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

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!

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.