Transient analysis of a M/M/{\infty} queue with discouragement and for the related embedded chain

Consider the following birth and death process with the following infinitesimal transition probabilities {\lambda}(k) ={\lambda}/(1+k) and {\mu}(k) = {\mu}k with {\lambda},{\mu}> 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.