This paper investigates multiprocessor scheduling with machine constraints, which has many applications in the flexible manufacturing systems and in VLSI chip design. Machines have different starting times and each machine can schedule at most k jobs in a period. The objective is to minimizing the makespan. For this strogly NP-hard problem, it is important to design near-optimal approximation algorithms. It is known that Modified LPT algorithm has a worst-case ratio of 3/2-1/(2m) for kequals2 where m is the number of machines. For k>2, no good algorithm has been got in the literature. In this paper, we prove the worst-case ratio of Modified LPT is less than 2. We further present an approximation algorithm Matching and show it has a worst-case ratio 2-1/m for every k>2. By introducing parameters, we get two better worst-case ratios which show the Matching algorithm is near optimal for two special cases.