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## Queuing Theory For Dummies

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**Queuing TheoryFor Dummies**Jean-Yves Le Boudec 1**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 • We will study this topic in two modules • 1. simple concepts (this module) • 2. queuing networks (later) 2**1. Deterministic Queuing**• Easy but powerful • Applies to deterministic and transient analysis • Example: playback buffer sizing 3**Solution of Playback Delay Pb**bits A(t) A’(t) D(t) d(t) (D2): r (t - d(0) - D) (D1): r(t - d(0) + D) time d(0) - D d(0) d(0) + D A. 5**2. Operational Laws**• Intuition: • Say every customer pays one Fr per minute present • Payoff per customer = R • Rate at which we receive money = N • In average λ customers per minute, N = λ R 6**Little Again**• 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 T=simulation duration • Both responseTimeCtr=0 and queueLengthCtr=0 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 7**Network Laws**10**Bottleneck Analysis**• Example • Apply the following two bounds (1) (2) 17 11**Bottlenecks**A 13**DASSA**• Intuition: within one busy period: to every departure we can associate one arrival with same number of customers left behind 14**Optimal Sharing**• Compare the two in terms of • Response time • Capacity 22**The Processor Sharing Queue**• Models: processors, network links • Insensitivity: whatever the service requirements: • Egalitarian 23**PS versus FIFO**• PS • FIFO 24**4. A Case Study**• Impact of capacity increase ? • Optimal Capacity ? 25**Methodology**26**4.2 Single Queue Analysis**Assume no feedback loop: 29**4.3 Operational Analysis**• A refined model, with circulating users • Apply Bottleneck Analysis ( = Operational Analysis ) waiting time 1/c Z/(N-1) -Z 30**Conclusions**• Queuing is essential in communication and information systems • M/M/1, M/GI/1, M/G/1/PS and variants have closed forms • Bottleneck analysis and worst case analysis are usually very simple and often give good insights • … it remains to see queuing networks 33