Batch-Service System. Consider a batch-service system as in Section 2.12 that processes items as follows. Items arrive to the station according to a Poisson process with rate λ and they enter a queue...

Batch-Service System. Consider a batch-service system as in Section 2.12 that processes items as follows. Items arrive to the station according to a Poisson process with rate λ and they enter a queue where they wait to be served. Items are processed in batches, and the number of items in a batch can be any number less than or equal to K (the service capacity). The service times of the batches are independent, exponentially distributed with rate μ independently of everything else. Only one batch can be served at a time and, during a service, additional arrivals join the queue. Batches are served when and only when the queue length is equal or greater than m (a control limit). In particular, if at the end of a service there are i ≥ m items in the queue, then a batch of i∧K items is served; and when the queue length is m − 1 and an arrival occurs, then a batch of size m is served. Let X_n
denote the queue length at the end of the nth service. Show that the probability of n arrivals during a service is qpn, where p = λ/(λ + μ) and q = 1 − p. Justify that X_n
is a Markov chain with