Hey guys,

As part of my objective function, Erlang delay formula is known as follows.

f(\rho)=[\sum_{i=0}^{c-1}{\frac{c!(1-\rho)}{i!(c\rho)^{c-i}}+1}]^{-1}

Previous works have proved that such formula is strictly increasing and convex in traffic intensity \rho\in(0,1) [1]. However, I failed to figure out a valid CVX expression for it based on disciplined convex programming rules. For instance, I don’t really know how to deal with the exponent of -1. By simulation, one may observe that the denominator is decreasing and convex, while the built-in function “inv_pos” only accepts concave arguments according to composition rule.

It’d be much appreciated if anyone could help me out. Thanks a lot.

Refs

[1] H. L. Lee, and M. A. Cohen, “A note on the convexity of performance measures of M/M/C queuing systems,” Journal of Applied Probability, vol. 20, no. 4, pp. 920-923, 1983.