Modeling the Bilateral Micro-Searching Behavior for Urban Taxi Services Using the Absorbing Markov Chain Approach

Transcription

1 Journal ofadvanced Transportation, Vol. 39, No. I, pp www. advanced-transport. corn Modeling the Bilateral Micro-Searching Behavior for Urban Taxi Services Using the Absorbing Markov Chain Approach K. I. Wong S. C. Wong M. G. H. Bell Hai Yang This paper develops a mathematical model that is based on the absorbing Markov chain approach to describe taxi movements, taking into account the stochastic searching processes of taxis in a network. The local searching behavior of taxis is specified by a logit form, and the O-D demand of passengers is estimated as a logit model with a choice of taxi meeting point. The relationship between customer and taxi waiting times is modeled by a double-ended queuing system. The problem is solved with a set of non-linear equations, and some interesting results are presented. The research provides a novel and potentially useful formulation for describing the urban taxi services in a network. Introduction In most large cities the taxi industry is subject to various types of regulation, such as entry restriction and price control, and many economists have examined the economic consequences of such regulatory restraints (e.g. Douglas, 1972). The general objective of such studies has been to understand the manner in which the demand for, and supply of, taxis are equilibrated in the presence of regulations, thereby providing information to governments to help them make decisions. The above studies use a highly aggregated model that was proposed by K.I. Wong and M.G.H. Bell, Centre for Transport Studies, Department of Civil and Environmental Engineering, Imperial College London, England, UK S.C. Wong, Department of Civil Engineering, The University of Hong Kong,, Hong Kong,P.R., China Hai Yang, Department of Civil Engineering, The Hong Kong University of Science and Technology, Hong Kong, P.R.,China

2 82 K.I. Wong, S.C. Wong, M.G.H. Bell, and H. Yang Douglas (1972) without any consideration of the spatial structure of the market. The aforementioned analytical approaches are based on abstract, simplified demand-supply models, and are useful in understanding the way in which taxi markets operate. However, in reality, the demand for, and supply of, taxi services takes place over space, and the equilibrium modeling of the problem should be conducted in connection with a detailed road network structure and customer origin-destination (0-D) demand pattern, as is the case in conventional network modeling. This has important implications for both the assessment of road traffic congestion due to taxi movements, and the precise understanding of the equilibrium nature of taxi services. In this respect, Yang and Wong (1998) made an initial attempt to characterize taxi movements for a given and fixed customer 0-D demand pattern in an uncongested road network, and the problem was formulated as a simultaneous equation system to describe the movements of both empty and occupied taxis. The model was further extended and reformulated as a simultaneous optimization model of two equilibrium sub-problems to incorporate congestion effects and customer demand elasticity (Wong et al., 200 l), for which one sub-problem is a combined network equilibrium model (CNEM) that describes the simultaneous movements of vacant and occupied taxis, as well as normal traffic in a user-optimal manner. The other sub-problem is a set of linear and nonlinear equations (SLNE) that ensure the satisfaction of the relationship between taxis and customer waiting times, and the relationship between customer demand and taxi supply. Furthermore, the potential applications of the model have been demonstrated by several case studies of the urban area of Hong Kong (Yang et al., 2001, 2002). The above network equilibrium studies describe vacant taxis that visit meeting points based on imperfect information about the potential locations of customers. They modeled the stochastic searching process in a macroscopic manner to characterize the long-term network equilibrium patterns of taxi movements in a general network. However, they neglected customer movements in response to a local variation in the level of taxi services and the microscopic searching behavior of taxis, in which taxi dnvers may be circulating locally in the network when searching for customers. In a more recent study, Yang et al. (2003) made a first attempt to study the bilateral search behavior of taxi drivers and customers. In their paper, the behavior of the drivers of vacant taxis and of customers searching, and the meeting of these two in a network were modeled with

3 Modeling the Bilateral logit models. The customer-taxi meeting and waiting times were considered using the characteristics of the production function of the service. However, the local circulation of vacant taxis that is not uncommon in urban streets was not modeled. This paper aims to formulate the taxi-customer bilateral searching and meeting behavior in a network, taking into account the stochastic micro-searching behavior of both taxis and customers when they are searching for each other. Both the mechanisms of taxis in search of customers and customers in search of taxis can be described using the absorbing Markov chain approach. Their matching mechanisms at meeting points can be formulated as a double-ended queuing model. This is useful for the local assessment of taxi service levels, the identification of local congestion that is induced by the circulation of vacant taxis, and the evaluation of the impact on taxi customers for short- to medium-term planning. Consider a road network with a set of nodes that are connected by a set of links. There are two principal characteristics that distinguish the taxi market from the idealized case of conventional transportation planning. Firstly, the behavior of taxi drivers who are searching for customers in a network is not destination based, i.e. they do not have a destination in the network, but search for a customer to maximize their profit. They may not travel on the shortest path, but on the path with the maximum probability of containing a customer. Secondly, customers who are looking for taxis may go and wait at the nearest taxi meeting point, or they may go to another meeting point that may have more vacant taxis. This bilateral search behavior of taxis and customers in a network is modeled with the absorbing Markov chain approach. As the Markov process approach obviates the need to generate the path set, it is appropriate for the modeling of the searching process of taxis, in which repeated cyclic flows and information imperfection usually coexist. For discussions of the Markov process and the convergence of this series, interested readers are referred to the work of Bell (1995), Akamatsu (1996), and Wong (1999). The determination of the transition probabilities (link choice probabilities) in the transition matrix can be specified by the meeting rates of each taxi-customer meeting point in a network. With the aforementioned micro-searching behavior, the matching mechanism at taxi meeting points, for example, taxi stands and roadside hailing, can be formulated as a double-ended queuing model. Conolly et al. (2002) discussed double-end queues, and considered the impatience of taxis and passengers, although close-form solutions are generally

4 84 K.I. Wong, S. C. Wong, M. G. H. Bell, and H. Yang difficult to obtain. With the arrival rates of taxis and customers, and the maximum physical capacities of both queues, the expected queue lengths and waiting times can be explicitly determined by the queuing theory. More recently, Matsushima and Kobayashi (2004) studied the matching externality of taxi spot markets using the double-ended queuing model. The thick-market externalities, namely, the expectations of both suppliers and consumers (supply and demand) will increase (and thus, waiting time will decrease) if both agents visit the spot markets more frequently. The paper is organized as follows. Afier introducing the notations and definitions, the basic assumptions are given. Both the customersearching behavior of vacant taxis and the taxi-searching behavior of customers in a network can be described as absorbing Markov processes. The meeting mechanism between taxis and customers at a taxi rank is formulated as a queuing model with double-end queues. A numerical example is used to demonstrate the effectiveness of the proposed me thodology. Taxi Movements in a Road Network Consider a road network G(V9A), where is the set of nodes and A is the set of links in the network. Let i I be the set of origin nodes and j be the set of destination nodes, and E K be the set of taxi- D.. customer meeting points in the network. Let IJ be the customer DC. demand from node i to node j, which can be split to 2kJ for the customer demand from node i to nodej via taxi meeting point k. Let Ok =ccdij ie'jgj be the number of customers boarding taxis at meeting D, = CD. 14' point k, JEJ be the number of customers, whose origin is node DQ = Dig i, who are walking to meeting point k, ~ E I be the number of D, =CDq customers going from meeting point k to destinationj, and ie I be the total number of customers being dropped off at destinationj.

5 Modeling the Bilateral Vacant Taxi Movements The absorbing Markov chain is employed to study the movements of vacant taxis in the network. The nodes in a network are considered as states, and the movements of vehicles between two incident nodes in a network correspond to events (transitions between states). The vehicles that are generated from an origin node repeatedly change their states. If they move to the state that corresponds to a destination node, then they are absorbed with a probability of 1. As the Markov chain approach requires that there should be no restrictions on the path set, it is appropriate for modeling the searching process of taxi movements, in which repeated cyclic flows and uncertainties usually exist. Suppose that a taxi dnver drops a customer in node i I. The driver will either stay at the same node, or move to an adjacent node to find the next customer, talung into consideration the availability of customers (which is indicated by the probability of joining a queue and the corresponding waiting time) and/or the profitability (expected profit per time) in these nodes. This process is repeated until a customer is found. The probabilistic movement of vehicles in an absorbing Markov chain can be fully described by the following matrix, whose elements denote the transition probability between two nodes:.=[i 01 R Q (1) where I denotes an identity matrix, the element of matrix R is the transition probability from a node in the set of non-destination nodes W to a destination node in J, and the element of matrix Q is the transition probability between a pair of nodes that are in W. Let p = [Pgl 7 r.. Q=[qgl, and R=[qj,.], then I/ is the probability of a taxi leaving the node/meeting point and successfully meeting a customer in node/meeting point j (absorbing state), and g is the probability of a taxi leaving nodelmeeting point and going to nodelmeeting point j in search of customers (transitions between states). The transition matrix has the following properties

6 86 K.I. Wong, S. C. Wong, M.G.H. Bell, and H. Yang The transition probabilities, '0 and '0, characterize the searching behavior of vacant taxis, their information about the network conditions, the degree of uncertainty of customer demand, and their searching strategy in the network. When a taxi arrives a meeting point k, it has a probability of 'kk of successfully meeting a passenger (conditioned by a specific waiting time), or alternatively the taxi may go to another adjacent meeting point j to search, with a probability of 46, jcji - where Ji is the set of meeting points that are directly connected to i. To model the fact that no vacant taxis on a road that is hailed by a passenger can refuse the request, a vehicle in node i cannot reach any nodes j e Ji directly without bypassing any of the adjacent nodes j Ji. ~n example of the transition graph that displays the transition probabilities for a 4-31 = node network is shown in Figure 1, in which {2,3) J2 = {1,3) 9, '3 = (1'2'4), '4 = {3), and k' represents the absorbing state of meeting point k, k = 1,2, 3,4. Let 'k be the total number of vacant taxis traversing meeting point k 0 = (&,k E K), and v = (Vk,k E K ) in search of customers. Denote Further, let 0" and v" be the corresponding row vectors of after n transitions. We have I 0 which is equivalent to O"+l = 0" + v"r R Q y"+' = v"q (5) We assume, under the initial conditions (i.e. n = 0), that the locations of the taxis are the destination pattern of the customers, and that there has been no meeting between taxis and customers yet, which corresponds to 0 the boundary conditions = and = O. The recursive application of the above two equations (4) and (5) can be simplified as 0 = D(R + QR + Q2R + Q3R +...)= D(I -Q)-'R - and (3) (4) (6)

7 Modeling the Bilateral v=d(i+q+q2+q3+ W v)=d(i-q)-' (7) 0 = lim 0" v= CV" where and n=o. As the row sums of the transition matrix Q must be less than unity, all of the eigenvalues fall within the unit circle in a complex plane. Hence, the matrix series in Eqs (6,7) is convergent. For futher discussions of the Markov chain, interested readers are referred to the work of Bell (1995; 2002), Akamatsu (1996), and Wong (1999). Consider an infinite random walk of taxis in the network. The vacant taxi movement can be described by a.. A =(I -Q)-'R 9 (8) where A and 0 is the probability of a taxi dropping off a customer in node i eventually meeting a customer in zone j. This also describes the way in which the initial taxi locations with pattern D travel in the network and pick up a passenger with pattern 0 eventually. Denote fa as the vacant taxi flow on link a that connects node i to node j, and we have Taxi Searching Behavior In this section, the determination of the transition probabilities g ' and '0 will be discussed. In the absorbing Markov chain approach, the probability of a taxi successfully meeting a passenger and leaving the system is defined as, where pi is the ratio of the number of taxi-passenger meetings to the number of taxis visiting the meeting point in equilibrium (which will be discussed later). Assuming that all customers can successhlly take a taxi, we have

8 88 K.I. Wong, S. C. Wong, M.G.H. Bell, and H. Yang Cvjrji = oi Assume that the intermediate return of meeting a customer in zone k is f,: =Pk[ zfi;tkj-piwk]-@?h:) 9 jej (12) t where frk is the expected profitability of visiting meeting point k if the PG = Db cdkj vacant taxi is in node i, kek is the probability of a passenger going to nodej from node k, and '4 is the corresponding collected fare, pi and P' 2 are the monetary values of waiting time and traveling time for taxis, respectively, hi^ is the travel time for taxis from t node i to node k, and wk is the taxi waiting time at node k. In the rest of the discussion, 'ki is assumed to be a linear function of the travel time. The first term on the right hand side of Eq. (12) is the expected gain, which takes into account the chance of meeting a passenger, the second term describes the time cost spent in the searching, and the third term accounts for the movement cost. The vacant taxis that cannot find a passenger at the meeting point will choose the next searching destination in accordance with the following distribution function: t where t where pk'j is the probability that a vacant taxis leaves meeting point J and goes to point k to search for customers. et is a non-negative

9 Modeling the Bilateral parameter that reflects the degree of uncertainty when taxis are searching for customers (e.g. the random arrival of customers and taxis at the meeting point). Eq. (14) assumes a logit-type distribution of taxi searching behavior, and Eq. (1 3) follows the conservation of transition matrix (2) of the absorbing Markov chain. Customers Searching For Taxis Customers that travel from origin i to destinationj can choose their taxi meeting point such that their total waiting time and taxi fare will be less. The probability of a customer going to a particular meeting point is specified as and mek kek i E I J E J (15) C where pf is the monetary value of walking time, p2 the monetary value C h&. of in-vehicle time, and p3 the monetary value of waiting time, is the hh.. travel time for customers going from node i to node k, is the travel time for taxis, 'Q the taxi fare between node k and nodej, and wk is the customer waiting time at taxi stand k. OC is a dispersion parameter that represents the uncertainties of passengers. Conservation of Taxi Movements Suppose that there are N cruising taxis operating in the network, and consider one unit period operation of taxis in the network with customer demand in a stationary state. From the principle of time conservation, the sum of total occupied taxi-hours and total vacant taxi-hours should be equal to the total taxi service time. Therefore, the following taxi service time constraint must be satisfied in view of the period modeled here (see Wong et al, 2001): C

10 90 K.I. Wong, S.C. Wong, M.G.H. Bell, and H. Yang c c Dkjh&. + c fata + C okw; = Nxl.0 jcjkck USA k d 7 (17) where fa is the vacant taxi flow and t' the travel time on link a A. Inter-Meeting Equilibrium of Taxis and Customers Consider a meeting point i, at which taxis are looking for passengers and passengers are looking for taxis. Arrivals accord with the Poisson p = v. process with mean rates of 1 for taxis and = Oi for passengers. A taxi that is joining the meeting point will reduce the passenger queue if there is one, or will otherwise increase the taxi queue by one, and vice versa, such that either a taxi queue or a passenger queue could be developed at any time. This problem can be formulated as a doubleended queuing model (see Kendall, 1951; Conolly et al., 2002; Matsushima and Kobayashi, 2004). We assume that there is a physical limit to the waiting queues for both taxis and passengers. A taxi can join in the queue as long as the taxi queue length is less than Lt, and passenger queues can be formed with a limit of L, although it is rare that the physical limit for passenger queues is binding, and in such case we can easily extend it to infinity (which will be discussed later). The passenger boarding time is assumed to be small and is thus neglected. Once a taxi or passenger joins a queue, then they will not leave until they have met a passenger or taxi. The queuing theory can be used to model the inter-meeting equilibrium of taxis and customers at a meeting point. Let, at any instant, the number of taxis that are waiting for customers be 1 IC t C Lt, and the number of customers that are waiting for taxis be Lc. If we define n as the net number of taxis in the queue such that t n = -Ic, then the probability of each state at the meeting point can be t denoted as pn, with -Lc 5 n 5 L. Consider a short interval At, we have, hat = Probability of one passenger arrival (or one taxi departure) - IAt = Probability of no passenger arrivals

11 Modeling the Bilateral PA' = Probability of one taxi arrival (or one customer departure) - )IAt = Probability of no taxi arrivals Neglecting the higher order terms of At (i.e. more than one arrival during the short interval), when time proceeds from to + At, then the transition of the system (as shown in Figure 2) becomes p,(t +4 =(I -mx@k-l(t) -41-mX1-@M) +(mx1 -@)etl(d - L~ +I 5 n I L' -1 (19) PLt (t + At)= (1 - hat)(pa)plt-,(t)+ (1 - hat)p,t (1) (20) The steady state solution of the system of transition in the long run gives n- dp (4-0 dt and 8(t)= pn, for an undersaturation system. Applying these conditions to Eqs (18)to(20) we have, correspondingly, - W-L. + hplc+l = 0 (21) C t Wn-I -(1 + p)pn + hpn+, = 0 n = -L + 1,..., L - 1, (22) WLq - hplt = 0 (23) Let p = ' p. Rearranging the equations, we have the balance equations of the system: - P-LC - PP-LC+l (24) C t (P, - PnJ = p(pn+l - P,) n = -L + 1,..., L - 1 Y (25) PLL1 = PPLt (26) Hence, we can show that L' -n Pn = P PLt n = -L',...,L' 7 (27)

12 92 K.I. Wong, S.C. Wong, M. G.H. Bell, and H. Yang The average queue length of taxis, If, and the average queue length of passengers,, can therefore be expressed as LC LC ic = c jp-j = PLt C jp [ 5 ] L'+j - plt p ~t j p j j=1 j=l j=1. (30) From Little's law, the waiting times of taxis and customers can be obtained by Wt =it (31) wc =jc/af 9 (32) where the actual arrival rates of taxis is iven by pf = ptl - 'L' J, that of passengers is given by y, and pf = " in the long run. This is also the actual inter-meeting rate of taxis and customers (i.e. the actual number of customers that successhlly meet a taxi, which is equal to the actual number of taxis that successfully meet a customer) at the meeting point, excluding those left due to the maximum limit of the queue. As in reality it is rare that there is a physical limit of the customer queue, we set Lc h' = A(1- qlc O0. With L' - - ', Eqs (3 1) and (32) become

13 Modeling the Bilateral These equations define the average waiting time of taxis and passengers at the meeting point in terms of the arrival rate of both taxis and passengers, and the physical limit of the taxi queues. It is worth noting that if the physical limit of the taxi queue is zero, i.e. if Lt = 0, then the system is reduced to an M/M/l queuing system in which taxis are the servers and passengers are the users. This is true for a case in which taxis pick up passengers on the roadside and taxi queues cannot be formed. Model Formulation In this section, we introduce the mathematical program for the incorporation of vacant taxi movements with the inter-meeting relationship between taxis and customers. It is formulated as a system of non-linear equations. vk =Ok 'Pk vk = Dk -t 'iqik i d (35) (36) mcjj (37) ici mek where

14 94 K.I. Wong, S.C. Wong, M.G.H. Bell, and H. Yang Eqs (35) to (38) represent the absorbing Markov chain problem with the transition matrix that is given by the logit model, and Eqs (40) to (43) determine the profitability of taxis, the travel cost of passengers, and the waiting times at each taxi-passenger meeting point. Eqs (35) and (36) are derived from Eqs (6) and (7); Eqs. (37) and (38) are from equations of Eqs (13) and (15); and Eq. (39) is the conservation of taxi service times, which is derived by substituting Eq. (9) into Eq. (17). It is noted that Eq. (36) has an alternative form of Vk = xbjkdk jej, where bjk is. an element of the matrix [I - QT'. The above problem is a set of non-linear equations in which the number of variables is equal to the number of equations. As Ok and Dk in Eqs (35) and (36) are over-specified, there is a redundancy in the set of equations, and Eq. (39) can rectify the system of equations. It is solved with the functions of the IMSL Math Library of FORTRAN, using a modified Powell hybrid algorithm and a finite-difference approximation to the Jacobian matrix. Numerical Example Consider the example network in Figure 1, and the demand matrices and travel time matrices that are shown in Table 1. Node 1 represents a business area in which many passengers are set down. Hence, there is generally a high supply of vacant taxis. Node 2 represents a residential area in which there is a high demand of taxi services to other nodes. Node 3 is restricted area in which taxis are not allowed to wait for 8' = 1.0. The taxi fare level is assumed to be 20 dollars per hour, i.e.,

15 Modeling the Bilateral % is 10 ($'hr). The passenger demands are higher from node 2 to other nodes than they are from node 1, as is shown in Table la. Table 1. Customer Demand Matrix and Travel Time Matrices (a) Customer demand matrix Dij (b) Travel time matrix for taxis hi (c) Travel time matrix for customers hc!i oo oo oo 1.oo

16 96 K.I. Wong, S.C. Wong, M.G.H. Bell, and H. Yang r11 P 0 k Meeting point k r22 0 r =a 934 q43 Figure 1. Example Network and the Corresponding Transition Graph for Taxi Searching Behavior

17 Modeling the Bilateral Moreover, there are also higher demands to node 1 than there are to node 2. Consequently, there is an oversupply of taxis at node 1, and an undersupply at node 2. The travel time matrix for taxis is different fi-om that of customers. There is a short-cut walk path between node 1 and node 2, which is not accessible to vehicles. The vehicle travel times from node 1 to node 3 and node 4 are smaller than those fi-om node 2. Hence, node 1 will attract the customers that originate from node 2. In the passenger demand matrix there are no taxi trips between node 1 and node 2, because the two nodes are within walking distance. Tables 2 to 5 show the numerical results of the example network. In Table 2, the passenger demand matrices show the dispersion of passengers in selecting their taxi meeting points. In Table 2a, we show that for nodes 1,3, and 4, most of the passengers stay in the same node to wait for taxis. However, at node 2, many passengers prefer to walk to node 1 to get a taxi to take advantage of the shorter walking time and the cheaper fare if they start their taxi journey from node 1. Because of the dispersion, the values in table of Db are different fi-om the customer demand matrix Dd in Table la. Table 2. Passenger Movement Matrices (a) Passenger movement matrix (walking trips) Dik (b) Passenger movement matrix (taxi trips)

18 98 K.I. Wong, S.C. Wong, M.G.H. Bell, and H. Yang The transition matrix is shown in Table 3, in which the elements are positive for those meeting point pairs that are connected by a link, and zero otherwise. Table 3. Transition Matrix 4ij In Table 4, the vacant taxi movement probabilities depict the fact that the diagonal elements generally have a higher value than the off-diagonal elements, because the vacant taxis have a higher chance of meeting passengers when they search more meeting points. Hence, the probability decreases with the distance and number of meeting points visited. However, there is the exception of the element (2,1), which indicates that a high percentage of taxis leave node 2 to go to node 1 to search for passengers, because there are more customers waiting for taxis at node 1 (44.8 passengersh) than at node 2 (15.2 passengersh), as is shown in Table 2b. The corresponding vacant taxi movement matrix is shown in Table 4b. The solution at each meeting point, that is, the total number of taxis that traverse a meeting point, the corresponding matching probabilities, and the waiting time of customers and taxis, is displayed in Table 5. It is noted that the taxi waiting time at node 3 is zero, because taxis are not allowed to wait for customers at that node. The sensitivity of the solution with respect to different model parameters is illustrated in the subsequent figures.

19 Modeling the Bilateral Table 4. Vacant Taxi Movement Probabilities and Vacant Taxi Movement Matrix (a) Vacant taxi movement probabilities aij (b) Vacant taxi movement matrix ag Di Table 5. Total Number of Taxis Traversing a Meeting Point, the Corresponding Matching Probabilities, and the Waiting Times of Customers and Taxis In Figure 3, the number of taxis is varied, and the probabilities of a taxi successfully picking up a passenger at meeting point k, Pk, and the number of vacant taxis that traverse meeting point "k are plotted for all nodes. Generally, the probabilities decrease with the size of the taxi fleet, and the vacant taxi flow increases with the number of taxis. This is because, as more taxis operate in the network, the taxi market becomes

20 100 K.I. Wong, S.C. Wong, M. G. H. Bell, and H. Yang more competitive. Hence, more customer searching is necessary to find a customer. P P P P P vuuuu h h h h h + Figure 2. Transition Graph of the Double-Ended Queuing System The changes in customer waiting time and taxi waiting time at the various nodes with respect to the dispersion parameters and et are shown in Figure 4. Generally, the waiting times are quite constant, except for node 2, at which the taxi waiting time increases sharply withe'. From Figure 4(a), we can see that when 0' increases, more customers move from node 2 to node 1 to wait for taxis, and the taxi waiting time at node 1 decreases, but the taxi waiting time at node 2 increases. Consequently, the customer waiting time at node 1 increases, and that at node 2 decreases. In Figure 4b, we show that when 0' increases, the taxi waiting time at node 4 increases, and the customer waiting time decreases. This is because node 4 is more profitable compared with the other nodes due to the taxi trips being longer from that node. Hence, when 0' increases, taxi drivers perceive more information, and more taxis visit node 4.

21 Vacant taxi flow \ / I Number of taxis X m c - Node 2 *Node 3 -c- Node 4 +Node 1 +Node 2 -A- Node 3

22 I 102 K.I. Wong, S.C. Wong, M.G.H. Bell, and H. Yung Figure 4. Customer Waiting Time and Taxi Waiting Time Against (a) the Dispersion Parameter for Customers, 0', the Dispersion Parameter for Taxis, r g -a- we. Node 2 +-WE - Node 3 +w-nodei I 0 02 OOOO I ec 0.015, 0 I b /--- ' I 0.02 OW 0.M) oa I I el

23 Modeling the Bilateral Conclusions A mathematical model was developed to describe taxi movements, taking into account the stochastic searching processes of taxis in a network. The absorbing Markov chain approach is employed to study the problem, the local searching behavior of taxis is specified by a logit form, and the 0-D demand of passengers is estimated as a logit model with a choice of taxi meeting point. A double-ended queuing system has been adopted to relate the customer and taxi waiting times at a meeting point. The problem has been solved with a set of non-linear equations, and some interesting results have been presented. The research provides a novel and potentially useful formulation for describing the urban taxi services in a network. For practical implementation of the model for solving realistic problems, the following issues need to be addressed in future studies, namely, the development of a more robust and efficient solution algorithm, incorporation of congestion effect in the network, and calibration and validation of real life network. Acknowledgements The first author gratefully acknowledges the financial support of the Fellowship scheme from the Croucher Foundation of Hong Kong. The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region (Project No: HKUST6107/03E and HKU7134/03E). References Akamatsu, T. (1 996) Cyclic flows, Markov process and stochastic traffic assignment. Transportation Research, 30B, Bell, M.G.H. (1995) Alternatives to Dial s logit assignment algorithm. Transportation Research, 29B, Bell, M.G.H. (2002) Transit network reliability: an application of absorbing Markov chains. In Michael A.P. Taylor (Ed.) Transportation and Traffic Theory in the 2 1 st Century: Proceedings of the Fifteenth International Symposium on Transportation and Traffic Theory, July, Adelaide, Australia, pp Conolly, B.W., Parthasarathy, P.R. and Selvaraju, N. (2002) Doubleended queues with impatience. Computers & Operations Research, 29,

24 104 K.I. Wong, S.C. Wong, M.G.H. Bell, and H. Yang Douglas, G.W. (1972) Price regulation and optimal service standards: The taxicab industry. Journal of Transport Economics and Policy, 20, Kendall D.G. (1951) Some problems in the theory of queues. Journal of the Royal Statistical Society, Series B, 13(2), Matsushima, K. and Kobayashi, K. (2004) Endogenous Market Formation with Matching Externality. In K. Kobayashi (Ed.) Structural Change in Transportation and Communications in the Knowledge Economy: Implications for Theory, Modeling and Data, Edward Elgar, pp Wong, K.I., Wong, S.C. and Yang H. (2001) Modeling urban taxi services in congested road networks with elastic demand. Transportation Research, 35B, Wong K.I., Wong S.C., Yang H. and Tong C.O. (2003) The effect of perceived profitability on the level of taxi service in remote areas. Journal of the Eastern Asia Society for Transportation Studies, 5, Wong, S.C. (1999) On the convergence of Bell s logit assignment formulation. Transportation Research, 33B, Yang, H. and Wong, S.C. (1998) A network equilibrium model of urban taxi services. Transportation Research, 32B, Yang, H., Leung, C.W.Y., Bell, M.G.H. and Wong, S.C. (2003) Characterizing the bilateral search behaviors of taxi drives and customers in networks. Proceedings of the Croucher Advanced Study Institute (ASI) on Advanced Modeling for Transit Supply and Passenger Demand, January, Hong Kong, pp. A2.1-A2.9. Yang, H., Wong, K.I. and Wong S.C. (2001) Modeling urban taxi services in road networks: progress, problem and prospect. Journal of Advanced Transportation, 35, Yang, H., Wong, S.C. and Wong, K.I. (2002) Demand-supply equilibrium of taxi services in a network under competition and regulation. Transportation Research, 36B,