- Email: [email protected]
Dynamical transitions in peak elevator traffic
Available online at www.sciencedirect.com
Physica A 333 (2004) 441 – 452
www.elsevier.com/locate/physa
Dynamical transitions in peak elevator tra c Takashi Nagatani∗ Division of Thermal Science, Department of Mechanical Engineering, Shizuoka University, Hamamatsu 432-8561, Japan Received 4 September 2003
Abstract We study the dynamical phase transitions in the elevator tra c during the morning peak period. We apply the coupled nonlinear map model to the dynamics of M elevators. We investigate the e+ect of elevator’s number M on the dynamical transition. It is shown that the dynamical transitions occur at two stages. The .rst transition occurs when an elevator carries the full load of passengers. Then, the second transition occurs if all the elevators carry the full load of passengers. The number of riding passengers 0uctuates with varying trips. This is due to the deterministic chaos. The motions of elevators depend on the loading parameter, the maximum capacity of an elevator, and the number of elevators. The dependence of the .rst and second transition points on the elevator’s number M is shown. c 2003 Elsevier B.V. All rights reserved. PACS: 89.40.+k; 05.45.−a; 82.40.Bj Keywords: Elevator; Transportation; Chaos; Dynamical transition; Fluctuation
1. Introduction Recently, tra c jams have attracted much attention in the .elds of physics [1–5]. The jams are a typical signature of the complex behavior of tra c 0ow. The interesting dynamical phase transitions have been found in the transportation system. The jamming transitions have been studied for the tra c 0ow, pedestrian 0ow, and bus-route problem [6–17]. The elevator tra c also exhibits severe congestion problems during the morning peak period. The maximum rate of serving passengers is achieved if each elevator carries a full load of passengers throughout its trip. When the rate is larger than the .rst critical ∗
Corresponding author. Fax: +81-53-4781048. E-mail address: [email protected] (T. Nagatani).
c 2003 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2003.10.001
442
T. Nagatani / Physica A 333 (2004) 441 – 452
value, the queuing occurs at the lobby 0oor. The queuing increases accordingly as the incoming rate of passengers at the lobby 0oor increases. Furthermore, the queuing diverges if the incoming rate is superior to the second critical value. The queuing is similar to the jam in the tra c 0ow. The dynamical transition to the queuing is interesting from the point of view of statistical mechanics and nonlinear dynamics. In the elevator tra c, there is the pioneer work by Poschel and Gallas [18]. They have found the dynamical jamming transition by varying the in0ow rate of down passengers into elevators by the use of the stochastic model. The application of the model is limited only to the evening peak tra c. In designing a building, the usual criterion for deciding the capacity of elevators is that one should be able to transport everyone in the building from the lobby 0oor to his destination within some period of time for the rush hour trips [19,20]. Another criterion used in elevator design is that a passenger’s waiting time should not exceed some speci.ed value. In the previous paper [21], we have presented a simpli.ed dynamical model to describe the motions of elevators in the morning peak tra c. We have studied the dynamical behavior of elevators induced by the interaction between elevators when the elevators shuttle between the 0oor lobby and upward 0oors, repeatedly. We have shown that the elevators exhibit the complex behavior induced by the deterministic chaos. In this paper, we investigate the dynamical phase transitions in the elevator tra c during morning peak period by using the coupled nonlinear map model. We show that the .rst and second dynamical transitions occur in the peak elevator tra c. Below the .rst transition point, all waiting passengers are able to board the coming elevator. However, above the .rst transition point, one or some elevators carry the full load of passengers and the other elevators are not full. Then, all waiting passengers cannot board the just arrived elevator. A part of passengers waiting at the lobby 0oor must wait for the next elevator. Above the second transition point, all the elevators carry full load of passengers and queuing at the lobby 0oor becomes longer and longer. In the mean time, the queuing diverges. We derive the dependence of the .rst and second transition points on elevator’s number M , loading parameter, and elevator’s capacity Fmax . 2. Model We apply the coupled nonlinear map model [21] to the system of M elevators serving N 0oors of a building during the morning peak period. The lobby 0oor is the only way to enter the building. Passengers enter the building from the lobby 0oor, board the elevators, and get o+ the elevators at some upward 0oors. The 0ow of passengers is in one way during the morning peak period. We describe the coupled nonlinear map model for a late convenience. We de.ne the number of passengers boarding elevator i at trip m by Bi (m). The parameter is the time it takes one passenger to board the elevator, so Bi (m) is the amount of time needed to board all the passengers at the lobby 0oor. The highest
T. Nagatani / Physica A 333 (2004) 441 – 452
443
0oor at which elevator i stops at trip m is de.ned by ni; max (m). So the moving time of elevator i is 2L=Vi (ni; max (m))=N where L is the height of the building and Vi is the mean speed of elevator i. The 0oor number at which elevator i stops at trip m is de.ned by ns; i (m), so the stopping time to leave the passengers is ns; i (m)Hts where Hts is the time stopping at a 0oor. The tour time equals the sum of these periods. Then, the arrival time ti (m + 1) of elevator i at the lobby 0oor at trip m + 1 is given by 2L ni; max (m) + ns; i (m)Hts for i = 1; 2; : : : ; M : ti (m + 1) = ti (m) + Bi (m) + Vi N (1) De.ne Wi (m) as the number of passengers waiting at the lobby 0oor just before elevator i arrives at the lobby 0oor at trip m. It is expressed by Wi (m) = Wi (m ) − Bi (m ) + (ti (m) − ti (m )) ;
(2)
where elevator i is that which arrived at the lobby 0oor just before elevator i in trip m; ti (m ) is the arrival time of elevator i at the lobby 0oor, and Bi (m ) is the number of passengers boarding elevator i . New passengers arrive at the lobby 0oor at rate . So (ti (m) − ti (m )) is the number of passengers who have arrived since the previous elevator i left the lobby 0oor. We de.ne the maximum capacity of elevator i as Fi; max . The passenger number Bi (m) boarding elevator i at trip m is given by Bi (m) = minFi; max ; Wi (m) :
(3)
When the number of passengers waiting at the lobby 0oor is higher than the maximum capacity, the number of passengers boarding the elevator is limited to the maximum capacity. The remaining passengers wait for the next elevator. We assume that the number of stops is proportional to the number of boarding passengers: ns; i (m) = Bi (m). We take ni; max (m) = ni; max + ni; max . By dividing actual time by the characteristic time 2Lni; max =(V0 N ), one obtains the following equation for the dimensionless arrival time of elevator i at the lobby 0oor: Ti (m + 1) = Ti (m) + Bi (m) + V0 =Vi + i (m) ;
(4)
Wi (m) = Wi (m ) − Bi (m ) + (Ti (m) − Ti (m ))
(5)
Bi (m) = minFi; max ; Wi (m) ;
(6)
with and where Ti (m) ≡
ti (m)V0 N ; 2Lni; max
and i (m) ≡
≡
ni; max (m) ni; max
( + Hts )V0 N ; 2Lni; max
≡
2Lni; max V0 N
444
T. Nagatani / Physica A 333 (2004) 441 – 452
is a dimensionless 0uctuation of highest 0oor. The mean and correlation are given by i (m)j (h) = Va2 ij mh : (7) We de.ne the root-mean-square 2i by Va . Thus, the dynamics of the elevators is described in terms of the simpli.ed map (4) with a white noise. The map is iterated simultaneously for M elevators. The dynamical property of the map is controlled by three parameters: loading parameter , the capacity Fi; max , and elevator number M . In the special case of a single elevator (M = 1), the preceding elevator i before the elevator i = 1 is itself at trip m − 1. Then, the dynamics of a single elevator system is given by i (m) = 0
and
2L n1; max (m) + ns; 1 (m)Hts ; V1 N
(1 )
W1 (m) = W1 (m − 1) − B1 (m − 1) + (t1 (m) − t1 (m − 1)) ;
(2 )
B1 (m) = minF1; max ; W1 (m) :
(3 )
t1 (m + 1) = t1 (m) + B1 (m) +
Thus, the dynamics of the single elevator tra c reduces to the one-dimensional map. The map has a stable .xed point HTf = 1=(1 − ) for ¡ 1 or F1; max + 1 for ¿ 1. The elevator moves at the constant value of the recurrent time. When is higher than one, the diverging queuing appears on the lobby 0oor. However, multiple elevator’s tra c is de.nitely di+erent from the single elevator tra c. The order of elevators changes with trip m for M elevators and the time headway between elevators changes from trip to trip because an elevator passes other elevators or is outstripped by other elevators. In the previous paper, we have shown that the dynamical system exhibits a deterministic chaos for the transportation system of two elevators. However, it was not studied how the number M of elevators a+ect the dynamical motion of elevators and the dynamical transitions. We study the e+ect of elevator’s number M on the dynamical transition in the following sections.
3. Simulation result We investigate the elevator behavior by the use of iterates of map (4)–(6). We study the 0uctuation of the number of riding passengers by varying the loading parameter for elevator’s number M = 2–4. Fig. 1(a) shows the plot of the riding passenger’s number B(m ) against loading parameter from su ciently large arrival m = 900–1000 for two elevators M = 2 where = 0:1; Fmax = 10, and Va = 0:0. Fig. 1(b) shows the plot of the remaining passenger’s number W (m ) − B(m ) against loading parameter from su ciently large arrival m = 900–1000 where the values of parameters are the same as Fig. 1(a). The riding passenger’s number 0uctuates and behaves irregularly irrespective of the deterministic case Va = 0:0.
T. Nagatani / Physica A 333 (2004) 441 – 452 40
1
Number of remaining passengers
Riding passenger’s number
20 2
10
0 0
(a)
445
6 Loading parameter
20 1 2 0 0
12
(b)
6 Loading parameter
12
Fig. 1. (a) Plot of the riding passenger’s number B(m ) against loading parameter from su ciently large arrival m = 900–1000 for M = 2, where = 0:1, Fmax = 10, and Va = 0:0. (b) Plot of the remaining passenger’s number W (m ) − B(m ) against loading parameter where the values of parameters are the same as Fig. 1(a).
When all the waiting passengers W (m ) can board the just arrived elevator, the remaining passenger’s number W (m ) − B(m ) becomes zero. Otherwise, one elevator or some elevators carry the full load of passengers and all the waiting passengers at the lobby 0oor cannot board the just arrived elevator. Some passengers remain at the lobby 0oor and the remaining passengers at the lobby 0oor wait for the next elevator. Then, W (m ) − B(m ) becomes a positive value. For low values of loading parameter , the number of riding passengers exhibits the localized distributions around three values. With increasing loading parameter , the localized distributions extend and become the two extended distributions. For 6:3 ¡ ¡ 7:7, the riding passenger’s number exhibits a single extended distribution. When loading parameter is higher than 7.7, the single extended distribution breaks into two distributions again. At 1; c = 7:3, the riding passenger’s number saturates at Fmax . An elevator carries a full load of passengers but the other elevators are not full. When loading parameter is higher than 10.0, number B(m) becomes 10.0. At 2; c = 10:0, all the elevators carry the full load. Therefore, the dynamical transitions occurs at two stages 1; c = 7:3 and 2; c = 10:0. We call these as the .rst and second transitions which are indicated by points 1 and 2. The .rst transition at 1; c = 7:3 is induced by a full load of a single elevator. The second transition at c = 10:0 is induced by a full load of all the elevators. The .rst transition is due to the chaotic motion of elevators passing each other freely. We note that the .rst transition does not occur in the single elevator tra c. Figs. 2 and 3 show, respectively, the distributions of riding passenger’s and remaining passenger’s numbers for M = 3 and 4 where the values of parameters except for M are the same as Fig. 1. With increasing M , the distribution of the riding passenger’s number extend. The elevators exhibit the extended distribution for a wide range of loading parameter’s value. The .rst and second transition points increase accordingly as the elevator’s number increases. The region between the .rst and second transitions also increases accordingly as the elevator’s number increases. In the region, partial elevators carry a full load of passengers. Therefore, the passenger’s number not boarding the just arrived elevator increases accordingly as the number of elevators increases.
446
T. Nagatani / Physica A 333 (2004) 441 – 452 40
1
Number of remaining passengers
Riding passenger’s number
20 2
10
0
1
2 0
0
(a)
20
8
0
16
(b)
Loading parameter
8
16
Loading parameter
Fig. 2. (a) Plot of the riding passenger’s number B(m ) against loading parameter from su ciently large arrival m = 900–1000 for M = 3, where = 0:1, Fmax = 10, and Va = 0:0. (b) Plot of the remaining passenger’s number W (m ) − B(m ) against loading parameter where the values of parameters are the same as Fig. 2(a).
40 1
2
10
0 0
(a)
10 Loading parameter
Number of remaining passengers
Riding passenger’s number
20
20 1 2 0 0
20
(b)
10 Loading parameter
20
Fig. 3. (a) Plot of the riding passenger’s number B(m ) against loading parameter from su ciently large arrival m = 900–1000 for M = 4, where = 0:1, Fmax = 10, and Va = 0:0. (b) Plot of the remaining passenger’s number W (m ) − B(m ) against loading parameter where the values of parameters are the same as Fig. 3(a).
We calculate the number of riding passengers with varying trips for the typical case of four elevators. Figs. 4(a) – (d) show the plots of the riding passenger’s number B(m) against arrival order m from m = 1000 to m = 1050 for the values of (a) = 7:0, (b) = 12:0, (c) = 16:0, and (d) = 20:0 and no noises Va = 0:0 where M =4, = 0:1, and Fmax = 10. The dotted line indicates the maximum capacity Fmax . The number B(m) of riding passengers cannot be superior to Fmax . For case (a) of a low value of loading parameter, the number of riding passengers changes alternately from zero to about seven. The riding passenger’s number varies irregularly with arrivals. When the elevators arrive simultaneously at the lobby 0oor, the time headway becomes zero. As a result, the riding passengers become zero because the waiting passengers are proportional to the time headway. With increasing loading parameter , the riding passenger’s number varies highly and irregularly from arrival to arrival. The probability that elevators arrive
T. Nagatani / Physica A 333 (2004) 441 – 452
20
Riding passenger’s number
Riding passenger’s number
20
10
10
0
0
1000
(a)
1050 Arrival order
(b)
1000
1050 Arrival order
20
Riding passenger’s number
Riding passenger’s number
20
10
10
0
0 1000
(c)
447
1050
Arrival order
1000
(d)
1050
Arrival order
Fig. 4. Plots of the riding passenger’s number B(m ) against arrival order m from m = 1000–1050 for M = 4 at values of (a) = 7:0, (b) = 12:0, (c) = 16:0, and (d) = 20:0 where = 0:1, Fmax = 10, and Va = 0:0.
simultaneously at the lobby 0oor becomes less and less with increasing loading parameter. When the loading parameter is higher than the .rst transition point, an elevator carries the full load of passengers (see Fig. 4b). Furthermore, when loading parameter increases, the elevators become full load frequently (see Fig. 4c). When loading parameter is superior to the second transition point, four elevators become always full load, the queuing occurs at the lobby 0oor, and the queuing diverges (see Fig. 4d). We study the mean Bav and root-mean-square Brms of the number B(m) of riding passengers, and the mean Wav − Bav of number W (m) − B(m) of remaining passengers. Fig. 5(a) shows the plots of the means Bav , Wav − Bav , and root-mean-square Brms against loading parameter for M = 2 where = 0:1, Fmax = 10, and Va = 0:0. Fig. 5(b) shows the plots of the means Bav , Wav − Bav , and root-mean-square Brms against loading parameter for M = 4 where = 0:1, Fmax = 10, and Va = 0:0. The mean value Bav increases accordingly as loading parameter , and saturates at the second transition point. Root-mean-square Brms increases with loading parameter until (a) = 8:0 and (b) = 15:0. Then, it decreases and becomes zero at the second transition point. The mean Wav − Bav is zero until the .rst transition point. When the
448
T. Nagatani / Physica A 333 (2004) 441 – 452 20
20
Average values
Average values
Wav-Bav Bav 10
Wav-Bav
10
Bav Brms
Brms
0
0 0
(a)
6 Loading parameter
0
12
(b)
10 Loading parameter
20
Fig. 5. (a) Plots of the mean values Bav and Wav − Bav of the riding passenger’s number and remaining passenger’s number against loading parameter for M = 2, where = 0:1, Fmax = 10, and Va = 0:0. Plot of the root-mean-square (rms) Brms of the riding passenger’s number for M = 2. (b) Plots of the mean values Bav and Wav − Bav of the riding passenger’s number and remaining passenger’s number against loading parameter for M = 4 where = 0:1, Fmax = 10, and Va = 0:0. Plot of the root-mean-square (rms) Brms of the riding passenger’s number for M = 4.
30 2nd transition point
1st and 2nd transition points
1st and 2nd transition points
15
10
5 1st transition point
0
15
1st transition point 0
0
(a)
2nd transition point
10 Capacity
0
20
(b)
10 Capacity
20
Fig. 6. Dependence of the transition points on the maximum capacity. (a) Plots of the .rst and second transition points 1; c , 2; c against maximum capacity Fmax for M = 2 where = 0:1 and Va = 0:0. The circles and solid line indicate, respectively, the .rst and second transition points. (b) Plots of the .rst and second transition points 1; c , 2; c against maximum capacity Fmax for M = 4 where = 0:1 and Va = 0:0.
loading parameter is larger than the .rst threshold, the mean Wav − Bav increases with loading parameter and then diverges at the second transition point. We study the e+ect of the maximum capacity on the .rst and second transition points. Fig. 6 shows plots of the .rst and second transition points 1; c , 2; c against maximum capacity Fmax for (a) M = 2 and (b) M = 4 where = 0:1 and Va = 0:0. The circles and solid line indicate, respectively, the .rst and second transition points. The .rst and second transition points increase accordingly as the maximum capacity increases. The di+erence between the .rst and second transition points also increases with the maximum capacity. The di+erence becomes large with increasing elevator number M . We study the dependence of the .rst transition point 1; c on the elevator’s
T. Nagatani / Physica A 333 (2004) 441 – 452
449
15
First transition point
M=4 10
M=3 M=2
5
M=1
0 0
10 Capacity
20
Fig. 7. The dependence of the .rst transition point 1; c on the elevator’s number M . Plot of the .rst transition point 1; c against maximum capacity Fmax for M = 1–4 where = 0:1 and Va = 0:0.
30 Second transition point
M=4 M=3 M=2
15
M=1
0 0
10 Capacity
20
Fig. 8. The dependence of the second transition point 2; c on the elevator’s number M . Plot of the second transition point 2; c against maximum capacity Fmax for M = 1–4 where = 0:1 and Va = 0:0.
number M . Fig. 7 shows plot of the .rst transition point 1; c against maximum capacity Fmax for M =1–4. With increasing elevator number M , the .rst transition point becomes higher but the increasing rate becomes lower with M . For M = 1, the .rst and second transition points are consistent each other. For comparison, we study the dependence of the second transition point 2; c on the elevator’s number M . Fig. 8 shows plot of the second transition point 2; c against maximum capacity Fmax for M = 1–4. With increasing elevator number M , the second transition point becomes higher. 4. Analytical result We present the analytical result for such a case that all the elevators take the same speed Vi = V0 and the same capacity Fi; max = Fmax . We study the dependence of the second transition point on the capacity and elevator’s number. When the loading
450
T. Nagatani / Physica A 333 (2004) 441 – 452
parameter is higher than the threshold 2; c , all elevators carry a full load of passengers. So the following relationship holds: Fmax = 2; c HT2; c ;
(8)
where HT2; c is the time headway at the threshold. If all elevators move with the same time headway, the following equation is obtained from Eq. (4) M HT2; c = 2; c HT2; c + 1 : One obtains the threshold from Eqs. (8) and (9) MFmax 2; c = : Fmax + 1
(9)
(10)
Fig. 8 shows the plot of threshold 2; c against capacity Fmax for M = 1–4 and = 0:1. The solid curve represents the theoretical result (10). The simulation result is consistent with the theoretical result. We derive the mean value of riding passengers below the second transition point. The mean tour time is given from Eq. (4) V0 : (11) HTi = Bi + Vi The passengers arrived at the origin over su ciently large time TL is given by TL . Below the second threshold, the number TL of passengers is consistent with those carried by all the elevators. The following relationship holds: M i=1
TL
Bi = TL : HTi
One obtains an approximation of riding passengers for ¡ 2; c ; M − B1 = for ¿ 2; c : Fmax
(12)
(13)
Fig. 9 shows the plots of mean number of riding passengers against loading parameters for elevator number M = 1–4 where = 0:1, Fmax = 10, and Va = 0:0. The solid curves represent the approximation (13). The simulation result Bav is consistent with the theoretical result B1 . 5. Linear stability analysis We consider the stability of the elevator tra c when all the elevators move with the same time headway. We apply the linear stability method to the elevator tra c model. For simplicity, we restrict the limiting case of Fmax → ∞. When all the elevators move with the same time headway, the following equation is obtained from Eqs. (4) and (5) M HT0; f = HT0; f + 1 ;
(14)
T. Nagatani / Physica A 333 (2004) 441 – 452
451
20
M=1
3
2
4
〈 B1
〈
10
0 0
10 Loading parameter
20
Fig. 9. Analytical result of the mean B1 of the number of riding passengers against loading parameter where = 0:1 and Va = 0:0.
where HT0; f is the time headway between an elevator and its predecessor. The .xed point is given by HT0; f = 1=(M − ) :
(15)
We study whether or not the .xed point is stable. Let yi (m) be a small deviation from the arrival time of elevator i at the .xed point: Ti (m) = T0; i (m) + yi (m) where T0; i+1 (m) − T0; i (m) = HT0; f . Then, the linearized equations are obtained from Eqs. (4) and (5) y1 (m + 1) = y1 (m) + (y1 (m) − yM (m − 1)) ;
(16)
yi (m + 1) = yi (m) + (yi (m) − yi−1 (m))
(17)
for i = 2; 3; : : : ; M :
By taking yi (m) = Yi e#m , the following equations of # are derived (e# − 1 − )M − (−)M e−# = 0 :
(18)
If # is a positive value, the .xed point is unstable. When # is a negative value, the .xed point is stable. In the special case of M = 1, one obtains # = 0 and # = ln(). Therefore, the .xed point is stable for ¡ 1 and unstable for ¿ 1. For the case of M =2, one obtains # = 0 and √ (1 + 2) ± 1 + 4 : # = ln 2 The maximum value is positive. Therefore, the .xed point is unstable for any value of . One .nds that the maximum value of # is positive for M ¿ 2. We .nd that the elevators moving with the same time headway are always unstable for any value of . Thus, the elevator tra c with multiple elevators is always unstable. Even if all the elevators move with the same time headway initially, the time headway between elevators increases or decreases by the instability. In the mean time, an elevator passes the other elevators or is outstripped by the other elevators. The instability induces the chaotic motion of the elevators.
452
T. Nagatani / Physica A 333 (2004) 441 – 452
6. Summary We have studied the dynamical phase transitions in M elevators tra c during the morning peak period by using the coupled nonlinear map model. We have shown that the number of passengers carried by an elevator exhibits the complex behavior with varying trips and the motion of elevators exhibits a deterministic chaos even if there are no noises. We have found that the dynamical transitions occur at the two stages: the .rst transition when an elevator carries a full load of passengers and the second transition if all the elevators carry a full load of passengers. Above the .rst transition point, a part of the passengers waiting at the lobby 0oor cannot board the just arrived elevator. Above the second transition point, the queuing of the waiting passengers is more and more with time. We have investigated the e+ect of elevator number M on the dynamical transition. We have clari.ed the dependence of the threshold (transition point) on the loading parameter, the elevator capacity, and the elevator number. It is important to decide the elevator number M and estimate the queuing of waiting passengers at the lobby 0oor in elevator design. The result obtained here will be useful for elevator designing in the building. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
T. Nagatani, Rep. Prog. Phys. 65 (2002) 1331. D. Helbing, Rev. Mod. Phys. 73 (2001) 1067. D. Chowdhury, L. Santen, A. Schadscheider, Phys. Rep. 329 (2000) 199. B.S. Kerner, Networks Spatial Econ. 1 (2001) 35. D. Helbing, H.J. Herrmann, M. Schreckenberg, D.E. Wolf (Eds.), Tra c and Granular Flow ‘99, Springer, Heidelberg, 2000. K. Nagel, M. Schreckenberg, J. Phys. I France 2 (1992) 2221. E. Ben-Naim, P.L. Krapivsky, S. Redner, Phys. Rev. E 50 (1994) 822. E. Tomer, L. Safonov, S. Havlin, Phys. Rev. Lett. 84 (2000) 382. M. Treiber, A. Hennecke, D. Helbing, Phys. Rev. E 62 (2000) 1805. H.K. Lee, H.-W, Lee, D. Kim, Phys. Rev. E 64 (2001) 056 126. O.J. O’loan, M.R. Evans, M.E. Cates, Phys. Rev. E 58 (1998) 1404. D. Chowdhury, R.C. Desai, Eur. Phys. J. B 15 (2000) 375. T. Nagatani, Phys. Rev. E 63 (2001) 036 116. H.J.C. Huijberts, Physica A 308 (2002) 489. S.A. Hill, Cond-Mat/0206008 (2002). T. Nagatani, Physica A 297 (2001) 260. T. Nagatani, Phys. Rev. E 66 (2002) 046 103. T. Poschel, J. Gallas, Phys. Rev. E 50 (1994) 2654. G.F. Newell, Transpn. Res. B 32 (1998) 583. T. Nagatani, Physica A 310 (2002) 67. T. Nagatani, Physica A 326 (2003) 556.
Physica A 333 (2004) 441 – 452
www.elsevier.com/locate/physa
Dynamical transitions in peak elevator tra c Takashi Nagatani∗ Division of Thermal Science, Department of Mechanical Engineering, Shizuoka University, Hamamatsu 432-8561, Japan Received 4 September 2003
Abstract We study the dynamical phase transitions in the elevator tra c during the morning peak period. We apply the coupled nonlinear map model to the dynamics of M elevators. We investigate the e+ect of elevator’s number M on the dynamical transition. It is shown that the dynamical transitions occur at two stages. The .rst transition occurs when an elevator carries the full load of passengers. Then, the second transition occurs if all the elevators carry the full load of passengers. The number of riding passengers 0uctuates with varying trips. This is due to the deterministic chaos. The motions of elevators depend on the loading parameter, the maximum capacity of an elevator, and the number of elevators. The dependence of the .rst and second transition points on the elevator’s number M is shown. c 2003 Elsevier B.V. All rights reserved. PACS: 89.40.+k; 05.45.−a; 82.40.Bj Keywords: Elevator; Transportation; Chaos; Dynamical transition; Fluctuation
1. Introduction Recently, tra c jams have attracted much attention in the .elds of physics [1–5]. The jams are a typical signature of the complex behavior of tra c 0ow. The interesting dynamical phase transitions have been found in the transportation system. The jamming transitions have been studied for the tra c 0ow, pedestrian 0ow, and bus-route problem [6–17]. The elevator tra c also exhibits severe congestion problems during the morning peak period. The maximum rate of serving passengers is achieved if each elevator carries a full load of passengers throughout its trip. When the rate is larger than the .rst critical ∗
Corresponding author. Fax: +81-53-4781048. E-mail address: [email protected] (T. Nagatani).
c 2003 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2003.10.001
442
T. Nagatani / Physica A 333 (2004) 441 – 452
value, the queuing occurs at the lobby 0oor. The queuing increases accordingly as the incoming rate of passengers at the lobby 0oor increases. Furthermore, the queuing diverges if the incoming rate is superior to the second critical value. The queuing is similar to the jam in the tra c 0ow. The dynamical transition to the queuing is interesting from the point of view of statistical mechanics and nonlinear dynamics. In the elevator tra c, there is the pioneer work by Poschel and Gallas [18]. They have found the dynamical jamming transition by varying the in0ow rate of down passengers into elevators by the use of the stochastic model. The application of the model is limited only to the evening peak tra c. In designing a building, the usual criterion for deciding the capacity of elevators is that one should be able to transport everyone in the building from the lobby 0oor to his destination within some period of time for the rush hour trips [19,20]. Another criterion used in elevator design is that a passenger’s waiting time should not exceed some speci.ed value. In the previous paper [21], we have presented a simpli.ed dynamical model to describe the motions of elevators in the morning peak tra c. We have studied the dynamical behavior of elevators induced by the interaction between elevators when the elevators shuttle between the 0oor lobby and upward 0oors, repeatedly. We have shown that the elevators exhibit the complex behavior induced by the deterministic chaos. In this paper, we investigate the dynamical phase transitions in the elevator tra c during morning peak period by using the coupled nonlinear map model. We show that the .rst and second dynamical transitions occur in the peak elevator tra c. Below the .rst transition point, all waiting passengers are able to board the coming elevator. However, above the .rst transition point, one or some elevators carry the full load of passengers and the other elevators are not full. Then, all waiting passengers cannot board the just arrived elevator. A part of passengers waiting at the lobby 0oor must wait for the next elevator. Above the second transition point, all the elevators carry full load of passengers and queuing at the lobby 0oor becomes longer and longer. In the mean time, the queuing diverges. We derive the dependence of the .rst and second transition points on elevator’s number M , loading parameter, and elevator’s capacity Fmax . 2. Model We apply the coupled nonlinear map model [21] to the system of M elevators serving N 0oors of a building during the morning peak period. The lobby 0oor is the only way to enter the building. Passengers enter the building from the lobby 0oor, board the elevators, and get o+ the elevators at some upward 0oors. The 0ow of passengers is in one way during the morning peak period. We describe the coupled nonlinear map model for a late convenience. We de.ne the number of passengers boarding elevator i at trip m by Bi (m). The parameter is the time it takes one passenger to board the elevator, so Bi (m) is the amount of time needed to board all the passengers at the lobby 0oor. The highest
T. Nagatani / Physica A 333 (2004) 441 – 452
443
0oor at which elevator i stops at trip m is de.ned by ni; max (m). So the moving time of elevator i is 2L=Vi (ni; max (m))=N where L is the height of the building and Vi is the mean speed of elevator i. The 0oor number at which elevator i stops at trip m is de.ned by ns; i (m), so the stopping time to leave the passengers is ns; i (m)Hts where Hts is the time stopping at a 0oor. The tour time equals the sum of these periods. Then, the arrival time ti (m + 1) of elevator i at the lobby 0oor at trip m + 1 is given by 2L ni; max (m) + ns; i (m)Hts for i = 1; 2; : : : ; M : ti (m + 1) = ti (m) + Bi (m) + Vi N (1) De.ne Wi (m) as the number of passengers waiting at the lobby 0oor just before elevator i arrives at the lobby 0oor at trip m. It is expressed by Wi (m) = Wi (m ) − Bi (m ) + (ti (m) − ti (m )) ;
(2)
where elevator i is that which arrived at the lobby 0oor just before elevator i in trip m; ti (m ) is the arrival time of elevator i at the lobby 0oor, and Bi (m ) is the number of passengers boarding elevator i . New passengers arrive at the lobby 0oor at rate . So (ti (m) − ti (m )) is the number of passengers who have arrived since the previous elevator i left the lobby 0oor. We de.ne the maximum capacity of elevator i as Fi; max . The passenger number Bi (m) boarding elevator i at trip m is given by Bi (m) = minFi; max ; Wi (m) :
(3)
When the number of passengers waiting at the lobby 0oor is higher than the maximum capacity, the number of passengers boarding the elevator is limited to the maximum capacity. The remaining passengers wait for the next elevator. We assume that the number of stops is proportional to the number of boarding passengers: ns; i (m) = Bi (m). We take ni; max (m) = ni; max + ni; max . By dividing actual time by the characteristic time 2Lni; max =(V0 N ), one obtains the following equation for the dimensionless arrival time of elevator i at the lobby 0oor: Ti (m + 1) = Ti (m) + Bi (m) + V0 =Vi + i (m) ;
(4)
Wi (m) = Wi (m ) − Bi (m ) + (Ti (m) − Ti (m ))
(5)
Bi (m) = minFi; max ; Wi (m) ;
(6)
with and where Ti (m) ≡
ti (m)V0 N ; 2Lni; max
and i (m) ≡
≡
ni; max (m) ni; max
( + Hts )V0 N ; 2Lni; max
≡
2Lni; max V0 N
444
T. Nagatani / Physica A 333 (2004) 441 – 452
is a dimensionless 0uctuation of highest 0oor. The mean and correlation are given by i (m)j (h) = Va2 ij mh : (7) We de.ne the root-mean-square 2i by Va . Thus, the dynamics of the elevators is described in terms of the simpli.ed map (4) with a white noise. The map is iterated simultaneously for M elevators. The dynamical property of the map is controlled by three parameters: loading parameter , the capacity Fi; max , and elevator number M . In the special case of a single elevator (M = 1), the preceding elevator i before the elevator i = 1 is itself at trip m − 1. Then, the dynamics of a single elevator system is given by i (m) = 0
and
2L n1; max (m) + ns; 1 (m)Hts ; V1 N
(1 )
W1 (m) = W1 (m − 1) − B1 (m − 1) + (t1 (m) − t1 (m − 1)) ;
(2 )
B1 (m) = minF1; max ; W1 (m) :
(3 )
t1 (m + 1) = t1 (m) + B1 (m) +
Thus, the dynamics of the single elevator tra c reduces to the one-dimensional map. The map has a stable .xed point HTf = 1=(1 − ) for ¡ 1 or F1; max + 1 for ¿ 1. The elevator moves at the constant value of the recurrent time. When is higher than one, the diverging queuing appears on the lobby 0oor. However, multiple elevator’s tra c is de.nitely di+erent from the single elevator tra c. The order of elevators changes with trip m for M elevators and the time headway between elevators changes from trip to trip because an elevator passes other elevators or is outstripped by other elevators. In the previous paper, we have shown that the dynamical system exhibits a deterministic chaos for the transportation system of two elevators. However, it was not studied how the number M of elevators a+ect the dynamical motion of elevators and the dynamical transitions. We study the e+ect of elevator’s number M on the dynamical transition in the following sections.
3. Simulation result We investigate the elevator behavior by the use of iterates of map (4)–(6). We study the 0uctuation of the number of riding passengers by varying the loading parameter for elevator’s number M = 2–4. Fig. 1(a) shows the plot of the riding passenger’s number B(m ) against loading parameter from su ciently large arrival m = 900–1000 for two elevators M = 2 where = 0:1; Fmax = 10, and Va = 0:0. Fig. 1(b) shows the plot of the remaining passenger’s number W (m ) − B(m ) against loading parameter from su ciently large arrival m = 900–1000 where the values of parameters are the same as Fig. 1(a). The riding passenger’s number 0uctuates and behaves irregularly irrespective of the deterministic case Va = 0:0.
T. Nagatani / Physica A 333 (2004) 441 – 452 40
1
Number of remaining passengers
Riding passenger’s number
20 2
10
0 0
(a)
445
6 Loading parameter
20 1 2 0 0
12
(b)
6 Loading parameter
12
Fig. 1. (a) Plot of the riding passenger’s number B(m ) against loading parameter from su ciently large arrival m = 900–1000 for M = 2, where = 0:1, Fmax = 10, and Va = 0:0. (b) Plot of the remaining passenger’s number W (m ) − B(m ) against loading parameter where the values of parameters are the same as Fig. 1(a).
When all the waiting passengers W (m ) can board the just arrived elevator, the remaining passenger’s number W (m ) − B(m ) becomes zero. Otherwise, one elevator or some elevators carry the full load of passengers and all the waiting passengers at the lobby 0oor cannot board the just arrived elevator. Some passengers remain at the lobby 0oor and the remaining passengers at the lobby 0oor wait for the next elevator. Then, W (m ) − B(m ) becomes a positive value. For low values of loading parameter , the number of riding passengers exhibits the localized distributions around three values. With increasing loading parameter , the localized distributions extend and become the two extended distributions. For 6:3 ¡ ¡ 7:7, the riding passenger’s number exhibits a single extended distribution. When loading parameter is higher than 7.7, the single extended distribution breaks into two distributions again. At 1; c = 7:3, the riding passenger’s number saturates at Fmax . An elevator carries a full load of passengers but the other elevators are not full. When loading parameter is higher than 10.0, number B(m) becomes 10.0. At 2; c = 10:0, all the elevators carry the full load. Therefore, the dynamical transitions occurs at two stages 1; c = 7:3 and 2; c = 10:0. We call these as the .rst and second transitions which are indicated by points 1 and 2. The .rst transition at 1; c = 7:3 is induced by a full load of a single elevator. The second transition at c = 10:0 is induced by a full load of all the elevators. The .rst transition is due to the chaotic motion of elevators passing each other freely. We note that the .rst transition does not occur in the single elevator tra c. Figs. 2 and 3 show, respectively, the distributions of riding passenger’s and remaining passenger’s numbers for M = 3 and 4 where the values of parameters except for M are the same as Fig. 1. With increasing M , the distribution of the riding passenger’s number extend. The elevators exhibit the extended distribution for a wide range of loading parameter’s value. The .rst and second transition points increase accordingly as the elevator’s number increases. The region between the .rst and second transitions also increases accordingly as the elevator’s number increases. In the region, partial elevators carry a full load of passengers. Therefore, the passenger’s number not boarding the just arrived elevator increases accordingly as the number of elevators increases.
446
T. Nagatani / Physica A 333 (2004) 441 – 452 40
1
Number of remaining passengers
Riding passenger’s number
20 2
10
0
1
2 0
0
(a)
20
8
0
16
(b)
Loading parameter
8
16
Loading parameter
Fig. 2. (a) Plot of the riding passenger’s number B(m ) against loading parameter from su ciently large arrival m = 900–1000 for M = 3, where = 0:1, Fmax = 10, and Va = 0:0. (b) Plot of the remaining passenger’s number W (m ) − B(m ) against loading parameter where the values of parameters are the same as Fig. 2(a).
40 1
2
10
0 0
(a)
10 Loading parameter
Number of remaining passengers
Riding passenger’s number
20
20 1 2 0 0
20
(b)
10 Loading parameter
20
Fig. 3. (a) Plot of the riding passenger’s number B(m ) against loading parameter from su ciently large arrival m = 900–1000 for M = 4, where = 0:1, Fmax = 10, and Va = 0:0. (b) Plot of the remaining passenger’s number W (m ) − B(m ) against loading parameter where the values of parameters are the same as Fig. 3(a).
We calculate the number of riding passengers with varying trips for the typical case of four elevators. Figs. 4(a) – (d) show the plots of the riding passenger’s number B(m) against arrival order m from m = 1000 to m = 1050 for the values of (a) = 7:0, (b) = 12:0, (c) = 16:0, and (d) = 20:0 and no noises Va = 0:0 where M =4, = 0:1, and Fmax = 10. The dotted line indicates the maximum capacity Fmax . The number B(m) of riding passengers cannot be superior to Fmax . For case (a) of a low value of loading parameter, the number of riding passengers changes alternately from zero to about seven. The riding passenger’s number varies irregularly with arrivals. When the elevators arrive simultaneously at the lobby 0oor, the time headway becomes zero. As a result, the riding passengers become zero because the waiting passengers are proportional to the time headway. With increasing loading parameter , the riding passenger’s number varies highly and irregularly from arrival to arrival. The probability that elevators arrive
T. Nagatani / Physica A 333 (2004) 441 – 452
20
Riding passenger’s number
Riding passenger’s number
20
10
10
0
0
1000
(a)
1050 Arrival order
(b)
1000
1050 Arrival order
20
Riding passenger’s number
Riding passenger’s number
20
10
10
0
0 1000
(c)
447
1050
Arrival order
1000
(d)
1050
Arrival order
Fig. 4. Plots of the riding passenger’s number B(m ) against arrival order m from m = 1000–1050 for M = 4 at values of (a) = 7:0, (b) = 12:0, (c) = 16:0, and (d) = 20:0 where = 0:1, Fmax = 10, and Va = 0:0.
simultaneously at the lobby 0oor becomes less and less with increasing loading parameter. When the loading parameter is higher than the .rst transition point, an elevator carries the full load of passengers (see Fig. 4b). Furthermore, when loading parameter increases, the elevators become full load frequently (see Fig. 4c). When loading parameter is superior to the second transition point, four elevators become always full load, the queuing occurs at the lobby 0oor, and the queuing diverges (see Fig. 4d). We study the mean Bav and root-mean-square Brms of the number B(m) of riding passengers, and the mean Wav − Bav of number W (m) − B(m) of remaining passengers. Fig. 5(a) shows the plots of the means Bav , Wav − Bav , and root-mean-square Brms against loading parameter for M = 2 where = 0:1, Fmax = 10, and Va = 0:0. Fig. 5(b) shows the plots of the means Bav , Wav − Bav , and root-mean-square Brms against loading parameter for M = 4 where = 0:1, Fmax = 10, and Va = 0:0. The mean value Bav increases accordingly as loading parameter , and saturates at the second transition point. Root-mean-square Brms increases with loading parameter until (a) = 8:0 and (b) = 15:0. Then, it decreases and becomes zero at the second transition point. The mean Wav − Bav is zero until the .rst transition point. When the
448
T. Nagatani / Physica A 333 (2004) 441 – 452 20
20
Average values
Average values
Wav-Bav Bav 10
Wav-Bav
10
Bav Brms
Brms
0
0 0
(a)
6 Loading parameter
0
12
(b)
10 Loading parameter
20
Fig. 5. (a) Plots of the mean values Bav and Wav − Bav of the riding passenger’s number and remaining passenger’s number against loading parameter for M = 2, where = 0:1, Fmax = 10, and Va = 0:0. Plot of the root-mean-square (rms) Brms of the riding passenger’s number for M = 2. (b) Plots of the mean values Bav and Wav − Bav of the riding passenger’s number and remaining passenger’s number against loading parameter for M = 4 where = 0:1, Fmax = 10, and Va = 0:0. Plot of the root-mean-square (rms) Brms of the riding passenger’s number for M = 4.
30 2nd transition point
1st and 2nd transition points
1st and 2nd transition points
15
10
5 1st transition point
0
15
1st transition point 0
0
(a)
2nd transition point
10 Capacity
0
20
(b)
10 Capacity
20
Fig. 6. Dependence of the transition points on the maximum capacity. (a) Plots of the .rst and second transition points 1; c , 2; c against maximum capacity Fmax for M = 2 where = 0:1 and Va = 0:0. The circles and solid line indicate, respectively, the .rst and second transition points. (b) Plots of the .rst and second transition points 1; c , 2; c against maximum capacity Fmax for M = 4 where = 0:1 and Va = 0:0.
loading parameter is larger than the .rst threshold, the mean Wav − Bav increases with loading parameter and then diverges at the second transition point. We study the e+ect of the maximum capacity on the .rst and second transition points. Fig. 6 shows plots of the .rst and second transition points 1; c , 2; c against maximum capacity Fmax for (a) M = 2 and (b) M = 4 where = 0:1 and Va = 0:0. The circles and solid line indicate, respectively, the .rst and second transition points. The .rst and second transition points increase accordingly as the maximum capacity increases. The di+erence between the .rst and second transition points also increases with the maximum capacity. The di+erence becomes large with increasing elevator number M . We study the dependence of the .rst transition point 1; c on the elevator’s
T. Nagatani / Physica A 333 (2004) 441 – 452
449
15
First transition point
M=4 10
M=3 M=2
5
M=1
0 0
10 Capacity
20
Fig. 7. The dependence of the .rst transition point 1; c on the elevator’s number M . Plot of the .rst transition point 1; c against maximum capacity Fmax for M = 1–4 where = 0:1 and Va = 0:0.
30 Second transition point
M=4 M=3 M=2
15
M=1
0 0
10 Capacity
20
Fig. 8. The dependence of the second transition point 2; c on the elevator’s number M . Plot of the second transition point 2; c against maximum capacity Fmax for M = 1–4 where = 0:1 and Va = 0:0.
number M . Fig. 7 shows plot of the .rst transition point 1; c against maximum capacity Fmax for M =1–4. With increasing elevator number M , the .rst transition point becomes higher but the increasing rate becomes lower with M . For M = 1, the .rst and second transition points are consistent each other. For comparison, we study the dependence of the second transition point 2; c on the elevator’s number M . Fig. 8 shows plot of the second transition point 2; c against maximum capacity Fmax for M = 1–4. With increasing elevator number M , the second transition point becomes higher. 4. Analytical result We present the analytical result for such a case that all the elevators take the same speed Vi = V0 and the same capacity Fi; max = Fmax . We study the dependence of the second transition point on the capacity and elevator’s number. When the loading
450
T. Nagatani / Physica A 333 (2004) 441 – 452
parameter is higher than the threshold 2; c , all elevators carry a full load of passengers. So the following relationship holds: Fmax = 2; c HT2; c ;
(8)
where HT2; c is the time headway at the threshold. If all elevators move with the same time headway, the following equation is obtained from Eq. (4) M HT2; c = 2; c HT2; c + 1 : One obtains the threshold from Eqs. (8) and (9) MFmax 2; c = : Fmax + 1
(9)
(10)
Fig. 8 shows the plot of threshold 2; c against capacity Fmax for M = 1–4 and = 0:1. The solid curve represents the theoretical result (10). The simulation result is consistent with the theoretical result. We derive the mean value of riding passengers below the second transition point. The mean tour time is given from Eq. (4) V0 : (11) HTi = Bi + Vi The passengers arrived at the origin over su ciently large time TL is given by TL . Below the second threshold, the number TL of passengers is consistent with those carried by all the elevators. The following relationship holds: M i=1
TL
Bi = TL : HTi
One obtains an approximation of riding passengers for ¡ 2; c ; M − B1 = for ¿ 2; c : Fmax
(12)
(13)
Fig. 9 shows the plots of mean number of riding passengers against loading parameters for elevator number M = 1–4 where = 0:1, Fmax = 10, and Va = 0:0. The solid curves represent the approximation (13). The simulation result Bav is consistent with the theoretical result B1 . 5. Linear stability analysis We consider the stability of the elevator tra c when all the elevators move with the same time headway. We apply the linear stability method to the elevator tra c model. For simplicity, we restrict the limiting case of Fmax → ∞. When all the elevators move with the same time headway, the following equation is obtained from Eqs. (4) and (5) M HT0; f = HT0; f + 1 ;
(14)
T. Nagatani / Physica A 333 (2004) 441 – 452
451
20
M=1
3
2
4
〈 B1
〈
10
0 0
10 Loading parameter
20
Fig. 9. Analytical result of the mean B1 of the number of riding passengers against loading parameter where = 0:1 and Va = 0:0.
where HT0; f is the time headway between an elevator and its predecessor. The .xed point is given by HT0; f = 1=(M − ) :
(15)
We study whether or not the .xed point is stable. Let yi (m) be a small deviation from the arrival time of elevator i at the .xed point: Ti (m) = T0; i (m) + yi (m) where T0; i+1 (m) − T0; i (m) = HT0; f . Then, the linearized equations are obtained from Eqs. (4) and (5) y1 (m + 1) = y1 (m) + (y1 (m) − yM (m − 1)) ;
(16)
yi (m + 1) = yi (m) + (yi (m) − yi−1 (m))
(17)
for i = 2; 3; : : : ; M :
By taking yi (m) = Yi e#m , the following equations of # are derived (e# − 1 − )M − (−)M e−# = 0 :
(18)
If # is a positive value, the .xed point is unstable. When # is a negative value, the .xed point is stable. In the special case of M = 1, one obtains # = 0 and # = ln(). Therefore, the .xed point is stable for ¡ 1 and unstable for ¿ 1. For the case of M =2, one obtains # = 0 and √ (1 + 2) ± 1 + 4 : # = ln 2 The maximum value is positive. Therefore, the .xed point is unstable for any value of . One .nds that the maximum value of # is positive for M ¿ 2. We .nd that the elevators moving with the same time headway are always unstable for any value of . Thus, the elevator tra c with multiple elevators is always unstable. Even if all the elevators move with the same time headway initially, the time headway between elevators increases or decreases by the instability. In the mean time, an elevator passes the other elevators or is outstripped by the other elevators. The instability induces the chaotic motion of the elevators.
452
T. Nagatani / Physica A 333 (2004) 441 – 452
6. Summary We have studied the dynamical phase transitions in M elevators tra c during the morning peak period by using the coupled nonlinear map model. We have shown that the number of passengers carried by an elevator exhibits the complex behavior with varying trips and the motion of elevators exhibits a deterministic chaos even if there are no noises. We have found that the dynamical transitions occur at the two stages: the .rst transition when an elevator carries a full load of passengers and the second transition if all the elevators carry a full load of passengers. Above the .rst transition point, a part of the passengers waiting at the lobby 0oor cannot board the just arrived elevator. Above the second transition point, the queuing of the waiting passengers is more and more with time. We have investigated the e+ect of elevator number M on the dynamical transition. We have clari.ed the dependence of the threshold (transition point) on the loading parameter, the elevator capacity, and the elevator number. It is important to decide the elevator number M and estimate the queuing of waiting passengers at the lobby 0oor in elevator design. The result obtained here will be useful for elevator designing in the building. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
T. Nagatani, Rep. Prog. Phys. 65 (2002) 1331. D. Helbing, Rev. Mod. Phys. 73 (2001) 1067. D. Chowdhury, L. Santen, A. Schadscheider, Phys. Rep. 329 (2000) 199. B.S. Kerner, Networks Spatial Econ. 1 (2001) 35. D. Helbing, H.J. Herrmann, M. Schreckenberg, D.E. Wolf (Eds.), Tra c and Granular Flow ‘99, Springer, Heidelberg, 2000. K. Nagel, M. Schreckenberg, J. Phys. I France 2 (1992) 2221. E. Ben-Naim, P.L. Krapivsky, S. Redner, Phys. Rev. E 50 (1994) 822. E. Tomer, L. Safonov, S. Havlin, Phys. Rev. Lett. 84 (2000) 382. M. Treiber, A. Hennecke, D. Helbing, Phys. Rev. E 62 (2000) 1805. H.K. Lee, H.-W, Lee, D. Kim, Phys. Rev. E 64 (2001) 056 126. O.J. O’loan, M.R. Evans, M.E. Cates, Phys. Rev. E 58 (1998) 1404. D. Chowdhury, R.C. Desai, Eur. Phys. J. B 15 (2000) 375. T. Nagatani, Phys. Rev. E 63 (2001) 036 116. H.J.C. Huijberts, Physica A 308 (2002) 489. S.A. Hill, Cond-Mat/0206008 (2002). T. Nagatani, Physica A 297 (2001) 260. T. Nagatani, Phys. Rev. E 66 (2002) 046 103. T. Poschel, J. Gallas, Phys. Rev. E 50 (1994) 2654. G.F. Newell, Transpn. Res. B 32 (1998) 583. T. Nagatani, Physica A 310 (2002) 67. T. Nagatani, Physica A 326 (2003) 556.