# 26 two station tandem queue in this system customers

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

Ask a homework question - tutors are online