Last night I started wonder about the fastest way to take a shopping trip with my university flat mates and was wonder about how we should queue for the check out. I have a feeling that queue theory plus some optimisation can answer my question, but don't know much about queue theory. So, here's my question:
Me and my friends are at the super market (lets say there are n of us). We are all ready to check out. Lets say there are m check outs. We can see how long the line is at each check out, it's l_i. We can calculate some statistical information (distribution of processing time for each check out).
Which queues should we join to get everyone out of the super market the quickest?
Denote the mean processing time of checkout $C_i$ by $\mu_i$ $\>(1\leq i\leq m)$. Then we have to minimize $$\max\bigl\{ \mu_i(\ell_i+x_i)\ \bigm|\ 1\leq i\leq m, \ x_i>0\bigr\}$$ under the constraints $$x_i\in{\mathbb N}_{\geq0} \ (1\leq i\leq m),\qquad\sum_{i=1}^m x_i=n\ .$$