In customer support contact centers, every service interaction involves a messaging dialogue between a customer and an agent; together, they exchange information, solve problems, and collectively co-produce the service. Because the service progression is shaped by the history of conversation so far, we propose a bivariate, marked Hawkes process cluster model of the customer-agent interaction. To evaluate our stochastic model of service, we apply it to an industry contact center dataset containing nearly 5 million messages. Through both a novel residual analysis comparison and several Monte Carlo goodness-of-fit tests, we show that the Hawkes cluster model indeed captures dynamics at the heart of the service and also surpasses classic models that do not incorporate the service history. Furthermore, in an entirely data-driven simulation, we demonstrate how this history-dependent model can be leveraged operationally to inform a prediction-based routing policy. We show that widely-used and well-studied customer routing policies can be outperformed with simple modifications according to the Hawkes model. Through analysis of a stylized model proposed in the contact center literature, we prove that service heterogeneity can cause this underperformance and, moreover, that such heterogeneity will occur if service closures are not carefully managed.