26 two station tandem queue in this system customers

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Unformatted text preview: stationary distribution ⇡ (n) for the M/M/s queue are unpleasant to write down for a general number of servers s, but it is not hard to use (4.26) to find the stationary distribution in a concrete cases: If s = 3, = 2 and µ = 1, then 1 X k=2 so P1 k=0 1 c 2X ⇡ (k ) = · 2 (2/3)j = 6c 2 j =0 ⇡ (k ) = 9c and we have 1 ⇡ (0) = , 9 2 ⇡ (1) = , 9 2 ⇡ (k ) = 9 ✓ ◆k 2 3 2 for k 2 Our next result is a remarkable property of the M/M/s queue. Theorem 4.8. If < µs, then the output process of the M/M/s queue in equilibrium is a rate Poisson process. Your first reaction to this should be that it is crazy. Customers depart at rate 0, µ, 2µ, . . ., sµ, depending on the number of servers that are busy and it is usually the case that none of these numbers = . To further emphasize the surprising nature of Theorem 4.8, suppose for concreteness that there is one server, = 1, and µ = 10. If, in this situation, we have just seen 30 departures in the last 2 hours, then it seems reasonable to guess that the server is busy and the...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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