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This thesis studies the optimal control $N$-policy of a $"non-
perfect"$ removable service station in the M/M/1 queueing
systems with infinite and finite capacities, respectively. A
$"non-perfect"$ removable service station means that the
removable service station is typically subject to unpredictable
breakdown. The form of the $N$-policy is that turn on the
service station (i.e. open the service station to provide
service) when $N$ customers are present in the system and turn
it off when the system is empty (i.e. no customers are in the
system), and this cycle is repeated. The service station can
break down only when it is turned on and there are at least one
customer in the system. First, we develop the steady-state
characteristics of the infinite capacity and finite capacity
systems, such as the probability distributions of the number of
customers in the system, the expected number of customers in
the system, and so on. Second, the results of these two systems
generalize (i) the $"perfect"$ M/M/1 queue with a removable
server; (ii) the $"non-perfect"$ M/M/1 queue; and (iii) the
$"standard"$ M/M/1 queue. Finally, we derive the total steady-
state expected cost function per unit time, and determine the
optimal value of the control parameter $N$, $N^\star$, in order
to minimize this function for these two systems.
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