Reference no: EM13909665
Let {N1(t); t > 0} be a Poisson counting process of rate λ. Assume that the arrivals from this process are switched on and off by arrivals from a second independent Poisson process {N2(t); t > 0} of rate γ .
rate λ= N1(t)
rate γ= N2(t)
NA(t)
Let {NA(t); t≥0} be the switched process, i.e., NA(t) includes the arrivals from {N1(t); t > 0} during periods when N2(t) is even and excludes the arrivals from {N1(t); t > 0} while N2(t) is odd.
(a) Find the PMF for the number of arrivals of the first process, {N1(t); t > 0}, during the nth period when the switch is on.
(b) Given that the first arrival for the second process occurs at epoch τ , find the conditional PMF for the number of arrivals of the first process up to τ .
(c) Given that the number of arrivals of the first process, up to the first arrival for the second process, is n, find the density for the epoch of the first arrival from the second process.
(d) Find the density of the interarrival time for {NA(t); t ≥ 0}. Note: This part is quite messy and is done most easily via Laplace transforms.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.
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