A store has $N$ counters. When you arrive there are $M>N$ customers waiting in front of you. Would you prefer they were lined up in a single queue or in $N$ queues? Follow-up: Does your answer change if you are the last customer for the day?
I have seen online discussions of whether one queue is in general more effective than multiple queues, and the general consensus seems to be that the expectation is the same while the variance changes. For example this thread seems to answer the same question. While I understand the intuitive explanation, I am still unable to formulate a coherent proof of this argument.
Another related answer is this blog. However, this seems to answer a slightly different problem, namely what is the average waiting time of all the customers before the queue is empty.
Back to the question at hand:
- The way I understand the original question, from my, the customer's, point of view, I shouldn't care at all about the average waiting time of the people in front of me. Rather I should only care for the expected time it will take me to get served.
- In the single queue model it seems to me that arrivals do not matter at all, since they will all queue behind me. In that sense my expected time to service should be the time it gets to serve the first $M-N$ people and the time it takes to serve one person from the last $N$ people to get served (since then you can go to his counter). So if we assume iid service times seems it should be something like $(M-N)\mathbb E[S]+\mathbb E[\min(S_1,\cdots,S_N)]$. Is this correct? If so, how could I account for different service performance?
- I have no idea how to actually model the multiple queues scenario, either with more customers arriving or not.
Can someone help set me on the right path? We can add assumptions if necessary to make this problem more reasonable.