Convexity Properties of the Erlang Loss Formula

We prove that the throughput of the M/G/x/x system is jointly concave in the arrival and service rates. We also show that the fraction of customers lost in the M/G/x/x system is convex in the arrival rate, if the traffic intensity is below some Ρ^* and concave if the traffic intensity is greater than Ρ^*. For 18 or less servers, Ρ^* is less than one. For 19 or more servers, Ρ^* is between 1 and 1.5. Also, the fraction lost is convex in the service rate, but not jointly convex in the two rates. These results are useful in the optimal design of queueing systems.