(a) Consider an M/M/1 queueing system with arrival rate λ, service rate μ, μ > λ. Assume that the queue is in steady state. Given that an arrival occurs at time t, find the probability that the system is in state i immediately after time t.
(b) Assuming FCFS service, and conditional on i customers in the system immediately after the above arrival, characterize the time until the above customer departs as a sum of rv s.
(c) Find the unconditional probability density of the time until the above customer departs. Hint: You know (from splitting a Poisson process) that the sum of a geometrically distributed number of IID exponentially distributed rv s is exponentially distributed. Use the same idea here.