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In this paper it is considered the queueing system, which consists of an infinite number of identical servers with FCFS buffer, which can store a queue of infinite length, with Poisson input flow of requests in a queueing system with intensity . Every request entering the queuing system, randomly selects and uses any of two servers of the system, and then is instantly sent to one of them, where a shorter queue. A share dynamics of servers in the system having the queue length is not less than can be described by infinite system of differential equations. It is possoble to formulate Tikhonov type Cauchy problem with initial conditions and small parameter for this infinite system of differential equations.A small parameter in the infinite system of differential equations allows to describe rapidly changing processes in large-scale queueing systems. The existence theoremis proved for the considered singularly perturbed Tikhon type Cauchy problem with initial conditions and small parameter.