# combinatorics – Optimal planning for mail box servicing and processing (for a message queue online consumer scheduling)

Consider I have n mail boxes {m1, m2, …. mn }. Messages come to each box at different rate/sec e.g., mail box n has an arrival rate of messages equal say 5 per second. In general, let the set a = {a1, a2, a3…, an} denotes the arrival rate of messages into each of the mail boxes respectively.

Now suppose that I have mail workers that can pick up and process messages at uniform rate of say x message/sec (suppose x = for example 10 message/sec)?

How many (minimum) mail workers I need so that no message waiting time in the system is more than say 2 seconds (generally s seconds)?

Constraint: every mail box should be assigned to only one mail worker, but a mail worker can be responsible of many mail boxes?

I beleive the problem looks similar to bin packing (online)? if that is true what is the best know approximation algorithm for the online and offline bin packing cases?

Thnak you.