What is the name of this public key encryption cryptosystem?
Maybe I miss something, but after some quick thoughts about that algorithm, I think it is not secure. Therefore it may not be used and be more like an example for teaching (with no name).
The problem with the algorithm is that you can derive the secret key q from the public key. When you have two integers a and b, where both are a multiple of q (like all q*z_i in your case), then a - b is also a multiple of q, that is much smaller.
So the private key q can be reconstructed within a fraction of a second for primes with few thousand bits. All you have to do it to take two q*z_i from the public key and compute the greatest (only) common divisor.