Simple math problem

Here is a problem I met during designing test case:

Module A is claimed to be able to process MaxPrcMsg messages per second. The buffer could store each message for at most 2 seconds. Once the message hasn’t been proceeded within 2 seconds, it will be expired. Input messages are rushed into Module A every 100ms. In case the testing duration is (n * 100ms). How to set the maximum number MaxInMsg of input messages to verify the performance of Module A with high load?

The author of this requirement has presented the result and brief calculation. I read about it and found it was interesting. We could solve it by primary school skills.

Firstly, please take a look at this simple math problem (water in and out):

Water is drained with stable velocity x L per second from tub, whose capacity is y L. At the same time, z L fresh water is poured in the tub every second. Give a duration t. How to set z as big as possible without breaking the maximum capacity of the tub.

See, so simple. Isn’t it?

The final result is

MaxInMsg = [2 + (n-1) * 0.1] * MaxPrcMsg

Assume the buffer is empty at the beginning, and MaxPrcMsg is 40.

  1. In case n is 1 (the duration is 100ms), MaxInMsg equals 80 as shown in above formula. It means at most 80 messages could be sent to module A within 100ms. But in the next 1.9 second, A cannot afford any more message at all.

  2. In case n is 3 (the duration is 300ms), MaxInMsg equals 88 as shown in above formula. It doesn’t mean we could send 88 messages at the first 100ms, or even (80+8) at the first and second messages. Since when n equals to 2, MaxInMsg should be 84. So the proper setting should be 80+4+4.

Would you please give a reasonable input for n = 100, to keep the buffer in always full state and module A always busy in the mean time?