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) - 61 -
( ) 16 1 6 1) 5 5g 16 7 17 1 31 18) 3 1) Amazon 16 1 Amazon 5 (.7kg) 3 Domino s Pizza 16 17 1 77 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) - 611 -
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= 4 4.1 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 + ) - 61 -
(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 + R4 19 3. (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.15.5.5 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 1-613 -
(14) ρ 1 s s min (λ/µ ) λ s min =. (35) µ s s min 4..1 4.1 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. 11.1.1. (3) (33) (a) (a, v, λ) = (.15, 5km/, 3 /) ( ) 1 = km = 6km 1km ( ) = km () s ( 3) = 1km 8 5 3 1 ( 3 1 ) 1 3 = 4km 1 13 = km 1 3 1 s = km R = 3km = 6km 3km. EW [ ] 6 5 4 3 1 =km =km = 1km =3km =4km =5km =6km 7 8 9 1 11 1 13 14 s [ ] 3 s EW (a, v, λ) = (.15, 5km/, 3 /) (b) a (, v, λ) = (1km, 5km/, 3 /) a 1 3.5.1.15..5 a a a s ( 4) a =.15( 9 ) 8 5 3 1 1 a =.( 1 ) 1 1 a =.1( 6 ) 8 43 EW [ ] 6 5 4 3 1 a=.1 a=.5 a=.15 a=. a=.5 5 6 7 8 9 1 11 1 s [ ] 4 a s EW (, v, λ) = (1km, 5km/, 3 /) - 614 -
(c) v (, a, λ) = (1km,.15, 3 /) v 1 4 5km/ 5km/ 75km/ 1km/ v s ( 5) v = 5km/ 8 5 3 1 1 v = 5km/ 1 1 v = 1km/ 8 54 R a R a v EW [ ] 6 5 4 3 1 v=1km/ v=75km/ v=5km/ v=5km/ 6 7 8 9 1 11 1 s [ ] 5 v s EW (, a, λ) = (1km,.15, 3 /) (d) λ λ λ λ λ (, a, v) = (1km,.15, 5km/) λ 1 5 15 / 3 / 45 / 6 / λ s ( 6) λ = 3 / 8 5 3 1 1 λ = 15 / 1 6 ( 4 ) ( ) λ = 6 / 1 18 8 λ v a EW [ ] 6 5 4 3 1 λ=15 / λ=3 / λ=45 / λ=6 / 5 1 15 s [ ] 6 λ s EW (, a, v) = (1km,.15, 5km/) (e) EW (, s) s EW (, s) ( 7) 7.5 1 5 4 s (a, v, λ) = (.15, 5km/, 3 /) s [ ] 16 14 1 1 8 EW =.5 EW =1 EW = EW =5 1 3 4 5 6 [km] 7 EW (, s) (a, v, λ) = (.15, 5km/, 3 /) - 615 -
( km ) s v a λ 4.. ELq [ ] 4 =km 3 =5km =6km =4km =3km 1 =km =1km 8 1 1 14 s [ ] 8 s EL q (a, v, λ) = (.15, 5km/, 3 /) EL [ ] 4 =km 3 1 =1km =km =3km =4km =5km =6km 8 1 1 14 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 = 1 4 11 w = 1 51 EW Π.4.3..1 =km =3km =1km =km =5km =4km =6km [ ] 1 1 8 6 4 w = 1 51 σ w = 1 46. 8 1 1 14 s [ ] 1 s Π (a, v, λ) = (.15, 5km/, 3 /) 4 6 8 1 1 w [ ] 11 (, a, v, λ, s) = (1km,.15, 5km/, 3 /, 9 ) - 616 -
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