Software testing is essential for software reliability improvement and assurance. However, software testing is subject to imperfect debugging in the sense that new defects may be introduced into the software under test while detected defects are removed. The quantitative effects of software testing on software reliability improvement are obscure. In this paper we propose a Markov usage model to explore the quantitative relationships between software testing and software reliability in the presence of imperfect debugging. Several interesting quantities for software reliability assessment are derived and the corresponding upper and lower bounds are obtained.