The formula for number of needed servers seems like a fine approximation; I'm very confused how you could use the pigeonhole principle. Perhaps to say that if you have N servers, one of them has at least total_users/N users? But assuming you have a decent methodology for balancing users across servers, and can rebalance to use a new server when added, this has basically the same effects as the formula given so I'm clearly missing why you pointed this out.