CVX Expressions for Erlang Delay Formula

Hey guys,

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


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.

[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.

Hello Sir, thank you so much for your helpful reply, and I think your calculation is absolutely correct.

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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

might provide an answer. This

is a newer and perhaps more readable version.

Thank you so much for your helpful reply.