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Unformatted text preview: r: (i) m = 0, n > 0, (ii) m > 0, n = 0, and (iii) m = 0, n = 0. In these
cases some of the rates are missing: (i) those in (a), (ii) those in (b), and (iii)
those in (a) and (b). However, since the rates in each group balance we have
⇡ Q = 0. 145 4.6. QUEUEING NETWORKS* Example 4.27. General two-station queue. Suppose that at station i:
arrivals from outside the system occur at rate i , service occurs at rate µi , and
departures go to the other queue with probability pi and leave the system with
probability 1 pi . 1 2 ? p1 µ1 ?
- µ2 p2
? p1 p2 Our question is: When is the system stable? That is, when is there a stationary
distribution? To get started on this question suppose that both servers are busy.
In this case work arrives at station 1 at rate 1 + p2 µ2 , and work arrives at
station 2 at rate 2 + p1 µ1 . It should be intuitively clear that:
(i) if 1 + p2 µ2 < µ1 and 2 + p1 µ1 < µ2 , then each server can handle their
maximum arrival rate and the system will have a stationary distribution.
(ii) if 1 + p2 µ2 > µ1 and 2 + p1 µ1 > µ2 , then there is positive probabi...
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