Convexity/Concavity of Average Number of Jobs in M/M/1 Queue?

Question

I am working on a problem involving the average number of jobs $L$ in an M/M/1 queue with arrival rate $\lambda$, service rate $\mu$. For traffic intensity $\rho = \frac{\lambda}{\mu}$,

$$ L = \frac{\rho}{1 - \rho} = \frac{\lambda}{\mu - \lambda}. $$

I wanted to assess if $L(\lambda, \mu)$ is jointly convex, so I started by finding the Hessian $H$ of the function.

$$ H = \begin{bmatrix} \frac{2\lambda}{(\mu-\lambda)^3} + \frac{2}{(\mu-\lambda)^2} & -\frac{2\lambda}{(\mu-\lambda)^3} - \frac{1}{(\mu-\lambda)^2} \\ -\frac{2\lambda}{(\mu-\lambda)^3} - \frac{1}{(\mu-\lambda)^2} & \frac{2\lambda}{(\mu-\lambda)^3} \end{bmatrix}. $$

As I understand, my next step should be to assess the value of the function $\vec{x}^TH\vec{x}$ where $\vec{x} = \begin{bmatrix}\lambda \\ \mu\end{bmatrix}$ over the feasible region (which is $\lambda, \mu \ge 0, \mu \gt \lambda$). However, through factoring and simplification I found that

$$ \begin{bmatrix}\lambda & \mu\end{bmatrix} \begin{bmatrix} \frac{2\lambda}{(\mu-\lambda)^3} + \frac{2}{(\mu-\lambda)^2} & -\frac{2\lambda}{(\mu-\lambda)^3} - \frac{1}{(\mu-\lambda)^2} \\ -\frac{2\lambda}{(\mu-\lambda)^3} - \frac{1}{(\mu-\lambda)^2} & \frac{2\lambda}{(\mu-\lambda)^3} \end{bmatrix}\begin{bmatrix}\lambda \\ \mu\end{bmatrix} = 0. $$

This indicates the matrix $H$ is both positive semidefinite and negative semidefinite which in turn means $L(\lambda, \mu)$ is both convex and concave.

Is this correct or am I missing something here? Either my computations or my intuitions are flawed because I thought only linear functions could be both convex and concave. And if it is correct, are there any implications that would disallow me from having $L(\lambda, \mu)$ as the objective in a convex optimization problem? Thanks!

Answer 1

Your calculations (factoring and simplification) are incorrect. $L$ is neither convex nor concave as a function of $\lambda$ and $\mu$.

This can be concluded by examining the eigenvalues of the Hessian of $L$ with respect to $\lambda$ and $\mu$. I used MAPLE to compute the Hessian, and then evaluate its eigenvalues at the point $\lambda = 0.5, \mu = 1$. The eigenvalues are 24.649 and -0.649. This shows that the Hessian is indefinite at that point, and therefore that $L$ is neither convex nor concave at that point, or in general.