Mathematical Model in the Form of an M/M/1 Retrial Queue with Non-Persistent Primary Customers

Let us consider a mathematical model in the form of a retrial queue (RQ), i.e., a single-server queueing system with a source of repeated calls. A Poisson arrival process with rate λ enters the system. A request that finds the server idle immediately occupies it for service, the service time being exponentially distributed with parameter μ. The arriving requests can be divided into two categories: primary and retrial requests. Primary requests are those that arrive at the system from an external source, while retrial requests are those that are located in the system orbit and repeatedly attempt to seize the server for service. If the server is busy, an arriving request joins the orbit with probability H, where it experiences a random delay whose duration is exponentially distributed with parameter σ, or leaves the system with probability 1−H. After the random delay, the request from the orbit makes a repeated attempt to access the server. If the server is idle, the retrial request occupies it for a random service time; if the server is busy, the request immediately returns to the orbit to initiate the next random delay.