This paper extends the k-out-of-n:G reliability system to a multi-server queue. We study a multi-server reliability-queuing model with the N-policy of repair. The queuing system considered here has n servers, each of which has identically and exponentially distributed service times with parameter $\mu$. Servers are subject to breakdown at an exponential rate $\gamma$. The repair process follows the N-policy of repair. Although these servers work independently of each other, service can be provided only when k functional servers are available in the system. We study the model in the steady state, using the matrix analytic method. We evaluate some associated performance measures and provide graphical/numerical illustrations. We consider an optimization problem, and the results of the study are presented.

In this paper, we consider the batch arrival of customers to a transport station. Customers belonging to each category is considered as a single entity according to a BMMAP. An Erlang clock of order m starts ticking when the transport vessel reaches the station. When the Lth stage of the clock is reached, an order for the next vessel is placed. The lead time for arrival of the vessel follows exponential distribution. There are two types of rooms in this system: the waiting rooms and the service rooms for customers in the transport station and in the vessel, respectively. The waiting room capacity for customers of type 1 is infinite whereas those for customer of type $2,\dots ,k$ are of finite capacities. The service room capacity $C_j$ for customer type $j,j=1,2,\dots ,k$ is finite. Upon arrival, customers of category j occupy seats designated for that category in the vessel, provided there is at least one vacancy belonging to that category. The total number of vessels with the operator is $h^{*}$. The service time of each vessel follows exponential distribution with parameter $\mu .$ Each group of customers belong to category j searches independently for customers of this category to mobilize passengers when the Erlang clock reaches $L_1$ where $L_1<L$. The search time for customers of category j follows exponential distribution with parameter ${\lambda }_j.$ The stability condition is derived. Some performance measures are estimated.