Queues with vacations and their applications Dieter Fiems and Herwig - - PowerPoint PPT Presentation

Queues with vacations and their applications

Dieter Fiems and Herwig Bruneel SMACS Research Group, Ghent University, Belgium {df,hb}@UGent.be

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Outline

• Vacations
• Mathematical model
• Queueing analysis
• Special cases
• Case study: Priority queues
• Conclusions

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Vacations – What?

• Queueing theory parlance for temporary server

unavailability

• Resource sharing
• Breakdowns
• Maintenance
• Errors
• Reconfiguration
• ...
• Correlation structure?

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Vacations – Resource Sharing

• Passive Optical Network

Optical Line Terminal Optical Network Unit 1 Optical Network Unit 2 Optical Network Unit 3

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Vacations – Resource Sharing

• Passive Optical Network

Optical Line Terminal Optical Network Unit 1 Optical Network Unit 2 Optical Network Unit 3

ONU 1 ONU 2 ONU 2

Available Vacation time Vantage point ONU 1

ONU 3

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Vacations – Resource Sharing

• Ethernet

... Bus

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Vacations – Resource Sharing

• Ethernet

... Bus

Vacation Back−off Back−off Station 1 Station 2 time time Collision Vacation

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Vacations – Resource Sharing

• Service differentiation

Class 1 Class 2

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Vacations – Resource Sharing

• Service differentiation

Class 1 Class 2

• Priority Queueing
• Weighted Round Robin
• Weighted Fair Queueing

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Vacations – Errors

• Go-Back-N ARQ
• ✁

Errors

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Vacations – Errors

• Go-Back-N ARQ

Errors

time time Error ACK Vacation S R

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Vacations – Non-telecom

Unsignalised intersection

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Vacations – Non-telecom

Unsignalised intersection Airplane queue

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Vacations – Models

• Desirable properties of vacation process
• Correlation between vacations
• Service interruptions
• Other dependences

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Vacations – Models

• Desirable properties of vacation process
• Correlation between vacations
• Service interruptions
• Other dependences
• Desirable properties of queueing system
• Realistic arrival process
• General service times

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Vacations – Models

• Desirable properties of vacation process
• Correlation between vacations
• Service interruptions
• Other dependences
• Desirable properties of queueing system
• Realistic arrival process
• General service times
• Approaches
• Analytic methods
• Numerical methods
• Simulation

IPS-MoMe, Warsaw, Poland – p.9/26

Vacations – Models

• Desirable properties of vacation process
• Correlation between vacations
• Service interruptions
• Other dependences
• Desirable properties of queueing system
• Realistic arrival process
• General service times
• Approaches
• Analytic methods
• Numerical methods
• Simulation

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Mathematical Model

• Discrete-time queueing system

slot k arrivals slot k+1 time departure synchronisation

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Mathematical Model

• Discrete-time queueing system

slot k arrivals slot k+1 time departure synchronisation

• Arrivals per slot, service times
• independent and identically distributed
• probability generating functions: E(z) and S(z)
• service times are bounded

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Mathematical Model

• Infinite capacity queue

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Mathematical Model

• Infinite capacity queue
• Single server system

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Mathematical Model

• Infinite capacity queue
• Single server system
• Vacation process

✲

i

vacation of n slots

j

state of the vacation process

✲

(k)

❄

queueing state    a during customer service b customer leaves non-empty system c customer leaves empty system d empty system Server in vacation state i and queueing state k takes a vacation of length n and goes to state j with probability b(k)

ij (n)

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Mathematical model

• Dealing with interrupted service

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Mathematical model

• Dealing with interrupted service
• Continue after interruption (CAI)

5 slots service time

✻ ❄ ✲

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Mathematical model

• Dealing with interrupted service
• Continue after interruption (CAI)

5 slots service time

✻ ❄ ✲

• Repeat after interruption (RAI)

✻ ❄ ✲

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Mathematical model

• Dealing with interrupted service
• Continue after interruption (CAI)

5 slots service time

✻ ❄ ✲

• Repeat after interruption (RAI)

✻ ❄ ✲

• Repeat after interruption with resampling (RAI,wr)

✻

service time resampled to 6 slots

❄ ✲

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Queueing Analysis

• Probability generating functions approach
• Matrices to deal with the finite state space of the

vacation process

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Queueing Analysis

• Probability generating functions approach
• Matrices to deal with the finite state space of the

vacation process

• Effective service times
• Defined as:

“the number of slots between the beginning

• f the slot where the customer is first served

until the end of the slot where the customer leaves the system”

• Effective service time analysis for the different
• peration modes
• Unified queueing analysis

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Queueing Analysis

• Effective service time for CAI

1 2 3 2 3 1 time Ω2 Ω1 1 T S

T = 1 +

S−1

• j=1

Ωj

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Queueing Analysis

• Effective service time for CAI

1 2 3 2 3 1 time Ω2 Ω1 1 T S

T = 1 +

S−1

• j=1

Ωj T(z) = z

∞

• n=1

s(n)Ω(z)j−1 Ω(z) = Ba(z)z

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Queueing Analysis

• Effective service time for RAI

1 2 3 S 1 2 1 2 3 1 T’ Γ T B

T =

• Γ + B + T ′ (int.)

S (no int.)

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Queueing Analysis

• Effective service time for RAI

1 2 3 S 1 2 1 2 3 1 T’ Γ T B

T =

• Γ + B + T ′ (int.)

S (no int.)

T(z) = X

k

s(k) “ IN − z[zBa(0) − IN]−1[(z Ba(0))k−1 − IN][Ba(z) − Ba(0)] ”−1 z (z Ba(0))k−1

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Queueing Analysis

• Queue content at departure epochs

System equations: Uk+1 =                            Uk − 1 +

Bb+T

• j=1

Ej if Uk > 0

Bc+T

• j=1

Ej − 1 if Uk = 0 and

Bc

• j=1

Ej > 0

˜ Bd

• j=1

˜ Ej +

T

• j=1

Ej − 1 if Uk = 0 and

Bc

• j=1

Ej = 0

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Queueing Analysis

• Queue content at departure epochs

Uk+1(z) = (Uk(z) − Uk(0)) Bb(E(z)) 1 z T(E(z)) + Uk(0) (Bc(E(z)) − Bc(e0)) 1 z T(E(z)) + Uk(0) Bc(e0) Λ(z) 1 z T(E(z)) Λ(z) =

• IN − ˜

Bd(e0) −1 ˜ Bd(E(z)) − ˜ Bd(e0)

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Queueing Analysis

• Queue content at departure epochs

Uk+1(z) = (Uk(z) − Uk(0)) Bb(E(z)) 1 z T(E(z)) + Uk(0) (Bc(E(z)) − Bc(e0)) 1 z T(E(z)) + Uk(0) Bc(e0) Λ(z) 1 z T(E(z)) Λ(z) =

• IN − ˜

Bd(e0) −1 ˜ Bd(E(z)) − ˜ Bd(e0)

• M/G/1 type queueing system!

U(z)Γ1(z) + U(0)Γ2(z) = 0

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Queueing Analysis

• Queue content at random slot boundaries
• Customer Delay

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Queueing Analysis

• Queue content at random slot boundaries
• Customer Delay
• Moments
• Moment generating property of pgfs
• Mean

E[X] = X′(1)

• Variance

Var[X] = X′′(1) − X′(1)2 + X′(1)

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Special cases

• Without interruptions of on-going service
• Exhaustive single/multiple vacation system
• Non-preemptive time-limited vacation system
• Number limited vacation system
• Pure limited vacation system
• Bernoulli schedule

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Special cases

• Without interruptions of on-going service
• Exhaustive single/multiple vacation system
• Non-preemptive time-limited vacation system
• Number limited vacation system
• Pure limited vacation system
• Bernoulli schedule
• With interruptions of on-going service
• Independent vacation process
• Preemptive time-limited vacation system

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Special cases

• Non-preemptive time-limited vacation system
• Timer follows geometric distribution
• Two states: timer not expired (1) or expired (2)
• During service: start in state 1, fixed probability to

go to state 2

• At the end of service: take a vacation if in state 2
• Queue empty: take a new vacation

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Special cases

• Non-preemptive time-limited vacation system
• Timer follows geometric distribution
• Two states: timer not expired (1) or expired (2)
• During service: start in state 1, fixed probability to

go to state 2

• At the end of service: take a vacation if in state 2
• Queue empty: take a new vacation

Ba(z) =  α 1 − α 1   ,Bd(z) =  

V (z) z

1   , Bb(z) = Bc(z) =  α + (1 − α)V (z) V (z)  

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Priority queueing

• Low-priority traffic of a preemptive priority model
• CAI ↔ preemptive resume
• RAI ↔ preemptive repeat identical
• RAI,wr ↔ preemptive repeat different

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Priority queueing

• Low-priority traffic of a preemptive priority model
• CAI ↔ preemptive resume
• RAI ↔ preemptive repeat identical
• RAI,wr ↔ preemptive repeat different
• Corresponding vacation model
• Independent vacation process
• Correlated high priority traffic
• High priority busy periods ↔ Vacations
• Functional equation for the busy periods

B(z) =

∞

• k=0

Wk [z B(z)]k

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Priority queueing

• Numerical results
• Two priority classes
• High priority class
• Bernoulli arrivals
• Geometrical packet lengths
• load: 20%
• High priority class
• Bernoulli arrivals
• Binomially distributed packet lengths, mean 10

slots

• load: ρ2

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Priority queueing

• Mean low-pri packet delay vs. mean high-pri packet

length

20 40 60 10 20 30 40 ρ2=0.5 ρ2=0.25 resume repeat diff. repeat id.

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Priority queueing

• Variance of the low-pri packet delay vs. mean high-pri

packet length

1000 2000 3000 4000 5000 10 20 30 40 ρ2 = 0.5 ρ2 = 0.25 resume repeat diff. repeat id.

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Conclusions and future work

• Conclusions
• Analytic results – probability generating functions

approach

• Large set of “classical vacation” systems
• Applications lead to heavily correlated vacation

systems

• Case study: priority queueing

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Conclusions and future work

• Conclusions
• Analytic results – probability generating functions

approach

• Large set of “classical vacation” systems
• Applications lead to heavily correlated vacation

systems

• Case study: priority queueing
• Future work
• Identify neglectable correlation
• Introduce other types of correlation
• Tools to work with matrices of pgfs

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Questions?

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