Mathematical Analysis of M/M/C Vacation Queueing Model with a Waiting Server and Impatient Customers

Abstract

In this study, the transient analysis of the M/M/C queueing system has been made under the provision of servers' single vacation and loss of impatient customers. Customers arrive in the system in the Poisson process and are served by multiple servers in an exponential distribution process. Customers are served in the chronological order of their arrival. The main purpose of this investigation is to derive (i) the probability distribution functions, (ii) the formulas for the expected number of the customers in the system as well as in queue in the explicit form, (iii) the expected sojourn time and the expected time spent in waiting in the queue. Moreover, the sensitiveness of performance measures due to the small change of vacation rate γ, impatient rate ξ, and server’s waiting rate η has also been shown graphically. To show the applicability of the model under study, ample numerical results have been illustrated. The error computations have also been cited during the vacation period and busy period. Queueing model understudy may have its applications in multichannel telecommunications, security systems in the airport, train stations, and the manufacturing system.