This paper focuses on flow control in high-speed and large-scale networks. Each node in the network handles its local traffic flow only on the basis of the information it knows. It is preferable, however, that the decision making of each node leads to high performance of the whole network. To this end, the relationship between local decision making and global performance of flow control is the essential object. We propose phenomenological models of flow control of high-speed and large-scale networks, and investigate the stability of these models.

Given an integer N, it is easy to determine whether or not N is prime, because a set of primes is in LPP. Then given a composite number N, is it easy to determine whether or not N is of a specified form? In this paper, we consider a subset of odd composite numbers +1MOD4 (resp. +3MOD4), which is a subset of odd composite numbers consisting of prime factors congruent to 1 (resp. 3) modulo 4, and show that (1) there exists a four move (blackbox simulation) perfect ZKIP for the complement of +1MOD4 without any unproven assumption; (2) there exists a five move (blackbox simulation) perfect ZKIP for +1MOD4 without any unproven assumption; (3) there exists a four move (blackbox simulation) perfect ZKIP for +3MOD4 without any unproven assumption; and (4) there exists a five move (blackbox simulation) statistical ZKIP for the complement of +3MOD4 without any unproven assumption. To the best of our knowledge, these are the first results for a language L that seems to be not random self-reducible but has a constant move blackbox simulation perfect or statistical ZKIP for L and without any unproven assumption.