This paper investigates resource scheduling in a wireless communication system operating with Energy Harvesting (EH) based devices and perfect Channel State Information (CSI). The aim is to minimize the packet loss that occurs when the buffer is overflowed or when the queued packet is older than a certain pre-defined threshold. We so consider a strict delay constraint rather than an average delay constraint. The associated optimization problem is modeled as Markov Decision Process (MDP) where the actions are the number of packets sent on the known channel at each slot. The optimal deterministic offline policy is exhibited through dynamic programming techniques, i.e. Value Iteration (VI) algorithm. We show that the gain in the number of transmitted packets and the consumed energy is substantial compared to: i) a naive policy which forces the system to send the maximum number of packets using the available energy in the battery, ii) two variants of the previous policy that take into account the buffer state, and iii) a policy optimized with an average delay constraint. Finally, we evaluate our optimal policy under imperfect CSI scenario where only an estimate of the channel state is available.