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Unformatted text preview: UNIVERSITY OF SOUTHERN CALIFORNIA, SPRING 2008 1 Lecture Notes 4 EE 549 — Queueing Theory Instructor: Michael Neely I. EXAMPLES OF STABILITY CONDITIONS Example: Consider a general arrival process and let N ( t ) represent the number of arrivals during [0 ,t ] . Suppose the following conditions hold: • N ( t ) t → λ as t → ∞ . • Packet lengths are B 1 ,B 2 ,B 3 .... (not necessarily i.i.d) • lim n →∞ 1 n ∑ n i =1 B i = B (w.p.1). Claim : The bit rate of this process is λ B . Proof: We have: X ( t ) = N ( t ) X i =1 B i Dividing by t and taking limits we have: lim t →∞ X ( t ) t = lim t →∞ 1 t N ( t ) X i =1 B i = lim t →∞ N ( t ) t ∑ N ( t ) i =1 B i N ( t ) = λ B completing the proof. If these arrivals feed into a single server queue with time average rate μ , then stability condition is λ B ≤ μ . II. SERVICE TIME DESCRIPTION Queueing systems may not involve bits. In such systems, the arrival process consists of jobs which have service times. The service time S i of job i represents the amount of time this job will take to complete when it gets to the server. How does this relate to our original concept (as shown in Fig 1)? Define an equivalent virtual system as follows: • A single server system with processing rate μ = 1 bit/s in which “packets” arrive exactly when the jobs arrive in the actual system....
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This note was uploaded on 12/21/2010 for the course EE 549 taught by Professor Neely during the Spring '08 term at USC.
- Spring '08