In the projective plane PG(2, q), upper bounds on the smallest size t2(2, q) of a complete arc are considered. For a wide region of values of q, the results of computer search obtained and collected in the previous works of the authors and in the present paper are investigated. For q ≤ 301813, the search is complete in the sense that all prime powers are considered. This proves new upper bounds on t2(2, q) valid in this region, in particular t2(2, q) < 0.998 3q ln q for 7 ≤ q ≤ 160001;t2(2, q) < 1.05 3q ln q for 7 ≤ q ≤ 301813;t2(2, q) < √ q ln 0.7295 q for 109 ≤ q ≤ 160001;t2(2, q) < √ q ln 0.7404 q for 160001 < q ≤ 301813.The new upper bounds are obtained by finding new small complete arcs in PG(2, q) with the help of a computer search using randomized greedy algorithms and algorithms with fixed (lexicographical) order of points (FOP). Also, a number of sporadic q's with q ≤ 430007 is considered. Our investigations and results allow to conjecture that the 2-nd and 3-rd bounds above hold for all q ≥ 109. Finally, random complete arcs in PG(2, q), q ≤ 46337, q prime, are considered. The random complete arcs