Multi-Server Queue with Batch Markovian Arrival Process, Heterogeneous Servers, Phase Type Distribution of Service Times and Customers Search in the Buffer

A multi-server queueing system are studied. Customer arrival to the system is described by a Batch Markovian arrival process. Servers of the system are heterogeneous and indexed. The service times have phase-type distributions with irreducible representation depending on the server's index. Server that becomes idle immediately starts searching for customers in the buffer and then provides service to the found customer. The search time has an exponential distribution with the rate dependent the index of a server and on the number of customers residing in the buffer. The dynamics of the system states is described by a multidimensional Markov chain that belongs to the class of the asymptotically quasi-Toeplitz Markov chain. Transparent ergodicity condition is proven. Computation of the stationary characteristics of the system is discussed.