At the bank near where I work, there is a central queue served by four assistants. From past experience, I know that the service times of the four assistants are exponentially distributed with means 2 minutes, 3 minutes, 5 minutes and 6 minutes. One day Anita enters the bank to discover that all four assistants are busy, but no one is waiting to be served. (a) Find the distribution of the time that Anita will have to wait before she can move forward for service. Show that her expected waiting time is 50 seconds. (b) Calculate the probability that Anita will have to wait more than three minutes until an assistant is free to serve her. (c) Two minutes after entering the bank, Anita is still waiting. Another customer, Elias, enters the bank and stands behind her in the queue. (So now the same four people are being served as at the start, and two people are standing in the queue.) State the distribution of Elias' waiting time before his service commences. Give a brief explanation for your answer. [3] [2] Hence calculate the mean and standard deviation of Elias' waiting time. (d) Suggest two random processes associated with the queueing system, one with a discrete time domain and one with a continuous time domain. In your examples, you should use the notation for random processes introduced in Part I of Book 2; for example. (x(t);120) or {X:n 0,1,...). Describe precisely what the random variables (X(t). X,, or whatever) represent, and state clearly the state space corresponding to each process. (See Activity 2.5 in Book 2 for the sort of thing that is required here.) [4] [4]