Performance Evaluation of a Single-Server Queueing System with Correlated Arrivals, Two-Tier Service Structure, Random Breakdowns and Phase-Type Repairs

A single-server queueing framework with infinite queueing capacity is formulated and analyzed, where customer arrivals are governed by a Markovian arrival process and both service and repair times are characterized through phase-type distributions. The service structure is two-tier: each incoming customer undergoes a mandatory primary service under a first-come, first-served discipline; then a secondary service becomes available on an optional basis, provided only at the customer’s request after the primary service is completed. When the system empties, the server initiates a shutdown process before entering a vacation period. Upon return from vacation, the server resumes service if customers are present; otherwise, successive vacations will be taken until demand arises. Random server failures can occur during either service mode, after which the server undergoes repair before restarting. The steady-state behavior of the system is analyzed using the matrix analytical method, from which the stability conditions, stationary probability vectors, and key performance metrics are derived. A cost analysis framework is also formulated to evaluate the economic implications of system operation. To substantiate these analytical findings and illustrate the practical applicability of the proposed model, a series of numerical experiments are performed.