We consider the problem of model-free reinforcement learning in the Markovian decision processes (MDP) under the PAC ("probably approximately correct") model. We introduce a new variant of Q-learning, called speedy Q-learning (SQL), to address the problem of the slow convergence in the standard Q-learning algorithm, and prove PAC bounds on the performance of SQL. The bounds show that for any MDP with n state-action pairs and the discount factor \gamma \in [0, 1) a total of O(n \log(n/\delta)/((1 − \gamma)^4\epsilon^2)) step suffices for the SQL algorithm to converge to an \epsilon-optimal action-value function with probability 1 − \delta. We also establish a lower-bound of \Omega(n \log(1/\delta)/((1 − \gamma)^2\epsilon^2)) for all reinforcement learning algorithms, which matches the upper bound in terms of \epsilon, \delta and n (up to a logarithmic factor). Further, our results have better dependencies on \epsilon and 1 −\gamma, and thus, are tighter than the best available results for Q-learning. SQL also improves on existing results for the batch Q-value iteration, so far considered to be more efficient than the incremental methods like Q-learning.