A Note on an Efficient Approach for Solving Quasi-Birth-and-Death Processes Exhibiting Special Structures

Key Points

• The aim is to develop an efficient approach for calculating stationary probability distributions in quasi-birth-and-death processes.
• Analyzed the structure of the infinitesimal generator of QBD processes.
• Derived explicit formulations for the eigenvalues of the rate matrix R.
• Showed how to obtain stationary probabilities using the unique eigenvalue without computing R.
• Found conditions under which the stationary probability vector can be directly determined.
• Demonstrated that the approach simplifies calculations for specific QBD structures.
• Proved the existence of a stationary probability distribution when conditions are met.