Consider a queuing system M/M/∞ with an unlimited number of servers. Assume that the system is operating under a stationary distribution. You are given the following information:
- Service requests arrive in such a way that the expected idle time is λ^(-1) = E[I] = 6 minutes.
- Service times are independently exponentially distributed with a mean equal to µ^(-1) = 15 minutes.
Assuming that the time variable below is measured in hours:
1. Given that the service operates 24 hours a day, find the expected queue length under the stationary distribution.
2. For the departure process, D = {D(t) : t ≥ 0}, find the covariance between counts of completed services within the time intervals (1 ≤ t ≤ 4) and (2 ≤ t ≤ 5), or Cov[(D(4) − D(1)), (D(5) − D(2))].