A Problem on Searching

Let the point with probability $p_k > 0,k = 1,2, \cdots ,n$, be located in a cell with the number $k;\sum _{k = 1}^n p_k = 1$. Only one cell is inspected per unit of time. If the point lies in the cell being inspectted, it can be discovered with a probability $p > 0$.
The results of such investigations are independent. Let us denote by $\alpha_t,1\leq\alpha_t\leq n$, the number of the cell investigated at time $t$ if the point was not discovered up to the time $t-1$. Let $\alpha=(\alpha_1,\alpha_2,\cdots,\alpha_t,\cdots)$ be the procedure of searching and $\tau_\alpha$ the time required for discovering the point. In this paper a procedure of searching $\alpha^*$ is determined so that
$$ {\mathbf M}\tau _{\alpha^*}=\mathop {\inf }\limits_\alpha{\mathbf M}\tau _a . $$