Queuing TheoryFor Dummies

Jean-Yves Le Boudec1

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 modules1. simple concepts (this module)

2. queuing networks (later)

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1. Deterministic QueuingEasy 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

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A(t) A’(t) D(t)

time

bits

d(0)d(0) - d(0) + (D

1): r

(t -

d(0)

+

) (D

2): 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

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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 / NbCustN = 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 += (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 thus

N = 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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Network Laws

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Bottleneck AnalysisApply the following two bounds

1.

2.

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Example

(1)

(2)

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Throughput Bounds

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Bottlenecks

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A

DASSA

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

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3. Single Server Queue

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

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2 4 6 8 10Requests per Second

0.5

1

1.5

2

2.5

Mean Response Time in seconds

Non Linearity of Response Time

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Impact of Variability

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0.2 0.4 0.6 0.8Utilization

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4

6

8

10

12

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Mean Response Time

Optimal SharingCompare the two in terms of

Response time

Capacity

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The Processor Sharing Queue

Models: processors, network links

Insensitivity: whatever the service requirements:

Egalitarian

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PS versus FIFO

PS FIFO

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4. A Case Study

Impact of capacity increase ?

Optimal Capacity ?

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Methodology

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4.1. Deterministic Analysis

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Deterministic Analysis

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4.2 Single Queue Analysis

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Assume no feedback loop:

4.3 Operational AnalysisA refined model, with circulating users

Apply Bottleneck Analysis ( = Operational Analysis )

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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

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