Unreliable queue model with multi-phases of services, delay repair and single vacation

In the present paper, we investigate a bulk queue model with unreliable server. The service is provided in multi phase; the first phase of service is essential for the entire arriving unit while remaining (l - 1) services are optional and are provided as per demand of unit. The second phase of service is followed by first phase, third is followed by second and so on. The server may stops working during any phase of service due to random failure. The repair work starts immediately as soon as server fails, provided the repair facility is available, otherwise there may be delay in repair. The server takes single vacation as soon system becomes empty. After completion of vacation the server may turns on with probability p if there are N customers accumulated in the system, otherwise it may remain idle with probability (1 - p). If there are more than N customers in the system then server immediately starts serving the waiting customer. The supplementary variable technique is applied to derive the server's state queue size distribution and queue size distribution at random epoch. The various performance measures of the system are obtained in explicit form. A numerical illustration is provided to verify the validity and sensitivity analysis of these performance measures.