Abstract In this paper, we consider a queuing model in which two types of customers, say, Types 1 and 2, arrive according to a Markovian arrival process (MAP). Type 1 customers have a buffer of capacity K and are served in groups of varying sizes ranging from a predetermined value L to a maximum size, K. The service times are exponentially distributed. Type 2 customers are served one at a time and the service times are assumed to be exponential. The system has two servers, of which one is totally dedicated to serving Type 2 customers. The other server can serve both Types 1 and 2 customers. Any arriving Type 1 customer finding the buffer full is considered lost. Any Type 2 customer not entering into service immediately orbit in an infinite space. These orbiting customers compete for service by sending out signals that are exponentially distributed. The steady state probability vector of this queuing model is of matrix-geometric type with a highly sparse rate matrix. This sparsity is exploited in the analysis and several interesting numerical examples are discussed. The Laplace-Stieltjes transform LST of the waiting time distribution of a Type 1 customer at an arrival epoch is derived.