Closing the Service: Contrasting Activity-Based and Time-Based Systematic Closure Policies

We examine different policies for systematic service closure in messaging service systems. The system is modeled as an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M/UHP/1$</tex> queue, where service times follow a history-based Hawkes cluster process. We propose and examine stopping-time rules that balance between queue length and the probability of prematurely closing conversations. In a simulation study, we compare two families of systematic closure policies: the first relies on predictive information regarding service progress, i.e., the conversation's activity levels, while the second relies on elapsed time without activity. When restricted to static threshold policies, both families provide similar performance. However, when allowing the threshold to vary with the system state, activity-level policies outperform the inactive-time policies. Moreover, a large difference is observed between static and dynamic threshold policies. We therefore conclude that state-dependent (i.e., dynamic) activity-based policy is the most promising candidate to achieve optimal closure rules.