Queues with vacations and their applications

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

IPS-MoMe, Warsaw, Poland – p.1/26

Outline • • • • • •

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


Vacations – What? •

•

Queueing theory parlance for temporary server unavailability • Resource sharing • Breakdowns • Maintenance • Errors • Reconfiguration • ... Correlation structure?


Vacations – Resource Sharing •

Passive Optical Network

Optical Network Unit 1

Optical Line Terminal Optical Network Unit 2




Passive Optical Network


Optical Line Terminal Optical Network Unit 2

Vantage point ONU 1 ONU 2

ONU 1

Available

Optical Network Unit 3 ONU 3

ONU 2

Vacation

time



Ethernet ...

Bus



Ethernet ...

Bus

Vacation Back−off Station 1 Back−off

time

Station 2 Collision Vacation

time IPS-MoMe, Warsaw, Poland – p.5/26


Service differentiation Class 1

Class 2



Service differentiation Class 1

Class 2

• • •

Priority Queueing Weighted Round Robin Weighted Fair Queueing IPS-MoMe, Warsaw, Poland – p.6/26

Errors

•

Vacations – Errors Go-Back-N ARQ


Errors

•

Vacations – Errors Go-Back-N ARQ

R time

Error

S ACK

time

Vacation


Vacations – Non-telecom

Unsignalised intersection


Vacations – Non-telecom

Unsignalised intersection

Airplane queue


Vacations – Models •

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



•

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



•

•

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


•

•

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

Mathematical Model •

Discrete-time queueing system synchronisation slot k

slot k+1 time arrivals

departure



Discrete-time queueing system synchronisation slot k

slot k+1 time arrivals

•

departure

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



Infinite capacity queue


Mathematical Model • •

Infinite capacity queue Single server system


Mathematical Model • • •

Infinite capacity queue Single server system Vacation process i

vacation of n slots

j

-

- state

of the vacation process

(k)  a during customer service ? customer leaves non-empty system queueing state bc customer leaves empty system  d empty system

Server in vacation state i and queueing state k takes a (k) vacation of length n and goes to state j with probability bij (n) IPS-MoMe, Warsaw, Poland – p.11/26

Mathematical model •

Dealing with interrupted service



Dealing with interrupted service • Continue after interruption (CAI)

5 slots service time -

6

?





6 •

?

Repeat after interruption (RAI) 6

?





6 •

?

Repeat after interruption (RAI) 6

•

?

Repeat after interruption with resampling (RAI,wr) service time resampled to 6 slots 6

? IPS-MoMe, Warsaw, Poland – p.12/26

Queueing Analysis • •

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


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 of 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 operation modes • Unified queueing analysis IPS-MoMe, Warsaw, Poland – p.13/26

Queueing Analysis •

Effective service time for CAI S T 1 Ω1

2

1

2

3 Ω2

T =1+ time

1

3 S−1 X

Ωj

j=1



Effective service time for CAI S T 1 Ω1

2

1

2

3 Ω2

T (z) = z

∞ X

3

T =1+ time

Ωj

j=1

1

s(n)Ω(z)j−1

S−1 X

Ω(z) = B a (z)z

n=1



Effective service time for RAI S

T 1

2

Γ

T =

(

1

B

1

2

3

1

2

3

T’

Γ + B + T 0 (int.) S (no int.)



Effective service time for RAI S

T 1

2

1

Γ

T = T (z) =

X k

B

(

1

2

1

3

2

3

T’

Γ + B + T 0 (int.) S (no int.)

“

a

s(k) IN − z[zB (0) − IN ]

−1

a

[(z B (0))

k−1

a

a

− IN ][B (z) − B (0)]

”−1

z (z B a (0))k−1



Queue content at departure epochs System equations:  BX b +T     U − 1 + E if U > 0 k j k     j=1     Bc c +T  BX X Ej − 1 if Uk = 0 and Ej > 0 Uk+1 =   j=1 j=1     ˜d B Bc T  X X X    ˜j +  E Ej − 1 if Uk = 0 and Ej = 0   j=1

j=1

j=1



Queue content at departure epochs 1 Uk+1 (z) = (Uk (z) − Uk (0)) B (E(z)) T (E(z)) z 1 c c + Uk (0) (B (E(z)) − B (e0 )) T (E(z)) z 1 c + Uk (0) B (e0 ) Λ(z) T (E(z)) z −1 ˜ d (E(z)) − B ˜ d (e0 ) ˜ d (e0 ) B Λ(z) = IN − B b



Queue content at departure epochs 1 Uk+1 (z) = (Uk (z) − Uk (0)) B (E(z)) T (E(z)) z 1 c c + Uk (0) (B (E(z)) − B (e0 )) T (E(z)) z 1 c + Uk (0) B (e0 ) Λ(z) T (E(z)) z −1 ˜ d (E(z)) − B ˜ d (e0 ) ˜ d (e0 ) B Λ(z) = IN − B b

•

M/G/1 type queueing system!

U (z)Γ1 (z) + U (0)Γ2 (z) = 0 IPS-MoMe, Warsaw, Poland – p.17/26

Queueing Analysis • •

Queue content at random slot boundaries Customer Delay


Queueing Analysis • • •

Queue content at random slot boundaries Customer Delay Moments • Moment generating property of pgfs • Mean E[X] = X 0 (1) •

Variance Var[X] = X 00 (1) − X 0 (1)2 + X 0 (1)


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


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


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


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 

B (z) =  a

α 1−α 0

1



 ,B (z) = d



V (z)  z

1

0



,

0   α + (1 − α)V (z) 0 b c   B (z) = B (z) = V (z) 0


Priority queueing •

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



•

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) =

∞ X

Wk [z B(z)]k

k=0



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



Mean low-pri packet delay vs. mean high-pri packet length resume repeat diff. repeat id.

60 ρ2=0.5 40

ρ2=0.25

20

10

20

30

40



Variance of the low-pri packet delay vs. mean high-pri packet length 5000 4000

resume repeat diff. repeat id.

3000 ρ2 = 0.5 2000 1000 0

ρ2 = 0.25 10

20

30

40


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


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 IPS-MoMe, Warsaw, Poland – p.25/26

Questions?
