We consider two variants of the standard multi-armed bandit problem, namely, the multi-armed bandit problem with transition costs and the multi-armed bandit problem on graphs. We develop block allocation algorithms for these problems that achieve an expected cumulative regret that is uniformly dominated by a logarithmic function of time, and an expected cumulative number of transitions from one arm to another arm uniformly dominated by a double-logarithmic function of time. We observe that the multi-armed bandit problem with transition costs and the associated block allocation algorithm capture the key features of popular animal foraging models in literature.