In this paper, we consider a queueing inventory system with batch arrival and batch service processes. Customers arrive in batches of sizes $1,2,\dots ,k$, according to a marked compound Poisson process. We call a batch of customers as belonging to j when there are j individual customers in that batch. The service facility has waiting rooms for each category of customers and also a room to serve them. Except for category 1, all other customers have finite waiting rooms. The service room has only a limited number of seats. These seats are arranged in such a fashion that customers belonging to category j have groups of seats, each with j seats for $j=1,2,\dots ,k$. Customers are taken for service according to the availability of seats designated to each category. A category j customer can be taken for service only if j items are available in the inventory. The service time of customers of category j is exponentially distributed with parameter depending on j for $j=1,2,\dots ,k$. The number of seats available in the service room for each category of customers is restricted to a finite number. The replenishment for items follows the $(s,S)$ policy: fill up to S at the time of replenishment. Lead time follows an exponential distribution. We analyze the system in the equilibrium state. Performance characteristics are evaluated and a number of numerical illustrations are provided.

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.