# CVX Expressions for Erlang Delay Formula

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.

We at Mosek think it is not possible. We do not have proof for that and would love to be proved wrong.

Here is the issue (and I write x instead of \rho). If c=2 then it evaluates to \frac{2x^2}{x+1} and you can write t(x+1)\geq 2x^2 with a rotated quadratic cone. If c=3 then you get \frac{9x^3}{3x^2+4x+2} and now the constraint is t(3x^2+4x+2)\geq 9x^3. That would be doable with a geometric mean if the quadratic polynomial factorized but it doesn’t. So we don’t know if that case can be done. All this assuming I didn’t make an error in my calculation.

Hello Sir, thanks a lot for taking time to review my problem.

1 Like

Hello Yackety, I want to know if you have solved this problem, how to solve it, I now have the same difficulty

My feeling is that using an approximation that is DCP representable is the best hope.

I no expert in this area but for instance

https://www.researchgate.net/publication/225314559_Improved_approximations_for_the_Erlang_loss_model