A logarithmic safety staffing rule for contact centers with call blending

We consider large contact centers that handle two types of jobs-inbound and outbound-simultaneously, a process commonly referred to as call blending. Inbound work arrives to the system according to an exogenous arrival process, whereas outbound work is generated by the contact center. We assume that there is an infinite supply of outbound work to process, and that inbound calls are prioritized over the outbound calls. We propose a logarithmic safety staffing rule, combined with a threshold control policy, ensuring that agents' utilization is very close to one at all times, but that there are practically always idle agents present. Specifically, we prove that it is possible to have almost all inbound calls answered immediately upon their arrival, in addition to satisfying a target long-run throughput rate of outbound calls, with at most a negligible proportion of those calls dropped. Simulation experiments demonstrate the effectiveness and accuracy of our analysis.