A particular telephone switch handles only automated junk voicemail calls that arrive as a Poisson process of rate λ = 100 calls per minute. Each automated voicemail call has duration of exactly one minute. Use the method of Problem 10.13.2 to estimate the expected number of calls E[M(t)]. Do your results differ very much from those of Problem 10.13.2?
Problem 10.13.2
For the telephone switch of Example 10.28, we can estimate the expected number of calls in the system, E[M(t)], after T minutes using the time average estimate
Perform a 600-minute switch simulation and graph the sequence 
1, 2,..., 600. Does it appear that your estimates are converging? Repeat your experiment ten times and interpret your results.
Example 10.28
Simulate 60 minutes of activity of the telephone switch of Example 10.4 under the following assumptions.
(a) The switch starts with M(0) = 0 calls.
(b) Arrivals occur as a Poisson process of rate λ = 10 calls/min.
(c) The duration of each call (often called the holding time) in minutes is an exponential (1/10) random variable independent of the number of calls in the system and the duration of any other call.
Example 10.4
Consider an experiment in which we record M(t), the number of active calls at a telephone switch at time t, at each second over an interval of 15 minutes. One trial of the experiment might yield the sample function m(t,s) shown in Figure 10.2. Each time we perform the experiment, we would observe some other function m(t,s). The exact m(t,s) that we do observe will depend on many random variables including the number of calls at the start of the observation period, the arrival times of the new calls, and the duration of each call. An ensemble average is the average number of calls in progress at t = 403 seconds. A time average is the average number of calls in progress during a specific 15-minute interval.