Heterogeneous queueing system $\mathrm{MR(S)/M(S)/}\infty$ with service parameters depending on the state of the underlying Markov chain

Data streams in information and communication systems include integrated heterogeneous streams, containing voice, text data and video. Since the service of different information units takes different time depending on their format, used protocols and so on, it is proposed to model such data transmission processes using heterogeneous queueing systems with services depending on the parameters of the incoming stream. In the paper, an infinite-server heterogeneous queueing system is considered. Arrivals are modeled as a Markov renewal process (MRP) with two states given by distribution functions of the interval lengths and by a transition probability matrix. The exponential distribution parameter of service time is determined by the state of the underlying Markov chain of the MRP at the moment when a customer arrives and does not change until the service completion. To study the system, the method of characteristic functions is used. Using their properties, analytical expressions are obtained for the initial moments of the first and the second order of the number of customers of each type present in the system in a steady-state regime. To analyze the relationship between the components of the process, a correlation moment is derived.