# Calculating Variance in Waiting Time for a Queueing Network

I'm working on a queueing network model that incorporates blocking and features two states. After defining the global balance equations, I solved them for my parameters arrival rates (λ), service rates (μ), and others.

To quantify the performance of the first queue in this network, I've successfully calculated the average number of customers (L) using the formula:

$$$$L = \sum_{(i, j) \in E} i \cdot \pi(i, j)$$$$

where $$\pi(i, j)$$ represents the steady-state probability of the system being in state (i,j), and E denotes the set of all possible states.

Also, I determined the variance in the number of customers ($$\sigma^2$$) with the equation:

$$$$\sigma^2 = \sum_{(i, j) \in E} (i - L)^2 \cdot \pi(i, j)$$$$

Based on these calculations, I applied Little's Law to compute the average waiting time in the queue (W) as:

$$$$W = L/\lambda$$$$

My Question: How can I calculate the variance in waiting time for this queueing network?

• if $\lambda$ is constant (& should be in steady-state, then Var (W) = Var (L) Commented Apr 9 at 2:51
• @SutanuMajumdar Is there a mathematical basis for this? W is normalized by $\lambda$. So, I think they will be different. The formulations for Var(L) and Var(W) are different for M/M/1 and M/M/c queues in the literature. Commented Apr 9 at 17:41