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Given an undirected graph G=(V,E) and a source vertex s@?V, the k-traveling repairman (KTR) problem, also known as the minimum latency problem, asks for k tours, each starting at s and together covering all the vertices (customers) such that the sum of the latencies experienced by the customers is minimum. The latency of a customer p is defined to be the distance traveled (time elapsed) before visiting p for the first time. Previous literature on the KTR problem has considered the version of the problem in which the repairtime of a customer is assumed to be zero for latency calculations. We consider a generalization of the problem in which each customer has an associated repairtime. For a fixed k, we present a (@b+2)-approximation algorithm for this problem, where @b is the best achievable approximation ratio for the KTR problem with zero repairtimes (currently @b=6). For arbitrary k, we obtain a (32@b+12)-approximation ratio. When the repairtimes of the customers are all the same, we present an approximation algorithm with a better ratio. We also introduce the bounded-latency problem, a complementary version of the KTR problem, in which we are given a latency bound L and are asked to find the minimum number of repairmen required to service all the customers such that the latency of no customer is more than L. For this problem, we present a simple bicriteria approximation algorithm that finds a solution with at most 2/@r times the number of repairmen required by an optimal solution, with the latency of no customer exceeding (1+@r)L, @r0.