Algorithmic Analysis of Queuing System with Varying Number of Servers, Phase-Type Service Time Distribution, and Changeable Arrival Process Depending on Random Environment

An MAP/PH/N-type queuing system functioning within a finite-state Markovian random environment is studied. The random environment’s state impacts the number of available servers, the underlying processes of customer arrivals and service, and the impatience rate of customers. The impact on the state space of the underlying processes of customer arrivals and of the more general, as compared to exponential, service time distribution defines the novelty of the model. The behavior of the system is described by a multidimensional Markov chain that belongs to the classes of the level-independent quasi-birth-and-death processes or asymptotically quasi-Toeplitz Markov chains, depending on whether or not the customers are absolutely patient in all states of the random environment or are impatient in at least one state of the random environment. Using the tools of the corresponding processes or chains, a stationary analysis of the system is implemented. In particular, it is shown that the system is always ergodic if customers are impatient in at least one state of the random environment. Expressions for the computation of the basic performance measures of the system are presented. Examples of their computation for the system with three states of the random environment are presented as 3-D surfaces. The results can be useful for the analysis of a variety of real-world systems with parameters that may randomly change during system operation. In particular, they can be used for optimally matching the number of active servers and the bandwidth used by the transmission channels to the current rate of arrivals, and vice versa.