Volume 35, Issue 1, 2026
DOI: 10.69980/03276716.2026.07
Modelling and Optimizing Psychological Service Systems Using Retrial Queue Models with Setup Times and Unreliable Servers
Abstract
In this paper, a single-server M/M/1 retrial queueing system is studied, which incor-porates therapist idle-state management, pre-session setup delays, and stochastic service disruptions with Bernoulli recovery, motivated from the operational dynamics of psychological service system like counseling centres, therapy clinics and psychiatric outpatient units. When there isn’t a waiting room, clients who are not served enter a virtual orbit and will†retry†after an exponentially distributed time period, consistent with the tendency to help seek behavior seen in mental health settings. To ensure it operates with minimal energy waste, the server is switched off when not in use, and activates after a period of set-up in preparation for service, similar to the pre-session preparation phase of a therapist. A server breakdown model is developed to model service disruptions that occur when the practitioner’s resources are exhausted, compassion fatigue occurs or the service is interrupted due to administrative issues, and a Bernoulli repair mechanism is added to distinguish between perfect and imperfect recovery. The steady-state probability distributions are derived by developing a continuous-time Markov chain framework to be used with probability generating functions. Closed form expressions for key performance metrics, such as the average orbit size and probabilities of the server state are derived. The findings demonstrate that higher repair effectiveness and reduced setup delays contribute significantly to system performance and lower overall operational cost, with direct implications for the design of responsive and sustainable mental health service systems.
Keywords
Retrial Queueing System, Psychological Service Systems, Setup Time, Unreliable Server, Bernoulli Repair Mechanism, Therapist Burnout, Cost Optimization.