ISSN 0005-1179, Automation and Remote Control, 2015, Vol. 76, No. 9, pp. 1673–1680. © Pleiades Publishing, Ltd., 2015. Original Russian Text © V.M. Bure, V.V. Mazalov, N.V. Plaksina, 2014, published in Upravlenie Bol’shimi Sistemami, 2014, No. 47, pp. 77–91.
LARGE SCALE SYSTEMS CONTROL
Estimating Passenger Traffic Characteristics in Transport Systems V. M. Bure∗ , V. V. Mazalov∗∗ , and N. V. Plaksina∗∗∗ ∗
∗∗
Saint Petersburg State University, St. Petersburg, Russia Institute of Applied Mathematical Research, Karelian Research Center, Russian Academy of Sciences, Petrozavodsk, Russia ∗∗∗ Petrozavodsk State University, Petrozavodsk, Russia e-mail:
[email protected],
[email protected], plaksina
[email protected] Received November 11, 2013
Abstract—This paper considers a statistical model of passenger flows between bus stops. By assumption, the intensities of incoming and outcoming passenger flows are known. We propose an estimation technique for the distribution of incoming passengers at a certain bus stop with respect to all successive bus stops. The simulation results based on natural experiments are also presented. DOI: 10.1134/S0005117915090131
1. INTRODUCTION Nowadays, the development of municipal microdistricts and the growing number of private vehicles bring to the problem of passenger traffic analysis and evaluation. In the sequel, we comprehend a passenger flow as the intensity of passenger traffic in one direction of a route. There exist forward and backward passenger flows. Passenger flows play an important role in defining the routes of urban transport and optimal traffic intervals of public transport facilities. Passenger flows are nonuniform, i.e., possess time-to-time variability (by hours, circadian periods, days of week, and seasons). The passenger traffic capacity of a separate district is determined depending on the number of residents, their mobility indices and the temporal nonuniformity coefficients of traffic. Different analysis techniques serve for acquiring information on the mobility indices of residents and the temporal nonuniformity coefficients of their traffic. Natural experiments guarantee higher accuracy, the error of such methods constitute approximately 5%, see [5]. Their major drawback consists in appreciable financial and human resources required, while data treatment consumes much time. Therefore, the results of natural observations may provide an inaccurate picture of real passenger traffic. Another approach to passenger traffic evaluation employs entropy-based methods [1]. Within the framework of the entropy model, passenger flows are calculated via demographic and socioeconomic data with information on the actual location of a district. The main disadvantage of the entropy-based method is the assumed stationarity of passenger flows: such models ignore their time-to-time variability (by hours, days of week, etc.). And so, natural experiments and demographic data are often combined with mathematical methods to obtain up-to-date information on passenger traffic [2, 4, 6, 7, 11]. These methods seem reasonable for constructing optimal (equilibrium) solutions in traffic planning by acquired information on passenger flows. Evaluated passenger traffic characteristics allow solving different applied problems of optimal urban transport control, viz., defining the number of buses required for passenger transportation considering the interests of passengers and carriers (cost-efficient routing). 1673
1674
BURE et al.
2. THE GENERAL MODEL In the general case, the transport model can be represented by a graph whose nodes act as stops and edges describe transport passageways. Choose a certain route of a transport vehicle, i.e., a sequence of stops connected by edges. Consider the case of K stops on a route. Suppose that there are forward passenger flows among the stops. We believe that the flows run according to the direction of the bus route. The problem is to find the shares of passengers in the total flow and their movement directions (the distributions of passenger flows). For instance, among ten passengers at a certain stop, some move to the end stop and the others leave the bus at intermediate stops. To solve the problem, conduct a series of r experiments, r = 1, . . . , N . An experiment corresponds to a single trip of the bus from the starting stop of the route to the end one. Our aim consists in fixing the number of incoming and outcoming passengers at each stop. Let xr1 , xr2 , . . . , xrK be the numbers of incoming passengers at stops i = 1, . . . , K, respectively, in experiment r (r = 1, . . . , N ). By analogy, r signify the numbers of outcoming passengers at stops j = 1, . . . , K, the quantities y1r , y2r , . . . , yK respectively, in experiment r (r = 1, . . . , N ). Obviously, xrK = y1r = 0 for any experiment r. By r the number of assumption, xri and yjr form observable quantities in the model. Denote by yij passengers incoming at stop i and outcoming at stop j. Actually, these are unobservable quantities to-be-estimated. For convenience, the information on passenger flows is described by Table 1. Table 1. Information on the number of incoming and outcoming passengers r r r r r y11 =0 y12 y13 ... y1K−1 y1K xr1 r r r r 0 y22 = 0 y23 ... y2K−2 y2K xr2 ... ... ... ... ... ... ... r r 0 0 0 ... yK−1,K−1 =0 yK−1,K xrK−1 0 0 0 ... 0 0 xrK = 0 r r r r r y1 = 0 y2 y3 ... yK−1 yK
Table 1 demonstrates the results of experiment r. All elements in the last column are the sums of elements from a corresponding row; similarly, the last row contains the sums of elements r indicates the (unobservable) number of passengers from a corresponding column. The quantity yij incoming at stop i and outcoming at stop j in experiment r: yjr
=
j−1 i=1
r yij .
(1)
In each experiment, we observe the elements of the last column and row. For all i and j, suppose r represent independent (for different r) identically distributed random that the passenger flows yij variables. Their mean value is the intensity of passenger traffic on an appropriate route from stop i to stop j. Designate by pij the share of passengers incoming at stop i and outcoming at stop j. It can be interpreted as the probability that a passenger entering at stop i leaves at stop j. Construct Table 2 from these values as follows. Table 2. Information on the distribution of passenger flows p11 = 0 p12 p13 . . . p1K−1 p1K 1 0 p22 = 0 p23 . . . p2K−1 p2K 1 ... ... ... ... ... ... ... 0 0 0 . . . pK−1,K−1 = 0 pK−1,K 1
In Table 2, the sum of elements in each row equals 1; this is the polynomial distribution [3] of the probabilities of going out for passengers incoming at an appropriate stop (whose number coincides with the index of row). AUTOMATION AND REMOTE CONTROL
Vol. 76
No. 9
2015
ESTIMATING PASSENGER TRAFFIC CHARACTERISTICS
1675
Consider experiment r and multiply row i from Table 2 by xri . Naturally, the resulting row r . Random corresponds to row i from Table 1, since xri pij is the mean value of the random variable yij variables in a same column of Table 1 appear mutually independent. Hence, their deviations from the mean value are random, independent and compensate each other in case of summation. Therefore, we have the following equality in a rough approximation: yjr ≈
j−1 i=1
xri pij .
(2)
If the number of experiments N is large, Eqs. (2) enjoy linear independence (the coefficients in different equations form mutually independent random variables). Consequently, these equations can be resolved in the unknown probabilities pij (here i < j, j = 1, . . . , K) by applying the least-squares technique (LST) [2] or the minimization method for the sum of absolute deviations (MSAD) [7]. 3. THE MODEL WITH K = 5 STOPS Consider the model with five stops and forward passenger flows. Let us analyze passenger flows on the route containing all these stops. According to the model, p12 reflects the passenger flow from stop 1 to stop 2, p13 describes the passenger flow from stop 1 to stop 3, and so on. For evaluating the characteristics of passenger flows, we have conducted 100 numerical experiments and compiled the following tables of passenger traffic. Tables 3–4 present the information on passenger flows for first 15 experiments. Table 3. Information on the number of incoming passengers Stop no. 1 2 3 4
Experiment no. 1 4 3 1 3
2 6 2 2 2
3 5 2 2 2
4 4 3 1 2
5 4 3 2 2
6 5 2 2 2
7 5 2 1 2
8 4 3 1 2
9 10 11 12 13 14 15 5 6 4 4 5 6 4 2 2 3 3 3 2 2 2 1 2 1 1 1 1 2 3 2 3 2 2 2
Table 4. Information on the number of outcoming passengers Stop no. 2 3 4 5
Experiment no. 1 3 3 3 2
2 3 5 3 1
3 3 4 3 1
4 3 4 1 2
5 2 3 4 2
6 3 4 3 1
7 2 4 2 2
8 3 4 1 2
9 10 11 12 13 14 3 2 3 3 3 3 3 5 3 3 3 4 3 2 4 2 3 3 2 3 1 3 2 1
15 4 2 2 1
To find the unknown variables pij (i < j, j = 1, . . . , K), employ the least-squares technique. The sum of squared deviations is defined by
s=
K N r=1 j=2
AUTOMATION AND REMOTE CONTROL
⎛ ⎝yjr −
Vol. 76
j−1 i=1
⎞2
xri pij ⎠ ,
No. 9
2015
(3)
1676
BURE et al.
where N = 100, K = 5. Then s=
5 100 r=1 j=2
+ (y3r
−
xr1 p13
⎛ ⎝y r − j
−
j−1 i=1
xr2 p23 )2
⎞2
xri pij ⎠ = +
(y4r
−
100 r=1
xr1 p14
(y2r − xr1 p12 )2 (4)
− xr2 p24 − xr3 p34 )2
+ (y5r − xr1 p15 − xr2 p25 − xr3 p35 − xr4 p45 )2 , with the constraints p12 + p13 + p14 + p15 = 1, p23 + p24 + p25 = 1, p34 + p35 = 1, pij 0, for any i and j. Using Lagrange’s method of multipliers [7], reduce the number of constraints by introducing a new variable S: S=
100 r=1
(y2r − xr1 p12 )2 + (y3r − xr1 p13 − xr2 p23 )2 + (y4r − xr1 p14 − xr2 p24 − xr3 p34 )2
(5)
+ (y5r − xr1 p15 − xr2 p25 − xr3 p35 − xr4 p45 )2 − λ1 (p12 + p13 + p14 + p15 − 1) −λ2 (p23 + p24 + p25 − 1) − λ3 (p34 + p35 − 1),
with the constraints pij 0, for any i and j. The problem is to minimize the function S in the unknown parameters pij , λt , for i < j, j = 1, . . . , K, t = 1, 2, 3. The function S achieves its minimum under the following parameter values: p12 = 0.48; p13 = 0.52; p14 = 0; p15 = 0; p23 = 0.67; p24 = 0.33; p25 = 0; p34 = 1; p35 = 0; p45 = 1. The total number of passengers at stops varies from 9 to 14. In the case of 11 passengers on the route, the mean passenger flows are distributed as shown by Table 5. Table 5. Information on the mean passenger flows obtained by the least-squares technique The mean number Stop no. 2 3 4 5 of passengers at a stop 1 2.4 2.6 0 0 5 2 0 2.01 0.99 0 3 3 0 0 1 0 1 4 0 0 0 2 2 The mean number 2.4 4.61 1.99 2 of passengers leaving at a stop
Now, find the unknowns pij by minimizing the sum of absolute deviations. The sum of the absolute deviations s1 for 5 stops and 100 experiments has the form j−1 5 100 r r xi pij s1 = y j − r=1 j=2 i=1
=
100 r=1
(6)
|y2r − xr1 p12 | + |y3r − xr1 p13 − xr2 p23 | + |y4r − xr1 p14 − xr2 p24 − xr3 p34 |
+ |y5r − xr1 p15 − xr2 p25 − xr3 p35 − xr4 p45 | , AUTOMATION AND REMOTE CONTROL
Vol. 76
No. 9
2015
ESTIMATING PASSENGER TRAFFIC CHARACTERISTICS
1677
with the constraints p12 + p13 + p14 + p15 = 1, p23 + p24 + p25 = 1, p34 + p35 = 1, pij 0, for any i and j. Introduce the variables
Zjr =
and
Wjr =
⎧ j−1 ⎪ ⎪ r− ⎪ ⎪ y xri pij , ⎪ j ⎨
if
i=1
⎪ ⎪ ⎪ ⎪ ⎪ ⎩0,
if
⎧ j−1 ⎪ ⎪ r ⎪ ⎪ xi pij − yjr , ⎪ ⎨
if
i=1
⎪ ⎪ ⎪ ⎪ ⎪ ⎩0,
if
yjr − yjr
−
i=1 j−1 i=1
yjr − yjr
j−1
−
xri pij 0 (7) xri pij
j−1 i=1 j−1 i=1
< 0;
xri pij 0 (8) xri pij
> 0.
It is clear that yjr − and
j−1 i=1
xri pij = Zjr − Wjr
(9)
r j−1 r y − xi pij = Zjr + Wjr . j i=1
(10)
And the problem lies in minimization of the function S1 : S1 =
5 100
(Zjr + Wjr ),
(11)
r=1 j=2
r r r subject to the constraints pij 0 for any i, j; 4j=i+1 pij = 1 for any i; yjr = j−1 i=1 xi pij + Zj − Wj , j = 2, . . . , 5; Zjr 0, Wjr 0, r = 1, . . . , N . Here the unknown parameters are the quantities pij , Zjr , and Wjr , where i < j 5, i = 1, . . . , 4. The function S1 achieves its minimum under the following parameter values: p12 = 0.5; p13 = 0.49; p14 = 0.01; p15 = 0; p23 = 0.68; p24 = 0.32; p25 = 0; p34 = 1; p35 = 0; p45 = 1. In the case of 11 passengers on the route, the mean passenger flows are distributed as shown by Table 6.
Table 6. Information on the mean passenger flows obtained by minimization of the sum of absolute deviations The mean number Stop no. 2 3 4 5 of passengers at a stop 1 2.5 2.45 0.05 0 5 2 0 2.04 0.96 0 3 3 0 0 1 0 1 4 0 0 0 2 2 The mean number 2.5 4.49 2.01 2 of passengers leaving at a stop AUTOMATION AND REMOTE CONTROL
Vol. 76
No. 9
2015
1678
BURE et al.
The accumulated information on passenger flows allows drawing conclusions on the amounts of passenger flows for different directions and on a whole route. The largest passenger flows exist between stops 1 and 2, as well as between stops 1 and 3, stops 2 and 3, and stops 3 and 4. The segments of the route corresponding to smallest passenger flows can be found by analogy. 4. THE MODEL WITH K = 10 STOPS In this section, consider the model with 10 stops and forward passenger flows. As previously, our intention is evaluating the characteristics of passenger flows by the parametric methods. We have conducted 100 numerical experiments and constructed the associated tables of passenger traffic. Tables 7 and 8 provide the information on the obtained passenger flows for first 15 experiments. Table 7. Information on the number of incoming passengers Stop no. 1 2 3 4 5 6 7 8 9
Experiment no. 1 5 3 2 3 3 3 1 3 1
2 6 3 1 2 4 3 1 2 2
3 6 2 1 3 3 3 2 2 2
4 4 3 2 3 3 3 1 2 1
5 6 2 2 3 3 4 2 2 1
6 5 3 1 3 5 3 1 2 1
7 6 2 1 3 5 4 1 2 2
8 4 2 2 3 3 3 1 3 2
9 10 11 12 13 14 15 4 4 5 6 4 4 6 3 2 3 2 3 2 2 2 2 2 2 2 2 1 3 3 3 3 3 3 3 4 5 4 3 4 3 5 4 3 4 4 4 3 4 1 2 2 2 1 1 1 2 3 2 2 2 3 2 2 1 2 1 2 2 2
Table 8. Information on the number of outcoming passengers Stop no. 2 3 4 5 6 7 8 9 10
Experiment no. 1 3 5 2 1 5 1 3 3 1
2 3 6 1 2 3 2 3 1 3
3 3 5 0 3 3 2 4 2 2
4 2 5 1 1 6 3 1 2 1
5 3 4 2 4 2 3 4 2 1
6 3 4 1 2 7 1 3 1 2
7 3 5 1 3 4 3 3 2 2
8 3 3 2 2 3 2 3 3 2
9 10 11 12 13 14 2 2 2 3 2 3 5 4 6 4 5 3 1 1 1 2 1 2 2 2 3 4 2 2 6 6 4 2 6 3 3 2 4 3 3 2 2 4 3 4 2 3 2 3 1 2 2 3 2 1 3 1 2 2
15 3 5 1 3 4 3 3 2 2
Based on the experimental data, we have evaluated the unknown parameters of passenger traffic by the least-squares technique and the minimization method for the sum of absolute deviations. Table 9 illustrates the obtained parameter values. Clearly, the both methods have yielded close values of the parameters and are hence applicable to passenger traffic evaluation. Therefore, by combining natural experiments with statistical methods, it is possible to accumulate information on most and least popular stops of passengers and, moreover, information on their traffic (where passengers enter and leave a bus, etc.). Such information is helpful in planning of AUTOMATION AND REMOTE CONTROL
Vol. 76
No. 9
2015
ESTIMATING PASSENGER TRAFFIC CHARACTERISTICS Table 9. Information on the distribution of obtained by the parametric methods LST MSAD LST MSAD p12 0.48 0.5 p29 0 0.01 p13 0.43 0.37 p210 0 0 p14 0.04 0.06 p34 0.77 0.65 p15 0 0.01 p35 0.09 0.1 p16 0.05 0.05 p36 0.1 0.13 p17 0 0.01 p37 0.02 0.06 p18 0 0.01 p38 0.03 0.05 p19 0 0.01 p39 0 0.01 p110 0 0 p310 0 0.01 p23 0.98 0.98 p45 0.66 0.61 p24 0.01 0.01 p46 0.2 0.21 p25 0.01 0 p47 0.04 0.08 p26 0 0.01 p48 0.09 0.1 p27 0 0 p49 0.01 0 p28 0 0 p410 0 0
1679
passenger flows
p56 p57 p58 p59 p510 p67 p68 p69 p610 p78 p79 p710 p89 p810 p910
LST MSAD 0.99 0.97 0 0.02 0.01 0.01 0 0.01 0 0 0.78 0.65 0.21 0.24 0.01 0.11 0 0 0.79 0.59 0.2 0.26 0.01 0.15 0.86 0.72 0.14 0.28 1 1
new routes and optimization of existing routes through proper increase or decrease of bus traffic intervals. The results of this research can be applied not only to bus service, but also to other public transport facilities (suburban, intertown, ground and underground railway services including monorail ones, water transport). However, today the most pressing problem concerns ground transport facilities; this observation follows from regular traffic jamming in many cities and governmental tenders announced at zakupki.gov.ru [8–10] and focused on urban public transport exploration and passenger traffic evaluation. And the present paper pays most attention to ground urban public transport. 5. CONCLUSIONS AND PROSPECTS This paper has explored the statistical model of passenger flows among stops. Using a series of natural experiments, we have constructed the mathematical model of the system. A passenger traffic evaluation procedure has been illustrated in the special cases of K = 5 and K = 10 stops. Passenger flows have been calculated through the least-squares technique and the minimization method for the sum of absolute deviations. Note that the proposed evaluation procedure for passenger flows admits easy realization in practice. For instance, in large cities passengers often enter buses in the first place and register their tickets in special automata. Passengers leave buses through other doors, where additional automata can be placed, too. ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research, project no. 13-0191158-GFEN a, the Mathematical Sciences Division Program of the Russian Academy of Sciences, and the Strategic Development Program of Petrozavodsk State University. REFERENCES 1. Artynov, A.P., Embulaev, V.N., Pupyshev, A.V., and Skaletskaya, V.V., Avtomatizatsiya upravleniya transportnymi sistemami (Automation of Transport Systems Management), Moscow: Nauka, 1984. 2. Bunday, B.D., Basic Optimization Methods, London: Edward Arnold, 1984. Translated under the title Metody optimizatsii. Vvodnyi kurs, Moscow: Radio i Svyaz’, 1988. AUTOMATION AND REMOTE CONTROL
Vol. 76
No. 9
2015
1680
BURE et al.
3. Borovkov, A.A., Teoriya veroyatnostei (Probability Theory), Moscow: Editorial URSS, 1999. 4. Bure, V.M. and Parilina, E.M., Teoriya veroyatnostei i matematicheskaya statistika: uchebnik (Probability Theory and Mathematical Statistics: Textbook), St. Petersburg: Lan’, 2013. 5. Varelopulo, G.A., Organizatsiya dvizheniya i perevozok na gorodskom passazhirskom transporte (Organization of Traffic and Passenger Service of Urban Passenger Transport), Moscow: Transport, 1981. 6. Gill, P.E., Murray, W., and Wright, M.H., Practical Optimization, London: Academic, 1981. Translated under the title Prakticheskaya optimizatsiya, Moscow: Mir, 1985. 7. Draper, N.R. and Smith, H., Applied Regression Analysis, New York: Wiley, 1981. Translated under the title Prikladnoi regressionnyi analiz, Moscow: Finansy i Statistika, 1986, vol. 1, 2nd ed. 8. Ofitsial’nyi sait Rossiiskoi Federatsii v seti Internet dlya razmeshcheniya informatsii o razmeshchenii zakazov na postavki tovarov, vypolnenie rabot, okazanie uslug (The Official Web-Site of the Russian Federation in the Internet Designated for Publication of Information on Placement of Orders for Goods, Works and Services), http://zakupki.gov.ru/pgz/public/action/orders/ info/common info/show?notificationId=6795209 (accessed December 12, 2013). 9. Ofitsial’nyi sait Rossiiskoi Federatsii v seti Internet dlya razmeshcheniya informatsii o razmeshchenii zakazov na postavki tovarov, vypolnenie rabot, okazanie uslug (The Official Web-Site of the Russian Federation in the Internet Designated for Publication of Information on Placement of Orders for Goods, Works and Services), http://zakupki.gov.ru/pgz/public/action/orders/ info/common info/show?source=epz¬ificationId=8275955 (accessed December 25, 2013). 10. Ofitsial’nyi sait Rossiiskoi Federatsii v seti Internet dlya razmeshcheniya informatsii o razmeshchenii zakazov na postavki tovarov, vypolnenie rabot, okazanie uslug (The Official Web-Site of the Russian Federation in the Internet Designated for Publication of Information on Placement of Orders for Goods, Works and Services), http://zakupki.gov.ru/pgz/public/action/orders/ info/common info//show?source=epz¬ificationId=5629611 (accessed December 25, 2013). 11. Postnikova, E., Kvantil’naya regressiya (Quantile Regression), http://allmath.ru/highermath/ probability/probability48/probability.htm (accessed December 12, 2013).
AUTOMATION AND REMOTE CONTROL
Vol. 76
No. 9
2015
