The $MMAP/M/R/0$ queueing system with reservation of servers operating in a random environment

We consider a multiserver queueing system without buffer, with customers of two types, operating in a random environment. The system is fed by a marked Markovian arrival process which depends on the environment state. Customers of the first type have absolute priority over customers of the second type. Instantaneous service rates are piecewise constant with parameters depending on the customer type and the current environment state. The system behavior is described by a continuous-time multivariate Markov chain. We present a generator of this chain in a block-tridiagonal form. We briefly describe the procedure for finding a stationary probability distribution of system states and obtain formulas for the main probabilistic characteristics of the system in terms of the stationary distribution. An algorithm for computing the Laplace–Stieltjes transform of the sojourn time for an arbitrary customer of the first type is obtained.