Optimal Deployment of Stations for a Car Sharing System with Stochastic Demands: a Queueing Theoretical Perspective

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1 Optmal Deployment of Statons for a Car Sharng System wth Stochastc Demands: a Queueng Theoretcal Perspectve Elsabetta Bond, Chara Boldrn, and Raffaele Bruno IIT-CNR Va G. Moruzz 1, 56124, Psa, ITALY {e.bond,c.boldrn,r.bruno}@t.cnr.t Abstract Car sharng holds a promse of reducng traffc congeston and polluton n ctes as well as of boostng the use of publc transport when used as a last-mle soluton n a multmodal transportaton scenaro. Despte ths huge potental, several problems related to the deployment and operatons of car sharng systems have yet to be fully addressed. In ths work, we focus on staton-based car sharng and we defne an optmzaton problem for the deployment of ts statons. The goal of ths problem s to fnd the mnmum cost deployment (n terms of number of statons and ther capacty) that can guarantee a pre-defned level of servce to the customers (n terms of probablty of fndng an avalable car/parkng space). Ths problem combnes nsghts from queueng theory (used to model the stochastc demand for cars/parkng spaces at the statons) wth a varant of the classcal set coverng problem. For ts evaluaton, we use a trace of more than 1, pckup and drop-off events at a free-floatng car sharng servce n The Netherlands, whch are used to model the nput demand of the car sharng system. Our results show that the proposed soluton s able to strke the rght balance between cost mnmsaton and qualty of servce, outperformng three alternatve schemes used as benchmarks. I. INTRODUCTION Car sharng systems are nnovatve moblty servces that are becomng ncreasngly popular n urban and sub-urban areas and have the potental to solve real-world problems of urban transports [1]. The prncple of a car sharng system s that customers can rent for lmted perod of tmes a car from a fleet of shared vehcles operated by a company or a publc organsaton. Nowadays, the most wdespread type of car sharng servce s one-way free-floatng car sharng [2]. In ths case, the return of the rented vehcle s possble at any parkng spot wthn the operatonal area of the car sharng servce [3]. The man advantage of ths approach s the great flexblty for car sharng members, who can pck up the nearest car and return t anywhere wthn a gven area. Examples of free-floatng car sharng servces are Car2go, DrveNow, and Enjoy. Whle the vast majorty of car sharng fleets around the world feature gasolne-powered cars, there s a general shft, both n car sharng and wthn the automotve sector n general, towards electrc cars. However, the combnaton of a free-floatng car sharng servce wth electrc vehcles *Ths work was partally funded by the ESPRIT project. Ths project has receved fundng from the European Unon s Horzon 22 research and nnovaton programme under grant agreement No s not straghtforward and t typcally reles on many and well-located chargng statons. Unfortunately, t s rarely the case that such an effcent and powerful nfrastructure s avalable, hence car sharng companes may run nto trouble. For example, n March 216 Car2go decded to replace ts electrc fleet n San Dego, US, wth gas-powered cars due to the lack of chargng statons n the cty [4]. An alternatve form of one-way car sharng s statonbased car-sharng (e.g., Autolb n France) n whch dedcated statons are deployed by the car sharng operator and users are requred to pck up and drop off shared vehcles only at one of the avalable statons. Clearly ths lmts the freedom of movement for the users. However, these statons can be easly equpped wth the necessary nfrastructure for rechargng electrc vehcles [5], whch makes staton-based car sharng systems a good canddate for electrc car sharng servces [6]. Another advantage of staton-based car sharng s that t provdes hgher relablty and predctablty of car locatons and parkng, the latter beng a partcularly attractve opton for cty centers, where fndng a parkng space can be a nghtmare for drvers of prvate cars. Whle the customers desre for flexblty can be addressed by smartly placng statons wthn the operatonal area, the huge drawback of staton-based car sharng s that t requres a sgnfcant captal nvestment to buld the necessary staton nfrastructure, makng t less fnancally attractve than a free-floatng servce. Thus, when deployng a staton-based car sharng system t s crucal to strke the rght balance between the costs for the operator and the qualty of servce provded to the customers. In fact, deployng fewer statons allows the operator to save money, but f the servce does not provde a good experence to the customer, the money saved on the nfrastructure s rapdly lost on mssed rentals and general customer dssatsfacton. Vce versa, an effectve deployment may ntally cost more but t may also ncrease rdershp and customer satsfacton n the end. Exstng plannng frameworks typcally rely on tme-space models, whch are models that assume a determnstc knowledge of the demand of vehcles at each tme nterval of the control perod [7], [8], [9]. However, n real-world scenaros the demand process of customers s stochastc and t exhbts seasonal patterns [1] as well as weather-dependent ones. Furthermore, n car sharng there s a complex nterdependence between travel demands and avalablty of vehcles

2 and parkng at each staton. In other words, the way users make rental requests affects n a complex manner the future avalabltes of cars and parkng spaces. Hence, modellng approaches that are able to take nto account a stochastc demand ([11], [12], [13]) look more suted to address ths complexty. However, to best of our knowledge, the lterature currently lacks a model that can consder smultaneously decsons related to the locaton and sze of statons whle takng nto account that customers demands, hence staton dynamcs, are stochastc and not determnstc. The contrbuton of ths paper s twofold. Frst, we develop an optmzaton model for the problem of determnng the number and locaton of the statons of a one-way car sharng systems n the presence of stochastc demand and congeston. We use a set-coverng model to formulate the problem of locatng statons and szng ther parkng capacty wth the am of mnmsng the deployment costs whle provdng a confgurable servce levels. Followng the approach adopted n [14], we model each canddate ste as a queueng-based system usng standard assumptons for the arrval and servce processes. Then, we develop a computatonally effcent, near-optmal soluton based on a greedy search method. As a second contrbuton, we apply our plannng methodology to a real case study usng a dataset contanng the pckup and drop-off events of a car sharng operator n The Netherlands wth a fleet of almost 4 shared vehcles. Our results show that the proposed optmzaton model s able to strke the rght balance between mnmsng the nfrastructure costs of the car sharng operator and mnmally affectng the servce provded to the customers (n terms of avalablty of shared cars and parkng spaces at statons). The remander of the paper s structured as follows. Secton II provdes an overvew of prevous related work. Secton III dscusses the prelmnares to our optmzaton problem,.e., how the demand wthn the operatonal area can be represented and how ths demand relates to the capacty of statons. In Secton IV we present the optmzaton problem for plannng the car sharng nfrastructure deployment n terms of locaton of statons and capacty of statons. Then, n Secton V, we evaluate the proposed optmzaton strategy usng as nput demand the pckup and drop-off events at a real-lfe car sharng system. Fnally, n Secton VI we draw our conclusons and present drectons for future research. II. RELATED WORK Faclty locaton s an optmzaton problem extensvely studed n the feld of logstcs and transportaton plannng. Faclty locaton models are often cast as set coverng problems, and the nterested reader s referred to [15] for a comprehensve survey on the topc. Prmarly nterestng for our work are the solutons proposed for the optmal deployment of statons for bke-sharng and car-sharng systems. Several works n the lterature ([16], [8], [9]) consder a demand process that s statc and determnstc. Ths lmts both the realsm (the actual demand s defntely not determnstc) and the power of the model. In fact, the practcal advantage of a modellng tool les n ts predctve power,.e., n ts capacty to capture the salent features of a complex dynamc system and, possbly, to represent t n a compact and synthetc way. Works that move towards ths drecton are [11], [12], [13]. The mathematcal tools that are used to acheve ths goal are queueng theory and flud models. In [11], a closed-queueng network s used to model a general rental system, and the closed-queueng framework s exploted for optmsng the number of shared vehcles n order to maxmse ther avalablty. The work n [12] focuses on the rebalancng problem n a sharng system assumng that vehcles are autonomous (.e., that they are drverless, hence they can autonomously reach under-served locatons f needed). The sharng system s modelled usng a fludc approach, where movng vehcles and arrvng/departng customers are seen as flows wthn the network. The goal s to fnd how large should be the flow of vehcles that autonomously move to statons that are under-served n order to rebalance the system. The same problem of rebalancng vehcles s nvestgated n [13], ths tme usng queueng theory. The authors argue that the latter model provdes the foundaton for the former. Our contrbuton falls wthn the same category of the works dscussed above, n that we explot queueng theory to model the dynamcs of the car sharng system. However, we depart from the state of the art n several ways. Dfferently from the related lterature, we combne the knowledge on the system dynamcs acqured through queueng theory wth a faclty locaton problem that ams at selectng the locaton of the statons as well as ther optmal capacty. The optmal capacty value s derved by treatng the staton as a queue, then estmatng the avalablty of vehcles at the statons as f they were jobs watng to be served at a queue. Dfferently from [11], [12], [13], we thus address the problem of statons wth fnte capacty K, whch s what s observed n real car sharng systems. Fnally, agan dfferently from the related lterature, we evaluate the proposed algorthm usng a trace of pckup and drop-off events at a real car sharng system. III. PRELIMINARIES In ths study, we assume that the study area s parttoned nto a set N of non-overlappng square cells. Each cell may contan a set of arrval and departure events of shared cars, or t can be empty. Each of these cells s a potental canddate locaton for the statons of the car sharng system. Wthout loss of generalty, we assume that statons are placed at the center of cells. We assume that users are wllng to travel at most a certan dstance n order to reach the nearest staton 1. We control ths wllngness usng parameter R. When R =users are very conservatve and want to fnd a statons exactly nsde the cell (let us assume ts locaton to be (x, y) n the grd) where they want to pck/drop a shared car. Wth R =1, users are wllng to explore at least the cells neghbourng (x±1,y±1) the one where they want to pck/drop a shared car. Wth R =2, they can addtonally explore the neghbours of the 1 For example, n the London bke sharng system a rule of thumb of 3m s used when choosng the dstance between statons (

3 neghbours (x ± 2,y± 2) of ther nterested cell, and so on and so forth. Snce users wll not explore beyond R, ther requests for cars/parkng spaces wll not be satsfed unless they fnd a staton wthn R. Thus, the coverage radus of each staton can be at most equal to R, snce each staton can serve requests that happen at most R cells away. The key dea of the proposed optmzaton model s that, gven the sets of arrval and departure events, the evoluton of the cars dropped off and pcked up at the car sharng statons can be modelled and, thus, predcted usng a queue-based model [17]. In fact, from the sets of arrval and departure events at each cell, a stochastc travel demand at the statons can be computed. More precsely, assumng a coverage radus R, let be the average rate of trps endng at staton (.e., the number of shared vehcles dropped off per unt tme), and µ the average rate of trps that depart from staton (.e., the number of vehcle pckups per unt tme). For the sake of model tractablty, smlarly to the related lterature [12], [11], [13], we assume that the arrval process s Posson, and that the tme between two consecutve pckups of parked cars at a staton s Exponental. Ths s clearly an approxmaton, snce pckups and drop-offs exhbt well-known daly and weekly trends n realty [18]. However, n Secton V we show that the model based on these assumptons stll mantans an mpressve predctve power. Under the above assumptons, f the statons had nfnte parkng spaces, they could be modelled as M/M/1 queues [17]. However, real statons have a fnte capacty, correspondng to the number of parkng spaces that they can ft n. In order to take nto account ths aspect, we model statons as M/M/1/K queues,.e., as queues wth Posson arrvals, Exponental servce tmes, and capacty (parkng spaces n our case) equal to K. Ths formulaton s very convenent, snce t allows us to descrbe wth closedform expressons the dynamcs of cars parked at statons. In fact, the probablty that, at steady-state, there are exactly, 1,...,K avalable shared cars at a staton s equal to the steady-state probablty of the M/M/1/K queue descrbng the staton. For the sake of clarty, we remnd that the steadystate probablty n that there are n cars parked n the system (or jobs, usng the queueng analogy) can be wrtten as follows: n = (1 ) n, (1) 1 K+1 wth n 2{, 1,...,K} and = µ 6=1(wth =1, n = 1 K+1, 8n [17]). From the car sharng operator s pont of vew there are two key aspects to consder when evaluatng the servce provded to the customer: the probablty that customers fnd an avalable car where they request one, and the probablty that customers fnd an avalable parkng space at the staton where ther trp ends. These two probabltes correspond, usng the queue analogy, to the probablty that the queue s not empty (1 ) and to the probablty that the queue s not full (1 K ), respectvely. Thus, when plannng the capacty of a staton, the goal of the car sharng operator s to choose the mnmum value of K such that these two probabltes are above some target performance levels, whch we denote as p car and p parkng, respectvely. Explotng Equaton 1 above, the followng mportant result can be derved. Lemma 1 (Optmal capacty): Assumng a coverage radus R, n order for the avalablty of cars and parkng spaces to be above the desred thresholds p car,p parkng 2 (, 1), the capacty K of a tagged staton should be equal to the mnmum soluton to the followng system of nequaltes: 8 < : K > log p car 1 p car 1 1 pparkng K > log 1 p parkng, where denotes the rato µ of the arrval and departure rates at staton when the coverage s R. Proof: The two nequaltes are obtaned by substtutng to 1 and 1 K the expressons for and K derved from Equaton 1, then nvertng to get K. Please note that the optmal capacty value K s only dependent on the stochastc demand at the staton ( = µ, whch n turn depends on the coverage R) and on the target avalablty of cars/parkng spaces p car and p parkng. Lemma 1 also tells us that the desred target performance cannot be always acheved. In fact, snce the argument of the logarthm must be postve, we have that (droppng subscrpt for the sake of clarty) p car < and, at the same tme, p parkng <1. It follows that, when <1, every p parkng s attanable for parkng space avalablty but only p car < s possble for the avalablty of cars. Vce versa, when >1, any p car s possble but only p parkng < 1 s attanable for parkng spaces. Intutvely, ths s due to the fact that, when <1, shared cars are pcked up quckly at statons once avalable (n fact, <1mples µ>,.e., a rate of departure greater than the rate of arrval). When cars are pcked up quckly, there s not much the operator can do about the avalablty of cars when plannng the staton nfrastructure, because smply ncreasng K has no effect on the number of avalable cars. In ths case, t s strategc that the car sharng operator performs vehcle redstrbuton (lke n [12]), whch would alter the value of, hence rebalancng the rato between pckups and drop-offs. The opposte holds true when >1: the staton tends to fll up quckly, hence customers wll most certanly fnd an avalable car but not necessarly a parkng space. IV. IMISING THE STATION INFRASTRUCTURE After havng lad out the buldng blocks of our model n the prevous secton, we now focus on a car sharng operator that has to decde how to deploy ts staton nfrastructure. Specfcally, the operator has to decde ) where to place the statons ) whch capacty to assgn to each staton. Ths problem s an nstance of a set cover problem [19]: we have to dentfy the nfrastructure that requres the mnmal captal nvestment and acheves the desred servce level n terms of coverage and avalablty of cars/parkng spaces. In order to approach ths set cover problem, we frst have to dentfy canddate statons. In prncple, statons could be placed at any cell of the grd. However, as we dscussed (2)

4 n Secton III, a staton can only serve those pckup/dropoffs happenng wthn R,.e., wthn the maxmum dstance that customers are wllng to tolerate. Hence, canddate statons should only be located n cells for whch there are pckup/drop-offs events wthn R. In the followng we denote the set of canddate statons as S. The control varables of our optmzaton problem are the capactes K, wth 2 S. Varable K s equal to the capacty of canddate staton f staton s actually selected, zero otherwse. When a staton s selected for deployment ts capacty K should satsfy the constrant n Lemma 1. In real lfe, the car sharng operator could be forced to keep the capacty of the statons below a certan value (e.g., because the muncpaltes may not be wllng to allocate too many parkng spaces to a sngle car sharng servce). We can ncorporate ths constrant nto our model by forcng the capacty of the staton to be smaller than a certan value K max, whch could be dfferent for dfferent statons (dependng on where they are postoned, for example). Here, wthout loss of generalty, we can assume K max s equal for all statons. Deployng statons s a bg cost for the operator. A fracton of ths cost s ndependent of the sze of the staton. We denote ths cost as C s and we refer to t as fxed cost. It corresponds to, e.g., the sum of costs for buldng the payment booth at the car sharng staton, mantenance, nsurance, etc. A fracton of the cost s nstead dependent on the sze of the staton. For example, we can magne that larger statons wll cost more to the car sharng operator n terms of cost for rentng the parkng space or for deployng a chargng spot at the parkng bay. We denote these latter costs wth C k and we call them varable costs. Relyng on the above dscusson, the optmzaton problem can now be cast nto an nteger programmng problem (IP) as n Problem 1. The objectve functon (Equaton 3) states that we want to mnmse both fxed costs and the varable costs (those proportonal to the capacty of the statons). We denote wth K > the ndcator functon, whch s equal to 1 when condton K > holds, equal to otherwse. Then, the frst constrant (Equaton 4) forces that each cell n whch drop-off/pckup events happen s served by at least one staton wthn the coverage radus R. We have denoted wth R j the set of potental statons that are wthn R from cell j. We also denote wth N the subset of N contanng only those cells n whch pckup/drop-off events take place. The second and thrd constrants (Equatons 5-6) are related to the target avalablty that the car sharng operator wants to acheve. The only dfference wth what we have dscussed n Secton III s that, n order to mantan the problem } and p () parkng = {p parkng, 1 }, wth arbtrarly smaller. Ths s necessary to keep the argument of the logarthms n Equaton 2 postve and t reflects what we dscussed n Secton III: assumng, e.g., <1, the only p car really attanable by staton corresponds to the smallest value between the desred p car and the of staton. The fourth always feasble, we redefne p () car =mn{p park, Problem 1 Infrastructure optmzaton X mnmze C s K> + C k K (3) K 2S subject to X K > 1, 8j 2N (4) 2R j K > log p () car 1 p () car! 1 (5) 1 p () parkng K > log 1 p () parkng (6) K apple K max (7) K 2 N (8) constrant (Equaton 7) forces the capacty to be smaller than or equal to the maxmum allowed capacty. Fnally, the ffth constrant (Equaton 8) states that K should be a natural number. A. Greedy approxmaton method Set covers problems lke the one n Problem 1 are known to be NP-hard. Luckly, greedy algorthms typcally work well n these cases and fnd a set cover not too far from the optmal cover [19]. Hence, we defne a greedy strategy for solvng Problem 1. Its pseudocode s provded n Algorthm 1. Ths greedy strategy works by pckng, at each step, the canddate staton that has the lowest cost per event covered (lne 1). The capacty of ths staton s equal to the value of K that satsfes Lemma 1 (lne 7). If ths value of K s greater than K max, we reduce the coverage radus of the staton and we repeat the search (lnes 11-16). Otherwse, the canddate staton s selected for deployment and we store the mappng between the selected staton and ts optmal capacty n K opt (lne 18). Then, we remove the staton from the canddate statons set (lne 19) and we update the set of events to be covered (lne 2). The latter update alters the demand at the remanng canddate statons, hence we also need to update the (Rj) j and K j at those canddate statons that are affected by the change n the set of events to be covered (lnes 21-24). When all events have been covered, the soluton to the optmzaton problem (K opt ) s returned (lne 27). V. EVALUATION In ths secton we evaluate the proposed optmzaton strategy aganst three benchmark algorthms. The demand for pckup and drop-off of vehcles s obtaned from the hstorcal data of a real-lfe free-floatng car sharng system operatng n The Netherlands. The dataset. It contans 1, 65 events correspondng to the pckup/drop-off tmes and GPS locatons for 349 vehcles of a real-lfe free-floatng car sharng fleet. The dataset covers the perod from May 17, 215 to July 1, 215 and t has a granularty of 1 mnute. We have dvded the servce area of the car sharng operator n 1m 1m cells, obtanng a grd wth 162 rows, 128 columns, and 2736 cells. In order to!

5 Algorthm 1 Greedy algorthm. E set of all pckup/drop-off events. N set of all cells n the scenaro 1: R = R. 8 2N 2: C (R) = {e 2E: dst(e, ) <R }. 8 2N 3: S = { 2N : C (R) \E 6= ;} 4: K opt = ; 5: for all statons 2S do 6: compute (R) 7: fnd K for (R) 8: end for 9: whle E6= ; do. based on events n C (R) \E accordng to Lemma 1 1: fnd staton 2S such that Cs+C kk C (R ) \E s maxmsed 11: f K >K max then 12: R = R 1 13: update C (R) 14: f C (R) = ; then. canddate staton removed 15: S = S {} 16: end f 17: else. canddate staton selected 18: K opt = K opt [{ 7! K } 19: S = S {} 2: E = E C (R) 21: for all statons j 2S: C (Rj) j \E has changed do 22: update (Rj) j 23: update K j for (Rj) j accordng to Lemma 1 24: end for 25: end f 26: end whle 27: return K opt run our algorthm, we need to compute the demand (, µ, ) at each canddate staton, based on the events that happen wthn ts coverage radus R. To ths am, we follow the smple method dscussed n [2]: for each potental staton, we count the number n a of arrvals, the number n d of departures, and we measure the busy tme T busy (.e., the tme the staton s not empty) durng the observaton perod T. Then, can be obtaned as na T, µ as n d T busy, and smply as µ. The benchmarks. We compare our plannng algorthm (whch s hereafter denoted as ) to three benchmarks. Frst, we consder a smple random strategy that places statons unformly at random and places as many of them as to guarantee that any event has a staton wthn dstance R. We denote ths strategy as. The capacty assgned to statons s fxed and equal to 6,.e, equal to the average capacty observed n the real staton-based car sharng system studed n [18]. The second random scheme we consder (denoted as RAND2) places as many statons as but ther locaton s selected unformly at random among all cells where pckup/drop-off events take place. RAND2 has an economc advantage wth respect to n that t uses an optmal number of statons (the number computed by ). We wll see later on that ths s not enough for meetng the requred qualty of servce: where statons are located and whch capacty s assgned to them s as mportant as ther number. The thrd benchmark (denoted as KMEANS) s a smarter algorthm that groups events based on k-means clusterng and then assgns a staton to each cluster center. The number of clusters,.e., the k of the k-means, s set equal to the number of statons placed by. KMEANS s oblvous to the staton coverage but groups events such that they can be served effectvely wth the desred number of statons k. Please note that k-means has been also used n [13] for staton plannng n ther evaluaton. For KMEANS and RAND2 there s no obvous way to select the capacty of statons. In order to make them compete on equal grounds wth, we use a smple strategy whereby we force them to have the same overall system capacty of ( P K ). Ths s obtaned by solvng the followng system of equatons: nfloor b K c + n cel d K e = P K n floor + n cel = S, (9) where K denotes the average capacty assgned by, and n floor and n cel the number of the largest prevous and the smallest followng nteger of K that should be assgned as capacty to statons. Model parameters. For our algorthm we need to select the target avalablty probablty for both shared cars and parkng spaces (p car and p parkng ). In ths evaluaton we set them to.8. We also have to set the maxmum dstance that customers are wllng to explore for fndng a car/parkng space, whch determnes the coverage radus of statons. We test the range R 2{1, 2, 3, 4, 5}, whch corresponds to a maxmum dstances between a cell and ts reference staton that goes from 28m (R =1) to 85m (R =5). Please note that we do not test case R =snce ths would mply that each cell should be equpped wth a staton: n fact, users would not be wllng to look for one outsde the cell where they want to drop/pck up the vehcle. In order to better evaluate the ablty of the proposed scheme to fne tune staton capacty, we wll gnore the constrant K <K max n ths evaluaton. Fnally, as for the cost values C s and C k, they wll be dscussed n Secton V-A.1. A. Results We consder separately two aspects: costs and performance n terms of servce provded to the customers. 1) Costs: We frst dscuss the performance of n terms of costs for the car sharng operator to deploy the nfrastructure. In order to provde a meanngful comparson, we contrast the costs for aganst the costs for, snce s the only benchmark for whch the number of statons deployed and the capacty of statons are completely ndependent of. We test three confguratons for. In the frst confguraton, we set C s =e and C k =1Ke. Ths corresponds to a case where there s no fxed cost for buldng a staton but every parkng space costs 1Ke. Wth ths confguraton, wll am at mnmsng the overall capacty n the system (.e., the sum of all capactes assgned

6 Cost (MEuro) Number of statons Fg. 1. R=1 R=3 R=5 Costs for deployng the nfrastructure Coverage radus R Fg. 2. Number of statons deployed. Cs= Cs=1KEuro Ck= Cs=5KEuro Cs= Cs=1KEuro Ck= Cs=5KEuro to the deployed statons) but not the number of statons. Vce versa, wth confguraton C s =5Ke and C k =e, wll try to mnme the number of statons, gnorng the capacty that t s assgned to them. Fnally, n the thrd confguraton, where C s =1Ke and C k =1Ke, tres to strke a balance between the number of statons deployed and ther sze. We observe n Fgure 1 that always outperforms, regardless of the specfc cost confguraton. Ths s due to a combnaton of factors. Frst, snce t prortses the coverage of areas where more events are takng place, s able to always use fewer statons than to cover the same set of events (Fgure 2). Second, s able to fne tune the capacty of the staton to the demand at the staton, hence t can use hgher or lower values of K dependng on the stuaton. On the contrary,, beng oblvous to the demand as well as to the costs, always uses the same capacty value, as shown n Fgure 3 It s nterestng to note the key role played by the coverage radus R n nfrastructure plannng. As Fgure 2 shows, the number of statons that we need to deploy (on the y- axs) decreases rapdly (regardless of the plannng strategy) % of statons R=1 R=3 R= Capacty assgned to statons [# parkng spaces] Fg. 3. Hstogram of capacty dstrbuton as we ncrease R, whch corresponds to the wllngness of customers to explore more ther surroundngs. Unfortunately, R s not somethng that the car sharng operator can control drectly. In a real lfe scenaro, the most realstc values of coverage are 1, 2, 3, whch correspond to dstances from roughly 3m to 6m. 2) Qualty of servce provded to customers: The goal of our optmsed plannng s to provde probablstc guarantees to the car sharng members n terms of avalablty of cars and parkng spaces. Wth the next set of plots we wll nvestgate whether ths objectve s acheved. In order to provde a far comparson, we use as benchmarks KMEANS and RAND2, whch, by defnton, use the same number of statons and the same total capacty of for a gven scenaro. Thus, the dfferent performance wll be due only to a dfferent dstrbuton of statons and capactes. Due to lack of space, we only show the results for C s =e and C k =1Ke, but the trend remans smlar wth the other cost confguratons. In Fgure 4 we plot the probablty that a parkng space s found. The black horzontal lne corresponds to the target probablty chosen for the avalablty of parkng spaces, equal to p parkng =.8. Each pont n the plot corresponds to one staton, and t s colored n red f the avalablty at the staton s below the desred threshold, n green otherwse. The vast majorty of ponts wth fall above the.8 threshold, wth an average of 96% of ponts n the realstc range R 2 [1, 3]. In ths same range, KMEANS features an average of 92% of statons correctly predcted, whle RAND2 shows the worst performance wth an average of 85%. It s also nterestng to notce that wth, for whch capacty values K depend on the at the statons, mssed predctons always affect extreme values of, ether close to zero or close to one. Ths s probably where the real system departs more sgnfcantly from the queueng theoretcal model. Next, we look at car avalablty. As shown n Fgure 5, all values at statons are n the range (, 1). Recallng the dscusson n Secton III, ths means that only p car < s attanable when optmsng K for car avalablty. Hence, we set p car =, wth =.1. Correct predctons wll thus le close to the bsector. As Fgure 5 shows, all algorthms perform qute well. However, clearly outperforms the others for small coverage values.

7 R=1 R=3 R=5 KMEANS RAND ρ Avalablty of parkng spaces Avalablty of parkng spaces <.8 >.8 Fg. 4. Observed probablty that drop-offs are mssed, as a functon of. R=1 R=3 R=5 KMEANS RAND ρ Avalablty of cars Avalablty of cars < ρ ε > ρ ε Fg. 5. Probablty that cars are found, as a functon of. VI. CONCLUSIONS In ths work we have consdered the problem of optmally deployng the staton nfrastructure of a car sharng system. From the car sharng operator s standpont, an optmal deployment should mnmse the cost for the statons (n terms of ther number and capacty) whle at the same tme meetng certan pre-defned levels n the qualty of servce provded to the customers. We have defned ths qualty of servce n terms of the probablty that a customer wll fnd a car/parkng space upon reachng a staton. Then, we have formulated an optmzaton problem that blends queueng theory wth a varant of a set cover problem. Specfcally, the optmal capacty of statons s obtaned descrbng the staton as an M/M/1/K queue, whle the optmal coverage s acheved through the set coverng problem. In order to solve ths NP-hard problem, we have then defned a greedy approxmaton algorthm, that, at each step, selects the canddate staton wth the smallest cost per covered event. Fnally, we have tested the proposed optmal algorthm usng a real-lfe trace of pckup and drop-off events at a freefloatng car sharng system. The results shown demonstrate that the proposed soluton sgnfcantly reduces the costs for staton deployment, whle at the same tme not mpactng the qualty of servce provded to the users, when compared aganst three benchmark approaches. We beleve that ths work has several avenues of future research. Frst, we want to nvestgate the valdty of alternatve queung models besdes M/M/1/K to nclude tmevarant arrval processes (e.g., for modellng peak and offpeak arrvals), of batch arrvals (e.g., for modellng relocaton schemes). 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