On Line Supplement to Strategic Customers in a Transportation Station When is it Optimal to Wait? A. Manou, A. Economou, and F.

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1 On Line Spplemen o Sraegic Comer in a Tranporaion Saion When i i Opimal o Wai? A. Mano, A. Economo, and F. Karaemen 11. Appendix In hi Appendix, we provide ome echnical analic proof for he main rel of he performance anali of he obervable model of a ranporaion aion and for he Lemma 7.1 and 7.2 ha concern he preervaion of he DMRL proper for diribion. Moreover, we dic briefl he cae where he inerarrival diribion F (x) i an IMRL diribion. Fir, we will prove Propoiion 3.2 and 3.3 ing an analic approach inpired b Kerner (28) (a probabiliic proof of Propoiion 3.3 i alo preened for comparion). The main idea i o e he o-called pplemenar variable echnie, i.e. o d he join diribion of he Markov proce {(N(), R()), }. Noe ha he econd variable i conino, o we conider he probabili deni fncion Pr[N () = n, R () (r, r + dr]] p (n, r; ) = lim, n, r, dr + dr of having n comer in he em and reidal ervice ime r a ime, when he comer follow a raeg. We alo conider he aniie p(n, r; ) = lim p (n, r; ), n, r, P n, () = e r p(n, r; )dr, n, ha correpond repecivel o he eilibrim (aionar, limiing) verion of p (n, r; ) and o he aociaed Laplace ranform (LT). Then he LST F n, () and he eilibrim probabiliie π n, ha we defined in Propoiion 3.2 and 3.3 are given b π n, = We hen have he following Lemma p(n, r; )dr = P n, (), n, (11.1) F n, () = P n, () π n,, n. (11.2) Lemma Conider he obervable model of a ranporaion aion, where he comer join he em according o a raeg = (, 1, 2,...). Then, he eilibrim deni (p(n, r; ), n, r ) aifie he following em of differenial eaion wih repec o r. p (, r; ) = λ p(, r; ) 1 f(r), r, (11.3) p (n, r; ) = λ n p(n, r; ) λ n 1 p(n 1, r; ), r, n 1. (11.4)

2 Proof. B conidering he evolion of he conino ime Markov proce {(N (), R ()), } in an inerval [, + d], we can eail ee ha p +d (, r; ) = p (, r + d; )(1 λ d) + p (n, ; )f(r)d + o(d), (11.5) n= p +d (n, r; ) = p (n 1, r + d; )λ n 1 d + p (n, r + d; )(1 λ n d) + o(d), n 1. (11.6) Taking he limi a in (11.5) and (11.6) ield p(, r; ) = p(, r + d; )(1 λ d) + p(n, ; )f(r)d + o(d), (11.7) n= p(n, r; ) = p(n 1, r + d; )λ n 1 d + p(n, r + d; )(1 λ n d) + o(d), n 1. (11.8) B rearranging he erm in (11.7) and dividing b d we obain ha p(, r + d; ) p(, r; ) d = λ p(, r + d; ) n= p(n, ; )f(r) + o(d). (11.9) d Taking he limi a d in (11.9) ield Noe, now, ha p (, r; ) = λ p(, r; ) f(r) n= p(n, ; ), r. (11.1) n= p(n, r; ) = f R (r) = 1 F (r), r, (11.11) he deni of he eilibrim reidal ervice ime diribion a r. In pariclar n= p(n, ; ) = 1/; h (11.1) redce o (11.3). Similarl, rearranging erm in (11.8), dividing b d and aking he limi a d ield (11.4). Now, we can provide an analic proof of Propoiion 3.3, ing Lemma Propoiion Analic proof. Inegraing boh ide of (11.4) ield which redce o p (n, r; )dr = λ n p(n, r; )dr λ n 1 p(n 1, r; )dr, n 1, (11.12) p(n, ; ) = λ n π n, λ n 1 π n 1,, n 1. (11.13) Smming (11.13), for n 1, and ing (11.11), we obain p(, ; ) = 1 λ π,. (11.14)

3 Now, mlipling boh ide of (11.4) b e λnr and rearranging erm ield (e λnr p(n, r; )) = λ n 1 e λnr p(n 1, r; ), n 1, r. (11.15) Inegraing (11.15), we obain ha which i wrien eivalenl a (e λnr p(n, r; )) dr = λ n 1 e λnr p(n 1, r; )dr, n 1, p(n, ; )) = λ n 1 Pn 1, (λ n ), n 1. (11.16) Plgging (11.13) ino (11.16) ield λ n π n, λ n 1 π n 1, = λ n 1 Pn 1, (λ n ), n 1 which give λ n π n, = λ n 1 π n 1, (1 F n 1, (λ n )), n 1. (11.17) Solving (11.17) for π n, ield (3.24), whenever n. Taking he limi in (3.24), a n, we dedce (3.23). To obain he iniial condiion (3.26) and (3.25) of he recrion cheme, we e a imilar line of argmen. More pecificall, we ar b mlipling (11.3) b e λ r and rearrange he erm o obain ha Inegraing (11.18) ield p(, ; ) = (e λ r p(, r; )) = 1 e λ r f(r), r. (11.18) (e λ r p(, r; )) dr = 1 Uing (11.14) and (11.19), we dedce ha e λ r f(r)dr = 1 F (λ ). (11.19) λ π, 1 = 1 F (λ ). (11.2) Solving (11.2) for π, ield (3.26), whenever. Taking he limi of (3.26), a, we dedce (3.25). I rn o ha Propoiion 3.3 can alo be hown ing a probabiliic approach. Thi alernaive proof can be fond below.

4 Propoiion Probabiliic proof. in he n-h waiing poiion of or em. We denoe b N (n) in he n-h waiing poiion, b λ (n) and b S (n) We will dedce (3.23)-(3.26) b appling Lile law he eilibrim nmber of comer he arrival rae of he comer a he n-h waiing poiion he waiing ime of a comer in he n-h waiing poiion. We have ha E[N (n) ] = π j,, (11.21) j=n λ (n) = λ n 1 π n 1,, (11.22) E[S (n) ] = E[R n 1,n 1 ]. (11.23) Indeed, he n-h waiing poiion i occpied if and onl if here are a lea n comer in he em and o we obain (11.21). Moreover, arrival in he n-h waiing poiion can occr onl when here are n 1 comer in he em and hen he arrival rae i λ n 1. The fracion of ime ha comer arrive in he n-h waiing poiion i hen π n 1, and we conclde ha he arrival rae λ (n) i given b (11.22). Finall, a comer ha join in he n-h waiing poiion ha een pon arrival n 1 comer in he em. Hence hi ojorn ime in he n-h waiing poiion coincide wih he condiional reidal ervice ime given ha here n 1 comer in he em and we obain (11.23). Appling Lile law ield π j, = λ n 1 π n 1, E[R n 1,n 1 ]. (11.24) j=n Conidering (11.24) for n = 1 and olving for π, ield π, = λ E[R, ]. (11.25) For = we obain immediael (3.25). For we bie E[R, ] from (3.22) and we obain (3.26). Now, for n 1, bracing (11.24) for n + 1, from (11.24) for n ield π n, = λ n 1 π n 1, E[R n 1,n 1 ] λ n π n, E[R n,n ]. (11.26) Solving for π n,, we obain ha π n, = λ n 1E[R n 1,n 1 ] π n 1,, n 1. (11.27) 1 + λ n E[R n,n ] For n = we obain (3.23). For n, plgging (3.2) ino (11.27) ield (3.24). Now, we can provide an alernaive proof of Propoiion 3.2.

5 Propoiion Analic proof. Mlipling boh ide of (11.4) b e r and inegraing wih repec o r ield or e r p (n, r; )dr = λ n e r p(n, r; )dr λ n 1 e r p(n 1, r; )dr, n 1, p(n, ; ) + P n, () = λ n Pn, () λ n 1 Pn 1, (), n 1. (11.28) Plgging (11.13), (11.2) ino (11.28) ield or λ n π n, λ n 1 π n 1, + π n, Fn, () = λ n π n, Fn, () λ n 1 π n 1, Fn 1, (), n 1, ( λ n )π n, Fn, () = λ n π n, + λ n 1 π n 1, (1 F n 1, ()), n 1. (11.29) For n, (11.29) ing (11.17) redce o ( λ n ) F Fn 1, (λ n ) n, () = λ F n 1, () n 1 F, n 1. (11.3) n 1, (λ n ) If λ n, we can ee eail ha (11.3) implie (3.12). Taking he limi of (3.12), a λ n, ield (3.13). On he oher hand, if n =, eaion (11.29) ing (3.23) ield which redce o (3.11). F n, () = 1 F n 1, (), n 1, E[R n 1, ] Following he ame line of argmen we dedce he iniial condiion (3.14)-(3.16). Indeed, mlipling boh ide of (11.3) b e r and inegraing wih repec o r ield Plgging (11.14) ino (11.31), we have From (11.2) and (11.32), we obain p(, ; ) + P, () = λ P, () 1 F (). (11.31) 1 + λ π, + P, () = λ P, () 1 F (). (11.32) If, eaion (11.33), ing (11.2), give ( λ )π, F, () = F (λ ) F (). (11.33) ( λ ) F F (λ ), () = λ F () 1 F. (11.34) (λ )

6 If λ, from (11.34) we dedce (3.15). Oherwie, aking he limi of (3.15) a λ, we obain (3.16). Now, if =, eaion (11.33), ing (3.25) we obain F, () = 1 F (), (11.35) which implie (3.14). Now, we give he proof of Lemma 7.1 and 7.2. Lemma Proof. Fir noe ha E[X x x x] = x F X ()d, (11.36) F X (x) where F X (x) = 1 F X (x) denoe he rvival fncion of X. Then, i i ea o ee ha he DMRL proper of X i eivalen o he condiion F Y ()d F Y () F X ()d, for, Y = X + ω, ω, (11.37) F X () or o he condiion ha F Y ()d F X ()d i increaing in, for, Y = X + ω, ω. (11.38) Uing (11.37) and (11.38), we alo obain ha he DMRL proper of X implie ha F Y () F X () F Y ()d, for, Y = X + ω, ω. (11.39) F X ()d We will now prove ha he condiional diribion of X T λ, given ha X T λ, i DMRL. In ligh of (11.38), i ffice o how ha or eivalenl ha F Y Tλ ()d F X Tλ ()d i increaing in, for, Y = X + ω, ω, (11.4) F X Tλ ()d F Y Tλ ()d F X Tλ ()d F Y Tλ ()d,, Y = X + ω, ω, (11.41) where in all place F Z (x) denoe he rvival fncion of he correponding random variable Z. Now, he rel can be proven b howing ha (11.39) implie (11.41). Indeed, noe ha F X Tλ ()d = F X () ( 1 e λ( )) d

7 and a imilar ideni hold for an Y = X + ω, ω, o afer a bi of algebraic maniplaion, he lef ide of (11.41) ame he form F X Tλ ()d F Y Tλ ()d [ F X (z)dz F Y () F X () [ F X (z)dz F Y () F X () F X Tλ ()d F Y Tλ ()d = ] F Y (z)dz λe ( λ( ) 1 e λ( )) dd ] F Y (z)dz λe ( λ( ) 1 e λ( )) dd, which i clearl nonnegaive, becae of (11.39). We now proceed o he proof of Lemma 7.2. Lemma Proof. Since X ha a DMRL diribion, we have ha F X ()d F X () I i alo ea o ee ha E[X T λ X T λ ] = F X ()d,. (11.42) F X () F X ( + )λe λ d d. (11.43) F X ()λe λ d Differeniaing (11.43) wih repec o λ and maniplaing a bi algebraicall ield d dλ E[X T λ X T λ ] = [ F X ()d F X () F X () F X ()d]e λ(+) λ 2 ( )dd ( F X ()λe λ d ) 2, which i nonnegaive, becae of (11.42). When he facili inerarrival diribion F (x) i IMRL, imilar rel wih he one ha are repored in Secion 7 hold. In pariclar, he obvio conerpar of Lemma 7.1 and 7.2 are valid. Therefore, R n,n are IMRL and E[R n,n ] are decreaing in n, for n =, 1,.... Moreover, if F (x) i ricl IMRL, hen he expeced ne benefi fncion Sn ob ( n ) i ricl increaing in n and coneenl here exi a mo one proper eilibrim joining probabili for an n. Finall, we can conclde ha he eilibrim joining raegie in hi cae are of hrehold pe. In a nhell, we can ae ha when F (x) i IMRL, imilar rel o he DMRL cae hold, he inealiie are revered and we have a kind of FTC iaion regarding he behavior of he comer. Noe, however, ha here i a difference: Unlike Theorem 7.1, here i no need o ame here ha he ranporaion facili ha nlimied capaci.