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notes3_EE549_2008

notes3_EE549_2008 - UNIVERSITY OF SOUTHERN CALIFORNIA...

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Unformatted text preview: UNIVERSITY OF SOUTHERN CALIFORNIA, SPRING 2008 1 Lecture Notes 3, Rate and Stability EE 549 — Queueing Theory Instructor: Michael Neely I. EXAMPLE RATE FUNCTION Here we give an example server rate process μ ( t ) that is stochastic in nature but has a well defined time average rate μ . Example: For t ≥ , define the stochastic process μ ( t ) as follows: μ ( t ) = R i whenever t ∈ [ i- 1 ,i ) where the { R i } variables are independent and identically distributed with the following distribution: • Pr[R = 1 kilobyte/sec] = 0.4 • Pr[R = 2 kilobyte/sec] = 0.4 • Pr[R = 3 kilobyte/sec] = 0.2 Claim : The μ ( t ) process of the above example has rate 1 . 8 kilobytes/sec. Proof: We have: Z t μ ( τ ) dτ = b t c X i =1 R i + Remainder ( t ) where b t c represents the largest integer less than or equal to t , and Remainder ( t ) represents the integrated portion of the ( b t c + 1) th timeslot (note that for all t , ≤ Remainder ( t ) ≤ 3 kilobytes). Dividing by t and taking limits, we have: lim t →∞ 1 t Z t μ ( τ ) dτ = lim t →∞ 1 t b t c X i =1 R i + lim t →∞ Remainder ( t ) t = lim t →∞ b t c t lim t →∞ 1 b t c b t c X i =1 R i + 0 = (1) ( E { R } ) = 1 . 8 kilobytes/sec (with prob. 1) where we have used the law of large numbers (LLN) in the final limit. 1 2 3 μ (t) 1 O 2 3 4 5 6 7 t R 1 R 2 R 3 R 4 R 5 R 6 R 7 Fig. 1. An example μ ( t ) sample path for the above example. UNIVERSITY OF SOUTHERN CALIFORNIA, SPRING 2008 2 busy busy Always Busy idle idle O t t * b Fig. 2. An example timing diagram illustrating busy and idle periods. II. EMPTYING FINITELY OFTEN Here we develop the notions of a queue of emptying infinitely often and emptying finitely often . Consider a single-server queue with input process X ( t ) , server process μ ( t ) , and corresponding unfinished work process U ( t ) ....
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notes3_EE549_2008 - UNIVERSITY OF SOUTHERN CALIFORNIA...

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