Please answer all a b c STOCHASTIC PROCESSES

Customers come to a service counter using a Poisson process of intensity v and line up in order of arrival if the counter is busy. The time of each service is independent of the others and has an exponential distribution of parameter 1. However, each customer who has just been served is not satisfied with the service received with probability p regardless of all stays and then returns to line up at the end of it. Determine: (a) the condition for the existence of a stationary distribution for the number of customers in the system (in line or in the process of being served); (b) this stationary distribution in the case where it exists; and in this case Kc) the proportion long-term average time that the server is busy. Customers come to a service counter using a Poisson process of intensity v and line up in order of arrival if the counter is busy. The time of each service is independent of the others and has an exponential distribution of parameter 1. However, each customer who has just been served is not satisfied with the service received with probability p regardless of all stays and then returns to line up at the end of it. Determine: (a) the condition for the existence of a stationary distribution for the number of customers in the system (in line or in the process of being served); (b) this stationary distribution in the case where it exists; and in this case Kc) the proportion long-term average time that the server is busy