# Cloud Database Analysis of Instant Messaging Apps on Mobile Devices
## 3 Problem Solution
### Dynamic Modeling of Instant Messaging Behavior  
The following variables are taken into account:  
* \(M(t)\): the total count of messages dispatched over time \(t\);
* \(U(t)\): the active user count at a given time \(t\);
* \(J(t)\): the rate at which new users are joining over time \(t\);
* \(C(t)\): the user churn rate over time \(t\);
* \(E(t)\): the user participation rate during the period \(t\).  
The number of messages exchanged and user interactions can be represented by the following system of differential equations:  
\[\frac{dM(t)}{dt}=\alpha U(t)E(t) \tag{1}\]  
Here, \(\alpha\) denotes a fixed value that reflects the typical number of messages exchanged per user per engagement.  
The user population \(U(t)\) changes over time due to the rates of users joining and leaving:  
\[\frac{dU(t)}{dt}=J(t)-C(t) \tag{2}\]  
The engagement rate can be influenced by external elements, such as new features or promotions, denoted by the term \(\beta(t)\):  
\[E(t)=E_{0}e^{-\gamma t}+\beta(t) \tag{3}\]  
The overall count of messages dispatched during the interval from \(t_{0}\) to \(t_{f}\) can be calculated as follows:  
\[M_{\text{total}}=\int_{t_{0}}^{t_{f}}\alpha U(t)E(t)\;dt \tag{4}\]  
The graphs illustrating the user population, the engagement rate and the total number of messages accumulated over time.  
The graphs in Figures 1-3 represent an assumption of how the app can behave after being launched on Google Play. If the marketing strategy goes well, this is the expected user growth in the next period.  
We will assume certain constants and model-user interactions and message exchanges over a specific period. The following is what we will simulate:  
**Count of active users**\(U(t)\)--influenced by the rate of joining \(J(t)\) and the rate of churning \(C(t)\).  
**Message frequency**\(M(t)\)--corresponding to the user count and their engagement levels.  
**Engagement frequency**\(E(t)\)--user fatigue leads to a decline over time, although external elements can cause occasional spikes.  
Figure 1: User population over time.  
Figure 3: Accumulated messages over time.  
Figure 2: Engagement rate over time.  
**Active users \(U(t)\)**--the count of active users steadily increases over time because the rate at which users join exceeds the rate at which they leave.  
**Engagement frequency \(E(t)\)**--user engagement initially declines over time but increases during the promotional period (days 10-15) due to external events that increase user activity.  
**Dispatched messages \(M(t)\)**--the overall quantity of messages exchanged increases progressively, with a significant uptick observed during the promotional period when user interaction is elevated.  
The forthcoming sections will utilize differential equations and integrals to characterize user growth, message frequency, and data traffic in mobile chat applications. These models aim to facilitate the analysis of both applications.  
Let \(U(t)\) represent the number of users at time \(t\). A conventional model for user growth is often described by a differential equation reflecting logistic growth:  
\[\frac{dU(t)}{dt}=rU(t)\bigg{(  
In Figure 5, the projected data traffic over time can be seen based on the volume of messages.  
The differential and integral models shed light on the temporal dynamics of chat applications. By incorporating real-world data, these models can be improved to precisely forecast user growth, message frequency, and data traffic.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Time \(t\) (Days)** & **Number of Users \(U(t)\)** & **Growth Rate \(\frac{dU}{dt}\)** \\ \hline
0 & 1000 & 500 \\
30 & 3000 & 1200 \\
60 & 7000 & 1400 \\
90 & 9500 & 500 \\
120 & 9900 & 200 \\ \hline \hline \end{tabular}
\end{table}
Table 1: User expansion over time.  
Figure 4: User expansion across time.  
Figure 5: Evolution of data traffic over time.