Abstract  

         Instead of individualized one to one assistance, here we describe systems that provide services for group of customers. This study introduces controllable arrival rates with vacation and interdependency of the system’s service and arrival processes. A faster and slower arrival rates are meant to be controllable arrivals, with Poisson (each time Poisson occurrence has one arrival) being the default assumption. Service begins only when the count of customers in the queue approaches or surpasses a and the capacity b (≥a ≥ 1). A vacation period defines when a server goes for performing other uninterruptible work when the system is idle. Then, all the steady-state equations are derived to find the system’s probabilities. 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 P_0,0(0). The expected number 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. There are works related to bulk service and vacation, but this is a new approach to give a bridge between bulk service and controllable arrival rates with vacation along with interdependency in the arrival and service process.

Keywords: M/M(a,b)/1 Queueing system, Bulk service, controllable arrival rates, steady states, interdependent model, stochastic processes, Vacation.