An M/G/2 queue where customers are served subject to a minimum violation of FCFS queue discipline

This article discusses the steady state analysis of the M/G/2 queuing system with two heterogeneous servers under new queue disciplines when the classical First Come First Served '(FCFS)' queue discipline is to be violated. Customers are served either by server-I according to an exponential service time distribution with mean rate μ or by server-II with a general service time distribution B(t). Sequel to some objections raised in the literature on the use of the classical FCFS queue discipline in heterogeneous service systems, two alternative queue disciplines (Serial and Parallel) are considered in this work with the objective that if the FCFS is violated then the violation is a minimum in the long run. Using the embedded method under the serial queue discipline and the supplementary variable technique under the parallel queue discipline, we present an exact analysis of the steady state number of customers in the system and most importantly, the actual waiting time expectation of customers in the system. Our work shows that one can obtain all stationary probabilities and other vital measures for this queue under certain simple but realistic assumptions.