Queuing Theory For 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

Use of Cumulative Functions 4

Solution of Playback Delay Pb 5 A(t) A’(t)D(t) time bits d(0)d(0) -  d(0) +  (D1): r(t - d(0) +  ) (D2): r (t - d(0) -  ) d(t) A.

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 / T where 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 += (t new - t old ). q(t old ) where q(t old ) is the number of customers in queue just before the event. responseTimeCtr += (t new - t old ). q(t old ) thus responseTimeCtr == queueLengthCtr and thus N = R. NbCust/T ; now NbCust/T is our estimator of 7

Other Operational Laws 8

The Interactive User Model 99

Network Laws 10

Bottleneck Analysis Apply the following two bounds  Example (1) (2) 17

Throughput Bounds 12

Bottlenecks 13 A

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

3. Single Server Queue 15

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

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Non Linearity of Response Time 20

Impact of Variability 21

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 PSFIFO 24

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

Methodology 26

4.1. Deterministic Analysis 27

Deterministic Analysis 28

4.2 Single Queue Analysis 29 Assume no feedback loop:

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

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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