Emergency Services Communication Systems (ESCS) are evolving into Internet Protocol based communication networks, promising enhancements to their function, availability, and resilience. This increase in complexity and cyber-attack surface demands better understanding of these systems' breakdown dynamics under extreme circumstances. Existing ESCS research largely overlooks simulation and the little work that exists focuses primarily on cybersecurity threats and neglects critical factors such as non-stationarity of call arrivals. This paper introduces a robust, adaptable graph-based simulation framework and essential mathematical models for ESCS simulation. The framework uses a representation of ESCSes where each vertex is a communicating finite-state machine that exchanges messages along edges and whose behavior is governed by a discrete event queuing model. Call arrival burstiness and its connection to emergency incidents is modeled through a cluster point process. Model applicability is demonstrated through simulations of the Seattle Police Department ESCS. Ongoing work is developing GPU implementation and exploring use in cybersecurity tabletop exercises.