Optimization of the capacity of the main storage in $G/M/1/K$ queueing system with an additional storage device

The problem of optimizing the capacity of the main storage device of a queuing system of the type $G/M/1/K$ with an additional storage device with the cost objective function is formulated taking into account the costs of the system associated with the loss of requests, storage of requests, maintenance of storage devices, and device downtime. A request arriving to the system is accepted into the main storage device if there is a free space; otherwise, according to the given probability distribution, it goes to the additional device if there is a free space. A request leaves the system (is lost) if both storage devices are full. If space is freed up in the main storage device, then one of the requests from the queue in the additional device immediately enters the queue in the main device. The unimodality of the objective function is proved and the procedure for finding the optimal capacity of the main storage device is proposed.