Abstract
Bike-sharing systems are becoming important for urban transportation. In these systems, users arrive at a station, pick up a bike, use it for a while, and then return it to another station of their choice. Each station has a finite capacity: it cannot host more bikes than its capacity. We propose a stochastic model of an homogeneous bike-sharing system and study the effect of the randomness of user choices on the number of problematic stations, i.e., stations that, at a given time, have no bikes available or no available spots for bikes to be returned to. We quantify the influence of the station capacities, and we compute the fleet size that is optimal in terms of minimizing the proportion of problematic stations. Even in a homogeneous city, the system exhibits a poor performance: the minimal proportion of problematic stations is of the order of the inverse of the capacity. We show that simple incentives, such as suggesting users to return to the least loaded station among two stations, improve the situation by an exponential factor. We also compute the rate at which bikes have to be redistributed by trucks for a given quality of service. This rate is of the order of the inverse of the station capacity. For all cases considered, the fleet size that corresponds to the best performance is half of the total number of spots plus a few more, the value of the few more can be computed in closed-form as a function of the system parameters. It corresponds to the average number of bikes in circulation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
For \(\rho =1\), \(\nu _{\rho }\) is the uniform distribution on \({0,\ldots,k}\). For \(\rho \ne 1\), \(\nu _{\rho }\) is geometric: \(\nu _{\rho }(k)=\rho ^k (Z(\rho ))^{-1}\) where \(Z(\rho )=(1-\rho ^{K+1})/(1-\rho )\) is the normalizing constant.
In our simulation, each station can host up to 30 bikes. Having more than 30 % of stations that are saturated occurs when there is a fleet of more than 25 bikes per station for the line and more than 30 bikes per station for the 2D grid.
To see this, the proportion \(s\) of bikes per station is the sum of two terms: the mean number of bikes per station \(\sum _{k=0}^Kk\nu _\rho (k)\) and the mean number of riding users per station. This term is the product of the effective arrival rate \(\lambda (1-\nu _\rho (0))\) times the mean riding time. The mean riding time is \(1/\mu +(1-\nu _\rho (K))\sum _{k=0}^{+\infty } \nu _\rho (K)^k k/\mu '\) because it is the sum of the mean trip time \(1/\mu\) and \(k/\mu '\) if the user returns at the \((k+1)\)th attempt, i.e. with probability \((1-\nu _\rho (K))\nu _\rho (K)^k\). It leads to the result.
References
Borgnat P, Abry P, Flandrin P, Robardet C, Rouquier J-B, Fleury E (2011) Shared bicycles in a city: a signal processing and data analysis perspective. Adv Complex Syst 14(03):415–438
Chemla D, Meunier F, Wolfler Calvo R (2013) Bike sharing systems: solving the static rebalancing problem. Discret Optim 10(2):120–146
Contardo C, Morency C, Rousseau L-M (2012) Balancing a dynamic public bike-sharing system. In: Technical report, CIRRELT, September 2012. https://www.cirrelt.ca/DocumentsTravail/CIRRELT-2012-09.pdf. Accessed April 2014
DeMaio P (2009) Bike-sharing: its history, models of provision, and future. In: Velo-city 2009 conference, May 2009
DeMaio P, Gifford J (2004) Will smart bikes succeed as public transportation in the United States? J Public Transp 7:1–16 (ISSN 1077–291X)
Fayolle G, Lasgouttes J-M (1996) Asymptotics and scalings for large product-form networks via the central limit theorem. Markov Process Relat Fields 2(2):317–348 (ISSN 1024–2953)
Fricker C, Servel N (2014) Two-choice regulation in heterogeneous closed networks. arXiv:1405.5997
Fricker C, Gast N, Mohamed A (2012) Mean field analysis for inhomogeneous bike sharing systems. In: AofA 2012, international meeting on probabilistic, combinatorial and asymptotic methods for the analysis of algorithms
Froehlich J, Neumann J, Oliver N (2008) Measuring the pulse of the city through shared bicycle programs. In: International workshop on urban, community, and social applications of networked sensing systems, UrbanSense08
Gast N, Gaujal B (2010) Mean field limit of non-smooth systems and differential inclusions. ACM SIGMETRICS Perform Eval Rev 38(2):30–32 (ISSN 0163–5999)
Gast N, Gaujal B (2012) Markov chains with discontinuous drifts have differential inclusion limits. Perform Eval 69(12):623–642
George DK, Xia CH (2010) Asymptotic analysis of closed queueing networks and its implications to achievable service levels. SIGMETRICS Perform Eval Rev 38(2):3–5
George DK, Xia CH (2011) Fleet-sizing and service availability for a vehicle rental system via closed queueing networks. Eur J Oper Res 211(1):198–207. doi:10.1016/j.ejor.2010.12.015 (ISSN 0377–2217)
Godfrey GA, Powell WB (2002) An adaptive dynamic programming algorithm for dynamic fleet management, I: single period travel times. Transp Sci 36(1):21–39
Guerriero F, Miglionico G, Olivito F (2012) Revenue management policies for the truck rental industry. Transp Res Part E Logist Transp Rev 48(1):202–214
Kaspi M, Raviv T, Tzur M (2014) Parking reservation policies in one-way vehicle sharing systems. Transp Res Part B Methodol 62:35–50. doi:10.1016/j.trb.2014.01.006. http://www.sciencedirect.com/science/article/pii/S0191261514000162 (ISSN 0191–2615)
Katzev R (2003) Car sharing: a new approach to urban transportation problems. Anal Soc Issues Public Policy 3(1):65–86 (ISSN 1530-2415)
Kurtz TG (1981) Approximation of population processes. In: Society for Industrial and Applied Mathmatics, vol 36
Malyshev VA, Yakovlev AV (1996) Condensation in large closed Jackson networks. Ann Appl Probab 6(1):92–115. doi:10.1214/aoap/1034968067 (ISSN 1050–5164)
Mitzenmacher M (2001) The power of two choices in randomized load balancing. IEEE Trans Parallel Distrib Syst 12(10):1094–1104
Nair R, Miller-Hooks E (2011) Fleet management for vehicle sharing operations. Transp Sci 45(4):524–540
Nair R, Miller-Hooks E, Hampshire RC, Bušić A (2013) Large-scale vehicle sharing systems: analysis of vélib’. Int J Sustain Transp 7(1):85–106
Raviv T, Kolka O (2013) Optimal inventory management of a bike-sharing station. IIE Trans 45(10):1077–1093
Raviv T, Tzur M, Forma I (2013) Static repositioning in a bike-sharing system: models and solution approaches. Eur J Transp Logist 2(3):187–229.
Schuijbroek J, Hampshire R, van Hoeve W-J (2013) Inventory rebalancing and vehicle routing in bike sharing systems. In: Technical report, Schuijbroek. http://repository.cmu.edu/tepper/1491/. Accessed May 2014
Shaheen S, Cohen A (2007) Growth in worldwide carsharing: an international comparison. Transp Res Rec J Transp Res Board 1992:81–89
Tibi D (2011) Metastability in communications networks. arXiv:1002.07/96v1
Waserhole A, Jost V (2012) Vehicle sharing system pricing regulation: transit optimization of intractable queuing network. In: Technical report, INRIA
Waserhole A, Jost V, Brauner N (2013) Pricing techniques for self regulation in vehicle sharing systems. Electron Notes Discret Math 41:149–156
Acknowledgments
This work is partially supported by the EU project QUANTICOL, 600708.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fricker, C., Gast, N. Incentives and redistribution in homogeneous bike-sharing systems with stations of finite capacity. EURO J Transp Logist 5, 261–291 (2016). https://doi.org/10.1007/s13676-014-0053-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13676-014-0053-5