Analysis of a Crowdsourcing Markovian Queue with Phase-Type and Imperfect Service, Working Vacations, Breakdown, and Repair

Analysis of a crowdsourcing Markovian queue with phase-type service is considered in this paper. In this model, a customer not only receives service but also assists in delivery. In other words, in a retail environment, while some customers shop in-store, others place orders online or by phone and require home delivery. Store management can utilize online customers as couriers to complete these deliveries. However, because not every customer may agree to take part, a probabilistic element is included to capture the chances of their participation. The model also incorporates imperfect service, reflecting cases where deliveries may fail or require rework, and working breakdowns, representing partial disruptions in service capacity rather than complete stoppages. To analyze the system under steady-state conditions, matrix-analytic methods are applied. Numerical examples illustrate the significant benefits of incorporating these dynamics into traditional queueing models.