Suppose we model a packet stream as a Poisson point process with rate lamba(t) packets per second at time t. Suppose each packet in the stream contains L bits in it, where L is a constant. Assume L = 10000 throughout this exercise. Suppose that lambda(t) = 100 packets/sec for 0 <= t <= 100, lambda(t) = 200 packets/sec for t > 200.
A. Find the mean (expected) value, as well as the standard deviation, of M(t) for all time t > 0.
B. Suppose we estimate the bit rate of the packet stream by averaging the instantaneous bit rate using a first order linear filter with time constant tau, as discussed in class. Let R_tau (t) be the estimate of the rate at time t using the time constant tau for the linear filter. Draw a typical waveform for R_tau (t) for 0 < t < 300, for each of the following cases: (i) tau = 0.1 second, (ii) tau = 10 seconds, (iii) tau = 0.01 seconds.
C. Suppose instead that there is a constant inter-arrival time of 0.01 seconds between each consecutive arriving packet. In this case, compute and plot the value of R_tau (t) versus time in steady state for the following cases: (i) tau = 0.1 second, (ii) tau = 10 seconds, (iii) tau = 0.01 seconds.