Consider $L$ queues in a discrete time system. At each time $n=0,1,2,\ldots$, one task would arrive at one of the queues with equal probability $\frac{1}{L}$. Immediately after that, a task scheduler would uniformly randomly pick up a queue, and schedule one task in that queue (if there is any). Tasks in queues are first come first served. Assume that working time of each task is 0, i.e., the task is gone immediately when it's scheduled. My question is, what is the probability that the task scheduler picks up an empty queue over the time? Would someone point me to any references if this is a known problem? Thanks.