It turns out that there is a chip available, the popular 555 timer integrated circuit (IC) that incorporates all of the functionality of your mystery circuit. There are several advantages to integrating these components onto a single integrated circuit:

• the mystery circuit is reduced to the size of an 8-pin chip;
• the circuit is much more reliable and less prone to interference from noise and stray capacitances and resistances since most of the connections are internal to the chip;
• an integrated circuit is able to operate much faster since most of these internal connections between components are much shorter, in the order of micrometers (um).

The 555 timer can perform a variety of timing functions. The timing, as you have discovered for the mystery circuit, is determined by the three external components, C, R4 and R5. These are also the only external components required to control the timing of the 555 timer. With additional components, the 555 timer can perform very sophisticated timing tasks. Some of these applications, as well as a nice introduction that you should be able to relate to, now that you have explored the mystery circuit, can be found here.