Serverless is a mainstream computing mode in modern cloud native systems. Different from traditional monolithic cloud, workloads for Serverless architecture are disaggregated into short-lived and fine-grained functions. In Serverless, functions are usually invoked with a bursty pattern, which means the system needs to deliver these functions at high throughput to meet SLA (Service-level agreement) requirements. To explore bursty traffic implications on Serverless platforms, in this paper, we propose a novel benchmarking suite for serverless systems - BBServerless. BBServerless is designed to capture end-to-end and system-level performance in bursty-traffic workloads which help reveal performance bottlenecks of Serverless platform and guide better architecture design for cloud systems. To demonstrate performance variations on Serverless platforms, we also design a traffic generating algorithm (based on Poisson distribution) for four mainstream cloud workloads, i.e. BigData, Stream processing (STREAM), Web Applications (WebApps), and Machine Learning Inference (MaLI). We conduct experiments with trace-driven simulations in a private cloud environment. With data collected from evaluations, we observe that the performance of time-localized components like CPU migration, branch prediction, and cache is highly correlated with end-to-end workload performance. and publicly available at Github (https://github.com/whoszus/BurstyServerlessBenchmark).