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  <front>
    <journal-meta />
    <article-meta>
      <title-group>
        <article-title>Simulating Car-Sharing Demand: A Data-Driven Approach Using Probability Distributions</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <string-name>Shkelqim Hajrulla</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Ejsi Veshaj</string-name>
          <xref ref-type="aff" rid="aff1">1</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Leonard Bezati</string-name>
          <xref ref-type="aff" rid="aff2">2</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Robert Kosova</string-name>
          <xref ref-type="aff" rid="aff3">3</xref>
        </contrib>
        <aff id="aff0">
          <label>0</label>
          <institution>Computer Engineering Department, Faculty of Engineering, Epoka University</institution>
          ,
          <addr-line>Tirana</addr-line>
          ,
          <country country="AL">Albania</country>
        </aff>
        <aff id="aff1">
          <label>1</label>
          <institution>Constructor University</institution>
          ,
          <addr-line>Campus Ring 1, 28759, Bremen</addr-line>
          ,
          <country country="DE">Germany</country>
        </aff>
        <aff id="aff2">
          <label>2</label>
          <institution>Department of Mathematics, Faculty of Technical Science, University of Vlora</institution>
          ,
          <country country="AL">Albania</country>
        </aff>
        <aff id="aff3">
          <label>3</label>
          <institution>Department of Mathematics, University Aleksander Moisiu</institution>
          ,
          <addr-line>Durres</addr-line>
          ,
          <country country="AL">Albania</country>
        </aff>
      </contrib-group>
      <abstract>
        <p>Car-sharing services face challenges in balancing fleet availability due to fluctuating demand. This paper simulates car-sharing demand using probability distributions, incorporating factors like daily and weekly patterns, seasonal trends, and external influences such as weather and public holidays. Model training considers convergence speed since it dictates how quickly algorithms arrive at solutions, which affects deployment time and computational resources. Using numerical analytical techniques and optimization frameworks, this research looks at how optimization is controlled in the dynamic field of machine learning. The simulation provides a flexible framework for demand analysis, aiding in more efficient fleet management and forecasting.</p>
      </abstract>
      <kwd-group>
        <kwd>eol&gt;Carsharing demand</kwd>
        <kwd>simulation</kwd>
        <kwd>fleet management</kwd>
        <kwd>time series</kwd>
        <kwd>artificial dataset</kwd>
      </kwd-group>
    </article-meta>
  </front>
  <body>
    <sec id="sec-1">
      <title>1. Introduction</title>
      <p>
        Car-sharing has become a crucial solution to urban mobility challenges, providing efficient
resource utilization and access to transportation. However, operators face fleet imbalance, where
the supply of cars does not align with fluctuating demand. Accurate demand prediction is essential
for addressing this issue, but there is often insufficient data available for car-sharing services, using
time series simulation tools [
        <xref ref-type="bibr" rid="ref9">10</xref>
        ] for assessing the effects of variable renewable energy generation
on power and energy systems.
      </p>
      <p>
        To overcome this limitation, this paper focuses on simulating car-sharing demand [
        <xref ref-type="bibr" rid="ref1">1</xref>
        ] by
developing a time series based on observed patterns. The simulation incorporates various
influencing factors such as time of day, day of the week, weather conditions, public holidays, and
fleet size [
        <xref ref-type="bibr" rid="ref2">2</xref>
        ]. Very important things are those reviews of vegetation phenological metrics extraction
using time-series. By generating simulated data, optimization [
        <xref ref-type="bibr" rid="ref9">10</xref>
        ] and simulation [11] of
carsharing, forecasting the carsharing service demand using multivariable models [
        <xref ref-type="bibr" rid="ref10">12</xref>
        ] this work
aims to provide a foundation for better demand forecasting and effective fleet management.
      </p>
    </sec>
    <sec id="sec-2">
      <title>2. Background</title>
      <p>2.1.</p>
      <sec id="sec-2-1">
        <title>Time series</title>
        <p>
          A time series consists of sequential observations recorded at uniform time intervals. These records
—whether measurements, counts, or other numerical data—are arranged in order of occurrence.
Examples include weather readings, financial market data, and heartbeat signals. The proliferation
of sensors and tracking systems has led to a surge in sequential data collection, making time series
data both widespread and vital. Such data helps in discovering causal relationships, identifying
trends and even prediction of next values in many applications such as, but not limited to,
medicine, weather-forecast, and economy deciding equivalences [
          <xref ref-type="bibr" rid="ref3">3</xref>
          ] among conjunctive aggregate
queries. Along with this, time series analysis in the area of data mining aims to extract order
pattern in data [
          <xref ref-type="bibr" rid="ref4">4</xref>
          ], a lot of interest has been around discovering trends from data which can
describe what has happened in the past and what can occur in the future hence guiding towards
carsharing revolution [
          <xref ref-type="bibr" rid="ref10">12</xref>
          ] giving a vital importance to time series.1 6th International Conference
RTA-CSIT on 22-24 MAY 2025 in Tirana, Albania.
2.2.
        </p>
      </sec>
      <sec id="sec-2-2">
        <title>Data simulation</title>
        <p>
          Data simulation is the process with which synthetic datasets are generated that resemble the
statistical properties and trends of real data. For instance, simulating visitors to a restaurant may
create a bimodal distribution that reflects lunch and dinner hours. This method is commonly used
in statistics, economics, finance, social sciences and engineering, to test a model or hypothesis
before using it on real data [
          <xref ref-type="bibr" rid="ref7">7</xref>
          ]. Sadly, though simulation is critical, we have little formal and
comprehensive guidance on how to simulate data, and it is not often offered as a course on its own.
This is particularly relevant for time series data, where differences in the timing of observations
make direct comparisons challenging. Simulation methods [
          <xref ref-type="bibr" rid="ref8">8</xref>
          ] are typically divided into two types:
parametric, which rely on predefined mathematical models [9], and non-parametric, which
generate data based on observed patterns. The simulation approach in this paper falls into the
latter category. While both data simulation and forecasting require forming hypotheses about
system dynamics, simulation uniquely accommodates qualitative data, scalable scenario testing,
and a more exploratory creative process during its early stages.
        </p>
      </sec>
    </sec>
    <sec id="sec-3">
      <title>3. Simulation Suite</title>
      <p>3.1.</p>
      <sec id="sec-3-1">
        <title>Initial Simulation phase</title>
        <p>
          Drawing on the analysis conducted by [
          <xref ref-type="bibr" rid="ref3">3</xref>
          ] for Car2Go in Vancouver, Canada, the following
illustrates the hourly demand:
        </p>
        <p>
          This scenario adopts a free-floating model; however, the simulation here does not distinguish
between one-way, round trip, and free-floating services. Instead, its primary objective is to
accurately replicate real-world behavior by incorporating all essential parameters for an effective
and scalable simulation model [
          <xref ref-type="bibr" rid="ref5">5</xref>
          ]. The figure above illustrates the number of travel requests per
hour over one week. Noticeable similarities emerge across days, with higher demand on Friday,
Saturday, and Sunday—a trend reflecting increased car-sharing usage over the weekend. This
consistent daily pattern allows the simulation to focus on modeling hourly demand within a
24hour period.
        </p>
        <p>
          The figure above isolates hourly demand trends for Monday and Tuesday, offering a closer look
at daily car sharing usage. Both days display similar patterns, indicating consistent behavior that
could impact operational planning and resource allocation [
          <xref ref-type="bibr" rid="ref6">6</xref>
          ]. Notably, two clear peaks appear on
each day one around 11 or 12 in the late morning/early afternoon, and another at about 5 or 6 in
the early evening, corresponding to periods of increased activity such as lunch breaks, errands, and
evening commutes in urban settings. Focusing on Monday, the 24-hour demand divides into three
segments, with the first segment (from midnight to around 6–7 a.m.) following a cosine-like
distribution, mathematically expressed as:
y
¿
A
¿
( ¿ f ¿ ¿ ¿+xo¿cs¿b+ ¿ ϕ ¿ )
(1)
Here, x denotes time, varying from 0 to a multiple of 2π to complete one cycle. By fine-tuning
parameters like amplitude (A) and bias (b), the simulation can shape the data as needed. While the
cosine function outputs values between -1 and 1, actual hourly demand falls within [0, 100], so
careful adjustments align the simulation with real-world (Car2Go) data.
Weather's impact on car-sharing usage continues to be a topic of debate among experts. While
some maintain that weather exerts a minimal influence on demand shifts, Figure 4 from [
          <xref ref-type="bibr" rid="ref6">6</xref>
          ] offers a
contrasting viewpoint.
In the referenced study, researchers systematically compared similar weekdays under different
weather conditions to reveal subtle influences on car-sharing usage. Their findings uncovered a
complex connection between weather patterns and demand dynamics. Specifically, heavy rain
between 6:30 am and 11:30 am was associated with a notable drop in demand compared to overcast
conditions, whereas light rain during the afternoon hours (from 1 pm to 5 pm) correlated with a
significant rise in demand. Using the insights from [9] small alterations to hourly demand were
performed. This simulation was calibrated to reflect these empirical relationships by determining
strong correlations between weather conditions and deviations in demand.
        </p>
      </sec>
      <sec id="sec-3-2">
        <title>3.3. Final Steps of the Simulation</title>
        <p>Our simulation is currently designed around a fixed fleet size of ~400 vehicles, which
considerably limits the solution space for type of operational scenarios. Therefore, to create a more
generic solution, we need to go beyond those limitations and enable variable fleet sizes. A proper
scalable simulation should allow the users to configure a fleet that best fits their requirements.
This will involve augmenting the model by incorporating fleet size as a time-varying parameter.
This central change reengineers the primary mechanics of the simulation, so that hourly demand
numbers directly react to the number of available vehicles. So, as the fleet expands/shrink, it will
definitively paint a more realistic picture of what is happening in the world (at least in the
simulation). The last result simulated dataset will have one additional column for fleet size now,
giving a complete picture of what vehicles mean in terms of their market presence, offering a
comprehensive view of how vehicle availability influences demand dynamics.</p>
      </sec>
    </sec>
    <sec id="sec-4">
      <title>4. Conclusions</title>
      <p>The final stage of the simulation has a notably unique feature that noticeably enhances the
simulation dynamics: mobility in fleet size, which encourages the simulation to be more realistic
according to the operational aspect. The upgrade allows stakeholders to investigate the impact of
altering fleet sizes on meeting demand and assist car-sharing companies optimize resource
allocation.</p>
      <p>We address this gap by developing a scalable, robust simulation model of car-sharing demand,
which is contextually tied to real-world parameters like hourly utilization profiles, weather effects,
and fleet size constraints. It then uses mathematical models, such as cosine and bimodal
distributions, to reflect daily and weekly demand variations in the simulation.</p>
      <p>Incorporating the probabilistic perspective gives a better representation of demand fluctuations
over longer periods. Moreover, the incorporation of weather data improves each model's realism,
accounting for the role of variable weather in the usage of car-sharing. To reflect monthly or
annual demand changes in the simulation, we can use mathematical models, such as sine and
modal probability distributions or even other models to the next future. These results contribute to
the recent literature on demand forecasting, provide guidance for operational planning, and
demonstrate how simulation can support decision-making in economic trends, or city-specific
mobility policies, to enhance predictive accuracy.</p>
    </sec>
    <sec id="sec-5">
      <title>Declaration on Generative AI</title>
      <p>The authors have not employed any Generative AI tools.
[11]
demand</p>
    </sec>
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