1. Queuing TheoryFor Dummies Jean-Yves Le Boudec 1

2. 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

3. 1. Deterministic Queuing • Easy but powerful • Applies to deterministic and transient analysis • Example: playback buffer sizing 3

4. Use of Cumulative Functions 4

5. 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

6. 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

7. 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

8. Other Operational Laws 8

9. The Interactive User Model 9 9

10. Network Laws 10

11. Bottleneck Analysis • Example • Apply the following two bounds (1) (2) 17 11

12. Throughput Bounds 12

13. Bottlenecks A 13

14. DASSA • Intuition: within one busy period: to every departure we can associate one arrival with same number of customers left behind 14

15. 3. Single Server Queue 15

16. i.e. which are event averages (vs time averages ?) 16

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20. Non Linearity of Response Time 20

21. Impact of Variability 21

22. Optimal Sharing • Compare the two in terms of • Response time • Capacity 22

23. The Processor Sharing Queue • Models: processors, network links • Insensitivity: whatever the service requirements: • Egalitarian 23

24. PS versus FIFO • PS • FIFO 24

25. 4. A Case Study • Impact of capacity increase ? • Optimal Capacity ? 25

26. Methodology 26

27. 4.1. Deterministic Analysis 27

28. Deterministic Analysis 28

29. 4.2 Single Queue Analysis Assume no feedback loop: 29

30. 4.3 Operational Analysis • A refined model, with circulating users • Apply Bottleneck Analysis ( = Operational Analysis ) waiting time 1/c Z/(N-1) -Z 30

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33. 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