Queuing Theory

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Queuing Theory

• for those who cannot wait

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All You Need to Know About Queuing Theory

• Queuing is essential to understand the behaviour
  of complex computer and communication systems
• In depth analysis of queuing systems is hard
• Fortunately, the most important results are easy

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1 Deterministic Queuing

• Easy but powerful
• Applies to deterministic and transient analysis
• Example playback buffer sizing

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Use of Cumulative Functions
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Solution of Playback Delay Pb
A.
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2 Stochastic Queuing SystemsClassical Results
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i.e. which are event averages (vs time averages ?)
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Non Linearity of Response Time
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Impact of Variability
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Optimal Sharing

• Compare the two in terms of
• Response time
• Capacity

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5.3 Operational Laws5.3.1 Littles Law
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Idea of Proof

• Consider a simulation where you measure R and N.
  You use two counters responseTimeCtr and
  queueLengthCtr. At end of simulation, estimate
  R responseTimeCtr / NbCust N
  queueLengthCtr / Twhere NbCust number of
  customers served and Tsimulation duration
• Both responseTimeCtr0 and queueLengthCtr0
  initially
• Q When an arrival or departure event occurs, how
  are both counters updated ?A queueLengthCtr
  (tnew - told) . q(told) where q(told) is the
  number of customers in queue just before the
  event. responseTimeCtr (tnew - told) .
  q(told)thus responseTimeCtr queueLengthCtr
  and thusN R . NbCust/T now NbCust/T is
  our estimator of ?

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Other Operational Laws
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The Interactive User Model
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4.2 Network Laws
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4.3 Bottleneck Analysis

• Apply the following two bounds

• Example

(1)
(2)
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Throughput Bounds
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Bottlenecks
A
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Case Study

• What is the impact on waiting time when capacity
  is doubled ?

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Application of Methodology
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Queuing Model
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Load (1) Transient Analysis
dmax
customers
dmax/2
B
A(t)
b(t)2ct
b0(t)ct
time
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Load (2) Stationary Analysis

• Assume no feedback loop

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Load (2) Stationary AnalysisA Refined Model,
with feedback loop

• Reduction of waiting time increases throughput
• A better model

c
Z
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Bottleneck Analysis
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• We also see that a key value isGive an
  interpretation for c
• The system should target c c

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Conclusion

• Queuing is essential in communication and
  information systems
• Bottleneck analysis and worst case analysis are
  usually very simple and often give good insights
• M/M/1, M/GI/1 and variants have closed forms