Consider the following birth and death process with the following infinitesimal transition probabilities λ(k) =λ/(1+k) and μ(k) = μk with λ,μ> 0. This process has known as a discouragement queue [5]. Although from the theoretical point of view the problem of the determination of the transition functions has been solved [4], the explicit form of them is not present in literature. We have solved this problem assuming that the solution is representable by a Taylor series, under the initial condition that the process starts to state zero. We discuss also the same problem for the embedded chain and using direct computation we obtain a recursive formula for the transient distribution.