L = λw d
L k = λ k j Fk j,j, c k = 0, 1,..., d 1, d L k k λ k k Fk,j c j 0 k j k + md m 0 λ k Fk, c L k k L(t) = c 0 λ(t)f c (t s, s) ds, 0 t < c, c d d d
25 number of patients 0 10 20 30 40 50 60 Estimated mean from data Estimated mean from simulation Sun Mon Tue Wed Thu Fri Sat time of a week L = λw
M t /GI/1 1
i i 0 j j 0 X {X i,j : i 0; j 0} X i,j i j 0 X Y i,j A i s=j X i,s i j j 0 Y i,0 = s=0 X i,s i Q i i Y i j,j = i A Y i j,j i j i A i j i 0 0/0 0 0
n λ k (n) 1 n Q k (n) 1 n Ȳ k,j (n) 1 n d A k+(m 1)d, m=1 Q k+(m 1)d = 1 n m=1 m=1 Y k+(m 1)d,j, j 0, m=1 ( k+(m 1)d Y k+(m 1)d j,j ), F k,j(n) c Ȳk,j(n) n λ k (n) = Y m=1 k+(m 1)d,j n A, j 0, m=1 k+(m 1)d W k (n) F k,j(n), c 0 k d 1, λ k (n) k 0 k d 1 n Qk (n) k Ȳk,j(n) k j F c k,j(n) k Wk (n) k n (A1) λk (n) λ k, n, 0 k d 1, (A2) (A3) c F k,j(n) Fk,j, c n, 0 k d 1, j 0, Wk (n) W k F c k,j n, 0 k d 1, d = 1 (A2)
k 0 ( ) k 0 d ( ) (A1) (A2) (A3) Q k (n) n L k L k = λ k j F c k j,j <, 0 k d 1, λ k Fk,j c (A1) (A2) ( ) d = 1 k = 0 L 0 lim Q0 (n) lim λ0 (n) = λ 0 λ k j F c k j,j = λ 0 F c 0,j = λ 0 W 0. L 0 = λ 0 W 0,
L λ W λ W L L = λw λ k (n) A1 λ k (n) k 0 k d 1 W k (n) A3 W k (n) k 0 k d 1 k 0 k d 1 A2 ( ) F c k,j(n) F c k,j d = 2 2 2 0 2 2 1 λ k W k λ 0 = 2 λ 1 = 0 W 0 = 2 W 1 = 0 lim Q0 (n) = 3 lim Q0 (n) = 2 L k L k L k
0 λ k j F c k j,j = k i=0 λ i l=0 F c i,k i+ld + i=k+1 L k L k (n) k λ i (n) i=0 l=0 F c i,k i+ld(n) + i=k+1 λ i λ i (n) l=1 l=1 F c i,k i+ld, 0 k d 1. F c i,k i+ld(n), 0 k d 1, λ i (n) F c i,j(n) n L k (n) Q k (n) ( ) lim L k (n) = L k 0 k d 1, L k (n) L k m > 1 X i,j = 0 j md (A3) (A2) L k Ē k (n) L k (n) Q k (n), Q k (n) L k (n) ( ) (A1) (A2) m u > 0 X i,j = 0 i 0 j dm u (A3) λ u > 0 A i λ u i 0 R n max 0 k d 1 {Ēk(n)} λ ud(m u + 2) 2, n m u, 2n Ēk(n)
(A3) (A2) X i,j = 0 i 0 j > dm u F k,j (n) = 0 0 k d 1 j dm u dm u W k (n) = j=1 F c k,j(n), 0 k d 1, F c k,j = 0 0 k d 1 j > dm u lim (A3) dm u W k (n) = lim dm u F k,j(n) c = lim dm u F k,j(n) c = F c k,j = W k, ( (A2) (A3)) (A2) (A3) (A3) (A2) D i i X i j,j i 0 i D i = Q i Q i+1 + A i+1, i 0. δ k (n) 1 D k+(m 1)d, n 0 k d 1. m=1
( ) δ k (n) n δ k δ k = λ k j f k j,j λ k j (F c k j,j Fk j,j+1) c 0 k d 1 λ k F c k,j f k,j F c k,j F c k,j+1 δ k (n) = 1 n D k+(m 1)d = 1 n m=1 (Q k+(m 1)d Q k+1+(m 1)d + A k+1+(m 1)d ) m=1 Q k (n) Q k+1 (n) + λ k+1 (n), 0 k < d 1, = Q d 1 (n) Q 0 (n + 1) + λ 0 (n + 1) + 1 n Q 0 1 n A 0, k = d 1. 1 lim n Q 1 0 = 0 lim n A 0 = 0 (A1) lim δ k (n) = L k L k+1 + λ k+1 = λ k j F c k j,j λ k+1 j F c k+1 j,j + λ k+1 = λ k j (F c k j,j fk j,j+1) c λ k+1 + λ k+1 = λ k j f k j,j. C Q (n) C L (n) nd 1 Q k+(m 1)d = Q i, m=1 k=0 m=1 k=0 i=0 (j + 1)X k+(m 1)d,j = m=1 k=0 nd 1 Y k+(m 1)d,j = i=0 Y i,j. C Q (n) n C L (n) n
35 1 C Q (n) C L (n) d = 5 n = 4 nd = 20 C Q(4) C L(4) Area(A) = C Q (n) = n k=0 k=0 Q k (n), Area(A B) = C L (n) = n λ k (n) W k (n) = n L k (n). k=0
C Q (n) C L (n) C L (n) = Y k+(m 1)d,j = n m=1 k=0 = n λ k (n) W k (n). k=0 k=0 Ȳ k,j (n) = n k=0 λ k (n) F k,j(n) c 0 k d 1 k=0 L k (n) = λ k (n) F k,j(n) c = λ k (n) W k (n). k=0 k=0 Ē(n) Ē k (n) = (B)/n 0 n, k=0 Ēk(n) 0 0
Q L i i Q L k (n) 1 n m=1 X i,j Y i,j Q L k+(m 1)d. ( ) Q L k (n) n L L k L L k = L k δ k 2λ k + λ k f k,0 L k 0 k d 1, L k λ k (A1) δ k f k,0 i i i i i i Q L i = Y i j,j+1 Y i,0 = Y i j,j X i j,j Y i,0 = Y i j,j X i j,j 2A i + X i,0 j=1 j=1 j=1 = Q i D i 2A i + X i,0, 0 Q L k (n) = 1 Q L k+(m 1)d = 1 (Q k+(m 1)d D k+(m 1)d 2A k+(m 1)d + X k+(m 1)d,0 ) n n m=1 m=1 = Q k (n) δ k (n) 2 λ k (n) + (Ȳk,0(n) Ȳk,1(n)). (A1) (A2) lim Ȳ k,j (n) = λ k F c k,j lim Q L k (n) = L k δ k 2λ k + λ k (F c k,0 Fk,1) c = L k δ k 2λ k + λ k f k,0.
A(t) t 1 A(t) Λ(t) t, Λ(t) = t 0 λ(u) du 0 < λ U λ(u) λ U <, u t λ c D(t) δ(t) ( ) c d d 0 d 0 L k L L k = δ k + 2λ k λ k f k,0 d λ d,k 0 δ d,k 0 d ( ) ( ) d d ( )
G t /GI t / L k λ k F c k,j k {Y n : n Z} Y n {Y nd+k,j : 0 k d 1; j 0} n n Y n (Z d ) Z d Z d
Z k k (n 1,..., n k ) Z (Y n1,..., Y nk ) d = (Y n1 +m,..., Y nk +m) m Z, = d {Y n : n Z} {(A nd+k, Q nd+k ) : n Z} A nd+k nd + k Q nd+k nd + k A nd+k Y nd+k,0 Q nd+k Y nd+k j,j. ({Y k,j : j 0}, A k, Q k ) λ k E[A k ] = E[Y k,0 ] L k E[Q k ] = E[Y k j,j ]. Fk,j c P (W k > j) W k k Fk,j c F k,j(n) c 0 < E[Y k,0 ] < k ( ) {Y n : n Z} 0 < λ k E[Y k,0 ] < 0 k d 1 k 0 k d 1 j 0 F c k,j(n) F c k,j E[Y k,j] E[Y k,0 ] n
L k E[Q k ] = λ k j Fk j,j. c Ȳ k,j (n) E[Y k,j ] = λ k F c k,j n k j F c k,j λ k E[Q k ] E[Y k j,j ] E[Q k ] = E[Y k j,j ] = λ k = λ k λ k Fk j,j. c F k,j G t /GI t / G t /GI t / Fk,j c P (W k j) k k ( G t /GI t / ) G t /GI t / Y {Y n : n Z} E[Y k,j ] = F c k,je[y k,0 ],
{W k,i : i 1} k m E[Y k,j ] = P (W k,i > j)p (A k = m) = m=1 i=1 mfk,jp c (A k = m) = Fk,jE[A c k ] = Fk,jE[Y c k,0 ]. m=1 GI t k k A (m) k k m 1 k 7 24 = 168 m 1 25 W (i) k k m k A (i) k W (m) k A (m) k > 0 r k 168 A (m) k r k = 25 25 > 0 m m=1 (A(m) k m=1 (A(m) k Āk (1/25) 25 i=1 A(m) k W k (1/25) 25 (m) Āk)(W k W k ) Āk) 25 2 (m) m=1 (W W (m) i=1 k k W k ) 2, m A (m) k = 0 168 0.056 0.21 (A (m) 36, W (m) 36 ) W (m) 36
i A (m) k = 0 k = 1, 2,, 7 24 = 168
corrrelation of num. of arrivals vs LoS 0.4 0.2 0.0 0.2 0.4 0.6 Sun Mon Tue Wed Thu Fri Sat time of a week (hour) Normal Q Q Plot Sample Quantiles 0.4 0.2 0.0 0.2 0.4 0.6 LoS (hour) 2 3 4 5 6 7 8 9 2 1 0 1 2 Theoretical Quantiles 4 6 8 10 12 14 16 number of arrivals in the hour number of hours with 0 arrival 0 2 4 6 8 10 Sun Mon Tue Wed Thu Fri Sat time of a week Q Q
λ(t) t N(t) N(s) [s, t] E[N(t)] E[N(s)] = t s λ(s) ds, λ(t) c W (t) t {W (t) : t R} Q {Q(t) : t R} c Q c {N s : s R} N c F c t,x P t (W (t) > x) P t c M t /GI/1 ( ) F c t s,s E[Q(t)] = 0 F c t s,sλ(t s) ds
2 A1 A2 A3 lim F k+ld,j(n) c lim Wk+ld (n) 0 k d 1 l 0 j 0 lim λk+ld (n) lim λ k+ld (n) λ k+ld = λ k, c lim F k+ld,j(n) F c k+ld,j = Fk,j, c lim Wk+ld (n) W k+ld = W k, λ k F c k,j W k λ k 1 λ k = lim λk (n) = lim n m=1 m=1 A k+(m 1)d 1 = lim n (A n 1 n 1 1 n 1 k + A (k+d)+(m 1)d ) = lim n n 1 m=1 A (k+d)+(m 1)d = λ k+d. Ȳk,j(n) = λ k (n) F c k,j(n) A1 A2 A3 lim Ȳ k,j (n) = λ k F c k,j 0 k d 1 j 0 λ k lim Ȳ k+d,j (n) = lim Ȳ k,j (n) 0 k d 1 j 0 F c c k+d,j = lim F k+d,j(n) = W k+d lim Ȳ k+d,j (n) lim λ k+d (n) = λ kf c k,j λ k = F c k,j, 0 k d 1, j 0. W k+d = F c k+d,j = F c k,j = W k 0 k d 1.
lim Lk (n) = λ k jf c k j,j k = 0, 1,, d 1 L lim Qk (n) = lim Lk (n) ϵ > 0 A1 N 1 n > N 1 sup 0 k d 1 λ k (n) λ k < ϵ A2 A3 lim F k,j(n) c Fk,j c = 0. N 2 n > N 2 max{n 1, N 2 } n > N 3 = = L k (n) i=0 λ k j Fk j,j c k λ i (n) k i=0 λ i l=0 l=0 F c i,k i+ld(n) + F c i,k i+ld i=k+1 k λ k j (n) F k j,j(n) c + k λ k j F c k j,j k λ k j F c k j,j(n) + m=1 j=1 k λ k j F c k j,j i=k+1 λ i m=1 j=1 l=1 λ i (n) l=1 F c i,k i+ld sup F k,j(n) c F k,j c < ϵ N 3 = 0 k d 1 F c i,k i+ld(n) λ d j (n) F d j,(m 1)d+j+k(n) c λ d j Fd j,(m 1)d+j+k c m=1 j=1 m=1 j=1 +ϵ ( k F k j,j(n) c + m=1 j=1 λ d j F c d j,(m 1)d+j+k(n) λ d j Fd j,(m 1)d+j+k c F d j,(m 1)d+j+k(n) ) c ( max λ k) ( k F k j,j(n) c F k j,j c + 0 k d 1 +ϵ ( k F k j,j(n) c + m=1 j=1 m=1 j=1 F d j,(m 1)d+j+k(n) ) c ( max λ k)dϵ + ϵd ( max W k + ϵ ). 0 k d 1 0 k d 1 F d j,(m 1)d+j+k(n) c Fd j,(m 1)d+j+k ) c
lim L k (n) λ k jf c k j,j = 0 0 k d 1 L k lim Qk (n) = lim Lk (n) Ē(n) Ē k (n) L k (n) Q k (n) 0 n. k=0 k=0 Ē(n) 0 n Q k (n) 1 n = 1 n = 1 n = 1 n = 1 n Q k+(m 1)d = 1 n m=1 m=1 m=1 m=1 m=1 m=1 k Y k j+(m 1)d,j + 1 n ( k+(m 1)d m=2 k Y k j+(m 1)d,j + 1 n 1 md n k Y k j+(m 1)d,j + 1 n m=1 j=1 n 1 m=1 j=1 k Y k j+(m 1)d,j + 1 n 1 n s n Y k+(m 1)d j,j ) k+(m 1)d j=k+1 Y k j+(m 1)d,j Y d j+(m 1)d,j+k Y d j+(m 1)d,j+k + 1 n 2 md n s=1 m=1 j=1 m=1 j=1 Y d j+(m 1)d,j+k+(s 1)d, Y d j+(m 1)d,j+k+d k L k (n) = Ȳ k j,j (n) + Ȳ d j,j+k+(s 1)d (n) = 1 n m=1 s=1 j=1 k Y k j+(m 1)d,j + 1 n Y d j+(m 1)d,j+k+(s 1)d. s=1 m=1 j=1 L k (n) Q k (n) Ē k (n) L k (n) Q k (n) = 1 n 1 Y d j+(m 1)d,j+k+(s 1)d + 1 n n = 1 n s=1 m=n s+1 j=1 m=1 j=1 s=n m+1 Y d j+(m 1)d,j+k+(s 1)d, s=n m=1 j=1 Y d j+(m 1)d,j+k+(s 1)d
k = 0, 1,, d 1 1 Ē(n) Ē k (n) = n = 1 n k=0 k=0 m=1 j=1 s=(n m)d m=1 j=1 s=n m+1 Y d j+(m 1)d,j+s. Y d j+(m 1)d,j+k+(s 1)d Ē(n) 0 n N 1, N 2 N 3 ϵ n > N 3 Ȳ k,j (n) λ k Fk,j c = λ k (n) F k,j(n) c λ k Fk,j c λ k λ k F c k,j(n) λ k Fk,j c + ϵ F k,j(n) c F c k,j(n) F c k,j + ϵ (W k + ϵ) λ k ϵ + ϵ (W k + ϵ). A1 A2 lim Ȳ k,j (n) = λ k F c k,j lim Ȳk,j(n) λ k Fk,j c = 0. ϵ > 0 λ kf c k,j k = 0, 1,, d 1 J k=0 j=j λ k F c k,j < ϵ. N 4 J/d x x n N 4 nd > J N 5 n > N 5 k=0 N 6 max{n 4, N 5 } n N 6 Ȳk,j(n) λ k Fk,j c < ϵ. k=0 j=n 6 d Ȳ k,j (n) 2ϵ.
N 7 N 6 /ϵ n > N 7 (n N 6 )/n > 1 ϵ n > max{n 7, 2N 6 } Ē(n) = 1 n = = m=1 j=1 s=(n m)d j=1 j=1 j=1 n N 1 6 n m=1 s=(n m)d n N 1 6 n 1 n m=1 s=n 6 d m=1 s=n 6 d j=1 s=n 6 d 2ϵ + 2ϵ + ϵ( Y d j+(m 1)d,j+s Y d j+(m 1)d,j+s + Y d j+(m 1)d,j+s + Y d j+(m 1)d,j+s + Ȳ d j,j+s (n) + k=0 j=1 s=0 λ k F c k,j + ϵ). j=1 j=1 j=1 1 n 1 n 1 n m=n N 6 +1 s=(n m)d m=n N 6 +1 s=0 m=n N 6 +1 s=0 (Ȳd j,s(n) n N 6 n Y d j+(m 1)d,s Y d j+(m 1)d,s Ȳ d j,s (n N 6 )) Y d j+(m 1)d,j+s s m N 6 N 7 2ϵ = j=1 s=0 j=1 s=0 j=1 s=0 2ϵ + ϵ( (Ȳd j,s(n) n N 6 n Ȳ d j,s (n N 6 )) (Ȳd j,s(n) (1 ϵ)ȳd j,s(n N 6 )) (Ȳd j,s(n) Ȳd j,s(n N 6 )) + ϵȳd j,s(n N 6 ) k=0 λ k F c k,j + ϵ). L k lim Qk (n) = lim Lk (n)
Ē k (n) = 1 n m=1 j=1 s=n m+1 Y d j+(m 1)d,j+k+(s 1)d. Y i,j = 0 i 0 j dm u n m u Ē k (n) = 1 n 1 n m u+1 j=1 m=n m u s=n m+1 m u+1 j=1 m=n m u s=n m+1 = 1 n λ ud (m u + 1)(m u + 2) 2 Y d j+(m 1)d,j+k+(s 1)d A d j+(m 1)d 1 n m u+1 j=1 m=n m u s=n m+1 λ ud(m u + 2) 2. 2n H = λg λ u
L = λw L = λw L = λw L = λw
L = λw H = λg L = λw L = λw 50 th L = λw L = λw L = λw L = λw
60% 23, 409 134 4 Mt T /GI t /
ˆσ 2 ˆm 1 T (d w ) = A + C d w + ϵ, T (d w ) A C d w ϵ N(0, σ 2 ) 25 7 = 175 ˆσ 2 = 202 ˆσ 2 / ˆm = 202/134 1.50 p(t) t t p(t)
Mt T /GI/
ˆp(t) p(t) ˆp(t) = 0.001082(x 13.5) 2 + 0.451996 x = ((t 1.5) mod 24) + 1.5 t [0, 24] 5 N N t ˆp(t) 1000 1000
1 25 L. (25) Q. (25)
arrival rate 0 5 10 15 20 1 7 13 19 25 Sun Mon Tue Wed Thu Fri Sat time of a week 10 a.m. 4 p.m. Fn(x) 0.0 0.2 0.4 0.6 0.8 1.0 Fn(x) 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 LoS (hour) 0 2 4 6 8 10 LoS (hour) 10 a.m. 4 p.m. average LoS (hour) 4.0 4.2 4.4 4.6 average LoS (hour) 3.2 3.4 3.6 3.8 4.0 5 10 15 20 25 sample size (weeks) 5 10 15 20 25 sample size (weeks)