The M/M/S/M+ Queuing System with Clients Abandonment

This paper develops a comprehensive analytical model for the M/M/S/M+ queueing system with customer abandonment. The transition rate diagram and associated transition probabilities are derived to characterize the system dynamics. Key performance metrics are obtained, including the mean number of customers in the system, the mean waiting time in the system, the probability of an empty system, the probability of finding all servers busy, and the abandonment probability. Specific focus is given to characterizing the abandonment behavior, deriving expressions for the mean abandonment time and the mean number of abandoned customers. The mathematical formulation provides insights into the impacts of arrival rates, service rates, customer impatience levels and system capacity on performance. Potential applications span various domains where customer abandonment significantly affects system design and resource provisioning, such as call centers, healthcare facilities, transportation services and e-commerce platforms. The analytical queueing model enables quantifying fundamental tradeoffs between service quality metrics like customer waiting times and the potential revenue losses from abandonments. The results can guide capacity planning, staffing optimization, and decision support for design and control of queueing systems with impatient customers.