Relocation in Car Sharing Systems with Shared Stackable Vehicles: Modelling Challenges and Outlook

One-way car sharing, in which customers are not forced to return the vehicle at the starting point of their journey, is the most popular among customers, due to the great flexibility it provides. One-way systems can be also classified into free-floating or station-based according to their parking restrictions. In fact, the former refers to a system in which the return of the rented vehicle is possible at any parking spot within the operational area of the car sharing service¹¹1For example, https://www.car2go.com/., while the latter requires that pickups or drop-offs of vehicles occur at designed parking stations deployed by the operator²²2For example, https://www.autolib.eu/en/.. Advantages of station-based systems is that they ensure higher reliability and predictability of car locations and parking, which make possible advance reservations from the users. One-way car sharing is not without drawbacks for the car sharing operators. With one-way car sharing, cars will follow the natural flows of people in a city, hence accumulating in commercial/business areas in the morning and in residential areas at night [3]. As a result, the availability of cars can become extremely unbalanced during the day, and certain areas may end up being underserved due to lack of available cars.

Previous research has proposed several approaches to solve the vehicle unbalance problem, including: user-based relocation, i.e., price incentives for the users to relocate the vehicles themselves [4]; operator-based relocation, i.e., workforce that moves vehicles from where they are not needed to where there is a significant demand [5, 6]; and optimal planning of station deployment to achieve better service accessibility and a more favourable distribution of vehicles [7]. It is important to point out that the relocation process is intrinsically inefficient: as one driver per car is needed, to relocate several cars a large workforce or many willing customers are necessary. This significantly complicates the relocation with respect to, e.g., bike sharing services, where a single worker with a van can redistribute a large amount of bicycles.

In order to address the above limitations, it is fundamental to reduce the ratio between the workforce size and the number of vehicles that can be relocated. Recently, innovative technologies have been proposed to enable more efficient vehicle rebalancing in CS systems. On the one hand, the rebalancing problem is solved in [8] using empty robotic vehicles autonomously driving between stations. On the other hand, new vehicle concepts with stackable capabilities have been recently released or are under development, which can be stacked into a train (through a mechanical and electric coupling) and/or folded together. Then, the train can be driven either by a car sharing worker (up to 8 vehicles) or by a customer (up to 2 vehicles). An illustration of this type of vehicle prototyped in the ESPRIT project [9] is provided in Fig. 1. Such stackable cars come with the promises of significantly improving the system manageability of future car sharing services. However, the evaluation and design of new car sharing services using these innovative stackable cars call for new modelling techniques able to accurately characterise their peculiar properties.

Various models for assessing the performance of one-way car sharing systems have been proposed in the literature. A class of modelling approaches relies on time-space models that describe the interactions between the operational decisions (i.e., movement of staff, relocation activities) and the number of vehicles at the station [5, 6]. The main drawbacks of this approach is the explosion in size of the state space, and the limited ability to deal with uncertain conditions due to the stochastic nature of customer arrivals. Thus, stochastic models have been recently proposed based on queueing theoretical approaches [10, 11] or fluid approximations [8]. However, how to model car sharing systems with stackable vehicles is still an open challenge.

Within this framework, the contributions of this paper can be summarised as follows:

• $\bullet$

  We validate a queueing theoretical model of one-way car sharing systems that was proposed in prior work [10] against a trace of real car sharing operational data, confirming the effectiveness of this modelling approach;

• $\bullet$

  We show, with a real-case study, that increasing the fleet size cannot solve the problem of vehicle shortages in hot spots because vehicle availability depends mostly on traffic patterns in the car sharing system;

• $\bullet$

  We present preliminary results about two simple heuristics for user-based relocation with stackable cars, showing that in some cases user-based relocation can increase the car availability at stations up to 200%;

• $\bullet$

  We develop a new queuing model to study the evolution of vehicle redistribution in a car sharing station under general user-based relocation policies for stackable cars and we derive an upper-bound on the relocation flow per station.

Individual queues representing the CS service centres are then networked together to reflect the CS network dynamics. To this aim, the probability matrix $\mathbf{P}=\{p_{ij}\}_{i,j\in\mathcal{S}}$ is introduced, whereby $p_{ij}$ denotes the probability that a customer leaving service centre $i$ with a car will head for service centre $j$. In addition, in order to model the fact that it takes a certain amount of time to go from service centre $i$ to service centre $j$, we introduce, as in [10], delay queues in the model. These delay queues are modelled as infinite-servers queues, and we denote their set with $\mathcal{I}$. There will be a delay queue between service centres $i$ and $j$ if $p_{ij}\neq 0$, thus $|\mathcal{I}|\leq|\mathcal{S}\times\mathcal{S}|$. Each server in an infinite-servers queue keeps a car for an exponential amount of time with mean $T_{ij}$ (where $T_{ij}$ is the expected travel time from $i$ to $j$)³³3The model does not consider traffic congestion, thus each vehicle travels (i.e. is served) independently and in parallel with the others. This implies that the overall service rate of the delay queue is proportional to the number of vehicles in the queue, as stated in Equation 1..

We can then summarise the characteristics of the CS queueing network as follows. We denote the number of shared cars in the CS system with $N$ and we let $\mathcal{K}=\mathcal{S}\cup\mathcal{I}$. The service rate of each queue $i\in\mathcal{K}$ is given by:

$$\mu_{i}(n_{i})=\begin{cases}\mu_{i}&\textrm{if }i\!\in\!\mathcal{S}\\ \frac{n_{i}}{T_{jk}}&\textrm{if }i\in\mathcal{I},j=o(i),k=d(i)\end{cases}$$

(1)

where $n_{i}\in\{0,1,\ldots,L\}$ denotes the number of vehicles at node $i$ and, for each $i\in\mathcal{I}$ $o(i)$ and $d(i)$ denote the upstream and downstream service centres of delay queue $i$ (corresponding to the origin and destination service centres of the CS trip). The routing matrix, which describe how vehicles move across queues, can then be obtained as follows:

$$r_{ij}=\begin{cases}p_{il}&\textrm{for }i\in\mathcal{S},j\in\mathcal{I},i=o(j),l=d(j),\\ 1&\textrm{for }i\in\mathcal{I},j\in\mathcal{S},j=d(i),\\ 0&\textrm{otherwise}\end{cases}\;.$$

(2)

The queueing network described above belongs to the category of single-class closed queueing networks, in particular it is a BCMP network [12]. For the sake of illustration, we provide a simple example in Fig. 2.

The first step in solving this model consists in solving the traffic equations [12], which in our case simplify to the following:

$$\begin{cases}e_{i}=\sum_{j\in S}e_{j}p_{ij}&\forall i\in\mathcal{S}\\ e_{ij}=e_{i}p_{ij}&\forall i,j\in\mathcal{S}\\ \end{cases},$$

(3)

where $e_{i}$ denotes the relative arrival rate at queues corresponding to CS stations and $e_{ij}$ the relative arrival rate to the delay queue linking station $i$ and station $j$. We can now exploit the results for BCMP networks, which are known to have a product form solution for the stationary distribution as follows:

$$P\left(\{n_{i}\}_{i\in\mathcal{S}},\{n_{j}\}_{i\in\mathcal{I}}\right)=\frac{1}{G(N)}\prod_{i\in\mathcal{S}}\left(\frac{e_{i}}{\mu_{i}}\right)^{n_{i}}\!\!\prod_{j\in\mathcal{I}}\left(\frac{e_{j}}{\mu_{j}}\right)^{n_{j}}\!\!\frac{1}{n_{i}!},$$

(4)

where $G(N)$ is a normalisation constant. $G(N)$ can be efficiently derived as described in [12] using the convolution algorithm. Important performance measures can then be obtained exploiting the normalisation constant as follows. The throughputs of both single-server queues (the CS service centres) and infinite-server queues (the delay queues) are given by:

$$\lambda_{i}=e_{i}\frac{G(N-1)}{G(N)},\quad i\in\mathcal{K}\;.$$

(5)

The throughput at CS service centres corresponds to the intensity of drop-offs. For single server queues, the average number of cars $\overline{N}_{i}$ parked at the service centres and the utilisation $\rho_{i}$ can be obtained as:

$$\overline{N}_{i}=\sum_{n=1}^{N}\left(\frac{e_{i}}{\mu_{i}}\right)^{n}\frac{G_{i}(N-n)}{G(N)},$$

(6)

$$\rho_{i}=\frac{e_{i}}{\mu_{i}}\frac{G(N-1)}{G(N)},$$

(7)

where $G_{i}(N-n)$ is the normalisation constant computed removing queue $i$ and considering $N-n$ jobs. Please note that the utilisation $\rho_{i}$ is a strategic metric for a car sharing network, since it gives the probability that there is at least one car available for pick up at the station.

Following the service centres strategy discussed in Section II, we have partitioned the study area into non-overlapping square cells. Each of these cells is then modelled as a -/M/1 queue. To this aim, we need to estimate their service rate $\mu_{i}$. We use the technique described in [13], whereby the service rate $\mu_{i}$ for queue $i$ is obtained as $\mu_{i}=\frac{n_{dep}-n_{init}}{T_{busy}}$, where $n_{dep}$ denotes the number of departures observed at the queue (corresponding to the number of pickups in our terminology), $n_{init}$ is the initial size of the queue (i.e., how many cars were parked at time $t_{0}$), and $T_{busy}$ is the time the queue has been busy (i.e., with at least one car parked). These quantities can be easily computed from the trace, and their distribution (in log-log scale) is shown in Fig. 3 for varying cell side length. For smaller cell side lengths, we observe several orders of magnitude of variability in the service rates, owing to the fact that cells are small and there can be very popular ones and very neglected ones. Vice versa, the behaviour is more homogeneous when cells are larger.

We next feed the service rates $\mu_{i}$ to the closed network model that we have described in Section II. We set the number $N$ of vehicles to $349$, as in our trace. The number of cells (i.e., the size of $\mathcal{S}$) is given by the number of active cells for each cell side length configuration. Then we derive the routing probabilities $p_{ij}$ by simply counting, for each service centre $i$, the fraction of trips from $i$ to any service centre $j$. For all the pairs of service centres for which the routing probability is non zero, we also compute the average duration $T_{ij}$ of trips from $i$ to $j$. The service rate of each server in the delay queue for $(i,j)$ is then given by the inverse of $T_{ij}$ (the MLE estimator of the rate of an exponential random variable is simply the inverse of the sample mean). For all cell side lengths, the resulting Markov chain is ergodic, hence a unique steady-state probability exists [12].

Since the closed queueing network is completely defined based on the car sharing trace, we can apply the formulas for the throughput, utilization, and average number of cars at the stations that we have derived in Section II. Due to the lack of space, we show only the results for cell side lengths 250m and 1000m (the results for the other cases are analogous). We observe that for the throughput and availability (Fig. 5 and 4) the predictions of the theoretical model are quite accurate, regardless of the size of the cell. In particular, it seems that predictions are only offset by a proportionality constant. A less accurate match is obtained for the average number of cars parked at the service centres when the utilisation of service centres is high, but the model still captures pretty closely the general trend of this metric (Fig. 6). Thus, overall, we can conclude that queueing-theoretical approaches can be safely used for modelling CS systems.

In order to preliminarily assess the impact of vehicle stackability on the relocation performance, in the following we evaluate two simple approaches: $i)$ a uniform relocation strategy in which each customer takes a second vehicle to his/her intended destination with a fixed probability $\alpha$; and $ii)$ a backpressure strategy in which a customer takes a second vehicle to his/her intended destination only if the number of parked vehicles at the destination station is smaller than at the origin station⁴⁴4This strategy is inspired by the backpressure routing algorithm, a method for directing traffic around a queueing network that achieves maximum network throughput [16].. The rationale behind the latter strategy is to use the redistribution to equalise the queue backlogs (i.e., the number of cars at each station waiting for customers to pick them up).

We simulate the car sharing system using a custom C++ simulator that tracks the evolution of the closed queueing network system described in Section II. For our evaluation, we use the same configuration as in Section III-A, i.e., cell side length 250m and a large fleet size of 1000 vehicles. The transient period before the system reaches a steady-state is discarded. We compare the relocation strategies in terms of the availability of vehicles that they can provide to each station (Fig. 8). Important observations can be derived from the results. First, relocation has a complex impact on the car availability. While increasing the fleet size produces a smoothed “scaling” effect on the car availability, relocating vehicles may cause stations with similar initial car availabilities to experience a different gain (corresponding to the dispersion of values in Fig. 8). Second, uniform relocation is not bringing about any significant performance gain, with most of the stations even having a reduced availability. On the contrary, a backpressure relocation scheme is effective in improving car availabilities since it smartly relocates vehicles where there might be a shortage. Third, with a backpressure relocation strategy the performance gain is higher for stations with low-to-medium availabilities. To quantify this trend, in Fig. 9 we show the availability variation (in percentage) for each station when using backpressure relocation. The results show that that are stations with very low availabilities that experience an availability increases up to $\sim 200\%$. However, the gains are highly variable and there are also stations that can suffer from degradation of car availability. This motivates the need for design and modelling tools that can allow to calculate optimal relocation probabilities between pairs of stations depending on their service characteristics.

As a first step, we consider the queue in isolation. As with legacy car sharing systems, customers arrive at stations and pick up vehicles for their journeys. With stackable vehicles, however, a customer can pick up up to $n$ vehicles, in case a redistribution is needed by the CS operator. For the sake of example, in the following we set $n=2$. As highlighted in Section IV, customers should not be always requested to perform relocation, though, because, e.g., if all customers headed towards service centre $j$ would always relocate vehicles, we might generate too large a flow towards station $j$, possibly emptying the origin station $i$. Hence, the relocation process from station $i$ to station $j$ is regulated by parameter $\alpha_{ij}$, which can be seen as the probability for customers headed for station $j$ from station $i$ to pick up an additional vehicle for relocation.

Let us now take a step back and rethink how the service rate has been modelled so far for single-server queues in Section II. There, we have used a unique service rate $\mu_{i}$ for each station $i$, regardless of the destination of customers picking up vehicles at station $i$. This approach is not suitable anymore, because, with the relocation queue, we want to distinguish between customers heading for different destinations, since they may operate redistribution with different probabilities $\alpha_{ij}$. Thus, we need to explode $\mu_{i}$ into its components $\mu_{ij}$, each describing the rate at which customers headed for station $j$ arrive at station $i$. Please note that, thanks to the properties of the exponential distribution, $\mu_{i}=\sum_{j}\mu_{ij}$. In fact, the interdeparture interval when the queue is busy is exponentially distributed, hence the arrival of customers headed towards different destinations can be handled as if it were a superposition of Poisson processes.

As future work we plan to investigate the behaviour of a network of relocation queues, and to exploit our queueing theoretical framework to derive optimal relocation policies. Furthermore, our model could be extended to also characterise operator-based relocation strategies in which a dedicated workforce is used to relocate train of vehicles, which we believe to be of both theoretical and practical importance.