## 2 Background and Motivation
### Scheduling Problem  
Scheduling strategy is critically important in FaaS environments and introduces unique challenges, as the provider is solely responsible for all scheduling decisions. Unlike virtual machine (VM) cloud provisioning, users in a serverless environment are not required to specify the resources they need. Therefore, the core of the scheduling problem lies in efficiently managing compute resources (i.e., containers) to handle dynamically changing workloads. As previously discussed, reliance on container-based execution introduces performance drawbacks, particularly the cold start issue. Consequently, a primary goal of any scheduling strategy is to minimize cold starts in order to maintain acceptable latency for function requests. Some providers address this by maintaining a pool of pre-warmed containers [11]. However, this approach can lead to underutilization and wasted resources during periods of low demand. Another common method involves assigning invocations to randomly selected workers without accounting for current load, which can result in poor resource provisioning, especially under bursty traffic conditions [12]. Thus, developing a scheduling approach that balances latency and resource efficiency is a key objective in the serverless computing paradigm. The scheduling problem becomes even more complex in multi-invoker scenarios, where selecting the most suitable invoker to process a user request is critical. Some baseline approaches, such as Round Robin, distribute the workload evenly across all invokers, regardless of actual need. While simple, this method can lead to underutilization of available resources. \begin{table}
\begin{tabular}{c c c} \hline
**ID** & **Function** & **Description** \\ \hline IMG & Image Resize & Resize image to 200 \(\times\) 200 px \\ FIL & File Compression & Compress a 10 MB file \\ ML & Complex ML Model & Load a pre-trained TensorFlow model, and predict sentiment \\ DB & Database Management & Update database \\ MM & Matrix Multiplication & Matrix size 500 \(\times\) 500 \\ \hline \end{tabular}
\end{table} 
Table 1: Tested the function specifications. Figure 1: Cold start overhead in tested applications. We conduct case studies to demonstrate how scheduling strategies directly impact resource utilization and latency in the serverless environment. Consider a scenario with four identical invokers, each with a capacity of 10 containers. The workload includes two phases: a steady phase with 20 invocations per minute for 5 min, and a burst phase with 200 invocations per minute for 5 min. Finally, four different scheduling strategies are used in this example as follows: Round Robin (RR), Random Selection (RS), Pre-Warmed (PW), which keeps five containers per invoker pre-initialized, and Greedy Packing (GP), which routes all invocations to a single invoker until it reaches full capacity. The results are presented in Table 2.  
The first observation is that all strategies except Pre-Warmed (PW) underutilize resources. For example, RR attempts to avoid overloading any single invoker by evenly distributing the workload across all invokers, but it fails to consolidate tasks for efficient resource use. RS shows similar results; its random assignment often creates \"hotspots\" where one invoker becomes overloaded while others remain idle, especially during the steady phase. This imbalance increases queuing time for the overloaded invoker and raises the frequency of cold starts, resulting in the worst latency (1200 ms). GP causes severe underutilization, with three invokers remaining idle throughout both phases and only 15% total CPU usage. On the other hand, PW--which resembles static provisioning--eliminates cold starts and achieves the best latency by using pre-warmed containers that absorb bursts of traffic. Its utilization results are significantly better than those of the other strategies. However, this comes at the cost of resource waste, as 20 pre-warmed containers remain unused during the steady phase.