In the literature, various authors have studied birth-death processes in which the birth/growth comes to a halt as the population size grows beyond certain threshold. Logistic growth is an example to such growth. Such processes might have potential applications in several fields such as biology, ecology, and energy harvesting in tethered high altitude platform system in communication systems. This study, considers birth-death processes with temporary birth halts. A stability condition under which the population size remains finite with probability 1 is obtained. An explicit expression has been obtained for the steady state distribution of the process. We also studied a truncated version of the above process, which might have potential applications in several fields such as tumor progression modelling and energy harvesting modelling. An explicit steady state distribution is obtained for the truncated process also. Dependence of the average steady state population size on various modelling parameters is studied numerically for finite and infinite processes.