Lining Theory For Dummies

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Lining Theory For Dummies Jean-Yves Le Boudec 1

All You Need to Know About Queuing Theory Queuing is fundamental to comprehend the conduct of complex PC and correspondence frameworks top to bottom investigation of lining frameworks is hard Fortunately, the most critical outcomes are simple We will concentrate this theme in two modules 1. straightforward ideas (this module) 2. lining systems (later) 2

1. Deterministic Queuing Easy yet capable Applies to deterministic and transient investigation Example: playback support estimating 3

Use of Cumulative Functions 4

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 each client pays one Fr for each moment introduce Payoff per client = R Rate at which we get cash = N In normal λ clients every moment, N = λ R 6

Little Again Consider a recreation where you measure R and N. You utilize two counters responseTimeCtr and queueLengthCtr. At end of reproduction, gauge R = responseTimeCtr/NbCust N = queueLengthCtr/T where NbCust = number of clients served and T=simulation length Both responseTimeCtr=0 and queueLengthCtr=0 at first Q: When a landing or flight occasion happens, how are both counters refreshed ? A: queueLengthCtr += (t new - t old ) . q(t old ) where q(t old ) is the number of clients in line just before the occasion. responseTimeCtr += (t new - t old ) . q(t old ) in this way responseTimeCtr == queueLengthCtr and along these lines N = R . NbCust/T ; now NbCust/T is our estimator of  7

Other Operational Laws 8

The Interactive User Model 9

Network Laws 10

Bottleneck Analysis Example Apply the accompanying two limits (1) (2) 17 11

Throughput Bounds 12

Bottlenecks A 13

DASSA Intuition: inside one occupied period: to each flight we can connect one entry with same number of clients abandoned 14

3. Single Server Queue 15

i.e. which are occasion midpoints (versus time midpoints ?) 16

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Non Linearity of Response Time 20

Impact of Variability 21

Optimal Sharing Compare the two regarding Response time Capacity 22

The Processor Sharing Queue Models: processors, organize joins Insensitivity: whatever the administration necessities: Egalitarian 23

PS versus FIFO PS FIFO 24

4. A Case Study Impact of limit increment ? Ideal Capacity ? 25

Methodology 26

4.1. Deterministic Analysis 27

Deterministic Analysis 28

4.2 Single Queue Analysis Assume no criticism circle: 29

4.3 Operational Analysis A refined model, with coursing clients Apply Bottleneck Analysis ( = Operational Analysis ) holding up time 1/c Z/(N-1) - Z 30

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Conclusions Queuing is basic in correspondence and data frameworks M/M/1, M/GI/1, M/G/1/PS and variations have shut structures Bottleneck investigation and most pessimistic scenario examination are typically exceptionally basic and frequently give great bits of knowledge … it stays to see lining systems 33