Consider the following multiple-queue single-server model of a packet network problem. At each discrete time $t=0,1,\ldots,n$, a packet may arrive at the server R with probability $1-\epsilon_1$. The arrived packet enters one of the queues $l=0,\ldots,L-1$ randomly with equal probability $\frac{1}{L}$. In the same time slot, one of the queues is then uniformly randomly chosen by the server. With probability $1-\epsilon_2$, a packet in the selected queue will be departed. The event that the selected queue is empty (and therefore no packet is departed) is referred to as *wasted*. Assume that before $t=0$ the system is empty. My questions is, what is the expected fraction of the number of wasted times, $W_n/n$, as a function of $L$, for given $n$, $\epsilon_1$ and $\epsilon_2$? I'm in particular interested in the order of the function.

I have performed some simulations. It is seen that

- $W_n/n$ is a monotonically increasing function of $L$.
- The order of the function seems to be different for $\epsilon_1>\epsilon_2$, $\epsilon_1=\epsilon_2$, and $\epsilon_1<\epsilon_2$.
- The function doesn't seem to change for different pairs of $\epsilon_1,\epsilon_2$ if $\frac{1-\epsilon_1}{1-\epsilon_2}$ is the same.

It is highly appreciated if someone can point me to some references of similar problems.