# Model for Determining Parking Demand Using Simulation-Based Pricing
## 3 Results and Data Analysis
### Mathematical Model of Demand
#### 3.1.2 Chaotic Demand for Parking Spaces  
In this model, chaotic demand behavior is not enforced through an explicit formula switch or threshold rule, instead, it emerges from the actual observed dynamics of the parking system. In cases where pre-price-change demand exceeded capacity, the release of suppressed demand after capacity becomes available can result in positive elasticity (\(\eta>0\)), contrary to the basic hypothesis (1), as seen in Figures 3-5. This outcome reflects the real-world behavior of the system and is consistent with the simulation results. The root cause lies in the insufficient number of parking spaces, as (1) holds only for parking demand in areas with enough free spaces. In addition to (1), the second hypothesis of our paper is that enough parking spaces can be ensured through price changes, which alter the composition of user categories in line with the initial basic condition (4) and requirement (5). Due to only one price change, for each hour on selected days, there are only two data pairs of prices and observed demand \((P,Q(P))\), before and after the price increase, shown in Figures 3-6. Therefore, the unknown function \(Q=Q(P)\) is approximated under the assumption that the change \(dQ\) is proportional to the price change \(dP\) at the demand value \(Q(P)\) for the current price \(P\). According to the collected real data, with an increase in price \(P\), the demand \(Q=Q(P)\) also increases. Hence, we now define an analogous proportional differential equation, where the right-hand side is positive, consistent with the observed increase in demand due to a positive price change:  
\[\frac{dQ}{dP}=\mu\cdot Q,\ \ \mu>0. \tag{13}\]Solving it is analogous to solving Equation (7):  
\[\int\frac{dQ}{Q}=\int\mu\cdot dP\]  
\[lnQ=\mu\cdot P+C\]  
where \(C\) is an arbitrary constant. Furthermore,  
\[e^{lnQ}=e^{\mu\cdot P+C}\]  
\[Q=e^{\mu\cdot P}\cdot e^{C}\]  
\[Q=e^{\mu\cdot P}\cdot K,\]  
where \(K=e^{C}\) is again an arbitrary but positive constant. The unknown chaotic demand function now becomes as follows:  
\[Q=K\cdot e^{\mu\cdot P}\cdot \tag{14}\]  
The parameters \(K\) and \(\mu\) are determined again from two known price-demand pairs:  
* \((P_{v},Q_{v}=Q(P_{v}))\)--higher price and (higher) demand at the higher price
* \((P_{m},Q_{m}=Q(P_{m}))\)--lower price and (lower) demand at the lower price  
The parameter \(\mu\), which gives the demand elasticity as per (1), is calculated by substituting the price-demand pairs into Formula (14) and solving the system of exponential equations:  
\[Q_{v}=K\cdot e^{\mu\cdot P_{v}}\]  
\[Q_{m}=K\cdot e^{\mu\cdot P_{m}},\]  
By dividing the equations, we eliminate the parameter \(K\):  
\[\frac{Q_{v}}{Q_{m}}=e^{\mu\cdot(P_{v}-P_{m})}\]  
\[\int\frac{dQ}{Q}=\int\mu\cdot dP\]0  
Which finally gives the following:  
\[\int\frac{dQ}{Q}=\int\mu\cdot dP\]1  
The parameter \(\mu\) is positive, as assumed in the differential Equation (13), in line with the data obtained for STU, DMP, and NMP user categories. Thus,  
\[\int\frac{dQ}{Q}=\int\mu\cdot dP\]2  
since \(P_{v}-P_{m}>0\). The elasticity function defined by (1) is now positive, since we get \(\eta=\mu\cdot\ P>0\). Since elasticity cannot be positive when parking capacity is sufficient, and we are changing the price precisely to ensure enough free parking spaces (4), parameter \(K\) is calculated from (14) by substituting \(-\mu\), opposite to the parameter \(\mu\) from (15).