Large language models (LLMs) have become an increasingly important workload for cloud providers catering to both enterprise and consumer applications. LLM inference requests from these applications have end-to-end latency SLOs that must be adhered to in production settings. However, existing LLM serving systems focus on optimization objectives such as request serving throughput or request execution latency rather than the end-to-end latency SLOs. Achieving end-toend SLOs for latency-sensitive requests is challenging due to head-of-line (HOL) blocking in the request queue, which results from bursty arrival rates and insufficient resources. To address the above challenge, we propose QLM, a multimodel queue management framework for LLM serving. QLM uses stochastic programming to orchestrate the actions of multiple LLM Serving Operations (LSOs) to reduce HOL blocking and maximize SLO attainment. Specifically, QLM uses the following LSOs: model swapping, request eviction, GPU-CPU state swapping, load balancing, and warm model start. Evaluation on heterogeneous GPU devices and models with real-world LLM serving dataset shows that QLM improves SLO attainment by 40–90% and throughput by 20–400% while maintaining or improving device utilization compared to other state-of-the-art LLM serving systems.