This paper deals with the optimal control policy of a single removable and unreliable server in an N-policy two-phase M X /M/1 queueing system without gating and server startups. The arrivals occur in batches according to a compound Poisson process and waiting customers receive batch service all at a time in the first phase and proceed to the second phase to receive individual service. The server is turned off each time the system empties, as and when the queue length reaches or exceeds N (threshold), the server is immediately turned on but is temporarily unavailable to serve the waiting batch of customers. The server needs a startup time before providing batch service in the first phase. The server is subject to breakdowns during individual service according to a Poisson process and repair times of the server follow an exponential distribution. The distribution of the system size is derived through probability generating functions and obtained other system characteristics. Finally, the expected cost per unit time is considered to determine the optimal operating policy at a minimum cost. The sensitivity analysis has been carried out to examine the effect of different parameters in the system.

The two-phase service models analyzed by several authors considered only the probabilistic nature of the queue parameters with fixed cost elements. But the queue parameters and cost elements will be in general are of both possibilistic and probabilistic in nature. Analyzing the performance of the queueing systems with fuzzy environment facilitates to investigate for the possibilistic interval estimates to the performance measures of a queueing system rather than point estimates. In this work, it is proposed to construct membership function of the fuzzy cost function to obtain confidence estimates for some performance measures of a controllable two-phase service single server Markovian gated queue with server startups and breakdowns under N-policy in which the queue parameters viz. arrival rate, startup rate, batch service rate, individual service rate, repair rate and cost elements are all defined as fuzzy numbers. Based on Zadeh’s extension principle and the α-cuts, a set of parametric nonlinear programming problems are developed to find the upper and lower bounds of the minimum total expected cost per unit time at the possibility level α. As the analytical solutions of the nonlinear programming problems developed for the proposed model are tedious, considering the system parameters and cost elements as trapezoidal fuzzy numbers, numerical results for the lower and upper bounds of the optimal threshold N* and the minimum total expected cost per unit time are computed using the nonlinear programming solver available in MATLAB.