On the transient behavior of a finite birth-death process with an application

The transient distribution of the number in the system and the distribution of the length of a busy period for a finite birth and death process is derived by solving the system of linear equations of Laplace-Transforms and finding the inversions through the properties of tridiagonal symmetric matrices. It is proved that the distributions of a busy period is hyperexponential. The steady-state solution of the number of customers in the system is verified without any difficulty. The numerical solution of the number of customers in the system and the busy period is possible by the use of a high speed computer for which a multi-server queueing system with balking and reneging services as an illustration. Some numerical comparisons are made with the randomization method.