Modeling Urban Taxi Services in Road Networks: Progress, Problem and Prospect Hai Yang K.I. Wong

Transcription

1 Journal of Advanced Transportation, Vol. 35 No. 3, pp www. advanced-transport. corn Modeling Urban Taxi Services in Road Networks: Progress, Problem and Prospect Hai Yang K.. Wong s. c. Wong Traditionally, many economists have examined the models and economics of urban taxi services under various types of regulation such as entry restriction and price control in an aggregate way. Only recently have we modeled urban taxi services in a network context. A realistic method has been proposed to describe vacant and occupied taxi movements in a road network and taxi drivers' search behavior for customers. A few extensions have been made to deal with demand elasticity and congestion effects together with development of efficient solution algorithms. Calibration and validation of the network taxi service models have been conducted towards their practical applications. This paper presents an overview of the research that has been carried out by the authors to develop network equilibrium models and solution algorithms for urban taxi services, and offers perspectives for future researches. Background n most large cities taxis are an important transportation mode that offers a speedy, comfortable and direct transportation service. For instance, in the urban area of Hong Kong, taxis currently form about 25% of the traffic stream overall, in some critical locations, taxis form as much as 50% to 60% of the traffic stream (Transport Department, ). Taxis make considerable demands on limited road space and contribute significantly to traffic congestion even when empty and cruising for Hai Yang is at the Department of Civil Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, P.R. China K.. Wong and S.C. Wong are at the Department of Civil Engineering, The University of Hong Kong, Pokfulam road, Hong Kong, P.R. China Received: October 2000; Accepted April 2001.

2 238 Hai Yang, K.. Wong and S. C. Wong customers. n general the taxi industry is subject to various types of regulation such as entry restriction and price control. n Hong Kong, taxi operations are subject to service area demarcation as well. The urban taxis operate throughout the territory, while the others are fundamentally confined to the rural areas of the New Territories and Lantau sland. Currently, annual taxi service surveys (surveys at sampled taxi stands and roadside observation points) have been conducted since 1986 to gather the information on customer/taxi waiting time, taxi utilization and taxi availability for the city of Hong Kong (Transport Department, ). These types of information have been effectively used for the evaluation of taxi services and the government decision-making with respect to the increase in the number of taxis andor adjustment of taxi fares. Furthermore, a simple modal split model of taxis has been developed in Hong Kong for the prediction of taxi person and vehicle origin-destination matrices, which has been incorporated as an important component in the comprehensive transport studies (Transport Department, 1993). Aggregate Modeling Many economists have examined the economic consequences of regulatory restraints such as entry restriction and price control in an aggregate way (Abe and Brush, 1976; Beesley, 1973; Beesley and Glaister, 1983; Douglas, 1972; De vany, 1975; Foerster and Gilbert, 1979; Frankena and Pautler, 1986; Hackner and Nyberg, 1995; Manski and Wright, 1976; Orr, 1969; Schroeter, 1983; Shrieber, 1975). Recent notable studies include Arnott (1996) and Cairns and Liston-Heyes (1996). The general objective of those studies aims to understand the manner in which the demand and supply are equilibrated in the presence of such regulations, thereby providing information for government decision-making in terms of such regulations. The general analytical framework on the economics of taxi can be described below: dd 80 D= D(F,W), -<O, -<O df aw

3 Modeling Urban Taxi Services in Road Networks: aw w= W(Y), -<o, av TC = c(q + Y ) (3) where D is the demand for taxi ride, Q occupied taxi-hours (equal to the product of average ride time and demand D), Y vacant taxi-hours, F the expected fare or money price of a taxi ride, W expected waiting time, TC total taxi costs, and c cost per taxi hour of service time. Equation (1) states that the demand for taxi rides is a decreasing hnction of the expected fare and the expected waiting time (a proxy of service quality); (2) states that expected waiting time is inversely proportional to the total vacant taxi-hours; and (3) states that the cost of operating a taxi is a constant per hour. This highly aggregate model was originally proposed by Douglas (1972) and has been adopted by subsequent studies on the economics of taxis, without consideration of the spatial structure of the market. t is commonly realized that there are two principal characteristics that distinguish the taxi market from the idealized market of conventional economic analyses: the role of customer waiting time and the complex intervening relationship shown in Figure 1 between users (customers) and suppliers (firms) of the taxi service. n the taxi market, the equilibrium quantity (total taxi-hours) of service supplied will be greater than the equilibrium quantity (occupied taxi-hours) demanded by a certain amount of slack (vacant taxi-hours). t is this amount of slack that governs the average customer waiting time. The expected customer waiting time is generally considered as an important value or quality of the services received by customers. This variable affects customers decision as to whether or not to take a taxi, and thus plays a crucial role in the determination of the price level and the resulting equilibrium of the market (De vany, 1975; Abe and Brush, 1976; Manski and Wright, 1976; Foerster and Gilbert, 1979). A reduction in expected waiting time increases the demand for taxi service. However, from the point of view of each individual taxi firm, expected customer waiting time is different from the quality of the typical product. n most markets where quality is a variable, each firm can decide what quality to produce. n most taxi markets, expected waiting time is not amenable to differentiation, but depends on the total number of vacant

4 240 Hai Yang, K.. Wong and S.C. Wong taxi-hours. An individual firm cannot offer customers an expected waiting time different from that offered by other firms, although a large firm may be able to affect expected waiting time (Frankena and Pautler, 1986). r 1 Exogenous Determinants of Demand 1 of Supply Entry Restriction Regulated Fare Structure Taxi Demand Taxi Availability Taxi Utilization Fig. 1. The Demand-Availability-Utilization-Supply Relation in a Taxi Market. Reproduced by permission, C.F. Manski and J.D. Wright, Nature of equilibrium in the market for taxi services, Transportation Research Record 6 19, Copyright 1976, Transportation Research Board, National Academy of Sciences, Constitution Avenue, N.W., Washington, D.C , USA. Furthermore, the demand for and supply of taxi services are interrelated through two intervening variables: taxi availability (as measured by expected customer waiting time) and taxi utilization (as measured by expected fraction of time a taxi is occupied). On the demand side, potential customers will consider taxi availability as well as fare in making their mode choice decisions. From the supply perspective, taxi firms will operate in response to taxi utilization rate as well as trip revenues and costs. Moreover, taxi availability, through its influence on the level of taxi use, indirectly affects the taxi utilization rate; the utilization rate, through its influence on the level of supply, in turn affects

5 Modeling Urban Taxi Sewices in Road Networky: taxi availability. This demand-availability-utilization-supply relation is shown in Figure 1 (Manski and Wright, 1976). Recently, Yang and Lau et al. (1998) developed an aggregate simultaneous equations model based on the Hong Kong taxi survey data (refer to Figure 2). Number of taxis, taxi fare and disposable income are used as exogenous variables; while passenger and taxi waiting times, taxi utilization in terms of the percentage of occupied taxis -on the roads, taxi availability in terms of vacant taxi headway and passenger demand are used as endogenous variables. The proposed nonlinear simultaneous equation model is found to be able to predict general outcomes of introducing new taxi policies, but the accuracy of prediction for certain variables needs to be enhanced. Furthermore, Xu et al. (1 999) applied a neural network approach for analysis of the taxi service situation found in the urban area of Hong Kong. They compared the performance of the neural network model and the simultaneous equations model, and found that both models can be used to predict the demand-supply equilibrium for given values of exogenous variables and thus can be used for policy sensitivity analysis. For example, the model can assist in the government to allocate the taxis efficiently to cover the entire city in order to attain a better level of service. Network Modeling Overview To precisely understand the equilibrium nature of urban taxi services and assess road traffic congestion due to (both vacant and occupied) taxi movements together with normal traffic, it is necessary and important to model taxi services in a network context. n this respect, Yang and Wong (1997, 1998) made an initial attempt to characterize taxi movements in a road network for a given and fixed customer origin-destination demand pattern. A simultaneous equations system is proposed to describe the movements of both empty and occupied taxis and solved by a fixed-point algorithm. A case study with the simple network model for the urban area on Hong Kong sland, Hong Kong, was conducted (Wong and Yang, 1998b; Wong et al., 1999). The model deals explicitly with the impact of taxi fleet size and the degree of uncertainty of taxi drivers on customer

6 2-12 Hai Yang, K.. Wong and S. C. Wong 1 Taxis J+J+ Passenger Waiting Time Time Disposable ncome L K 1 Tasi Passenger Demand '-j7\yca.iulncrrricced.r/.. 1 Taxi Waiting Time 1 j y Total Population i i- Fig. 2. Postulated Synchronous Relationships Among the Endogenous and Exogenous Variables in the Simultaneous Equation Model utilization and taxi availability at equilibrium. t can thus help provide information for government decision-making about taxi regulations. Wong

7 1Lllodeeling Urban Taxi Services in Road Nehvorks: 213 and Yang (1 998c) reformulated the taxi service problem in networks as an optimization problem that leads to a more efficient and convergent iterative balancing algorithm for the case without traffic congestion. Yang ef al (2001) and Wong and Yang (l998a) further extended the simple network model of urban taxi services proposed by Yang and Wong ( 1997, 1998). The extensions include incorporation of congestion effects, customer demand elasticity, reformulation of the problem as two simultaneous optimization problems and development of a new solution algorithm. n the proposed model, normal traffic is included and assumed to follow conventional equilibrium routing behavior. nstead of the characterization of pure taxi movements in a network by a system of nonlinear equations, a simultaneous optimization of two equilibrium problems is proposed for taxi movements in congested road networks (Wong et al., 2001). One problem is a combined network equilibrium model that describes simultaneous movements of vacant and occupied taxis as well as normal traffic in a user-optimal manner for given total customer generation from each origin and total customer attraction to each destination. The other problem is a set of linear and nonlinear equations ensuring that the relation between taxi and customer waiting times and the relation between customer demand and taxi supply are satisfied. The model can be used for the assessment of road traffic congestion due to both taxi and normal traffic movements and evaluation of the level of taxi services in a net\\ ork context. Taxi movement in a road network Consider a road network G( V,A) where V is the set of nodes and A is the set of links n the network. Suppose stationary taxi movements and customer demands exist in the network. n any given hour, the number of customers demanding taxi rides from origin zone i to destination zone] is D,, (tripshour), D,, is assumed to be a function of customer waiting time at origin zone i. the monetary price and travel time cost for taxi ride from zone i to zone j. Let and J be the sets of customer origin and destination zones, respectively, we have the folloning trip end constraints

8 244 Hai Yang, K.. Wong and S. C. Wong Let t, (v,) be the travel time on link a E A and hi be the congested travel time on route k E R,, hl; = zaea t, (v,)sp;" where R, is the set of routes between 0-D pair (i,j) and 6Fk = 1 if route k between 0-D pair (i, j) uses link a and 0 otherwise. For simplicity, 1, (v,) is assumed to be an increasing hnction of total vehicular flow v, (including both taxis and normal traffic) on link a E A due to traffic congestion. Further let h, be the travel time via the shortest route from origin i E to destination j E J, hl, = min (hi, k E R,, ), all times here are measured as a fraction of an hour. Suppose that there are N cruising taxis operating in the network and consider one unit period (one hour) operations of taxis in the network with the customer demand in a stationary state. The total occupied time of all taxis is the taxi-hours required to complete all = D,,, i E, j E J trips, and is thus given by C,,c Th,,, where T,? is the occupied EJ taxi movements (vehh) from zone i E to zone j E J along the shortest congested route. Furthermore, the total unoccupied time consists of the moving times of vacant taxis from zone to zone and waiting (searching) times withn zones. This time is given by cjej clcl T\( {h,, + S}, where qr is the vacant taxi movements (vehh) from zone j E J to zone i E along the shortest congested route, s,, i E is the taxi waitingkearch time at zone i E 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 onehour period modeled here: The destination choice of empty taxis depends on the behavior of taxi drivers. t is assumed here that, once a customer ride is completed, a taxi

9 Modeling Urban Taxi Services in Road Networks: becomes vacant and cruises either at the same zone or goes to other zones to seek its next customer. n doing this, each taxi driver is assumed to attempt to minimize individual expected search time required to meet the next customer. The expected search time in each zone is a random variable due to variations in perceptions of taxi drivers and the random arrival of customers. This random variable is assumed to be identically distributed with a Gumbel density function. With these behavior assumptions, the probability that a vacant taxis originating at zone j E J meets a customer eventually at zone i E is specified by the following logit model: where q,, when i = j represents the probability of a taxi that takes a customer to zone] to search and meet the next customer in the same zone, 0 is a nonnegative parameter that can be calibrated from observational data (Wong and Yang, 1998b; Wong et al., 1999). The value of 0 (h- ) reflects the degree of uncertainty on customer demand and taxi services in the whole market from the perspective of individual taxi drivers. Note that the assumption of minimization of expected search time may not hold in certain situations. For example, taxi drivers do receive requests from call stations to pick up customers, and the taxi drivers do coordinate so that no more than one taxi driver is sent to the customer. Also, a taxi driver may try to maximize profitability, rather than minimize the expected search time when choosing certain locations for a potential customer such as airports. The profitability for a taxi service at an airport depends on the expected customer trip length from the airport that is generally much longer than the average length of taxi rides in the urban area. Thus, different search behaviors of taxi drivers may exist and a mixed search behavior model should be adopted in reality. n a stationary equilibrium state, the movements of vacant taxis over the network should meet the customer demands at all origin zones, or every customer will eventually receive a taxi service after the waiting and

10 246 Hai Yang, K.1. Wong and S. C. Wong searching process. Since there are DJ taxis to complete senices at destination zone j E J per hour. we thus have We consider separate demand fbnction for each 0-D pair (ij) and assume that customer demand depends on customer waiting time, the monetary price and travel time cost for taxi ride as given below: where is the expected customer waiting time at zone i 1, El, i E, j E J is the monetary cost that customer pays for a taxi ride from zone i to zonej, and hv is the travel time of the taxi ride from zone i to zonej. Dll is assumed to be monotonically decreasing with respect to each of W,, f, and hll. To simplie our analysis, the fare of a taxi ride E, is assumed to comprise a flag-fall charge of F, dollars (conunon for all 0-D pairs) and a variable component proportional to the trip time h, (say, T dollars per unit ride time): F,] = F, + ~ h,, i E, j E J. (1 1) A1 tematively D, =fill exp{-y[fo +th, +v,h, +v,t], r~,~ej (12)

11 Modeling Urban Taxi Services in Road Networks: where Dv is the potential customer demand from zone i E to zone j E J, v, and v, are the monetary values to the customer of unit waiting time and in-vehicle travel time, respectively, y is a scaling parameter which indicates the sensitivity of demand to full trip price. Note that we have used a slightly altered fare form in (13) of a taxi ride c, = Fo (1 + Kh,, ), where K is a proportionate constant (K = T/F~). Assumption of constant K = T/F, in this altered form means that the flag-fall charge and the charge rate per unit travel time will be changed in the same proportion. This is true in Hong Kong where the two components have been adjusted nearly at the same percentage over the past 10 years (Transport Department, ) (the model still holds for other places where this proportionate adjustment is not necessarily true). Thus it is enough to alter F, alone to investigate the impacts of taxi fare on the equilibrium outcome. Customer waiting time, an endogenous variable of the model, varies across zones. Within each zone the expected waiting time of customer can be described as a function of the density of vacant taxis in the subarea W; = W, (N,,sl), i E (14) where N, is the number of vacant taxis per hour that meet customers at zone i 1 (note that N1 = 0, at equilibrium), and s, is the expccted average (cruising and/or waiting) time that a vacant taxi spends in zone i E in finding a customer (s, is a variable to be determined from network equilibrium of taxi movements). Note that specification of the customer waiting time function (14) depends on the distribution of taxi stands over individual zones. n the case with a continuous taxi stand distribution (taxi can pick up customer anywhere on the streets), we can assume that the vacant taxis move randomly through the street, and the expected average customer waiting time is proportional to the area of the zone and inversely proportional to the (cruising) vacant taxi hours:

12 248 Hai Yang, K.1. Wong and S.C. Wong where A, is the area of zone i E and 71 is a common model parameter to all zones. This approximate distribution of waiting times can be derived theoretically (Douglas, 1972; Yang et nl., 2001). Algorithmic development When congestion is not taken into account and customer 0-D demand is given and fixed, the problem is described by a simultaneous equations system (4)-(9) only for the movements of both empty and occupied taxis. n this case the problem is simple and can be solved by a fixed-point algorithm (Yang and Wong, 1997, 1998). Nevertheless, the fixed-point algorithm for solving the taxi traffic problem cannot be guaranteed to be convergent and thus may be unsuitable for large-scale real applications. Wong and Yang (1998~) reformulated the whole movement of all empty and occupied taxis as an optimization model, from which a gravity-type distribution of empty taxis is derived. As a result, the taxi movement model is solved efficiently by the established iterative balancing method and can be incorporated into any standard transportation planning packages. To solve the general network equilibrium problem of urban taxi services with congestion effects and demand elasticity together with normal traffic, Wong et 01. (2001) proposed a simultaneous optimization model of two equilibrium problems. One problem is a combined network equilibrium model that describes simultaneous movements of vacant and occupied taxis as well as normal traffic in a user-optimal manner for given total customer generation from each origin and total customer attraction to each destination. The other problem is a set of linear and nonlinear equations ensuring that the relation between taxi and customer waiting times and the relation between customer demand and taxi supply are satisfied. The network equilibrium problem can be solved by the conventional multi-class combined trip distribution and assignment algorithm, whereas the linear and nonlinear equation problem is solved by a Newtonian algorithm (Press et al., 1992) with a line search. Calibration and Validation - A Case Study Here we present a case study with the city of Hong Kong (shown in Figure 3) to demonstrate the practical application of the taxi model. This

13 Modeling Urban Taxi Services in Road Networks: case study is restricted to the fixed-demand case without consideration of congestion effect and hence the equations (lo)-( 15) of the elastic demand model are irrelevant. The model parameters were calibrated by determining the most likely values such that the discrepancies between modeled and observed results were minimized. The Hong Kong city, which is formed by Hong Kong sland and Kowloon (Urban), and New Territories (Rural), was divided into 274 zones. The year of 1992 was used as the base year for the analysis. To assess the level of taxi service in Hong Kong, information on screen-lines and cordon counts were taken from the Annual Traffic Census data (ATC, Transport Department, 1992). Passenger and taxi waiting times at selected operative taxi stands and roadsides were also collected in the same year 1992 from the Level of Taxi Services surveys (LTS, Transport Department, 1992). The selected sites on Hong Kong covered by the surveys in 1992 were also reported in the survey report (LTS, Transport Department, 1992). The taxi passenger demand pattern was obtained by a comprehensive travel characteristics survey conducted in 1992 (TCS, Transport Department, 1993). The taxi travel times between traffic zones were determined by means of a network model using EMME/2 package (NRO, 1997), which were calibrated also based on the travel characteristics survey. With the demand pattern and network model, the key model parameter of 8 was calibrated such that the difference between modeled and observed screen-line and cordon counts was minimized. The calibration and validation process was done for the morning peak period (7:45-9:30 a.m.) as suggested from the Second Comprehensive Transport Study (CTS-2, Transport Department, 1993). The taxi fleet size in Hong Kong in 1992 was vehicles. t was assumed that 90% of taxis were in service during the morning peak period and the value of time for the taxi driver was 40 (HK$/h) for the toll charge (TCS, Transport Department, 1993). The tolls in all charged tunnels in Hong Kong in 1992 were used in the analysis; for example, the cross-harbor toll is HK$lO for taxis in From the analysis, it was found that the value of 8 at minimum error was 11.0 (h-). A bar chart for the comparison of the modeled and observed taxi movement across the screenlines and cordons is shown in Figure 4 (from the ATC report, the occupied and vacant taxis cannot be separated, and therefore only the total observed taxi counts are shown for the comparison). A good agreement between modeled and observed results is obtained for the whole Hong Kong city.

14 250 Hai Yang, K.. Wong and S.C. Wong M C 2 M C 0 O n......

15 ~ Ovacant ~ Ooccupied 2- Comparison of modeled (occupied and vacant) and observed no. of taxi across screenlines and cordons 2 m g ~ m c c 0 0 lml00 observed (modclcd) 1 (modeled) SOM) 0 Screenlines and Cordons Figure 4. Thc nar Chart of C'ompiirisoll hctwccn Modclcd and Ohscrvcd Taxi Movcnicnts across Scrccn-lines and Cordons in tlic City 01' long Kong T & n n 4

16 252 Hai Yang, K.. Wong and S. C. Wong t can be seen that while Hong Kong External Cordon, Cross Harbor Cordon, Screen-lines S-S, R-R and T-T contain a small proportion of vacant taxis, the Central Kowloon areas (as seen from Kowloon External Cordon and Screen-line C-C) are subject to a high percentage of vacant taxi movements. n particular, the very small proportion of vacant taxis on the Cross Harbor Cordon is attributed to the high toll charge for crossharbor traffic, which produces a large frictional effect for vacant taxi movements. ndeed in Hong Kong, very few taxi drivers drive an vacant taxi from Kowloon to the Hong Kong sland to meet a customer and vice versa, because a driver has to a pay a cross-harbor tunnel toll in doing so. The results obtained are in good consistence with recent observations of taxi movements in Hong Kong (CTS-2 & TCS, Transport Department, 1993). n the taxi model, the taxi search time consists of two components: the time for a vacant taxi circulating within a zone and the time the vacant taxi spent waiting (say at taxi stands). While the taxi waiting times were observed regularly from The Level of Taxi Service surveys at different locations in Hong Kong every year, it is generally difficult to obtain the circulating time in a particular zone especially for large zones. t is envisaged that the circulating time depends on the size of the zone concerned and the highway network topography within the zone. To confirm whether the modeled taxi search time reflects reality, the following relationship is assumed and checked for validity from statistical analysis: where si, wi and Ri are respectively the taxi search time, taxi waiting time and zone size for the zone i, u is a model parameter relating the circulating time and zone size. While the first term of the right hand side in equation (16) represents the taxi waiting time, the second term represents the circulating time in a zone. To check for the validity of the established relationship in equation (1 6), a regression analysis was carried out and the results are shown in Table 1. The coefficient u for the whole Hong Kong city was found to be 8.84 (mih), with R2 of and t-statistics of These statistics show a good confidence for the anticipated relationship in equation (1 6). Therefore, the taxi model seems to produce reasonable estimates of the

17 Modeling Urban Taxi Services in Road Networks: taxi search times consistent with the recent direct observations. t was also found that there are two sets of clusters of data in the sampling points, one for the urban area covering Hong Kong sland and Central Kowloon and the other for rural area of Hong Kong including New Territories. Separate analyses of these data sets were also conducted and the results are also shown in Table 1. The coefficient u was found to be (mi-) for urban area and 6.16 (minlkm) for rural area. They may be explained by the fact that the travel speed in rural area is generally higher than that in urban area, due to the different network topography in these areas (urban area generally has higher junction density as compared with the rural area and hence lower circulating speed). The statistics have also shown strong relationships for the separate spatial models. Area Coefficient u (midun) The whole city of Hong Kong 8.84 Urban Rural 6.16 R square t-statistics Future Research Opportunities Although the up-to-date research provides some important progresses in network modeling of urban taxi services and model calibration and validation, by no means is this problem well solved. There are many open research avenues; two of them are described below. Modeling service area repulation Existing economic analyses of the (de)regulation of taxi services have overwhelmingly focused on price and entry control. There remains one important issue to be examined: service area regulation of multi-class taxis. For instance, in Hong Kong New Territories taxis can only take customers to as far as Tsuen Wan (the western part of Kowloon) or Shatin (the northern part of Kowloon), but cannot pick up customers in the urban area, whereas urban taxis can pick up customers in both urban areas and New Territories. The objective of this service area regulation is to guarantee the taxi service quality in terms of taxi availability in rural

18 254 Hai Yang, K.. Wong and S. C. Wong areas. Evidently, in this case spatial structure of the taxi market is essential, and the existing single class network equilibrium models for taxi services have to be extended to the multi-class taxi service situation to describe how various kinds of taxis (vacant and occupied taxis for both urban and New Territories taxis) cruise in a road network to search for various kinds of customers and provide transportation services. Modeling multi-period taxi service equilibrium with endogenous service intensity One of the important characteristics of taxi mode that distinguishes from other public transportation services is its flexibility and availability for the whole day. n particular, taxis may be the only public transportation services available from middle night to early morning in many large cities such as Hong Kong. Therefore, one has to take into account taxi service quality and profitability during different period of a day. n general, customer demand for taxi services and taxi service intensity (number of taxis in service) exhibits substantial time of day changes. A typical hourly change pattern of taxis for the three Hong Kong cross-harbor tunnels is depicted in Figure 5. t is thus interesting and meaningful to develop multi-period taxi service models with endogenous taxi service and utilization intensity. One possible way to model time of day taxi services is to divide the whole service period (one whole day) into a number of sub-periods. During each sub-period, taxi supply and demand characteristics are assumed to be uniform. Customer demand is period specific and described as a function of waiting time for a given fare structure. Taxi operating cost consists of two components: one component being a function of total service time and the other component being period dependent. Each taxi driver can choose which period(s) and how much portion of a selected period to provide service. The demand-supply equilibrium is characterized by that drivers cannot increase their individual profits by extending total service time or shifting service periods. The model can thus ascertain at equilibrium the intensity of use of taxis, utilization rate for taxi and level of service quality in each period. This information is useful for prediction of the effects of alternative regulations such as time of day differentiation of service charge on the taxi industry.

19 Modeling Urban Taxi Services in Road Networks: 255 / 0 L / :

20 256 Hai Yang, K.. Wong and S. C. Wong Acknowledgements This study was substantially supported by a research grant from the Research Grants Council of the Hong Kong Special Administrative Region (Project No. HKU E). References Abe, M.A. and Brush, B.C. (1976) On the regulation of price and service quality: The taxicab problem. Quarterly Review of Economics and Business, Autumn 16, Arnott, R. (1996) Taxi travel should be subsidized. Journal of Urban Economics 40, Beesley, M.E. (1973) Regulation of taxis. Economic Journal 83, Beesley, M.E. and Glaister, S. (1983) nformation for regulation: The case of taxis. The Economic Journal 93, Cairns, R.D. and Liston-Heyes, C. (1996) Competition and regulation in the taxi industry. Journal ofpublic Economics 59, De vany, A.S. (1975) Capacity utilization under alternative regulatory constraints: An analysis of taxi markets. Journal of Political Economy 83, Douglas, G.W. (1972) Price regulation and optimal service standards: The taxicab industry. Journal of Transport Economics and Policy 20, Foerster, J.F. and Gilbert, G. (1979) Taxicab deregulation: Economic consequences and regulatory choices. Transportation 8, Frankena, M.W. and Pautler, P.A. (1986) Taxicab regulation: An economic analysis. Research in Law and Economics 9, Hackner, J. and Nyberg, S. (1995) Deregulating taxi services: A word of caution. Journal of Transport Economics and Policy 29, NRO (1997) ME12 User 's Manual. NRO Consultants nc. Manski, C.F. and Wright, J.D. (1976) Nature of equilibrium in the market for taxi services. Transportation Research Record 619, Orr, D. (1969) The taxicab problem: A proposed solution. Journal of Political Economy 77, Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (1992) Numerical Recipes in Fortran 77: The Art of Scient~c

21 Modeling Urban Taxi Services in Road Networks: _ Computing. Second Edition, Cambridge University Press, Cambridge, New York. Schroeter, J.R. (1983) A model of taxi service under fare structure and fleet size regulation. Bell Journal of Economics 14, Shrieber, C. (1975) The economic reasons for price and entry regulation of taxicabs. Journal of Transport Economics and Policy 9, Transport Department ( ) The level of taxi services. TTSD Publication Series, Hong Kong Government, Hong Kong. Transport Department (1993) Updating of second comprehensive transport studyjinal report. Hong Kong Government, Hong Kong. Transport Department (1998) The annual trafjic census Hong Kong Government, Hong Kong. Transport Department with MVA Asia (1993) Travel characteristics survey: Final Report. Hong Kong Government, Hong Kong. Yang, H., Lau, Y.W., Wong, S.C. and Lo, H.K. (1998) A macroscopic taxi model for passenger demand, taxi utilization and level of services. Transportation 27: Yang, H. and Wong, S.C. (1997) Modeling urban taxi services: A network equilibrium approach. Proceedings of the Second Conference of Hong Kong Society for Transportation Studies, The Hong Kong Polytechnic University, 6 December 1997, pp Yang, H. and Wong, S.C. (1998) A network model of urban taxi services. Transportation Research 32B, Yang, H., Wong, S.C. and Wong, K.. (2001) Demand-supply equilibrium of taxi services in a network under competition and regulation. Transportation Research B (in press). Wong, K.., Wong, S.C. and Yang, H. (2001) Modeling urban taxi services in congested road networks with elastic demand. Transportation Research 35B, Wong, K.., Wong, S.C. and Yang, H. (1999) Calibration and validation of a network equilibrium taxi model for Hong Kong. Proceeding of the Fourth Conference of Hong Kong Society for Transportation Studies, Hong Kong, 4 December 1999, pp Wong, S.C. and Yang, H. (1998a) Modeling the level of taxi services in congested road networks. Proceedings of Tristan ZZZ, Volume ZZ, Puerto Rico, June 17-23, Wong, S.C. and Yang, H. (1998b) Calibration of a network equilibrium model for urban taxi services. Proceedings of the First Asia Pacijc

22 258 Hai Yang, K.. Wong and S. C. Wong Conference on Transportation and the Environment, Singapore, May 13-15, 1998, pp Wong, S.C. and Yang, H. (1998~) Network model of urban taxi services: improved algorithm. Transportation Research Record 1623, Xu, J.M., Wong, S.C., Yang, H. and Tong, C.O. (1999) Modeling the level of urban taxi services using a neural network. Journal of Transportation Engineering, ASCE, Vol. 125, No.3,