149 (Newell [5]) Newell [5], [1], [1], [11] Li,Ryu, and Song [2], [11] Li,Ryu, and Song [2], [1] 1) 2) ( ) ( ) 3) T : 2 a : 3 a 1 :

Transactions of the Operations Research Society of Japan Vol. 58, 215, pp. 148 165 c ( 215 1 2 ; 215 9 3 ) 1) 2) :,,,,, 1. [9] 3 12 Darroch,Newell, and Morris [1] Mcneil [3] Miller [4] Newell [5, 6], [1] Newell [6] Mcneil [3] Darroch,Newell, and Morris [1] Miller [4] Miller [4] 148

149 (Newell [5]) Newell [5], [1], [1], [11] Li,Ryu, and Song [2], [11] Li,Ryu, and Song [2], [1] 1) 2) ( ) ( ) 3) 2 3 4 5 2. 2.1. 1 T : 2 a : 3 a 1 :

15 4 a 2 : 5 b 1 : 6 b 2 : 7 c : c kc (1 k)c k 8 λ 1 λ 4 : 9 µ 1 µ 4 : a 1 = a 1 (1 k)c, b 1 = b 1+kc, a 2 = a 2 (1 k)c, b 2 = b 2 + kc, a = a + (1 k)c 1: 2.2. : a 1 = b 2, a 2 = b 1 T = a 1 + b 1 + = a 2 + b 2 + λ i T µ i b 1 (i = 1, 2), λ i T µ i b 2 (i = 3, 4) a 1, a 2, b 1, b 2, T (2.1)

2.3. 151 2.3.1. t A 1 (t) D 1 (t) A 1 (t) = λ 1 t ( t T ) ( t < a 1 + ) D 1 (t) = µ 1 t µ(a 1 + ) (a 1 + t < t ) λ 1 t (t t T ) (2.2) 2 t n 2 t t = µ 1(a 1 + ) µ 1 λ 1 (2.3) 2: W 1 W 1 S 1 W 1 = S 1 = T A 1 (t)dt T D 1 (t)dt = λ 1µ 1 (a 1 + ) 2 2(µ 1 λ 1 ) W 2 = λ 2µ 2 (a 1 + ) 2 2(µ 2 λ 2 ) (2.4) (2.5)

152 2.3.2. t A 2 (t) D 2 (t) 2.3.1 A 2 (t) = λ 3 t + λ 3 (T a 1 a ) ( t T ) ( t < a ) µ 3 (t a ) (a t < t ) D 1 (t) = λ 3 t + λ 3 (T a 1 a ) (t t < a 1 + a ) λ 3 T (a 1 + a t T ) (2.6) 3 t n 3 t t = λ 3(T a 1 a ) + µ 3 a µ 3 λ 3 (2.7) 3: W 3 = S 2 = T A 2 (t)dt T D 2 (t)dt = λ 3µ 3 (T a 1 ) 2 2(µ 3 λ 3 ) (2.8) W 4 = λ 4µ 4 (T a 1 ) 2 2(µ 4 λ 4 ) (2.9)

2.3.3. 153 W W q W = W 1 + W 2 + W 3 + W 4 = λ 1µ 1 (a 1 + ) 2 + λ 2µ 2 (a 1 + ) 2 2(µ 1 λ 1 ) 2(µ 2 λ 2 ) W W q = (λ 1 + λ 2 + λ 3 + λ 4 )T + λ 3µ 3 (a 2 + ) 2 2(µ 3 λ 3 ) + λ 4µ 4 (a 2 + ) 2 2(µ 4 λ 4 ) (2.1) (2.11) (2.11) (2.1 (2.2) (2.6) 2.4. 2.2 2.3 Min W q s.t. T = a 1 + a 2 + λ i T µ i a 2 (i = 1, 2), λ i T µ i a 1 (i = 3, 4) a 1, a 2, T (2.12) a 1, a 2, T a, λ i, µ i a 3. 3.1. 3.1 (2.12) a 2 a 1 T ρ i = λ i µ i (i = 1, 2, 3, 4) a 1 T (2.12) (3.1) max(ρ 3, ρ 4 )T a 1 T max(ρ 1, ρ 2 )T 2a (3.1) T a 2 + T a 1 (2.12) (3.2) 2 Min W q s.t. max(ρ 3, ρ 4 )T a 1 T max(ρ 1, ρ 2 )T T (3.2) a 1 T ρ i, a

154 3.2. (3.2) W q ( [7] Step [Step 1] W q (Hessian matrix) : 2 W q a 2 1 2 W q = 2 W q T a 1 = 2 W q a 1 T 2 W q T 2 2(A + B) T 2[A(a 1 + ) + Ba 1 ] T 2 2[A(a 1 + ) + Ba 1 ] [A(a 1 + ) 2 + Ba 2 1] T 2 T 3 ( 1 λ1 µ 1 A = + λ ) 2µ 2, 2(λ 1 + λ 2 + λ 3 + λ 4 ) µ 1 λ 1 µ 2 λ ( 2 1 λ3 µ 3 B = + λ ) 4µ 4 2(λ 1 + λ 2 + λ 3 + λ 4 ) µ 3 λ 3 µ 4 λ 4 (3.3) [Step 2] W q det(a λe) = λ : λ = (C + D) ± (C + D) 2 4(CD E 2 ) 2 (3.4) C = 2(A + B), D = 2[A(a 1 + ) 2 + Ba 2 1], E = 2[A(a 1 + ) + Ba 1 ] 2 T 3 T 2 [Step 3] : 1 = (C + D) 2 4(CD E 2 ) = (C D) 2 + 4E 2 > 2 (CD E 2 ) = 16a2 AB > (A >, B > ) < (C + D) (C >, D > ) T 4 3 1 2 [Step 4] W q 2 (positive semi-definite) W q 3.3. 3.3 Step

155 [Step 1] T W q a 1 (a 1 ) W q a 1 = W q a 1 = NT M (3.5) M = [λ 1 µ 1 (µ 2 λ 2 ) + λ 2 µ 2 (µ 1 λ 1 )](µ 3 λ 3 )(µ 4 λ 4 ), N = [λ 3 µ 3 (µ 4 λ 4 ) + λ 4 µ 4 (µ 3 λ 3 )](µ 1 λ 1 )(µ 2 λ 2 ) (3.2) T (T ) W q a 1 (a 1 ) max(ρ 3, ρ 4 )T T max(ρ 1, ρ 2 )T a 1 4 T a 1 1 N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )] T T T 1 a 1 1 = max(ρ 3, ρ 4 )T T T 1 a 1 3 = a 1 = NT M 2 N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )] T T T 2 a 1 2 = T max(ρ 1, ρ 2 )T T T 2 a 1 3 = a 1 = NT M N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )] N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )] 4: T a 1 a 1 a 1 a 1 ( 5)

156 5 1 a 1 max(ρ 3, ρ 4 )T a 1 1 = max(ρ 3, ρ 4 )T W q max(ρ 3, ρ 4 )T NT M 2a (3.6) T T M [1 max(ρ 3, ρ 4 )]N max(ρ 3, ρ 4 )M (N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )]) T (N[1 max(ρ 3, ρ 4 )] > M[1 max(ρ 1, ρ 2 )]) 2 a 1 T max(ρ 1, ρ 2 )T a 1 2 = T max(ρ 1, ρ 2 )T W q T max(ρ 1, ρ 2 )T NT M 2a T T (N[1 max(ρ 3, ρ 4 )] < M[1 max(ρ 1, ρ 2 )]) T N [1 max(ρ 1, ρ 2 )]M max(ρ 1, ρ 2 )N (N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )]) (3.7) (3.8) (3.9) 3 max(ρ 3, ρ 4 )T a 1 T max(ρ 1, ρ 2 )T a 1 3 = a 1 W q max(ρ 3, ρ 4 )T NT M T max(ρ 1, ρ 2 )T 2a T T T M [1 max(ρ 3, ρ 4 )]N max(ρ 3, ρ 4 )M N [1 max(ρ 1, ρ 2 )]M max(ρ 1, ρ 2 )N (N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )]) (N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )]) (3.1) (3.11)

157 5: a 1 a 1

158 [Step 2] a 1 a 1 W q T 1 T W q T (T ) 6 T 6 N[1 max(ρ 3, ρ 4 ) M[1 max(ρ 1, ρ 2 )] N[1 max(ρ 3, ρ 4 ) M[1 max(ρ 1, ρ 2 )] 6: W q T 1 a 1 max(ρ 3, ρ 4 )T a 1 1 = max(ρ 3, ρ 4 )T W q H = W q T = HI [max(ρ 3, ρ 4 )] 2 ( ) 2 T 2 + HJ [1 max(ρ 3, ρ 4 )] 2 > (3.12) 1 2(λ 1 + λ 2 + λ 3 + λ 4 ), I = λ 1µ 1 µ 1 λ 1 + λ 2µ 2 µ 2 λ 2, J = λ 3µ 3 µ 3 λ 3 + λ 4µ 4 µ 4 λ 4 W q T T 1 = (N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )]) T (N[1 max(ρ 3, ρ 4 )] > M[1 max(ρ 1, ρ 2 )]) (3.13)

159 2 a 1 T max(ρ 1, ρ 2 )T a 1 2 = T max(ρ 1, ρ 2 )T W q W q T = HJ [max(ρ 1, ρ 2 )] 2 ( ) 2 T 2 + HI [1 max(ρ 1, ρ 2 )] 2 > (3.14) W q T T (N[1 max(ρ 3, ρ 4 )] < M[1 max(ρ 1, ρ 2 )]) T 2 = (N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )]) (3.15) 3 max(ρ 3, ρ 4 )T a 1 T max(ρ 1, ρ 2 )T a 1 3 = a 1 W q [ W q T = HI N ( ) ] 2 + HJ M 2 2 2a 1 > (3.16) () 2 T W q T M T 3 = [1 max(ρ 3, ρ 4 )]N max(ρ 3, ρ 4 )M (N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )]) N T 3 = [1 max(ρ 1, ρ 2 )]M max(ρ 1, ρ 2 )N (N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )]) T 3 a 1 { max(ρ 3, ρ 4 )T (N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )]) a 1 3 = T 3 max(ρ 1, ρ 2 )T 3 (N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )]) (3.17) (3.18) [Step 3] W q a 1 T 1 N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )] 6 (3.18) T = T 1 T 3 a 1 1 a 1 3 a 1 = max(ρ 3, ρ 4 )T (3.13) (3.17) W q T (3.19) 2 N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )] a 1 = max(ρ 3, ρ 4 )T 2a T = (3.19)

16 T = T 2 T 3 a 1 1 a 1 3 a 1 = T max(ρ 1, ρ 2 )T (3.15) (3.17) W q T (3.2) a 1 = T max(ρ 1, ρ 2 )T = max(ρ 3, ρ 4 )T 2a (3.2) T = 1 2 ( (2.12) (3.2)) a 1 a 2 T (3.21) a 1 = b 2 = max(ρ 3, ρ 4 )T a 2 = b 1 = max(ρ 1, ρ 2 )T T = b 2 = max(ρ 3, ρ 4 )T kc b 1 = max(ρ 1, ρ 2 )T kc 2[a + (1 k)c] T = T b 1 b 2 4 1 N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )] b 2 = max(ρ 3, ρ 4 )T kc ( ) T M [1 max(ρ 3, ρ 4 )]N max(ρ 3, ρ 4 )M b 1 = T max(ρ 3, ρ 4 )T kc ( T M [1 max(ρ 3, ρ 4 )]N max(ρ 3, ρ 4 )M b 2 = NT M b 1 = MT N kc kc (T (T M [1 max(ρ 3, ρ 4 )]N max(ρ 3, ρ 4 )M ) M [1 max(ρ 3, ρ 4 )]N max(ρ 3, ρ 4 )M ) 2 N[1 max(ρ 3, ρ 4 )] M[1 max(ρ 1, ρ 2 )] b 2 = T max(ρ 1, ρ 2 )T kc ( ) T N [1 max(ρ 1, ρ 2 )]M max(ρ 1, ρ 2 )N b 1 = max(ρ 1, ρ 2 )T kc ( T N [1 max(ρ 1, ρ 2 )]M max(ρ 1, ρ 2 )N b 2 = NT M b 1 = MT N kc kc (T (T N [1 max(ρ 1, ρ 2 )]M max(ρ 1, ρ 2 )N ) N [1 max(ρ 1, ρ 2 )]M max(ρ 1, ρ 2 )N ) ) ) (3.21) (3.22) (3.23) (3.24)

161 3.4. a k, c [8] T = 2c 1 [max(ρ 1, ρ 2 ) + max(ρ 3, ρ 4 )]/.9 (3.25) (3.22) (3.25) 2c 1 [max(ρ 1, ρ 2 ) + max(ρ 3, ρ 4 )]/.9 = 2[a + (1 k)c] (3.26) a = ( + k 1)c (3.27) W q a k, c = 1 [max(ρ 1, ρ 2 ) + max(ρ 3, ρ 4 )] 1 [max(ρ 1, ρ 2 ) + max(ρ 3, ρ 4 )]/.9 (3.28) 4. 4.1. 8 9 1 (T = 6 ) (b 1 = 27 ) (b 2 = 21 ) (a = 3 ) (c = 3 ) 2 3 4 2 λ 1 λ 4 3 4 µ 1 µ 4 2 4 λ 1 =.83, λ 2 =.53, λ 3 =.55, λ 4 =.8 µ 1 =.227, µ 2 =.136, µ 3 =.19, µ 4 =.157 (4.1) ρ 1 =.366, ρ 2 =.39, ρ 3 =.289, ρ 4 =.51

162 4.2. 4.1 (3.22) b 2 = max(ρ 3, ρ 4 )T kc = 6.6 b 1 = max(ρ 1, ρ 2 )T kc = 9.44 2[a + (1 k)c] T = = 28.4 (4.2) k =.5 W q (4.2) W q { W q = 13.62 W q = 8.47 (4.3) W q 4.1 1 (4.1) (2.11) W q W q 38% 5., [1] 1)2 4 (ρ i ) 2)

163 ( ) ( ) a c k 3), [1] 3) OR Newell [5], [1] (2.1) 4 A 2424154 [1] J.N. Darroch, G.F. Newell, and R.W.J. Morris: Queues for a Vehicle-Actuated Traffic Light. Operations Research, 12-6 (1964), 882 895. [2] M. Li, Y. Ryu, and Y. Song: Some Results on Opimal Signal Switching Time at an Intersection Based on Arrival of Vehicles. Proceedings of Asian Conference of Management Science and Applications, (213), 122 126. [3] D.R. McNeil: A Solution to the Fixed-Cycle Traffic Light Problem for Compound

164 Poisson Arrivals. Journal of Applied Probability, 5-3 (1968), 624 635. [4] A.J. Miller: Settings for Fixed-Cycle Traffic Signals. Operations Research, 14-4 (1963), 373 386. [5] G.F. Newell: Queues for a Fixed-Cycle Traffic Light, The Annals of Mathematical Statistics, 31-3 (196), 589 597. [6] G.F. Newell: Approximation Methods for Queues with Application to the Fixed-Cycle Traffic Light. SIAM Review OR, 7-2 (1965), 223 24. [7] : (, 21). [8] : (, 1993). [9] : IR, 18 19 (27). [1], : (, 21), 191 194. [11], : -OR,, 14 (212), 43 47. 814-18 8-19-1 E-mail: lmz@fukuoka-u.ac.jp

165 ABSTRACT THE STUDY ON THE OPTIMAL SIGNAL CYCLE AND THE SWITCHING TIME AT AN INTERSECTION Mingzhe Li Yuchao Zhang Fukuoka University In a city, traffic congestion usually occurs because of population and heavy traffic, and it causes a variety of problems including the economic loss, environmental problems etc. One of the biggest reasons of traffic congestion on a general road is because of traffic signals at intersections. In this paper, we focus on the signal control at an intersection, and theoretically find out the optimal signal cycle and the optimal switching time at an objective intersection so as to reduce traffic congestion. Here, we assume that the vehicles arrive in the objective intersection continuously with a constant ratio, and each car arrives in the objective intersection within a signal cycle can pass though it during green light time of the cycle. In the case, we also explore some properties, such as the relationship between the simultaneous red light and the yellow light. Finally, we verify our model though a case study by considering a real intersection in fukuoka.