In that case h rather depends on amount of peers polled or on the slowest peer to respond than on amount of tips.
Why?
Amount of tips is supposed to be small, so amount of data transfered is small. So the time to send a request and receive an answer should depend mostly on network and peers latency. If all requests are performed in parallel, than the slowest to respond peer determines the time. If requests are sent serially, then the time is roughly proportional to requests amount.