Fundamentals of Queueing Theory by Thompson James M. Gross Donald Harris Carl M. Shortle John F

Author:Thompson, James M., Gross, Donald, Harris, Carl M., Shortle, John F.
Language: eng
Format: epub
Publisher: John Wiley & Sons
Published: 2011-08-26T16:00:00+00:00

We now show that this guess for xj satisfies the condition for Theorem 1.2. From the matrix P,

Now,

(5.28)

So,

[This implies that the expected single-step displacement of the chain from state i > 0 is ρ – 1 < 0. This is because ∑jpij destination state starting from i, and ∑jpij = (1 – ρ) ∑pijxj = i – (1 – ρ).]

Also,

Hence it follows that the chain possesses identical stationary and long-run distributions when ρ < 1.

The proof of the necessity of ρ < 1 for ergodicity arises directly from the existence of the generating function Π(z) over the interval |z| < 1,

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