queuing for dummies - performance evaluationperfeval.epfl.ch/printme/queuing.pdf · product form 49...
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Queuing TheoryFor Dummies
Jean‐YvesLeBoudec1
All You Need to Know About Queuing Theory
Queuingisessentialtounderstandthebehaviour ofcomplexcomputerandcommunicationsystemsIndepthanalysisofqueuingsystemsishardFortunately,themostimportantresultsareeasy
Wewillstudyonlysimpleconcepts
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1. Deterministic QueuingEasybutpowerfulAppliestodeterministicandtransientanalysisExample:playbackbuffersizing
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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(t)
A.
2. Operational Laws
Intuition:SayeverycustomerpaysoneFrperminutepresentPayoffpercustomer=RRateatwhichwereceivemoney=NInaverageλcustomersperminute,N=λR
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Little Again
ConsiderasimulationwhereyoumeasureRandN.YouusetwocountersresponseTimeCtr andqueueLengthCtr. Atendofsimulation,estimate
R= responseTimeCtr /NbCustN=queueLengthCtr /T
whereNbCust =numberofcustomersservedandT=simulationduration
BothresponseTimeCtr=0 andqueueLengthCtr=0 initiallyQ:Whenanarrivalordepartureeventoccurs,howarebothcountersupdated?A: queueLengthCtr +=(tnew ‐ told). q(told)whereq(told)isthe
numberofcustomersinqueuejustbeforetheevent.responseTimeCtr +=(tnew ‐ told). q(told)
thusresponseTimeCtr == queueLengthCtr andthus
N=RxNbCust/T;nowNbCust/Tisourestimatorof
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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 AnalysisApplythefollowingtwobounds1.2.
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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:withinonebusyperiod:toeverydeparturewecanassociateonearrivalwithsamenumberofcustomersleftbehind
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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
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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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Mean Response Time
Optimal SharingComparethetwointermsof
ResponsetimeCapacity
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The Processor Sharing Queue
Models:processors,networklinks
Insensitivity:whatevertheservicerequirements:
Egalitarian
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PS versus FIFO
PS FIFO
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4. A Case Study
Impactofcapacityincrease?OptimalCapacity?
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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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Assumenofeedbackloop:
4.3 Operational AnalysisArefinedmodel,withcirculatingusers
ApplyBottleneckAnalysis(=OperationalAnalysis)
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Z/(N-1)
-Z
1/c
waiting time
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5. Networks of QueuesStability
QueuingnetworksarefrequentlyusedmodelsThestabilityissuemay,ingeneral,beahardone
Necessaryconditionforstability(NaturalCondition)
serverutilization<1
ateveryqueue
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Instability Examples
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Poissonarrivals ;jobsgothrough stations1,2,1,2,1then leaveAjobarrivesastype1,thenbecomes 2,then 3etcExponential,independentservicetimeswith meanmi
Priority schedulingStation1:5>3>1Station2:2>4
Q:What is thenaturalstability condition?A: λ (m1 +m3 + m5 )<1
λ (m2 + m4) < 1
λ =1m1 =m3 =m4 = 0.1m2 =m5 = 0.6Utilizationfactors
Station1:0.8Station2:0.7
Networkisunstable!
Ifλ (m1 +… +m5 )<1networkisstable;why?
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Bramson’s Example 1: A Simple FIFO Network
Poissonarrivals;jobsgothroughstationsA,B,B…,B,AthenleaveExponential,independentservicetimes
Steps2andlast:meanisLOthersteps:meanisS
Q:Whatisthenaturalstabilitycondition?A: λ (L +S )<1
λ ((J‐1)S +L )<1Bramsonshowed:maybeunstablewhereasnaturalstabilityconditionholds
Bramson’s Example 2A FIFO Network with Arbitrarily Small Utilization Factor
m queues2typesofcustomersλ =0.5eachtyperoutingasshown,…=7visitsFIFOExponentialservicetimes,withmeanasshown
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L LS L LS S S S S S S
Utilization factorat every station≤4λ SNetworkis unstable forS ≤0.01L ≤S8m =floor(‐2(logL )/L)
Take Home Message
Thenatural stability conditionis necessary butmay notbe sufficient
Thereis aclassofnetworkswhere this never happens.ProductFormQueuingNetworks
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Product Form Networks
Customershaveaclass attributeCustomersvisitstationsaccordingtoMarkovRouting
Externalarrivals,ifany,arePoisson
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2StationsClass=step,J+3classes
Canyou reduce thenumberofclasses?
Chains
Customerscanswitchclass,butremaininthesamechain
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ν
Chains may be open or closed
Openchain=withPoissonarrivals.CustomersmusteventuallyleaveClosedchain:noarrival,nodeparture;numberofcustomersisconstant
ClosednetworkhasonlyclosedchainsOpennetworkhasonlyopenchainsMixednetworkmayhaveboth
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3Stations4classes1openchain1closed chain
ν
Bramson’s Example 2A FIFO Network with Arbitrarily Small Utilization Factor
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L LS L LS S S S S S S
2StationsMany classes2openchainsNetworkis open
Visit Rates
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2Stations5classes1chainNetworkis open
Visit ratesθ11 =θ13= θ15 =θ22 =θ24 = λθsc =0otherwise
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ν
Constraints on Stations
Stationsmustbelong toarestricted catalog ofstationsSee Section8.4forfulldescriptionWe will give commonly used examplesExample 1:GlobalProcessorSharing
OneserverRateofserveris shared equally among allcustomers presentServicerequirements forcustomers ofclassc aredrawn iid from adistributionwhich depends ontheclass(andthestation)
Example 2:DelayInfinite number ofserversServicerequirements forcustomers ofclassc aredrawn iid from adistributionwhich depends ontheclass(andthestation)Noqueuing,servicetime=servicerequirement =residence time
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Example 3:FIFOwith BserversB serversFIFOqueueingServicerequirements forcustomers ofclassc aredrawn iid from anexponentialdistribution,independent oftheclass (butmay depend onthestation)
Example ofCategory 2(MSCCCstation):MSCCCwith BserversB serversFIFOqueueing with constraintsAt most onecustomer ofeach classis allowed inserviceServicerequirements forcustomers ofclassc aredrawn iid from anexponentialdistribution,independent oftheclass (butmay depend onthestation)
Examples 1and2areinsensitive (servicetimecan be anything)Examples 3and4arenot(servicetimemustbe exponential,same forall)
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Saywhichnetworksatisfiesthehypothesesforproductform
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A
B (FIFO, Exp)C (Prio, Exp)
The Product Form Theorem
Ifanetworksatisfies the« ProductForm »conditionsgiven earlierThestationary distrib ofnumbers ofcustomers can be written explicitlyItis aproduct ofterms,where each term depends only onthestationEfficientalgorithms exist tocompute performancemetrics foreven very largenetworks
ForPSandDelaystations,servicetimedistributiondoes notmatter other thanthrough its mean (insensitivity)
Thenatural stability conditionholds
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Conclusions
Queuingisessentialincommunicationandinformationsystems
M/M/1,M/GI/1,M/G/1/PSandvariantshaveclosedforms
BottleneckanalysisandworstcaseanalysisareusuallyverysimpleandoftengivegoodinsightsQueuingnetworksmaybeverycomplextoanalyzeexceptifproductform–beabletorecognizethem!
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