CVX Expressions for Erlang Delay Formula


#1

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.


(Erling D.Andersen) #2

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


(Michal Adamaszek) #3

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.


#4

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


#5

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