Proposal for optimal placement platform of bikes using queueing networks
 Shinya Mizuno^{1}Email authorView ORCID ID profile,
 Shogo Iwamoto^{2},
 Mutsumi Seki^{3} and
 Naokazu Yamaki^{3}
Received: 28 May 2016
Accepted: 18 November 2016
Published: 3 December 2016
Abstract
In recent social experiments, rental motorbikes and rental bicycles have been arranged at nodes, and environments where users can ride these bikes have been improved. When people borrow bikes, they return them to nearby nodes. Some experiments have been conducted using the models of Hamachari of Yokohama, the Niigata Rental Cycle, and Bicing. However, from these experiments, the effectiveness of distributing bikes was unclear, and many models were discontinued midway. Thus, we need to consider whether these models are effectively designed to represent the distribution system. Therefore, we construct a model to arrange the nodes for distributing bikes using a queueing network. To adopt realistic values for our model, we use the Google Maps application program interface. Thus, we can easily obtain values of distance and transit time between nodes in various places in the world. Moreover, we apply the distribution of a population to a gravity model and we compute the effective transition probability for this queueing network. If the arrangement of the nodes and number of bikes at each node is known, we can precisely design the system. We illustrate our system using convenience stores as nodes and optimize the node configuration. As a result, we can optimize simultaneously the number of nodes, node places, and number of bikes for each node, and we can construct a base for a rental cycle business to use our system.
Keywords
Closed queueing network Optimization Gravity model Cloud computingBackground
Examples of rental bicycle systems
City  Name of rental bicycle  Number of nodes  Number of bikes 

Niigata, Japan  Niigata Rental Cycle (4/2003–present)  20  164 
Edogawaku, Japan  ECycle social experiment (9/2009–present)  3  400 
Yokohama, Japan  Yokohama city community cycle (10/2009–11/2009)  10  100 
Toyama, Japan  Aville (3/2011–6/2011)  15  150 
Sakai, Japan  Sakai community cycle (9/2011–present)  4  450 
Barcelona, Spain  Bicing (12/2007–present)  250  3000 
Paris, France  Velib (7/2007–present)  1500  20,000 
London, UK  London cycle Hire scheme (7/2010–present)  400  6000 
Lyon, France  Vélo’v (5/2005–present)  250  3400 
Madison, WI, USA  Madison Bcycle (3/2013–present)  33  300 
In this paper, we attempt to solve problems such as where to place nodes, how many nodes should be prepared, and how many bikes should be available for each node. We formulate a model using the queueing network (Miyazawa 1993, 2006), and this model is calculated using mathematical analysis.
Previous fundamental research on queueing networks includes that of Gordon and Newell (1967), who proposed improvements to Jacksonstyle closed queueing networks (Jackson 1957). Baskett et al. (1975) developed BCMP networks, which are a general queueing model with complex classes and arbitrary service distribution. We can build a flexible model using these approaches. It is also important to compute a characteristic value for a closed queueing network; we often use the convolution algorithm (Buzen 1973) and mean value analysis (Reiser and Lavenberg 1980).
Applied works include George and Xia (2011) and Waserhole and Jost (2016), who describe vehicle rental systems using closed queuing networks. In particular, Waserhole and Jost discuss optimization over nonstationary demands. Their model needs some additional work, because it does not include the ability to set vehicles at a station. Therefore, we construct a model with vehicle capacity. In another approach to bikesharing systems, Etienne and Latifa (2014) looked at mobility patterns using modelbased clustering methodology and analyzed 2,500,000 trip data points. Boyac et al. (2015) proposed an optimization framework for car sharing in Nice, France. It would be interesting to analyze their data using the approach of Etienne and Latifa.
Similar bike rental systems are now being used all over the world. In Japan, the Ministry of Economy, Trade and Industry has authorized a plan to utilize electrically assisted bicycledrawn carts for delivery businesses (Ministry of Economy, Trade and Industry 2014). There are various such plans as social experiments in Japan (Yamakawa 1992; Abe and Kawashima 2003; Miida 2002; Kawamoto 2007). Zhang et al. (2016) analyzed China’s model (in Ningbo, Hangzhou, and Beijing) and described the rental station planning of bicycle sharing systems, as well as the allocation, operation, and dispatch of public bicycles (Zhang et al. 2016). Aeschbach et al. (2015) examined London’s Barclays Cycle Hire. Here, we consider improving these models with a generic rental bike system that does not depend on specific areas.
Using our model, we can easily visualize the settings of the system as they change with time. Our method for designing bike distribution systems does not depend on the country or the area being deployed. Thus, as an example, we use convenience stores as nodes to distribute bikes. Moreover, we use the Google Maps application program interface (API) to obtain parameters, such as transit time and distance between nodes.
Modeling using closed queueing networks
 1.
Our distribution system contains one class of bikes.
 2.
The system contains K nodes.
 3.
N is the total number of bikes in this system. N is limited. The number of bikes at node k is denoted by \(n_k\), where \(N = \sum _{k=1}^K n_k\).
 4.
The service period at node k follows an exponential distribution and has a mean of \(\frac{1}{\mu _k}\).
 5.
\(\alpha _k\) is the arrival rate of bikes that have reached node k from other nodes in the system.
 6.
\(p_{i,j}\) is the probability that a bike served at node i travels to node j, such that \(1\,\le\,i,\,j\,\le\,K\), \(p_{i,j}\,\ge\,0\), \(\sum _{j=1}^K p_{i,j} = 1\).

\(f_{i,j}\): Movement from area i to j, which we obtain from (3),

\(q_{i,j}\): Total movement from area i to area j,

\(r_{j,i}\): Total movement from area j to area i,

\(s_{i,j}\): Absolute value of the elevation difference between nodes i and j,

\(d_{i,j}\): Distance between areas i and j,

C: Constant value of the gravity normalization model,
Note that \(q_i\) and \(r_j\) are both asymmetric: riders tend to go to more popular nodes from less popular one. So, \(q_i\) and \(r_j\) need information about direction of movement. We also often need to consider the elevation of each node. If this is of no concern in an area, we set \(\gamma\) to 0. If we consider \(d_{i,j}\) to be important, then we increase the distance parameter \(\eta\). These parameters indicate what we emphasize, either distance or population, for the transition probability in this model.

\(L_k\): Number of bikes in the system at node k,

\(CP_k\): Capacity of node k,
We know that the problem of bikes tending to converge at a node occurs. We must transport bikes to distribute them at each node. The objective function (4) indicates that bikes are distributed as efficiently as possible.
Configuration of the proposed system
Initial settings
Example of postal code data used by our system
Postal code  State name  Region name  City name  Town name  Population 

4300805  Shizuoka  Koutou District  Nakaku Hamamatsushi  Aioicho  858 
4338111  Shizuoka  Hagioka District  Nakaku Hamamatsushi  Aoinishi  9814 
4338114  Shizuoka  Hagioka District  Nakaku Hamamatsushi  Aoihigashi  2159 
4328043  Shizuoka  Kousai District  Nakaku Hamamatsushi  Asadacho  813 
Example of information data for the nodes used by our system
ID  Node name  Postal code, address  Latitude, longitude  Service rate  Capacity 

1  Hamamatsu training school of information  4300929, Nakaku Hamamatsushi, Shizuoka  34.7071129, 137.7409474  5.0  10 
2  Thanks Hamamatsu Act Street  4300928, Nakaku Hamamatsushi, Shizuoka  34.7084987, 137.7340032  5.0  10 
3  Thanks Hamamatsu Sumiyoshi  4300906, Nakaku Hamamatsushi, Shizuoka  34.7309325, 137.7241608  5.0  10 
4  Thanks Hamamatsu Wagou  4338125, Nakaku Hamamatsushi, Shizuoka  34.7392201, 137.7079812  5.0  10 
Example for information data for the nodes used in our system
ID  From ID  To ID  Distance(m)  Time (s) 

1  1  2  6781  747 
2  1  3  7314  868 
3  1  4  8289  1314 
4  1  5  16,396  1366 
5  1  6  9175  1228 
6  1  7  8002  1476 
7  1  8  14,962  1041 
Optimization parameters
Settings parameters for GA
Gene item  Value 

Number of genes  100 
Number of generations  1000 
Intersection  Partially matched crossover 
Selection pressure  0.7 
Sudden generation  Insertion mutation 
Sudden incidence  0.03 
Parallelization method  Master–slave parallelization 
Parameters for the closed queueing network
Parameter  Value 

Number of bikes  100 
Total number of nodes  20 
Service rate  5.0 
Capacity of the number of bikes at each node  10 
Optimization procedure
We may need to analyze a varying number of fixed nodes. After we select the fixed nodes, we select the best nodes from the dynamic nodes to compute the objective function using a closed queueing network. Thus, we obtain the average queue length at each node. We should confirm the value of the objective function because we want to verify that the GA converges to plot the change of value for the objective function. Next, we display the results on Google Maps.
Numerical example
We use regional information from Hamamatsu, Japan. In this region, there are 466 postal codes. In addition, we register as nodes the 304 convenience stores in Hamamatsu. In this case, we ignore the elevation of each node, to simplify the computation. We take the populationlocated nodes i and j as \(q_{i,j}\) and \(r_{j,i}\).
The settings of the GA used in this example are shown in Table 5. Several parallel computing techniques for GAs have been proposed (Darrell 1994). In this example, we adopt the master–slave parallelization approach because it is easily implemented in Google Maps using the PHP language. In Table 6, we show the parameters for the gravity model and closed queueing network.
The condition of capacity event \(A_k\) for each node in (4) is as follows: the difference between the number of bikes and the capacity of a node is not larger than twice the node capacity, and is not less than 1/10 of its capacity. How to satisfy a condition such as this can be selected based the particular target model.

For \(k = 1, 2, \ldots ,K\),

if \(L_k > 2 \cdot CP_k\), then add \(PT_K = 1000\) to the objective function,

else if \(L_k < CP_k \cdot 0.1\), then add \(PT_k = 200\) to the objective function,

else add \(\left L_k  CP_k \right\) to the objective function.
Decision for the gravity parameter
Parameters for the gravity model
Parameter  Value 

Population parameter of the gravity model \(\alpha\)  1.0 
Population parameter of the gravity model \(\beta\)  1.0 
Population parameter of the gravity model \(\gamma\)  0.0 
Distance parameter of the gravity model \(\eta\)  0.5 
Computing the initial number of fixed nodes
Optimization results for 20 nodes
Optimization results for 20 nodes
Node ID  Number of bikes  Element of the objective function for each node 

15  5.722361  4.277639 
23  2.039668  7.960332 
25  6.334638  3.665362 
34  2.590727  7.409273 
42  12.10129  2.101291 
48  8.77132  1.22868 
56  6.257035  3.742965 
57  1.888292  8.111708 
65  3.077918  6.922082 
79  4.900534  5.099466 
81  7.030343  2.969657 
112  1.693075  8.306925 
171  1.141893  8.858107 
175  18.00817  8.008174 
179  0.752507  200 
209  7.62364  2.37636 
213  7.060485  2.939515 
232  0.425842  200 
247  0.794523  200 
294  1.785734  8.214266 
The 17 optimized nodes obtained after removing the penalty nodes
Node ID  Number of bikes  Element of the objective function for each node 

15  6.61131  3.38869 
23  2.252811  7.747189 
25  6.852832  3.147168 
34  2.709976  7.290024 
42  16.18106  6.181063 
48  10.05988  0.05988 
56  7.637594  2.362406 
57  1.978323  8.021677 
65  3.124947  6.875053 
79  5.043066  4.956934 
81  8.121089  1.878911 
112  1.529695  8.470305 
171  1.140414  8.859586 
175  11.52231  1.522305 
209  7.805536  2.194464 
213  5.894068  4.105932 
294  7.981989  2.018011 
Optimization results for 14 nodes
Node ID  Number of bikes  Element of the objective function for each node 

15  6.053414  3.946586 
48  7.028729  2.971271 
23  2.57486  7.42514 
57  2.271532  7.728468 
25  8.127648  1.872352 
34  3.183185  6.816815 
42  15.97184  5.971837 
56  9.570738  0.429262 
79  4.771828  5.228172 
81  11.44458  1.444584 
65  3.630369  6.369631 
175  11.32166  1.321662 
209  9.896153  0.103847 
213  4.153461  5.846539 
For an actual rental cycle system, administrators often transport bikes to other nodes by truck because of converging bikes at a specific node. In this model, if the optimal node is not chosen, bikes will converge at one node. We use the objective function effectively and we succeed in distributing bikes. Thus, we obtain better results by adjusting the optimization procedure. We suggest that this approach should be followed when designing a bike distribution system.
Conclusion
Currently, in various places, social experiments are being conducted concerning the sharing of regular or batteryassisted bicycles. To increase the effectiveness of these bike distribution systems, it is important to carefully arrange the distribution nodes of the system. In this study, we optimized the arrangement of nodes of Hamamatsu, Japan. If provided with regional information, such as postal codes, region names, and node information, our approach can be applied to other locations. Moreover, using the Google Maps API, we can compute the required parameters in a timely fashion.
In this rental bike system, we have the problem that bikes converge at a specific node. To resolve this problem, we prepared the node candidate information exceeding 300 and 422 regional information. From our numerical computation, we found that we obtained better results when using as few fixed nodes as possible. If we use a greater number of nodes, such as 20 nodes, we obtain a better result than GA when removing a node according to specific conditions. Then we consider that such a calculation condition is required to perform the optimal placement and arrangement of rental bikes.
Our proposed approach has several unique features. First, all parameter computations can be performed using cloud computing and the Google Maps API. Next, using the gravity model, we can compute the transition probabilities through population and distance. It is important to determine the transition probability of the queueing network, thus we conclude that it is effective to use the gravity model. Finally, we performed effective analysis using a queuing network. Based on the results, we are confident that our proposed approach can be used to generate effective arrangements of bike distribution nodes.
We have several directions for future work. In our optimization problem, we assumed that the capacity and service rate of each node have the same values. As such, we investigated realistic node data, for example, elevation, to determine appropriate parameters for computing the transition probability, and we expect the optimization to be more representative of actual results. We need to interview field staff and gather real data to improve the model. The model also does not include the travel time between nodes. In this study, we used a GA for optimization. A more precise calculation is likely needed to compare computing time with utility. We also consider a simulation of this model to be needed because of the amount of information, such as a comparison of the results of this model and simulation including travel time. We aim next to develop a more realistic model coupled with simulation data.
Declarations
Authors’ contributions
SM and NY have a conception and design of the study. SI construct this analysis platform to analyze and interpretation of data and MS has Collection and assembly of data and drafting of the article. SM has the critical revision of the article for important intellectual content. All authors read and approved the final manuscript.
Acknowledgements
Advice and comments given by Yasuyuki Muramatsu has been a great help in this research.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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