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We consider a wireless sensor network with n sensor nodes. The sensed data needs to be transferred in a multi-hop fashion to a common processing center. We consider the standard data sampling/sensing scheme where the sensor nodes have a sampling process independent of the transmission scheme. In this paper, we study the problem of optimizing the end-to-end delay in a multi-hop single-sink wireless sensor network. We prove that the delay-minimization objective function is strictly convex for the entire network. We then provide a distributed optimization framework to achieve the required objective. The approach is based on distributed convex optimization and deterministic distributed algorithm without feedback control. Only local knowledge is used to update the algorithmic steps. Specifically, we formulate the objective as a network level delay minimization function where the constraints are the reception-capacity and service-rate probabilities. Using the Lagrangian dual composition method, we derive a distributed primal-dual algorithm to minimize the delay in the network.We further develop a stochastic delay control primal-dual algorithm in the presence of noisy conditions. We also present its convergence and rate of convergence. The proposal is extensively evaluated by analysis and simulations.