This paper analyze a finite capacity Markovian feedback queue
with discouraged arrivals, reneging and retention of reneged customers
in which the inter-arrival and service times follow exponential
distribution. The transient solution of the system, with results in terms of
the eigenvalues of a symmetric tri-diagonal matrix. Feedback in queuing
literature represents customer dissatisfaction because of inappropriate
quality of service. In case of feedback, after getting partial or incomplete
service, customer retries for service . After the completion of service ,
each customer may rejoin the system as a feedback customer for
receiving another regular service with probability 1  or he can leave the
system with probability 1 q ( 1 1  q 1 ). A reneged customer can be
retained in many cases by employing certain convincing mechanisms to
stay in queue for completion of service. Thus, a reneged customer can be
retained in the queuing system with probability 2 q or he may leave the
queue without receiving service with probability 2  ( 2 2  q 1 ).
Expressing the Laplace transforms of the system of governing equations
in matrix form and using the properties of symmetric tri-diagonal
matrices, the steady state probabilities are derived and some important
queuing models are derived as special cases of this model.