem, H., A. Beuselinc. 00. Reducing passenge boading time in aiplanes: A simulation appoach. Euopean J. of Opeations Reseach, 4,