Queueing Queueing Systems Systems Modeling and Performance Evaluation Modeling and Performance Evaluation with Computer Science

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Queueing

Systems

_Systems

Modeling and Performance Evaluation

with Computer Science

Spring, 2003 Spring, 2003 Dr. Eric Hsiao

Dr. Eric Hsiao--kuangkuang WuWu http://

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What is going to be covered?

(

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Course Outline

•

• ProbabilityProbability

–

– Discrete/Continuous random variableDiscrete/Continuous random variable –

– Conditional ProbabilityConditional Probability

•

• Queuing ModelingQueuing Modeling

–

– M/M/1/kM/M/1/k –

– Bulk Service, Bulk ArrivalBulk Service, Bulk Arrival –

– M/G/1M/G/1 –

– G/G/1G/G/1

•

• Case Studies:Case Studies:

–

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Lecture Progress

(February, 2003)

•

• QueueingQueueing SystemsSystems

–

– System FlowSystem Flow –

– Specification and Measure of Specification and Measure of QueueingQueueing System

System

•

• Notation and Structure for Basic Notation and Structure for Basic Queueing

Queueing SystemsSystems •

• Probability Z transformProbability Z transform •

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Daily Experiences

•

• Waiting in Line:Waiting in Line:

–

– Waiting for breakfastWaiting for breakfast –

– Stopped at a traffic lightStopped at a traffic light –

– Slowed down on the freewaysSlowed down on the freeways –

– Delayed at the entrance to parking facilityDelayed at the entrance to parking facility –

– Queued for access to an elevatorQueued for access to an elevator –

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Systems of Flow

•

• QueueingQueueing Systems Systems

–

– Systems of flowSystems of flow

•

• A flow system is one in which some A flow system is one in which some

commodity flows, moves, or is transferred

through one or more finite

through one or more finite--capacity channels capacity channels in order to go from one point to another

in order to go from one point to another

•

• Commodity: (produce the demand)Commodity: (produce the demand)

–

– Such as packet massage, telephone message, Such as packet massage, telephone message, automobiles

automobiles

•

• Channel: (provide the service)Channel: (provide the service)

–

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Service and Demand

the arrival rate R the service rate (or capacity) C

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Steady and Unsteady Flow

•

• Whether the flow is steady or Whether the flow is steady or unsteady?

unsteady?

–

– Steady: those systems in which the flow Steady: those systems in which the flow proceeds in a predictable fashion

proceeds in a predictable fashion

–

– If R<C, a reliable and smooth fashionIf R<C, a reliable and smooth fashion –

– If R>C, the mean capacity is less than the If R>C, the mean capacity is less than the average flow requirements, chaotic

average flow requirements, chaotic

congestion occur

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History of Computer Using

•

• Single UserSingle User •

• BatchBatch •

• TimeTime--SharingSharing •

• Sharing Communication lineSharing Communication line •

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Modeling

Mathematical Model of Real World Real World Verification Approximate solution Solution to the Model

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Resource Sharing

•

• A resource is a device that can do A resource is a device that can do works for you at a finite time

works for you at a finite time

–

– e.g. A communication Channele.g. A communication Channel –

– e.g. A computere.g. A computer

•

• A demand requires work from resourceA demand requires work from resource

–

– e.g. message e.g. message –

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User Behavior

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Bursty

Asynchronous

Demands

•

• You cannot predict exactly You cannot predict exactly whenwhen they they will demand access

will demand access

•

• You cannot predict exactly You cannot predict exactly how muchhow much they will demand access

they will demand access

•

• Most of timeMost of time they do not need access to they do not need access to resource

resource

•

• When they ask for it, they want When they ask for it, they want immediate

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Typical Traffic

Interactive Traffic

Short Response time Reliable Transmit

Real Time traffic

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Resource Sharing

•

• Type1: Everyone use his resource Type1: Everyone use his resource singlely

singlely (not efficient).(not efficient). •

• Type2: Using Pool of resource sharing Type2: Using Pool of resource sharing those resources (by switching) plus the

those resources (by switching) plus the

cost of switch

•

• Type3: Using a large resource (as an Type3: Using a large resource (as an unit).

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Law of Large Number

•

• The first resource sharing principleThe first resource sharing principle •

• Although each member of a Large population Although each member of a Large population may behave in a Random fashion, the

may behave in a Random fashion, the

population as a whole behave in a

predictable fashion.

–

– This is the This is the ““smoothing smoothing ““ effect of large populationeffect of large population –

– The predictable The predictable fahsionfahsion presents a total demand presents a total demand equal to the sum of the average demands of each

equal to the sum of the average demands of each

member

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Conflict Resolution

•

• QueueingQueueing: one gets severed, others : one gets severed, others wait

wait

•

• Splitting: Each get a piece of resourceSplitting: Each get a piece of resource •

• Blocking: One get served, all others are Blocking: One get served, all others are refused

refused

•

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Response Time

•

• When the throughput and capacity go When the throughput and capacity go up, the response time will go down

up, the response time will go down

•

• Economy of ScaleEconomy of Scale

–

– The second resource sharing principleThe second resource sharing principle –

– if you scale up throughput and capacity by if you scale up throughput and capacity by some factor F, then you reduce response

some factor F, then you reduce response

time by the factor

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Economy of Scale

DATA

Original: B Block/sec Cbit/sec Scale: NB Block/sec NC bit/sec

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Throughput, Efficiency,

Response time

•

• If you scale the capacity more slowly If you scale the capacity more slowly

than throughput while holding response

time constant, then efficiency will

increase

•

• Key tradeoff among:Key tradeoff among:

–

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System of Flow

•

• Flow of a commodity (demand) through Flow of a commodity (demand) through a finite

a finite--capacity channel (resource)capacity channel (resource)

–

– Steady FlowSteady Flow –

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Steady Flow

•

• Demand are known, constant smooth: Demand are known, constant smooth: predictable

predictable

•

• Single Channel:Single Channel:

–

– R = Arrival Rate (Cans/Sec)R = Arrival Rate (Cans/Sec) –

– C = Capacity (Cans/Sec)C = Capacity (Cans/Sec) –

– if R <= C Fineif R <= C Fine –

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Network of Channels

•

• MaxMax--Flow MinFlow Min--Cut TheoremCut Theorem

•

• R < C for each channelR < C for each channel •

• MaxmumMaxmum Flow , label the node, find a Flow , label the node, find a path

path Taipei

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Unsteady Flow(I)

•

• Arrival time of Demand: Arrival time of Demand: UnpredicatbleUnpredicatble •

• SiseSise (Service time) of Demand: (Service time) of Demand: Unpredictable

Unpredictable

•

• Single Channel:Single Channel:

–

– Queue LengthQueue Length –

– Waiting TimeWaiting Time –

– Sever UtilizationSever Utilization –

– Throughput Throughput –

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Unsteady Flow(II)

•

• Network of ChannelNetwork of Channel

–

– capacitycapacity –

– throughputthroughput –

– Response TimeResponse Time –

– EfficiencyEfficiency –

– design

Combinatonics and probablities kill you

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General

Queueing

_Queueing

System

_System

Queueing System

•

• How to improve the system How to improve the system performance

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Review of

Queueing

_Queueing

•

• QueueingQueueing SystemsSystems

–

– NotationNotation –

– MarkovianMarkovian Queue, BirthQueue, Birth--andand--deathdeath –

– M/M/1 M/M/1 --> M/M/k/m> M/M/k/m –

– Stage Stage --> > ErlangianErlangian distributiondistribution –

– ParallelParallel –

– Network of QueueNetwork of Queue –

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limited resouce (fixed number of queue size buffer N

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How often they arrive

how long they will stay

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What we are interested ?

•

• How long we are going to wait ?How long we are going to wait ? •

• How big the queue size should How big the queue size should be ?

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Observation 1

•

• Each customer could be Each customer could be

characterized as the following:

–

– how often the traffic produced ?how often the traffic produced ? –

– how many service it may require ?how many service it may require ?

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Observation 2

•

• Some users might be in the queue ?Some users might be in the queue ?

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Observation

•

• Current State depends on Previous Current State depends on Previous State

State

Ν(τ)

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Computer Queue System

•

• MarkovianMarkovian Chain:Chain:

–

– current state depends on previous one current state depends on previous one state only

state only

–

– time domaintime domain

•

• discretediscrete •

• continuouscontinuous

–

– state domain:state domain:

•

• dsicretedsicrete •

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Birth

-

_-

Death Process

_{Death Process}

•

• Transitions are Transitions are allowdallowd between between neighbors:

neighbors:

–

– P(k) to P(k+1) P(k) to P(k+1)

•

• birth happen (arrival)birth happen (arrival)

–

– P(k) to P(kP(k) to P(k--1)1)

•

• death happen (death)death happen (death) •

• PossionPossion and and ExponetialExponetial Distributions Distributions are

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Μ/Μ/1 0 1 2 3 4 λ λ λ λ µ _µ _µ _µ _µ

Number of buffers <-> Number of Customers

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Format

•

• M / M / 1 / 2M / M / 1 / 2

number of server

Number of buffer size

Amount of service a customer require B(x) = P[service time <=x]

Arrival Time

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Probability

• • Sum of P(k) = 1Sum of P(k) = 1 • • P(k) <= 1P(k) <= 1 •

• E[N] = Sum of E[N] = Sum of K P(k)K P(k) •

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General

Queueing

_Queueing

System

_System

•

• C(n) nth customer to enter the systemC(n) nth customer to enter the system •

• N(t) number of customer in the system at N(t) number of customer in the system at time t

time t

•

• a(n) arrival time for C(n)a(n) arrival time for C(n) •

• t(n) t(n) interarrvalinterarrval time between C(ntime between C(n--1) and C(n)1) and C(n) •

• x(n) service time for C(n)x(n) service time for C(n) •

• w(n) waiting time for C(n)w(n) waiting time for C(n) •

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Time

-

_-

diagram notation

_{diagram notation}

Servicer Queue S(n) W(n) C(n) C(n-1) C(n) _C(n+1) t(n) C(n)

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Classical M/M/1

Queueing

_Queueing

•

• Single Server QueueSingle Server Queue •

• Poisson Arrival ProcessPoisson Arrival Process •

• Exponential Distribution for service timeExponential Distribution for service time •

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M/M/1 Analysis

•

• StateState--transitiontransition--rate diagramrate diagram

a a

0 1 2 n

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What you should need for

Queueing

modeling

_modeling

•

• Probability (such as arrival rate, service Probability (such as arrival rate, service rate)

rate)

•

• Transform (zTransform (z--transform, transform, LaplaceLaplace transform)