Consider a single-server exponential system in which customers arrive at a rate $\lambda$ and have a regular service rate $\mu$. When a customers arrives and the system is busy, the customer joins the queue. However, when the queue length reaches a given threshold, say N, the service rate is increased to $\alpha\mu. (\alpha>1)$
a) Give a condition on $\lambda, \mu, \alpha$ which guarantees that the system is stable. (i.e the queue length does not go to infinity).
b) Determine the long term expected queue length.
c) Determine the mean time spent in the system for an arriving customer.