M/M(a,b)/1 Model Of Interdependent Queueing With Controllable Arrival Rates

Objectives: Instead of only providing individualized one-on-one assistance, some studies in the literature on queueing theory describe systems that provide services in batches. This study introduces controllable arrival rates and interdependency in such a system’s service and arrival processes and then obtains the queueing system’s probabilities and characteristics. It also verified the obtained results numerically. Methods: Controlling the arrival rates by faster and slower arrival rates are expected for the input, with Poisson (each time Poisson occurrence has one arrival) being the default assumption. The general bulk service rule dictates that the service be delivered in batches. Service begins only when the count of customers in the queue approaches or surpasses a and the capacity b (a  1). For brevity, a batch’s service time distribution is assumed to be exponential and is not dependent on the batch size. Then, all the steady-state probabilities are derived using a recursive approach. Findings: We used M/M(a,b)/1 as the notation. For this model, steady-state solutions & characteristics are derived and explored. All the probabilities are expressed in terms of P0;0(0). The expected count of customers and waiting time depends on the interdependency, service rate, faster arrival rate, and slower arrival rate. According to each parameter, all the results are verified. Novelty: There are works related to bulk service in queuing theory, but this is a new approach to give a bridge between bulk service and controllable arrival rates along with interdependency in the arrival and service process.