Input-Output System with Batch Arrivals. Let X(t) denote the quantity of items in an input-output system that operates as follows. Batches of items arrive at times that form a Poisson process with...

Input-Output System with Batch Arrivals. Let X(t) denote the quantity of items in an input-output system that operates as follows. Batches of items arrive at times that form a Poisson process with rate λ, and the batch sizes are i.i.d. with geometric distribution (1 − α)^αn−1. The items are served by a single server and the service times are independent exponentially distributed with rate μ > λ/(1−α). Justify that X(t) is a CTMC and specify its transition rates. Show that its stationary distribution is