A computer system has three terminals that are connected to a
CPU that can handle at most two active users simultaneously. If a
person logs in at a terminal and requests service when two other users are active, then the request is held in a buffer until one of the users is finished. Customers arriving when there are no free terminals leave and never return. The arrival rate for users follow a Poisson process with rate λ and the usage times are independent exponential (µ). Let X(t) be the number of users of the computer system at time t, including requests held in the buffer.
(a) Determine the long run probability that the CPU is fully utilised. From now on, assume λ = 4 and µ = 3.
(b) Find the long run proportion of time when the system is idle.
(c) If the system is idle at time t = 0, find the mean time needed for the number of customers to reach 3.
(d) Compute P(1) to 3 decimal point accuracy.