公益社団法人日本都市計画学会都市計画論文集 Vol.53 No 年 10 月 Journal of the City Planning Institute of Japan, Vol.53 No.3, October, 2018 A queueing model for goods d
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- うまじ りゅうとう
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1 A queueing model for goods delivery service by drones In many depopulated rural districts in Japan, it as been ard to run regional retail stores and many of tem ave closed down. As te result, tere as been appeared many residents wo feel difficulties to purcase commodities for teir own. Tey are called kaimono-jakusa in Japan. To solve tis social problem ome delivery services by drones are proposed and several social experiments ave been carried out. In tis time we developed M/G/s queueing models for te drone system depending on te metods of urban operations researc. Finally, we derive te number of drones to acieve te expected delivery efficiency. Keywords : drone, ome delivery service, M/G/s queueing model, distance distribution M/G/s Osamu Kurita 1 food desert 8,1,17) ( ) () (a) (b) (c),6,14,15,16) (1) () (3) 7,9,1,11) 3,4,5) () (5 ) AED( ) M/M/s 19) M/G/s 1),3) ) (Keio University)
2 ( ) ) 5 5g ) 3 1) Amazon 16 1 Amazon 5 (.7kg) 3 Domino s Pizza FAA( ) AED AED 1kg 1 s 1 1 Amazon 5 a 1 = ( ), (1) a = ( ), () a = a 1 + a, (3) τ = ( ). (4) t t = a 1 + a + τ = a + τ (5) a 1 a ( a ) λ a + τ/ ( ) w t L ( ) t L = a + τ/ + w. (6) a 1 x f(x) = (x ) (x min x x max ) (7) (x min x max x ) x g(t) v τ = x/v (5) t = a + x/v (8) dx = v dt g(t) g(t) = v ( v ) f (t a) (9) ( a + x min t a + x ) max v v t σ t c x t = a + x, (1) v ( ) σt ( = x x ), (11) v c = σ t t. (1)
3 σ t t x x x s λ µ ρ µ = 1 t, (13) ρ = λ sµ (14) M/G/s M/G/s ( ) s 5) 1 ( ) 3 M/M/s ( 5) ) M/G/s EW (M/G/s) 3,4,5) 5) EW (M/G/s) 1 + c c + 1 c R(s, ρ) EW (M/M/s). (15) c t (1) ρ (14) EW (M/M/s) M/G/s M/M/s EW (M/M/s) { (sρ) s s 1 (sρ) k = s!sµ(1 ρ) k! k= } 1 + (sρ)s (16) s!(1 ρ) (15) R(s, ρ) R(s, ρ) = 1 {1 + ϕ(s, ρ)ψ(s, ρ)}, (17) ϕ(s, ρ) = (1 ρ)(s 1)( 4 + 5s ), (18) [ ψ(s, ρ) = 1 exp 16sρ (s 1) (s + 1)ϕ(s, ρ) ]. (19) (15) M/G/1 M/M/s M/D/s (M/D/s 5) ) s ρ 1 (s, ρ) 1% 4) EL q EL EL q (M/G/s) EL(M/G/s) Little EL q (M/G/s) = λew (M/G/s), () EL(M/G/s) = EL q (M/G/s) + sρ. (1) Π() 4) M/M/s { s 1 } 1 Π (sρ)s (sρ) k + (sρ)s. () s!µ(1 ρ) k! s!(1 ρ) k= R P ( 1) F x = FP P R O x (i) R F P R O (ii) R < 1 P F x. x f(x) (i) R (ii)r < 7,9,1) (i) R f(x) = x R ( x R ) x πr arccos x + R x (R < x R + ) x F (3) (ii) R < f(x) = x πr arccos x + R (4) x ( R < x + R). x ( ) (i) R x R ( x R ) 1 F (x) = + x {x πr arccos R + x (5) x 1 (δ + R arctan + R x )} x δ (R < x R + )
4 (ii) R < F (x) = 1 + x {x πr arccos R + x x 1 (δ + R arctan + R x )} x δ ( R < x + R). (6) δ = ( + R + x)( + R x)( R + x)( + R + x). (7) x 7,9,1) (i) R [{ x = 4R ( ) } 7 + E( 9π R R ) { ( ) } ] 4 1 K( R R ), (8) (ii) R < [{ x = 4 ( ) } 7 + E( R 9π R ) { ( ) ( ) } ] R + 3 K( R R ). (9) K(k) 1 E(k) 13) K(k) = E(k) = π/ π/ dθ 1 k sin θ, (3) 1 k sin θ dθ. (31) 1,11) (i) < R x 3 R + R 4 3R 3, (3) (ii) R x + R 8 + R (33) (3) = % = R.3% (33) = R.14% x R x = R +. (34) 4. R λ s R = 3km a v λ 1 s (4..1 ) a v λ 1 km 6km a v 5km/ ( 3 6km/) 5km/ 1km/ λ 3 / 15 / 6 / R = 3km 3 /km () = π 3 3/ 45 1 (1 ) = 45/ 1 1: 16: 6 λ = 1/6 = 35 / (4..1 ) (4.. ) 1 ( ) km, 1,, 3, 4, 5, 6 (a),(e).5,.1,.15, a.,.5 (b) v km/ 5, 5, 75, 1 (c) λ / 15, 3, 45, 6 (d) ρ 1 ρ
5 (14) ρ 1 s s min (λ/µ ) λ s min =. (35) µ s s min x x (1) (11) t σ t (13) (14) µ ρ (16) (19) (15) ( EW ) x (8) (9) Fortran C Matematica Matlab Maple Matematica Ver (3) (33) (a) (a, v, λ) = (.15, 5km/, 3 /) ( ) 1 = km = 6km 1km ( ) = km () s ( 3) = 1km ( 3 1 ) 1 3 = 4km 1 13 = km s = km R = 3km = 6km 3km. EW [ ] =km =km = 1km =3km =4km =5km =6km s [ ] 3 s EW (a, v, λ) = (.15, 5km/, 3 /) (b) a (, v, λ) = (1km, 5km/, 3 /) a a a a s ( 4) a =.15( 9 ) a =.( 1 ) 1 1 a =.1( 6 ) 8 43 EW [ ] a=.1 a=.5 a=.15 a=. a= s [ ] 4 a s EW (, v, λ) = (1km, 5km/, 3 /)
6 (c) v (, a, λ) = (1km,.15, 3 /) v 1 4 5km/ 5km/ 75km/ 1km/ v s ( 5) v = 5km/ v = 5km/ 1 1 v = 1km/ 8 54 R a R a v EW [ ] v=1km/ v=75km/ v=5km/ v=5km/ s [ ] 5 v s EW (, a, λ) = (1km,.15, 3 /) (d) λ λ λ λ λ (, a, v) = (1km,.15, 5km/) λ / 3 / 45 / 6 / λ s ( 6) λ = 3 / λ = 15 / 1 6 ( 4 ) ( ) λ = 6 / λ v a EW [ ] λ=15 / λ=3 / λ=45 / λ=6 / s [ ] 6 λ s EW (, a, v) = (1km,.15, 5km/) (e) EW (, s) s EW (, s) ( 7) s (a, v, λ) = (.15, 5km/, 3 /) s [ ] EW =.5 EW =1 EW = EW = [km] 7 EW (, s) (a, v, λ) = (.15, 5km/, 3 /)
7 ( km ) s v a λ 4.. ELq [ ] 4 =km 3 =5km =6km =4km =3km 1 =km =1km s [ ] 8 s EL q (a, v, λ) = (.15, 5km/, 3 /) EL [ ] 4 =km 3 1 =1km =km =3km =4km =5km =6km s [ ] 9 s EL (a, v, λ) = (.15, 5km/, 3 /) s ( 8 9 1) s 4.3 [, 1] u x F (x)( (5) (6)) u = F (x) (36) x x = x (8) x t = a + x /v NTT S 4 Simulation System( ) Ver. 5 (, a, v, λ, s) = (1km,.15, 5km/, 3 /, 9 ) 11 1: 16:(6 ) (1 ) EW = w = 1 51 EW Π =km =3km =1km =km =5km =4km =6km [ ] w = 1 51 σ w = s [ ] 1 s Π (a, v, λ) = (.15, 5km/, 3 /) w [ ] 11 (, a, v, λ, s) = (1km,.15, 5km/, 3 /, 9 )
8 5 ( ) ( ) n 1 i q i i x i Q Q = q 1 + q + + q n (37) T λ = Q/T x = 1 n q i x i, x = 1 n q i x i (38) Q Q i=1 i=1 (1) (11) 4 6 x x min (> ) x min 4 ( ) (4..1 (d) ) ( ) ( ) 1) ( ) [ttp://diamond.jp/articles/-/1184] ) (15) 3) (1988a) M/D/s A Vol.71 A No.3 pp ) (1988b) M/G/s pp. 5) (16) 6) (15) 7) (1986) ( ) pp ) (14) No.1 pp ) (4) 1) (13) 11) (1989) 1 B pp ) (1) Vol. 45, No. 3, pp ) (1956) I () 14) (17) 15) (17) Vol. 59 No. 11 pp ) (17) 17) (14) Vol. 49 No. 3 pp ) Yomiuri Online ( ) [ttp:// OYT1T518.tml] ) Boutilier, J.J., S.C. Brooks, A. Janmoamed, A. Byers, J.E. Buick, C. Zan, A.P. Scoellig, S. Ceskes, L.J. Morrison and T.C.Y. Can(17): Optimizing a drone network to deliver automated external defibrillators, Circulation, Vol. 135, pp ) Goodcild, A. and J. Toy(18): Delivery by drone: An evaluation of unmanned aerial veicle tecnology in reducing CO emissions in te delivery service industry, Transportation Researc Part D, Vol. 61, pp ) Grippa, P., D.A. Berens, F. Wall and C. Bettstetter(18): Drone delivery systems: Job assignment and dimensioning, Autonomous Robots, Vol. 18 [Publised online]. ) Hong, I., M. Kuby and A.T. Murray(18): A rangerestricted recarging station coverage model for drone delivery service planning, Transportation Researc Part C, Vol. 9, pp ) Murray, C.C. and A.G. Cu(15): Te flying sidekick travelling salesman problem: Optimization of droneassisted parcel delivery, Transportation Researc Part C, Vol. 54, pp
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