A lower bound for the Erlang C formula in the Halfin-Whitt regime

One of the classical models of queueing theory is the M/M/s queue or Erlang delay model. This model has s homogeneous servers working in parallel. Customers arrive according to a Poisson process with arrival rate λ, and the service times are independent and exponentially distributed with mean 1/μ. Let the offered load be a = λ/μ and assume a < s to have a proper steady-state distribution. The most important performance characteristic for this system is the probability that a customer is delayed when arriving at the system in steady state. This probability is known as the Erlang C formula, given by (with ρ = a/s)