Example
Example 5.25 (The Output Process of an Infinite Server Poisson Queue) It turns out that the output process of the M/G/∞ queue—that is, of the infinite server queue having Poisson arrivals and general service distribution G—is a nonhomogeneous Poisson process having intensity function λ(t) = λG(t). To verify this claim, let us first argue that the departure process has independent increments. Towards this end, consider nonoverlapping intervals O1 , … , Ok ; now say that an arrival is type i, i = 1, … , k, if that arrival departs in the interval Oi .
5.4 Generalizations of the Poisson Process
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By Proposition 5.3, it follows that the numbers of departures in these intervals are independent, thus establishing independent increments. Now, suppose that an arrival is “counted” if that arrival departs between t and t + h. Because an arrival at time s, s < t + h, will be counted with probability G(t−s + h)−G(t−s), it follows from Proposition 5.3 that the number of departures in (t, t + h) is a