## 3 Adaptive Task Traffic-Shaping Algorithm
### Task Request Queueing Theory Model  
In a GIS service system, users send requests through various terminal access devices, the system treats all requests equally and provides services based on the FCFS principle. In order to model each single server (considered to have only one service counter) of the GIS service system as a queue system, referencing the queueing theory model, the following conditions need to be satisfied [35]:  
(1) During the operation of the GIS service system, user requests arrive at the service counter one at a time and each request is independent of the others; the interarrival time follows an exponential distribution with the average arrival rate \(\lambda\), and arriving user requests can be modeled as a Poisson process (very common in this type of phenomenon).  
(2) The user requests are processed by the service node (contains a request queue) following an FCFS discipline. The processing time for each task is assumed to follow an exponential distribution with rate \(\mu\).  
Based on the above assumptions, the M/M/1 queueing model [36; 37] can be used to model a single-server cloud service system. Call \(\rho=\frac{\lambda}{\mu}\) the traffic intensity, and \(n\) the mean number of tasks in the queue. If the system reaches a steady state (in other words, if the rate at which tasks arrive is slower than the rate at which they are processed), \(L_{s}\) is used to indicate the steady state and denoted as  
\[L_{s}=\sum\
olimits_{n=1}^{\infty}n(1-\rho)\rho^{n}=\left(\rho+2\rho^{2}+3 \rho^{3}+\ldots\right)-\left(\rho^{2}+2\rho^{3}+3\rho^{4}+\ldots\right)=\frac {\rho}{1-\rho}=\frac{\lambda}{\mu-\lambda} \tag{1}\]  
It can be seen that Equation (1) describes the relationship between \(\lambda\), \(\mu\), and \(L_{s}\), where \(L_{s}\) can be obtained by real-time monitoring in the system, and if \(\mu\) can be calculated dynamically, the system expects the average arrival rate of the task \(\lambda\). As a result, the token fill rate can be adjusted according to \(\lambda\), so as to keep the system in a healthy and stable state.  
Figure 3: Overall flowchart.  
a greater impact on the system load, so their weights should be higher. In this case, the indicators can be sorted according to their actual values and assigned corresponding weights. Based on extensive experiments, the paper proposes the following set of weight values: {0.5, 0.4, 0.1}.
4. When one indicator falls within the [0.75, 1] range and the other two indicators fall within the [0, 0.75] range, it is assumed that this indicator is the current system bottleneck and its weight should be very high. Based on extensive experiments, the paper proposes the following set of weight values: {0.9, 0.07, 0.03}.
5. When two indicators fall within the [0.75, 1] range and one indicator falls within the [0, 0.75] range, it is assumed that these two indicators are the current system bottleneck and their weights should be very high. In this case, the indicators can be sorted according to their actual values and assigned corresponding weights. Based on extensive experiments, the paper proposes the following set of weight values: {0.5, 0.47, 0.03}.