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Heavy-traffic analysis of the M/PH/1 discriminatory processor sharing queue with phase-dependent weights Maaike Verloop (CWI) Urtzi Ayesta (CNRS-LAAS & BECAM) Sindo N u nez-Queija (University of Amsterdam & CWI) Heavy-traffic

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SLIDE 1

Heavy-traffic analysis of the M/PH/1 discriminatory processor sharing queue with phase-dependent weights

Maaike Verloop (CWI) Urtzi Ayesta (CNRS-LAAS & BECAM) Sindo N´ u˜ nez-Queija (University of Amsterdam & CWI)

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SLIDE 2

Heavy-traffic analysis of the M/PH/1 discrimina- tory processor sharing queue with phase-dependent weights

  • Discriminatory Processor Sharing

vs Egalitarian processor Sharing

  • Dynamics in heavy traffic
  • Proof for phase type distributions
  • R. N´

u˜ nez-Queija 1

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SLIDE 3

Model description

Class k

  • Poisson arrivals, λk = Λpk
  • Service Bk(x) := P(Bk ≤ x)
  • Load ρk = λkβk
  • # customers Nk
  • Service rate

wk

K

j=1 Njwj

Stability ρ = K

k=1 ρk < 1

  • R. N´

u˜ nez-Queija 2

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SLIDE 4

Model description

Class k

  • Poisson arrivals, λk = Λpk
  • Service Bk(x) := P(Bk ≤ x)
  • Load ρk = λkβk
  • # customers Nk
  • Service rate

wk

K

j=1 Njwj

Stability ρ = K

k=1 ρk < 1

  • R. N´

u˜ nez-Queija 2-1

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SLIDE 5

Model description

Class k

  • Poisson arrivals, λk = Λpk
  • Service Bk(x) := P(Bk ≤ x)
  • Load ρk = λkβk
  • # customers Nk
  • Service rate

wk

K

j=1 Njwj

Stability ρ = K

k=1 ρk < 1

  • R. N´

u˜ nez-Queija 2-2

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SLIDE 6

Model description

Class k

  • Poisson arrivals, λk = Λpk
  • Service Bk(x) := P(Bk ≤ x)
  • Load ρk = λkβk
  • # customers Nk
  • Service rate

wk

K

j=1 Njwj

Stability ρ = K

k=1 ρk < 1

  • R. N´

u˜ nez-Queija 2-3

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SLIDE 7

Model description

Class k

  • Poisson arrivals, λk = Λpk
  • Service Bk(x) := P(Bk ≤ x)
  • Load ρk = λkβk
  • # customers Nk
  • Service rate

wk

K

j=1 Njwj

Stability ρ = K

k=1 ρk < 1

  • R. N´

u˜ nez-Queija 2-4

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SLIDE 8

Model description

Class k

  • Poisson arrivals, λk = Λpk
  • Service Bk(x) := P(Bk ≤ x)
  • Load ρk = λkβk
  • # customers Nk
  • Service rate

wk

K

j=1 Njwj

Stability ρ = K

k=1 ρk < 1

Applications: time-shared computing systems, TCP, ADSL

  • R. N´

u˜ nez-Queija 2-5

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SLIDE 9

Literature

  • Kleinrock [1967] (‘Priority Processor-shared Model’)
  • Fayolle, Mitrani & Iasnogorodski [1980]
  • Grishechkin [1992, 1994]
  • Rege & Sengupta [1994, 1996]
  • Borst, Van Ooteghem & Zwart [2003]
  • Altman, Jimenez & Kofman [2004]
  • Avrachenkov, Ayesta, Brown, N-Q [2005]
  • R. N´

u˜ nez-Queija 3

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SLIDE 10

Discriminatory PS versus (Egalitarian) PS

PS: Mean queue lengths are entirely “insensitive” as op- posed to non-preemptive disciples like FCFS

ENk =

ρk 1 − ρ For DPS:

ENk are all finite under the usual stability condition

(regardless of the higher-order moments of the service re- quirements) Other insensitivity properties of PS only carry over to DPS in asymptotic regimes

  • R. N´

u˜ nez-Queija 4

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SLIDE 11

Discriminatory PS versus Egalitarian PS

PS: Expected sojourn time for jobs of given size

ETk(x)

x = 1 1 − ρ DPS: true in the limit as x → ∞ [Fayolle, Mitrani & Iasno- gorodski] and the ”bias” is also insensitive lim

x→∞

  • ETk(x) −

x 1 − ρ

  • =
  • j λj(1 − wk

wj )E((Bj)2)

2(1 − ρ)2 .

  • R. N´

u˜ nez-Queija 4-1

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SLIDE 12

Discriminatory PS versus Egalitarian PS

Tails of the sojourn time distributions PS and DPS: For regularly varying service requirement distributions with finite variance (conditions can be relaxed):

P{Tk > x} P{Bk > (1 − ρ)x} → 1,

as x → ∞ Again, the “scaling factor” 1 − ρ is insensitive and common to all classes

  • R. N´

u˜ nez-Queija 4-2

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SLIDE 13

Discriminatory PS versus Egalitarian PS

Time-scale separation (1): Class k operates on a much faster time scale than class k + 1, for all k = 1, 2, . . . , K − 1

  • arrival rates λkfk(r)
  • service requirements of class k distributed as Bk/fk(r)
  • with fk+1(r)/fk(r) → 0 as r → ∞
  • R. N´

u˜ nez-Queija 4-3

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SLIDE 14

Discriminatory PS versus Egalitarian PS

Time-scale separation (2): The limiting distribution of the slow class is geometric

Pr{N2 = n2} → (1 −

ρ2 1 − ρ1 )( ρ2 1 − ρ1 )n2 and

Pr{N1 = n1|N2 = n2}

→ Γ(n1 + n2w2

w1

+ 1) Γ(n1 + 1)Γ(n2w2

w1

+ 1)ρn1

1 (1 − ρ1)

n2w2 w1 +1

For PS all limits can be replaced with equalities

  • R. N´

u˜ nez-Queija 4-4

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SLIDE 15

Dynamics

  • R. N´

u˜ nez-Queija 5

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Dynamics

  • R. N´

u˜ nez-Queija 5-1

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SLIDE 17

Dynamics

  • R. N´

u˜ nez-Queija 5-2

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SLIDE 18

Dynamics

  • R. N´

u˜ nez-Queija 5-3

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SLIDE 19

Dynamics (2)

  • R. N´

u˜ nez-Queija 6

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SLIDE 20

Dynamics (2)

  • R. N´

u˜ nez-Queija 6-1

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SLIDE 21

Heavy traffic: Theorem

For phase-type distributions (1 − ρ)(N1, N2, . . . , NK) d → E · ( ¯ ρ1 w1 , ¯ ρ2 w2 , . . . , ¯ ρK wK ) where E is exponential with mean

  • k pkE[(Bk)2]/

k pkEBk

  • k 1

wk ¯

ρkE[(Bk)2]/EBk State-space collapse:

“In heavy traffic (1 − ρ)N1, . . . , (1 − ρ)NK are proportional to a common exponentially distributed random variable”

  • R. N´

u˜ nez-Queija 7

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SLIDE 22

Heavy traffic: Theorem

For phase-type distributions (1 − ρ)(N1, N2, . . . , NK) d → E · ( ¯ ρ1 w1 , ¯ ρ2 w2 , . . . , ¯ ρK wK ) where E is exponential with mean

  • k pkE[(Bk)2]/

k pkEBk

  • k 1

wk ¯

ρkE[(Bk)2]/EBk State-space collapse:

“In heavy traffic (1 − ρ)N1, . . . , (1 − ρ)NK are proportional to a common exponentially distributed random variable”

  • R. N´

u˜ nez-Queija 7-1

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SLIDE 23

Heavy traffic: Interpretation

(1 − ρ)(N1, N2, . . . , NK) d → E · ( ¯ ρ1 w1 , ¯ ρ2 w2 , . . . , ¯ ρK wK ) where E is exponential with mean

  • k pkE[(Bk)2]/

k pkEBk

  • k 1

wk ¯

ρkE[(Bk)2]/EBk Assume that Ni/Nj = ni/nj for some constants nk (as in the exponential case), then wjnj

J

i=1 wini

= µ

J

  • i=1

p0iaij normalizing J

i=1 wini = 1 gives the result up to a multi-

plicative factor

  • R. N´

u˜ nez-Queija 8

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SLIDE 24

Heavy traffic: Interpretation

(1 − ρ)(N1, N2, . . . , NK) d → E · ( ¯ ρ1 w1 , ¯ ρ2 w2 , . . . , ¯ ρK wK ) where E is exponential with mean

  • k pkE[(Bk)2]/

k pkEBk

  • k 1

wk ¯

ρkE[(Bk)2]/EBk Assume that N′

i/N′ j = ni/nj for some constants nk (as in the

exponential case), then wjnj

J

i=1 wini

= µ

J

  • i=1

p0iaij normalizing J

i=1 wini = 1 gives the result up to a multi-

plicative factor Work-conserving and non-idling:

EV =

ρ 2(1 − ρ)

  • k

pkE[(Bk)2]/

  • k

pkEBk

  • R. N´

u˜ nez-Queija 8-1

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SLIDE 25

Heavy traffic: Interpretation

(1 − ρ)(N1, N2, . . . , NK) d → E · ( ¯ ρ1 w1 , ¯ ρ2 w2 , . . . , ¯ ρK wK ) where E is exponential with mean

  • k pkE[(Bk)2]/

k pkEBk

  • k 1

wk ¯

ρkE[(Bk)2]/EBk Assume that N′

i/N′ j = ni/nj for some constants nk (as in the

exponential case), then wjnj

J

i=1 wini

= µ

J

  • i=1

p0iaij normalizing J

i=1 wini = 1 gives the result up to a multi-

plicative factor Work-conserving and non-idling:

EV =

ρ 2(1 − ρ)

  • k

pkE[(Bk)2]/

  • k

pkEBk

  • R. N´

u˜ nez-Queija 8-2

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SLIDE 26

Phase-type service requirements (1)

Recall

  • Poisson arrivals, λk = Λpk
  • Service Bk(x) := P(Bk ≤ x)
  • Load ρk = λkβk
  • # customers Nk
  • Service rate

wk

K

j=1 Njwj

Stability ρ = K

k=1 ρk < 1

  • R. N´

u˜ nez-Queija 9

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SLIDE 27

Phase-type service requirements (2)

Class k has mk service phases total # service phases: K

k=1 mk := J

p0i = probability that arriving customer is in phase i µi = service rate in phase i gi = service weight in phase i pij = phase transition probabilities pi0 = probability of completing service after phase i gi(¯ n) := nigi

K

k=1 nkgk

, if ¯ n = ¯

10

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SLIDE 28

Phase-type service requirements (2)

Class k has mk service phases total # service phases: K

k=1 mk := J

p0i = probability that arriving customer is in phase i µi = service rate in phase i gi = service weight in phase i pij = phase transition probabilities pi0 = probability of completing service after phase i gi(¯ n) := nigi

K

k=1 nkgk

, if ¯ n = ¯

10-1

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SLIDE 29

Phase-type service requirements (2)

Class k has mk service phases total # service phases: K

k=1 mk := J

p0i = probability that arriving customer is in phase i µi = service rate in phase i gi = service weight in phase i pij = phase transition probabilities pi0 = probability of completing service after phase i gi(¯ n) := nigi

K

k=1 nkgk

, if ¯ n = ¯

10-2

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SLIDE 30

Phase-type service requirements (2)

Class k has mk service phases total # service phases: K

k=1 mk := J

p0i = probability that arriving customer is in phase i µi = service rate in phase i gi = service weight in phase i pij = phase transition probabilities pi0 = probability of completing service after phase i gi(¯ n) := nigi

K

k=1 nkgk

, if ¯ n = ¯

10-3

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SLIDE 31

Phase-type service requirements (2)

Class k has mk service phases total # service phases: K

k=1 mk := J

p0i = probability that arriving customer is in phase i µi = service rate in phase i gi = service weight in phase i pij = phase transition probabilities pi0 = probability of completing service after phase i gi(¯ n) := nigi

K

k=1 nkgk

, if ¯ n = ¯

10-4

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SLIDE 32

Phase-type service requirements (2)

Class k has mk service phases total # service phases: K

k=1 mk := J

p0i = probability that arriving customer is in phase i µi = service rate in phase i gi = service weight in phase i pij = phase transition probabilities pi0 = probability of completing service after phase i gi(¯ n) := nigi

K

k=1 nkgk

, if ¯ n = ¯

10-5

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SLIDE 33

Phase-type service requirements (3)

[Λ +

J

  • i=1

gi(¯ n)µi]P(¯ n) =

J

  • i=1

[Λp0iδniP(¯ n − ¯ ei) + gi(¯ n + ¯ ei)µipi0P(¯ n + ¯ ei) +

J

  • j=1

gi(¯ n + ¯ ei − ¯ ej)µipijδnjP(¯ n + ¯ ei − ¯ ej)] Transformation: R(¯ n) = P(¯ n)

J

j=1 njgj

, if ¯ n = ¯ p(¯ z) =

  • zn1

1 . . . znJ J P(¯

n) = E[zN′

1

1

. . . zN′

J

J ]

r(¯ z) =

  • zn1

1 . . . znJ J R(¯

n) = E

 zN1

1

. . . zNJ

J

J

i=1 Nigi

; J

j=1 Nj > 0

 

p(¯ z) =

J

  • i=1

gizi ∂r ∂zi + 1 − ρ

  • R. N´

u˜ nez-Queija 11

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SLIDE 34

Phase-type service requirements (3)

[Λ +

J

  • i=1

gi(¯ n)µi]P(¯ n) =

J

  • i=1

[Λp0iδniP(¯ n − ¯ ei) + gi(¯ n + ¯ ei)µipi0P(¯ n + ¯ ei) +

J

  • j=1

gi(¯ n + ¯ ei − ¯ ej)µipijδnjP(¯ n + ¯ ei − ¯ ej)] Transformation: R(¯ n) = P(¯ n)

J

j=1 njgj

, if ¯ n = ¯ p(¯ z) =

  • zn1

1 . . . znJ J P(¯

n) = E[zN′

1

1

. . . zN′

J

J ]

r(¯ z) =

  • zn1

1 . . . znJ J R(¯

n) = E

 zN1

1

. . . zNJ

J

J

i=1 Nigi

; J

j=1 Nj > 0

 

p(¯ z) =

J

  • i=1

gizi ∂r ∂zi + 1 − ρ

  • R. N´

u˜ nez-Queija 11-1

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SLIDE 35

Phase-type service requirements (3)

[Λ +

J

  • i=1

gi(¯ n)µi]P(¯ n) =

J

  • i=1

[Λp0iδniP(¯ n − ¯ ei) + gi(¯ n + ¯ ei)µipi0P(¯ n + ¯ ei) +

J

  • j=1

gi(¯ n + ¯ ei − ¯ ej)µipijδnjP(¯ n + ¯ ei − ¯ ej)] Transformation: R(¯ n) = P(¯ n)

J

j=1 njgj

, if ¯ n = ¯ p(¯ z) =

  • zn1

1 . . . znJ J P(¯

n) = E[zN′

1

1

. . . zN′

J

J ]

r(¯ z) =

  • zn1

1 . . . znJ J R(¯

n) = E

  zN′

1

1

. . . zN′

J

J

J

i=1 N′ igi

; J

j=1 N′ j > 0

  

p(¯ z) =

J

  • i=1

gizi ∂r ∂zi + 1 − ρ

  • R. N´

u˜ nez-Queija 11-2

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SLIDE 36

Phase-type service requirements (4)

Partial differential equation Λ(1 − ρ)(1 −

J

  • j=1

p0jzj) =

J

  • i=1

{µigi(pi0 +

J

  • j=1

pijzj − zi) − Λgizi(1 −

J

  • j=1

p0jzj)} ∂r ∂zi Can be used for analysis of

  • Moments of the queue length distribution

[Van Kessel, N-Q, Borst, 2004]

  • Heavy traffic (today)
  • R. N´

u˜ nez-Queija 12

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SLIDE 37

Phase-type service requirements (4)

Partial differential equation Λ(1 − ρ)(1 −

J

  • j=1

p0jzj) =

J

  • i=1

{µigi(pi0 +

J

  • j=1

pijzj − zi) − Λgizi(1 −

J

  • j=1

p0jzj)} ∂r ∂zi Can be used for analysis of

  • Moments of the queue length distribution

[Van Kessel, N-Q, Borst, 2004]

  • Heavy traffic (today)
  • R. N´

u˜ nez-Queija 12-1

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SLIDE 38

Heavy-traffic scaling

Notation

  • Change of variables zi = e−si
  • e−(1−ρ)¯

s := (e−(1−ρ)s1, . . . , e−(1−ρ)sJ)

Goal lim

ρ↑1 p(e−(1−ρ)¯ s)

:= lim

ρ↑1 E(e−(1−ρ)s1N1 · . . . · e−(1−ρ)sJNJ)

=

E(e−s1 ˆ

N1 · . . . · e−sJ ˆ NJ)

Investigate ˆ r(¯ s) = E

 1 − e−s1 ˆ

N1 · . . . · e−sJ ˆ NJ

J

j=1 ˆ

Njgj · 1(J

j=1 ˆ

Nj>0)

 

  • R. N´

u˜ nez-Queija 13

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SLIDE 39

Heavy-traffic scaling

Notation

  • Change of variables zi = e−si
  • e−(1−ρ)¯

s := (e−(1−ρ)s1, . . . , e−(1−ρ)sJ)

Goal lim

ρ↑1 p(e−(1−ρ)¯ s)

:= lim

ρ↑1 E(e−(1−ρ)s1N1 · . . . · e−(1−ρ)sJNJ)

=

E(e−s1 ˆ

N1 · . . . · e−sJ ˆ NJ)

Investigate ˆ r(¯ s) = E

 1 − e−s1 ˆ

N1 · . . . · e−sJ ˆ NJ

J

j=1 ˆ

Njgj · 1(J

j=1 ˆ

Nj>0)

 

  • R. N´

u˜ nez-Queija 13-1

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SLIDE 40

Heavy traffic analysis

Investigate ˆ r(¯ s) = E

 1 − e−s1 ˆ

N1 · . . . · e−sJ ˆ NJ

J

j=1 ˆ

Njgj · 1(J

j=1 ˆ

Nj>0)

 

Lemma The function ˆ r(¯ s) satisfies the following partial differential equation: 0 =

J

  • i=1

Fi(¯ s)∂ˆ r(¯ s) ∂si ∀ ¯ s ≥ 0, where Fi(¯ s) = gi

 µi(−si +

J

  • j=1

pijsj) + ˆ λ

J

  • j=1

p0jsj

  ,

with ˆ λ equal to the limiting arrival rate

  • R. N´

u˜ nez-Queija 14

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SLIDE 41

Heavy traffic analysis

Investigate ˆ r(¯ s) = E

 1 − e−s1 ˆ

N1 · . . . · e−sJ ˆ NJ

J

j=1 ˆ

Njgj · 1(J

j=1 ˆ

Nj>0)

 

Lemma The function ˆ r(¯ s) satisfies the following partial differential equation: 0 =

J

  • i=1

Fi(¯ s)∂ˆ r(¯ s) ∂si ∀ ¯ s ≥ 0, where Fi(¯ s) = gi

 µi(−si +

J

  • j=1

pijsj) + ˆ λ

J

  • j=1

p0jsj

  ,

with ˆ λ equal to the limiting arrival rate

  • R. N´

u˜ nez-Queija 14-1

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SLIDE 42

Heavy traffic analysis

Investigate ˆ r(¯ s) = E

 1 − e−s1 ˆ

N1 · . . . · e−sJ ˆ NJ

J

j=1 ˆ

Njgj · 1(J

j=1 ˆ

Nj>0)

 

Lemma The function ˆ r(¯ s) satisfies the following partial differential equation: 0 =

J

  • i=1

Fi(¯ s)∂ˆ r(¯ s) ∂si ∀ ¯ s ≥ 0, where Fi(¯ s) = gi

 µi(−si +

J

  • j=1

pijsj) + ˆ λ

J

  • j=1

p0jsj

  ,

with ˆ λ equal to the limiting arrival rate

  • R. N´

u˜ nez-Queija 14-2

slide-43
SLIDE 43

State space collapse in heavy traffic

The function ˆ r(¯ s) is constant on the J − 1 dimensional set Hc := {¯ s ≥ ¯ 0 :

J

  • j=1

ˆ ρj gj sj = c}, c > 0, hence ( ˆ N1, ˆ N2, . . . , ˆ NJ) d =

  • ˆ

ρ1 g1 , ˆ ρ2 g2 , . . . , ˆ ρJ gJ

  • · X,

15

slide-44
SLIDE 44

State space collapse in heavy traffic

The function ˆ r(¯ s) is constant on the J − 1 dimensional set Hc := {¯ s ≥ ¯ 0 :

J

  • j=1

ˆ ρj gj sj = c}, c > 0, hence ( ˆ N1, ˆ N2, . . . , ˆ NJ) d =

  • ˆ

ρ1 g1 , ˆ ρ2 g2 , . . . , ˆ ρJ gJ

  • · X,

15-1

slide-45
SLIDE 45

Residual service requirements in heavy traffic

For phase-type distributed service requirements lim

ρ↑1 E

  • e− K

l=1 sl(1−ρ)Nl−K l=1

yl

h=1 sl,hBr l,h

  • = E
  • e− K

l=1 sl ˆ

Nl

  • ·

K

  • l=1

yl

  • h=1

E

  • e−sl,hBfwd

l

  • for yl ∈ {0, 1, . . .} and sl,h, sl > 0, l = 1, . . . , K, h = 1, . . . , yl

For PS, the limit can be replaced with an equality (for all loads)

  • R. N´

u˜ nez-Queija 16

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SLIDE 46

Residual service requirements in heavy traffic

For phase-type distributed service requirements lim

ρ↑1 E

  • e− K

l=1 sl(1−ρ)Nl−K l=1

yl

h=1 sl,hBr l,h

  • = E
  • e− K

l=1 sl ˆ

Nl

  • ·

K

  • l=1

yl

  • h=1

E

  • e−sl,hBfwd

l

  • for yl ∈ {0, 1, . . .} and sl,h, sl > 0, l = 1, . . . , K, h = 1, . . . , yl

For PS, the limit can be replaced with an equality (for all loads)

  • R. N´

u˜ nez-Queija 16-1

slide-47
SLIDE 47

Concluding remarks

  • Joint queue length distribution of DPS in heavy traffic

⋄ Closed form analysis ⋄ State space collapse ⋄ Sensitive to second moments, but not ”too much” (see paper)

  • Product form for residual service requirements
  • More

⋄ Size based scheduling ⋄ Monotonicity with respect to the weights

  • R. N´

u˜ nez-Queija 17

slide-48
SLIDE 48

Heavy-traffic analysis of the M/PH/1 discriminatory processor sharing queue with phase-dependent weights

Maaike Verloop (CWI) Urtzi Ayesta (CNRS-LAAS & BECAM) Sindo N´ u˜ nez-Queija (University of Amsterdam & CWI)