Solving Erlang C systems with customer abandonment and server unavailability using matrix-analytic methods
Queueing systems with customer abandonment and server unavailability appear in various service settings, such as call centers and healthcare operations. Traditional queueing models, including Erlang-C, Erlang-B, and Erlang-A, assume servers are always available, making them unsuitable for analyzing systems with stochastic server unavailability. In this study, we analyze such systems in steady-state using the framework of level-dependent quasi-birth-and-death (LDQBD) processes.
Computing steady-state probabilities for LDQBD processes is challenging because it typically involves calculating rate matrices expressed as an infinite sum. Existing numerical methods often require an exogenous truncation level and lack error guarantees. Recent advancements, such as the algorithm proposed by Rastpour et al. [1], overcome this challenge by leveraging the physical interpretation of rate matrices. However, these methods have not been applied to queueing systems with both customer abandonment and server unavailability.
We first model the system as an LDQBD system with infinite levels and finite phases, where the phases are two-dimensional. Then, we prove that the rate matrices of the LDQBD process exhibit a property where they do not increase element-wise beyond a certain level. This property allows us to modify and extend existing numerical methods to efficiently compute steady-state probabilities for the system, within any desired error tolerance.
By adapting and expanding upon the algorithm proposed by [1], we develop an efficient numerical algorithm to compute steady-state probabilities without requiring an exogenous truncation level. This enables us to derive key performance metrics, including the expected number of customers in the system, expected waiting time, abandonment probability, probability of delay, and server utilization, with any desired accuracy. These measures provide valuable managerial insights into system performance, allowing decision-makers to assess the impact of server unavailability on service quality.
