text: 9 1.0 ongoing study. Here we use a model studied in Ref. [72], C(cid:144)C0 where entanglement dynamics was examined in a per- Fw(cid:144)F0 0.8 Ec(cid:144)E0 turbative model of non-Markovian noise. Considering an entangled spin pair subjected to random frequency 0.6 ﬂuctuations [73], a Gaussian noise model was adopted, which is essentially non-Markovian, and has a well de- 0.4 ﬁned Markov limit. Following Ref. [72], we include this memory dependent noise model to propose a global mas- 0.2 ter equation of the form (cid:80) (cid:16) (cid:0) (cid:1)(cid:17) 0.0 0 1 2 3 4 5 ý t dρ(cid:80)= (cid:16) (cid:16) (cid:17)(cid:17) dt † − † † G (t)(N + 1) A ρA 1 A A ρ + ρA A αβ αβ α β 2 α β α β FIG. 4: (Colour online) Concurrence EC/EC(0), Fisher in- + G (t)N A† ρA − 1 A A† ρ + ρA A† , formation FW/FW(0) and coherence C/C(0) with respect to αβ αβ α β 2 α β α β γt for the Werner state with x = 1/2, with a as a parameter. (5.1) We have taken a = 1/2. Here we are normalizing all the mea- where sures by their magnitude at initial time t = 0. The red (large (cid:0) (cid:1) dashed) plot is for coherence, green (bold line) plot is for Fisher information and blue (small dashed) plot is for entan- − − glement. We see here that entanglement decays fastest, fol- Gαβ(t) = γαβ 1 e Λt . (5.2) lowed by Fisher information. Coherence sustains for a much longer period. The environmental memory information is encoded in the time dependent coeﬃcient G , with t = 1/Λ as the αβ c → environmental memory characteristic time. t 0 is the dipole interactions [34], does not aﬀect the ground and c Markov limit, for which G (t) = γ at all times. Let excited states. Only the symmetric and anti-symmetric αβ αβ us now consider a speciﬁc case of depolarizing channel, states are aﬀected by it. The incoherent part of the for which dipole-dipole interaction is basically the global part of the dynamical map with the strength γ . Therefore, the 1∗2 global interaction aﬀects only ρ23 and ρ23, which are the A = σ− ⊗ I ; A = I ⊗ σ−. (5.3) components of the symmetric and anti-symmetric states. 1 2 We intend to ﬁnd an exact solution of the master equa- tion for this particular case in order to study the dy- V. GLOBAL MASTER EQUATION WITH TIME namics of coherence and correlation for special kind of VARYING PARAMETERS X states as described by equation (2.21). We once again take the initial state as the Werner state (3.3). This In this section, we further generalize our global master is basically an initial value problem of coupled ﬁrst or- equation by considering the memory eﬀect of the bath. In der ordinary diﬀerential equation (ODE) with variable a practical situation, an environment usually has mem- coeﬃcients. We know that they are not exactly solv- ory. In an experiment, a composite quantum system can able for all cases. But fortunately for our case, an an- be exposed to various kind of noises such as vacuum alytical solution can be obtained. In a linear system of noise, phase noise, thermal noise as well as a mixture ODE such as Y˙ (t) = A(t)Y (t), if the coeﬃcient matrix (cid:82) of diﬀerent kind of noises. Diﬀerent noise models has commutes with itself for any two arguments, that is, if been proposed in recent years to model solvable approx- A(t)A(s) = A(s)A(t) for all s and t, then we can write (cid:48) (cid:48) t imate master equations [69, 70]. Correlation dynamics B(t) = A(t )dt and observe B(t)A(s) = A(s)B(t). 0 in the Markovian (no memory) regime has been exten- For this we have a formal solution of the coupled set of sively studied [71]. However, in practice, an environment diﬀerential equations as Y (t) = Y (0) exp(B(t)). By this is more likely to be non-Markovian. A systematic in- procedure we can solve the master equation (5.1) exactly vestigation of the dynamics of quantum coherence and for the case of two qubit X states. The components of correlation in the presence of non-Markovian noise is an the density matrix can be expressed as >0C(cid:144)C ,0WF(cid:144)WF ,0cE(cid:144)cE: ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: ongoing

text: study.

text: Here

text: we

text: use

text: a

text: model

text: studied

text: in

text: Ref.

text: where

text: entanglement

text: dynamics

text: was

text: examined

text: in

text: a

text: per-

text: turbative

text: model

text: of

text: non-Markovian

text: noise.

text: Considering

text: an

text: entangled

text: spin

text: pair

text: subjected

text: to

text: random

text: frequency

text: ﬂuctuations

text: a

text: Gaussian

text: noise

text: model

text: was

text: adopted,

text: which

text: is

text: essentially

text: non-Markovian,

text: and

text: has

text: a

text: well

text: de-

text: ﬁned

text: Markov

text: limit.

text: Following

text: Ref.

text: we

text: include

text: this

text: memory

text: dependent

text: noise

text: model

text: to

text: propose

text: a

text: global

text: mas-

text: ter

text: equation

text: of

text: the

text: form

text: ý

text: t

text: dt

text: G

text: (t)(N

text: A

text: ρA

text: A

text: A

text: ρ

text: ρA

text: A

text: αβ

text: αβ

text: α

text: β

text: α

text: β

text: α

text: β

text: FIG.

text: (Colour

text: online)

text: Concurrence

text: Fisher

text: in-

text: G

text: (t)N

text: A†

text: ρA

text: A

text: A†

text: ρ

text: ρA

text: A†

text: formation

text: and

text: coherence

text: with

text: respect

text: to

text: αβ

text: αβ

text: α

text: β

text: α

text: β

text: α

text: β

text: γt

text: for

text: the

text: Werner

text: state

text: with

text: x

text: with

text: a

text: as

text: a

text: parameter.

text: We

text: have

text: taken

text: a

text: Here

text: we

text: are

text: normalizing

text: all

text: the

text: mea-

text: where

text: sures

text: by

text: their

text: magnitude

text: at

text: initial

text: time

text: t

text: The

text: red

text: (large

text: dashed)

text: plot

text: is

text: for

text: coherence,

text: green

text: (bold

text: line)

text: plot

text: is

text: for

text: Fisher

text: information

text: and

text: blue

text: (small

text: dashed)

text: plot

text: is

text: for

text: entan-

text: glement.

text: We

text: see

text: here

text: that

text: entanglement

text: decays

text: fastest,

text: fol-

text: Gαβ(t)

text: γαβ

text: e

text: Λt

text: lowed

text: by

text: Fisher

text: information.

text: Coherence

text: sustains

text: for

text: a

text: much

text: longer

text: period.

text: The

text: environmental

text: memory

text: information

text: is

text: encoded

text: in

text: the

text: time

text: dependent

text: coeﬃcient

text: G

text: with

text: t

text: as

text: the

text: αβ

text: c

text: environmental

text: memory

text: characteristic

text: time.

text: t

text: is

text: the

text: dipole

text: interactions

text: does

text: not

text: aﬀect

text: the

text: ground

text: and

text: c

text: Markov

text: limit,

text: for

text: which

text: G

text: (t)

text: γ

text: at

text: all

text: times.

text: Let

text: excited

text: states.

text: Only

text: the

text: symmetric

text: and

text: anti-symmetric

text: αβ

text: αβ

text: us

text: now

text: consider

text: a

text: speciﬁc

text: case

text: of

text: depolarizing

text: channel,

text: states

text: are

text: aﬀected

text: by

text: it.

text: The

text: incoherent

text: part

text: of

text: the

text: for

text: which

text: dipole-dipole

text: interaction

text: is

text: basically

text: the

text: global

text: part

text: of

text: the

text: dynamical

text: map

text: with

text: the

text: strength

text: γ

text: Therefore,

text: the

text: global

text: interaction

text: aﬀects

text: only

text: and

text: which

text: are

text: the

text: A

text: σ−

text: I

text: A

text: I

text: σ−.

text: components

text: of

text: the

text: symmetric

text: and

text: anti-symmetric

text: states.

text: We

text: intend

text: to

text: ﬁnd

text: an

text: exact

text: solution

text: of

text: the

text: master

text: equa-

text: tion

text: for

text: this

text: particular

text: case

text: in

text: order

text: to

text: study

text: the

text: dy-

text: V.

text: GLOBAL

text: MASTER

text: EQUATION

text: WITH

text: TIME

text: namics

text: of

text: coherence

text: and

text: correlation

text: for

text: special

text: kind

text: of

text: VARYING

text: PARAMETERS

text: X

text: states

text: as

text: described

text: by

text: equation

text: We

text: once

text: again

text: take

text: the

text: initial

text: state

text: as

text: the

text: Werner

text: state

text: This

text: In

text: this

text: section,

text: we

text: further

text: generalize

text: our

text: global

text: master

text: is

text: basically

text: an

text: initial

text: value

text: problem

text: of

text: coupled

text: ﬁrst

text: or-

text: equation

text: by

text: considering

text: the

text: memory

text: eﬀect

text: of

text: the

text: bath.

text: In

text: der

text: ordinary

text: diﬀerential

text: equation

text: (ODE)

text: with

text: variable

text: a

text: practical

text: situation,

text: an

text: environment

text: usually

text: has

text: mem-

text: coeﬃcients.

text: We

text: know

text: that

text: they

text: are

text: not

text: exactly

text: solv-

text: ory.

text: In

text: an

text: experiment,

text: a

text: composite

text: quantum

text: system

text: can

text: able

text: for

text: all

text: cases.

text: But

text: fortunately

text: for

text: our

text: case,

text: an

text: an-

text: be

text: exposed

text: to

text: various

text: kind

text: of

text: noises

text: such

text: as

text: vacuum

text: alytical

text: solution

text: can

text: be

text: obtained.

text: In

text: a

text: linear

text: system

text: of

text: noise,

text: phase

text: noise,

text: thermal

text: noise

text: as

text: well

text: as

text: a

text: mixture

text: ODE

text: such

text: as

text: Y˙

text: (t)

text: A(t)Y

text: (t),

text: if

text: the

text: coeﬃcient

text: matrix

text: of

text: diﬀerent

text: kind

text: of

text: noises.

text: Diﬀerent

text: noise

text: models

text: has

text: commutes

text: with

text: itself

text: for

text: any

text: two

text: arguments,

text: that

text: is,

text: if

text: been

text: proposed

text: in

text: recent

text: years

text: to

text: model

text: solvable

text: approx-

text: A(t)A(s)

text: A(s)A(t)

text: for

text: all

text: s

text: and

text: t,

text: then

text: we

text: can

text: write

text: t

text: imate

text: master

text: equations

text: Correlation

text: dynamics

text: B(t)

text: A(t

text: )dt

text: and

text: observe

text: B(t)A(s)

text: A(s)B(t).

text: in

text: the

text: Markovian

text: (no

text: memory)

text: regime

text: has

text: been

text: exten-

text: For

text: this

text: we

text: have

text: a

text: formal

text: solution

text: of

text: the

text: coupled

text: set

text: of

text: sively

text: studied

text: However,

text: in

text: practice,

text: an

text: environment

text: diﬀerential

text: equations

text: as

text: Y

text: (t)

text: Y

text: exp(B(t)).

text: By

text: this

text: is

text: more

text: likely

text: to

text: be

text: non-Markovian.

text: A

text: systematic

text: in-

text: procedure

text: we

text: can

text: solve

text: the

text: master

text: equation

text: exactly

text: vestigation

text: of

text: the

text: dynamics

text: of

text: quantum

text: coherence

text: and

text: for

text: the

text: case

text: of

text: two

text: qubit

text: X

text: states.

text: The

text: components

text: of

text: correlation

text: in

text: the

text: presence

text: of

text: non-Markovian

text: noise

text: is

text: an

text: the

text: density

text: matrix

text: can

text: be

text: expressed

text: as

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: [FG12] Dario Fiore and Rosario Gennaro. Publicly veriﬁable delegation of large polynomials and matrix computations, with applications. In ACM CCS, pages 501–512, 2012. [Fid72] Charles M. Fiduccia. Polynomial evaluation via the division algorithm: The fast fourier trans- form revisited. In STOC, pages 88–93, 1972. [GGP10] Rosario Gennaro, Craig Gentry, and Bryan Parno. Non-interactive veriﬁable computing: Out- sourcing computation to untrusted workers. In CRYPTO, pages 465–482, 2010. [GH98] Oded Goldreich and Johan Ha˚stad. On the complexity of interactive proofs with bounded communication. Inf. Process. Lett., 67(4):205–214, 1998. [GKR08] Shaﬁ Goldwasser, Yael Tauman Kalai, and Guy N. Rothblum. Delegating computation: Inter- active proofs for muggles. J. ACM, 62(4):27, 2015. Original in STOC’08. [GMR85] Shaﬁ Goldwasser, Silvio Micali, and Charles Rackoff. The knowledge complexity of interac- tive proof-systems (extended abstract). In STOC, pages 291–304, 1985. [GP14] Joshua A. Grochow and Toniann Pitassi. Circuit complexity, proof complexity, and polynomial identity testing. In FOCS, pages 110–119, 2014. [GR15] Tom Gur and Ron D. Rothblum. Non-interactive proofs of proximity. In Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS 2015, pages 133– 142, 2015. [GVW02] Oded Goldreich, Salil P. Vadhan, and Avi Wigderson. On interactive proofs with a laconic prover. Computational Complexity, 11(1-2):1–53, 2002. [Hor72] Ellis Horowitz. A fast method for interpolation using preconditioning. Inf. Process. Lett., 1(4):157–163, 1972. [IP01] Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367–375, 2001. [IPS13] Russell Impagliazzo, Ramamohan Paturi, and Stefan Schneider. A satisﬁability algorithm for sparse depth two threshold circuits. In FOCS, pages 479–488, 2013. [IPZ01] Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512–530, 2001. [JMV13] Hamidreza Jahanjou, Eric Miles, and Emanuele Viola. Local reductions. Technical Report TR13-099, Electronic Colloquium on Computational Complexity, July 2013. [JMV15] Hamid Jahanjou, Eric Miles, and Emanuele Viola. Local reductions. In ICALP, pages 749– 760, 2015. [Kar82] Richard M Karp. Dynamic programming meets the principle of inclusion and exclusion. Op- erations Research Letters, 1(2):49–51, 1982. [KI04] Valentine Kabanets and Russell Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity, 13(1-2):1–46, 2004. [KRR14] Yael Tauman Kalai, Ran Raz, and Ron D. Rothblum. How to delegate computations: the power of no-signaling proofs. In STOC, pages 485–494, 2014. 13

label: reference

text: Dario

text: Fiore

text: and

text: Rosario

text: Gennaro.

text: Publicly

text: veriﬁable

text: delegation

text: of

text: large

text: polynomials

text: and

text: matrix

text: computations,

text: with

text: applications.

text: In

text: ACM

text: CCS,

text: pages

text: Charles

text: M.

text: Fiduccia.

text: Polynomial

text: evaluation

text: via

text: the

text: division

text: algorithm:

text: The

text: fast

text: fourier

text: trans-

text: form

text: revisited.

text: In

text: STOC,

text: pages

text: Rosario

text: Gennaro,

text: Craig

text: Gentry,

text: and

text: Bryan

text: Parno.

text: Non-interactive

text: veriﬁable

text: computing:

text: Out-

text: sourcing

text: computation

text: to

text: untrusted

text: workers.

text: In

text: CRYPTO,

text: pages

text: Oded

text: Goldreich

text: and

text: Johan

text: Ha˚stad.

text: On

text: the

text: complexity

text: of

text: interactive

text: proofs

text: with

text: bounded

text: communication.

text: Inf.

text: Process.

text: Lett.,

text: Shaﬁ

text: Goldwasser,

text: Yael

text: Tauman

text: Kalai,

text: and

text: Guy

text: N.

text: Rothblum.

text: Delegating

text: computation:

text: Inter-

text: active

text: proofs

text: for

text: muggles.

text: J.

text: ACM,

text: Original

text: in

text: Shaﬁ

text: Goldwasser,

text: Silvio

text: Micali,

text: and

text: Charles

text: Rackoff.

text: The

text: knowledge

text: complexity

text: of

text: interac-

text: tive

text: proof-systems

text: (extended

text: abstract).

text: In

text: STOC,

text: pages

text: Joshua

text: A.

text: Grochow

text: and

text: Toniann

text: Pitassi.

text: Circuit

text: complexity,

text: proof

text: complexity,

text: and

text: polynomial

text: identity

text: testing.

text: In

text: FOCS,

text: pages

text: Tom

text: Gur

text: and

text: Ron

text: D.

text: Rothblum.

text: Non-interactive

text: proofs

text: of

text: proximity.

text: In

text: Proceedings

text: of

text: the

text: Conference

text: on

text: Innovations

text: in

text: Theoretical

text: Computer

text: Science,

text: ITCS

text: pages

text: Oded

text: Goldreich,

text: Salil

text: P.

text: Vadhan,

text: and

text: Avi

text: Wigderson.

text: On

text: interactive

text: proofs

text: with

text: a

text: laconic

text: prover.

text: Computational

text: Complexity,

text: Ellis

text: Horowitz.

text: A

text: fast

text: method

text: for

text: interpolation

text: using

text: preconditioning.

text: Inf.

text: Process.

text: Lett.,

text: Russell

text: Impagliazzo

text: and

text: Ramamohan

text: Paturi.

text: On

text: the

text: complexity

text: of

text: k-SAT.

text: J.

text: Comput.

text: Syst.

text: Sci.,

text: Russell

text: Impagliazzo,

text: Ramamohan

text: Paturi,

text: and

text: Stefan

text: Schneider.

text: A

text: satisﬁability

text: algorithm

text: for

text: sparse

text: depth

text: two

text: threshold

text: circuits.

text: In

text: FOCS,

text: pages

text: Russell

text: Impagliazzo,

text: Ramamohan

text: Paturi,

text: and

text: Francis

text: Zane.

text: Which

text: problems

text: have

text: strongly

text: exponential

text: complexity?

text: J.

text: Comput.

text: Syst.

text: Sci.,

text: Hamidreza

text: Jahanjou,

text: Eric

text: Miles,

text: and

text: Emanuele

text: Viola.

text: Local

text: reductions.

text: Technical

text: Report

text: Electronic

text: Colloquium

text: on

text: Computational

text: Complexity,

text: July

text: Hamid

text: Jahanjou,

text: Eric

text: Miles,

text: and

text: Emanuele

text: Viola.

text: Local

text: reductions.

text: In

text: ICALP,

text: pages

text: Richard

text: M

text: Karp.

text: Dynamic

text: programming

text: meets

text: the

text: principle

text: of

text: inclusion

text: and

text: exclusion.

text: Op-

text: erations

text: Research

text: Letters,

text: Valentine

text: Kabanets

text: and

text: Russell

text: Impagliazzo.

text: Derandomizing

text: polynomial

text: identity

text: tests

text: means

text: proving

text: circuit

text: lower

text: bounds.

text: Computational

text: Complexity,

text: Yael

text: Tauman

text: Kalai,

text: Ran

text: Raz,

text: and

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text: D.

text: Rothblum.

text: How

text: to

text: delegate

text: computations:

text: the

text: power

text: of

text: no-signaling

text: proofs.

text: In

text: STOC,

text: pages


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text: Kiran

text: S.

text: Kedlaya

text: and

text: Christopher

text: Umans.

text: Fast

text: polynomial

text: factorization

text: and

text: modular

text: compo-

text: sition.

text: SIAM

text: J.

text: Comput.,

text: Carsten

text: Lund,

text: Lance

text: Fortnow,

text: Howard

text: Karloff,

text: and

text: Noam

text: Nisan.

text: Algebraic

text: methods

text: for

text: interactive

text: proof

text: systems.

text: JACM,

text: Daniel

text: Lokshtanov,

text: Da´niel

text: Marx,

text: and

text: Saket

text: Saurabh.

text: Lower

text: bounds

text: based

text: on

text: the

text: exponential

text: time

text: hypothesis.

text: Bulletin

text: of

text: the

text: EATCS,

text: J.

text: C.

text: Lagarias

text: and

text: Andrew

text: M.

text: Odlyzko.

text: Computing

text: pi(x):

text: An

text: analytic

text: method.

text: J.

text: Algorithms,

text: Bryan

text: Parno,

text: Jon

text: Howell,

text: Craig

text: Gentry,

text: and

text: Mariana

text: Raykova.

text: Pinocchio:

text: Nearly

text: practical

text: veriﬁable

text: computation.

text: In

text: IEEE

text: Symposium

text: on

text: Security

text: and

text: Privacy,

text: pages

text: Pavel

text: Pudla´k

text: and

text: Russell

text: Impagliazzo.

text: A

text: lower

text: bound

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text: algorithms

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text: (prelimi-

text: nary

text: version).

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text: SODA,

text: pages

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text: R.

text: Pratt.

text: Every

text: prime

text: has

text: a

text: succinct

text: certiﬁcate.

text: SIAM

text: J.

text: Comput.,

text: Liam

text: Roditty

text: and

text: Virginia

text: Vassilevska

text: Williams.

text: Fast

text: approximation

text: algorithms

text: for

text: the

text: diam-

text: eter

text: and

text: radius

text: of

text: sparse

text: graphs.

text: In

text: STOC,

text: pages

text: Herbert

text: John

text: Ryser.

text: Combinatorial

text: mathematics.

text: Mathematical

text: Association

text: of

text: America,

text: The

text: Carus

text: mathematical

text: monographs.

text: Adi

text: Shamir.

text: IP

text: PSPACE.

text: Journal

text: of

text: the

text: Association

text: for

text: Computing

text: Machinery,

text: A.

text: Shen.

text: IP

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text: proof.

text: J.

text: ACM,

text: P.

text: M.

text: Spira.

text: On

text: time

text: hardware

text: complexity

text: tradeoffs

text: for

text: boolean

text: functions.

text: In

text: Proceedings

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text: Fourth

text: Hawaii

text: International

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text: Sciences,

text: pages

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text: Thaler.

text: Time-optimal

text: interactive

text: proofs

text: for

text: circuit

text: evaluation.

text: In

text: CRYPTO,

text: pages

text: Leslie

text: Valiant.

text: Completeness

text: classes

text: in

text: algebra.

text: In

text: STOC,

text: pages

text: Leslie

text: G.

text: Valiant.

text: Reducibility

text: by

text: algebraic

text: projections.

text: In

text: Logic

text: and

text: Algorithmic:

text: an

text: Interna-

text: tional

text: Symposium

text: held

text: in

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text: de

text: l’Enseign.

text: Math.

text: Virginia

text: Vassilevska

text: Williams.

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text: of

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text: problems:

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text: time

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text: International

text: Symposium

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text: Exact

text: Computation,

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text: There has been much work on bounding the communication between the prover and veriﬁer. For in- stance, this is not the ﬁrst time that Merlin and Arthur have led to an unexpected square-root speedup: Aaronson and Wigderson [AW09] gave a√n MA communication protocol for computing the inner product of two n-length vectors which runs in O˜( n) time. Their protocol uses a nice bivariate encoding of vec- tors, although it is somewhat different from ours (which is univariate). Gur and Rothblum [GR15] obtain a similar square-root speedup for checking sums in the “non-interactive property testing” setting. Goldreich and Hastad [GH98] and Goldreich, Vadhan, and Wigderson [GVW02] studied interactive proofs which seek to minimize the number of bits sent from Merlin to Arthur. The “small bits” case is of course even more restrictive than the “small rounds” case. The latter reference shows that for any language L that has an interactive proof with b bits of communication, there is an O(1)-round interactive proof for L that uses only exp(b) communication. The authors also conjectured an “Arthur-Merlin ETH” that #SAT does not have a 2o(n)-time AM-proof system with O(1) rounds. What we report in this paper is rather far from disproving this “AMETH” conjecture, but it is interesting that some non-trivial progress can be made. Goldwasser, Kalai, and Rothblum [GKR08] study what they call delegating computation, proving (for example) that for all logspace-uniform NC circuits C, one can prove that C(x) = 1 on an input x of length n with O˜(n) veriﬁcation time, O(logn) space, and poly(logn) communication complexity between the prover and veriﬁer. Despite the amazingly low running time and space usage, the protocols of this work are highly non-interactive: they need poly(logn) rounds between the prover and veriﬁer as well. Relating our work to proof complexity, Grochow and Pitassi [GP14] introduced a new algebraic proof system based on axioms satisﬁed by any Boolean circuit that solves the polynomial identity testing prob- lem. The proofs in their system can be efﬁciently veriﬁed by running a polynomial identity test, implying they can be viewed as proof of a Merlin-Arthur type. An intriguing property of their proof system is that super-polynomial lower bounds for it would prove lower bounds for the Permanent. The area of veriﬁable computation (e.g. [PHGR13]) is a new subject in cryptography, and is certainly related to our work. However, in crypto the work appears to be either very speciﬁc to particular functions, or it relies on very heavy machinery like probabilistically checkable proofs, or it relies on cryptographic hardness assumptions. In our setting, we want non-interactive proofs for batch computations that are shorter than the compu- tation time, with the typical “perfect completeness” and “low error soundness” conditions preserved, and which work unconditionally. 3 Fast Multipoint Circuit Evaluation (With Merlin and Arthur) In this section, we give the proof system for multipoint arithmetic circuit evaluation: Theorem 3.1. For every prime power q and ε > 0, MULTIPOINT CIRCUIT EVALUATION for K points in (Fq)n on an arithmetic circuit C of n inputs, s gates, and degree d has an MA-proof system where: • · · Merlin sends a proof of O(K d log(Kqd/ε)) bits, and • Arthur tosses at most log(K·qd/ε){ coin}s, ou·tputs (C(a1),·...,C(aK)) incorrectly with probability at most ε, and runs in time (K max d,n + s poly(logs)) poly(log(Kqd/ε)). We have stated the theorem at this level of generality because we need good bounds on the parame- ters to obtain certain consequences. For example, in our proof system for quantiﬁed Boolean formulas (Theorem 1.3), the parameters s, K, q, and d are all various exponentials in n. · · Because instances of MULTIPOINT CIRCUIT EVALUATION have length O((K n + slogs) logq), the running time of Theorem 3.1 is essentially linear in the input length, up to the factor of d in Merlin’s proof (in general, d could be much larger than n). So Theorem 3.1 is extremely powerful for arithmetic circuits of low degree. 5 ##LTLine## ##LTLine##

text: There

label: paragraph

text: has

text: been

text: much

text: work

text: on

text: bounding

text: the

text: communication

text: between

text: the

text: prover

text: and

text: veriﬁer.

text: For

text: in-

text: stance,

text: this

text: is

text: not

text: the

text: ﬁrst

text: time

text: that

text: Merlin

text: and

text: Arthur

text: have

text: led

text: to

text: an

text: unexpected

text: square-root

text: speedup:

text: Aaronson

text: and

text: Wigderson

text: gave

text: a√n

text: MA

text: communication

text: protocol

text: for

text: computing

text: the

text: inner

text: product

text: of

text: two

text: n-length

text: vectors

text: which

text: runs

text: in

text: O˜(

text: n)

text: time.

text: Their

text: protocol

text: uses

text: a

text: nice

text: bivariate

text: encoding

text: of

text: vec-

text: tors,

text: although

text: it

text: is

text: somewhat

text: different

text: from

text: ours

text: (which

text: is

text: univariate).

text: Gur

text: and

text: Rothblum

text: obtain

text: a

text: similar

text: square-root

text: speedup

text: for

text: checking

text: sums

text: in

text: the

text: “non-interactive

text: property

text: testing”

text: setting.

text: Goldreich

text: and

text: Hastad

text: and

text: Goldreich,

text: Vadhan,

text: and

text: Wigderson

text: studied

text: interactive

text: proofs

text: which

text: seek

text: to

text: minimize

text: the

text: number

text: of

text: bits

text: sent

text: from

text: Merlin

text: to

text: Arthur.

text: The

text: “small

text: bits”

text: case

text: is

text: of

text: course

text: even

text: more

text: restrictive

text: than

text: the

text: “small

text: rounds”

text: case.

text: The

text: latter

text: reference

text: shows

text: that

text: for

text: any

text: language

text: L

text: that

text: has

text: an

text: interactive

text: proof

text: with

text: b

text: bits

text: of

text: communication,

text: there

text: is

text: an

text: interactive

text: proof

text: for

text: L

text: that

text: uses

text: only

text: exp(b)

text: communication.

text: The

text: authors

text: also

text: conjectured

text: an

text: “Arthur-Merlin

text: ETH”

text: that

text: #SAT

text: does

text: not

text: have

text: a

text: AM-proof

text: system

text: with

text: rounds.

text: What

text: we

text: report

text: in

text: this

text: paper

text: is

text: rather

text: far

text: from

text: disproving

text: this

text: “AMETH”

text: conjecture,

text: but

text: it

text: is

text: interesting

text: that

text: some

text: non-trivial

text: progress

text: can

text: be

text: made.

text: Goldwasser,

text: Kalai,

text: and

text: Rothblum

text: study

text: what

text: they

text: call

text: delegating

text: computation,

text: proving

text: (for

text: example)

text: that

text: for

text: all

text: logspace-uniform

text: NC

text: circuits

text: C,

text: one

text: can

text: prove

text: that

text: C(x)

text: on

text: an

text: input

text: x

text: of

text: length

text: n

text: with

text: O˜(n)

text: veriﬁcation

text: time,

text: O(logn)

text: space,

text: and

text: poly(logn)

text: communication

text: complexity

text: between

text: the

text: prover

text: and

text: veriﬁer.

text: Despite

text: the

text: amazingly

text: low

text: running

text: time

text: and

text: space

text: usage,

text: the

text: protocols

text: of

text: this

text: work

text: are

text: highly

text: non-interactive:

text: they

text: need

text: poly(logn)

text: rounds

text: between

text: the

text: prover

text: and

text: veriﬁer

text: as

text: well.

text: Relating

text: our

text: work

text: to

text: proof

text: complexity,

text: Grochow

text: and

text: Pitassi

text: introduced

text: a

text: new

text: algebraic

text: proof

text: system

text: based

text: on

text: axioms

text: satisﬁed

text: by

text: any

text: Boolean

text: circuit

text: that

text: solves

text: the

text: polynomial

text: identity

text: testing

text: prob-

text: lem.

text: The

text: proofs

text: in

text: their

text: system

text: can

text: be

text: efﬁciently

text: veriﬁed

text: by

text: running

text: a

text: polynomial

text: identity

text: test,

text: implying

text: they

text: can

text: be

text: viewed

text: as

text: proof

text: of

text: a

text: Merlin-Arthur

text: type.

text: An

text: intriguing

text: property

text: of

text: their

text: proof

text: system

text: is

text: that

text: super-polynomial

text: lower

text: bounds

text: for

text: it

text: would

text: prove

text: lower

text: bounds

text: for

text: the

text: Permanent.

text: The

text: area

text: of

text: veriﬁable

text: computation

text: (e.g.

text: is

text: a

text: new

text: subject

text: in

text: cryptography,

text: and

text: is

text: certainly

text: related

text: to

text: our

text: work.

text: However,

text: in

text: crypto

text: the

text: work

text: appears

text: to

text: be

text: either

text: very

text: speciﬁc

text: to

text: particular

text: functions,

text: or

text: it

text: relies

text: on

text: very

text: heavy

text: machinery

text: like

text: probabilistically

text: checkable

text: proofs,

text: or

text: it

text: relies

text: on

text: cryptographic

text: hardness

text: assumptions.

text: In

text: our

text: setting,

text: we

text: want

text: non-interactive

text: proofs

text: for

text: batch

text: computations

text: that

text: are

text: shorter

text: than

text: the

text: compu-

text: tation

text: time,

text: with

text: the

text: typical

text: “perfect

text: completeness”

text: and

text: “low

text: error

text: soundness”

text: conditions

text: preserved,

text: and

text: which

text: work

text: unconditionally.

text: Fast

text: Multipoint

text: Circuit

text: Evaluation

text: (With

text: Merlin

text: and

text: Arthur)

text: In

text: this

text: section,

text: we

text: give

text: the

text: proof

text: system

text: for

text: multipoint

text: arithmetic

text: circuit

text: evaluation:

text: Theorem

text: For

text: every

text: prime

text: power

text: q

text: and

text: ε

text: MULTIPOINT

text: CIRCUIT

text: EVALUATION

text: for

text: K

text: points

text: in

text: (Fq)n

text: on

text: an

text: arithmetic

text: circuit

text: C

text: of

text: n

text: inputs,

text: s

text: gates,

text: and

text: degree

text: d

text: has

text: an

text: MA-proof

text: system

text: where:

text: Merlin

text: sends

text: a

text: proof

text: of

text: O(K

text: d

text: log(Kqd/ε))

text: bits,

text: and

text: Arthur

text: tosses

text: at

text: most

text: log(K·qd/ε){

text: coin}s,

text: ou·tputs

text: incorrectly

text: with

text: probability

text: at

text: most

text: ε,

text: and

text: runs

text: in

text: time

text: (K

text: max

text: d,n

text: s

text: poly(logs))

text: poly(log(Kqd/ε)).

text: We

text: have

text: stated

text: the

text: theorem

text: at

text: this

text: level

text: of

text: generality

text: because

text: we

text: need

text: good

text: bounds

text: on

text: the

text: parame-

text: ters

text: to

text: obtain

text: certain

text: consequences.

text: For

text: example,

text: in

text: our

text: proof

text: system

text: for

text: quantiﬁed

text: Boolean

text: formulas

text: (Theorem

text: the

text: parameters

text: s,

text: K,

text: q,

text: and

text: d

text: are

text: all

text: various

text: exponentials

text: in

text: n.

text: Because

text: instances

text: of

text: MULTIPOINT

text: CIRCUIT

text: EVALUATION

text: have

text: length

text: O((K

text: n

text: slogs)

text: logq),

text: the

text: running

text: time

text: of

text: Theorem

text: is

text: essentially

text: linear

text: in

text: the

text: input

text: length,

text: up

text: to

text: the

text: factor

text: of

text: d

text: in

text: Merlin’s

text: proof

text: (in

text: general,

text: d

text: could

text: be

text: much

text: larger

text: than

text: n).

text: So

text: Theorem

text: is

text: extremely

text: powerful

text: for

text: arithmetic

text: circuits

text: of

text: low

text: degree.

text: ##LTLine##

text: ##LTLine##


text: In the kth round of interaction for k = 2,...,c, the honest prover sends the 2n/(c+1)+o(n)-degree polyno- mial Qk(y) := ∑ P(Ψ1(r1),...,Ψ n (rk−1),Ψ1(y),...,Ψ n (y),Ψ1( jc+1)...,Ψ n ( jc+1)). c+1 c+1 c+1 ∈{ jk+1,...,jc+1 − } 0,1,...,2n/(c+1) 1 ∈ F The veriﬁer again chooses a random r . k p ∈ F Finally in the cth round, after the prover has sendt Qc(y) and the veriﬁer has chosen rc p at random, − the remaining computation is to compute the sum ∑2i=n/0(c+1) 1 Qc( ji), and to verify that ∑ P(Ψ1(r1),...,Ψ n (r1),...,Ψ1(rc),...,Ψ n (rc),Ψ1( jc+1),...,Ψ n ( jc+1)) = Qc(rc). c+1 c+1 c+1 j ∈{ c+1 − } 0,1,...,2n/(c+1) 1 ≤ − In each of the c rounds, the chance of picking a “bad” ri is at most 2c+n1+o(n)/p exp( Ω(n)). Thus, with ω(1) rounds of interaction, Arthur and Merlin can compute #SAT in 2o(n) veriﬁcation time and communication. 3.2 Univariate Polynomial Identity Testing and the Nondeterministic SETH A nice aspect of Theorem 3.2 and its corollaries is that the randomness is low: for example, the obvious derandomization strategy of simulating all O(cid:63)(2n/2) coin tosses recovers a nondeterministic O(cid:63)(2n) time algorithm for counting SAT assignments modulo 2. The proof system itself motivates the following problem. Let univariate polynomial identity testing (UPIT) be the problem of testing identity for two arithmetic circuits with one variable, degree n, and O(n) wires, over a ﬁeld of order poly(n). The following corollary is immediate from the proofs of Theo- rems 1.1, 3.3, and the above observations: ∈ − Corollary 3.1. If UPIT NTIME[n2 ε] for some ε > 0, then #Circuit-SAT for o(n)-depth circuits is − computable in nondeterministic 2n(1 ε/2)+o(n) time. By [Wil11, JMV15, CGI+15], this further implies that ENP does not have 2o(n)-size sublinear-depth circuits. In particular, the randomized veriﬁcation task of Arthur in the protocol of Theorem 3.3 directly reduces to solving UPIT on two univariate circuits of degree 2n/2+o(n) and size 2n/2+o(n). Hence, assuming the − hypothesis of Corollary 3.1, Arthur’s veriﬁcation can be performed deterministically in 2n(1 ε/2)+o(n) time. This is an intriguing example of how derandomization within polynomial time can imply strong circuit lower bounds: it is easy to see that UPIT is solvable in O˜(n) time with randomness, and in O˜(n2) time deterministically, by efﬁcient interpolation on n+1 distinct points (Theorem 2.2). In all other cases we are aware of (such as [KI04, Wil10]), the necessary derandomization problem is only known to be solvable in deterministic exponential time. Thus, the Nondeterministic SETH predicts that the exponent of the simple O˜(n2) algorithm for UPIT cannot be improved, even with nondeterminism. 4 Quantiﬁed Boolean Formulas 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

text: the

text: kth

text: round

text: of

text: interaction

text: for

text: k

text: the

text: honest

text: prover

text: sends

text: the

text: polyno-

text: mial

text: Qk(y)

font: HYQPUW+StandardSymL

text: n

text: n

text: n

text: F

text: The

text: veriﬁer

text: again

text: chooses

text: a

text: random

text: r

text: k

text: p

text: F

text: Finally

text: in

text: the

text: cth

text: round,

text: after

text: the

text: prover

text: has

text: sendt

text: Qc(y)

text: and

text: the

text: veriﬁer

text: has

text: chosen

text: rc

text: p

text: at

text: random,

text: the

text: remaining

text: computation

text: is

text: to

text: compute

text: the

text: sum

text: Qc(

text: ji),

text: and

text: to

text: verify

text: that

text: n

text: n

text: n

text: Qc(rc).

text: j

text: In

text: each

text: of

text: the

text: c

text: rounds,

text: the

text: chance

text: of

text: picking

text: a

text: “bad”

text: ri

text: is

text: at

text: most

text: exp(

text: Ω(n)).

text: Thus,

text: with

text: rounds

text: of

text: interaction,

text: Arthur

text: and

text: Merlin

text: can

text: compute

text: #SAT

text: in

text: veriﬁcation

text: time

text: and

text: communication.

text: Univariate

text: Polynomial

text: Identity

text: Testing

text: and

text: the

text: Nondeterministic

text: SETH

text: A

text: nice

text: aspect

text: of

text: Theorem

text: and

text: its

text: corollaries

text: is

text: that

text: the

text: randomness

text: is

text: low:

text: for

text: example,

text: the

text: obvious

text: derandomization

text: strategy

text: of

text: simulating

text: all

text: coin

text: tosses

text: recovers

text: a

text: nondeterministic

text: time

text: algorithm

text: for

text: counting

text: SAT

text: assignments

text: modulo

text: The

text: proof

text: system

text: itself

text: motivates

text: the

text: following

text: problem.

text: Let

text: univariate

text: polynomial

text: identity

text: testing

text: (UPIT)

text: be

text: the

text: problem

text: of

text: testing

text: identity

text: for

text: two

text: arithmetic

text: circuits

text: with

text: one

text: variable,

text: degree

text: n,

text: and

text: O(n)

text: wires,

text: over

text: a

text: ﬁeld

text: of

text: order

text: poly(n).

text: The

text: following

text: corollary

text: is

text: immediate

text: from

text: the

text: proofs

text: of

text: Theo-

text: rems

text: and

text: the

text: above

text: observations:

text: Corollary

text: If

text: UPIT

text: ε]

text: for

text: some

text: ε

text: then

text: #Circuit-SAT

text: for

text: o(n)-depth

text: circuits

text: is

text: computable

text: in

text: nondeterministic

text: time.

text: By

text: this

text: further

text: implies

text: that

text: ENP

text: does

text: not

text: have

text: sublinear-depth

text: circuits.

text: In

text: particular,

text: the

text: randomized

text: veriﬁcation

text: task

text: of

text: Arthur

text: in

text: the

text: protocol

text: of

text: Theorem

text: directly

text: reduces

text: to

text: solving

text: UPIT

text: on

text: two

text: univariate

text: circuits

text: of

text: degree

text: and

text: size

text: Hence,

text: assuming

text: the

text: hypothesis

text: of

text: Corollary

text: Arthur’s

text: veriﬁcation

text: can

text: be

text: performed

text: deterministically

text: in

text: time.

text: This

text: is

text: an

text: intriguing

text: example

text: of

text: how

text: derandomization

text: within

text: polynomial

text: time

text: can

text: imply

text: strong

text: circuit

text: lower

text: bounds:

text: it

text: is

text: easy

text: to

text: see

text: that

text: UPIT

text: is

text: solvable

text: in

text: O˜(n)

text: time

text: with

text: randomness,

text: and

text: in

text: time

text: deterministically,

text: by

text: efﬁcient

text: interpolation

text: on

text: distinct

text: points

text: (Theorem

text: In

text: all

text: other

text: cases

text: we

text: are

text: aware

text: of

text: (such

text: as

text: the

text: necessary

text: derandomization

text: problem

text: is

text: only

text: known

text: to

text: be

text: solvable

text: in

text: deterministic

text: exponential

text: time.

text: Thus,

text: the

text: Nondeterministic

text: SETH

text: predicts

text: that

text: the

text: exponent

text: of

text: the

text: simple

text: algorithm

text: for

text: UPIT

text: cannot

text: be

text: improved,

text: even

text: with

text: nondeterminism.

text: Quantiﬁed

text: Boolean

text: Formulas

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 Deﬁnition 3: A rate allocation mechanism is said to be memoryless if the service rate of each user at time t is 𝑓𝑙𝑜𝑤 𝑎 independent of its service rate for all times prior to t, [3]. Deﬁnition 4: (CM4 fairness) A rate allocation matrix R(t) is said to satisfy Constrained Multi-user Multi-server Max- 𝑓𝑙𝑜𝑤 𝑏 𝑆1 1.8 𝑀𝑏𝑝𝑠 Min (CM4) fairness, when increasing the normalized service rate of any user results in decreasing the normalized service rate of some other user(s) with smaller or equal normalized 𝑓𝑙𝑜𝑤 𝑐 𝑆 1.8 𝑀𝑏𝑝𝑠 2 service rate. It should be mentioned that CM4 fairness is just the general deﬁnition of max-min fairness which has been applied 𝑓𝑙𝑜𝑤 𝑑 𝑆 1 𝑀𝑏𝑝𝑠 3 to the rate allocation in the constrained multi-user multi- server queuing system [1], [9]. Since CM4 fair rate allocation is deﬁned in terms of instantaneous rates, it is obviously Fig. 3. A sample queuing system, the weight of all users are assumed to { } { } memoryless. b{e e}qu{al. }Here there are two clusters, C1 = ( a,b,c , s1,s2 ), C2 = ( d , s3 ). The normalized service rate of users in cluster 1 is rC1 = 1.2 Theorem 1: Consider a ﬂuid ﬂow system conforming to Mbps, and the normalized service rate of user d in cluster 2 is rC2 = 1 the system model of Fig. 2. A rate allocation matrix R(t) Mbps. satisﬁes CM4 fairness, if and only if statement (1) holds for any backlogged user j: { } clustering consisting of M clusters is deﬁned as C (t) M , m m=1 where C (t) = (I (t), S (t)). if r (t) > 0 and π = 1 ⇒ rj(t) ≥ ri(t) (1) Deﬁnitmion 5: Supmpose thme rate allocation matrix R(t) satis- i,k j,k φj φi ﬁes CM4 fairness when B(t) = B is the set of backlogged { } users at time t. Clustering C (t) M is said to be a Fairness m m=1 Theorem 1, that is a concise statement of Theorem 2 in Oriented Clustering (FOC) in conjunction with B if: ∈ [9], offers a necessary and sufﬁcient condition on R(t) to be a) For any cluster C (t), any user i I (t) has the same m m fair in conjunction with CM4 fairness deﬁnition. Since the normalized service rate, ri(t)/ϕi = rCm(t). We deﬁne proof follows the same line of arguments as what provided for rCm(t) as the cluster service rate. (cid:54) Theorem 2 in [9], it is omitted here. To get a better intuition b) For any two distinct clusters C (t) and C (t), m = l, (cid:54) m l on the concept of CM4 fairness, we consider an example. rCm(t) = rCl(t). Example 2.1: Consider the queuing system shown in Fig. 3. c) Server k belongs to cluster Cm(t) with∈rCm(t) > 0, if Assume trafﬁc stream of users to be ﬂuid ﬂow. According to and only if r (t) > 0 for some user i I (t). i,k m CM4 fairness deﬁnition, the fair service rate of user d equals Lemma 1: For given set of backlogged users and service to 1 Mbps, while the fair service rate of users a, b and c rates of servers, there exists a unique FOC. equals to 1.2 Mbps. To achieve such fair service rates, the For a given set of backlogged users at time t, B(t) = B, total service rate of server 3 should be allocated to user d. it may be possible that more than one rate allocation matrices However, there exist several ways for allocating the service satisfying CM4 fairness exist. However, Lemma 1 implies rate of server 1 and server 2 to users a, b and c, while each that all of them result in the same unique FOC. The proof of of them gets a service rate of 1.2 Mbps. Lemma 1 can be found in the appendix. A similar clustering The reader can verify that in this example there are more method referred to as rate clustering is presented in [9]. In- than one rate allocation matrices satisfying CM4 fairness. deed, the FOC satisﬁes the rate clustering property. However, However, all of such matrices share some common properties. the rate clustering does not result in a unique clustering for a First, in all cases users a, b and c get the same service rate. given set of backlogged users. Second, in all of them users with higher service rate are not Fact 1: For given service rate of servers, FOC at time t 0 given service on the server allocated to the user with lower could be viewed as a function of the set of backlogged users service rate. Based on these observations, the set of users and B = B(t ), i.e., {C (B )}M . servers could be classiﬁed into two clusters (Fig. 3), satisfying 0 0 m 0 m=1 It is straightforward to show that for any two clusters m, l the following conditions. I) Users in each cluster have the such that rCm > rCl, any user i in Cl can not be eligible for same fair service rates. II) All users in a cluster are only getting service from servers k in cluster C (i.e., π = 0). m i,k given service by servers in that cluster. In the following, the This is formally stated in the following corollary. concept of clustering is deﬁned in general for any arbitrary { } Corollary 1: Let C (t) M be the FOC which corre- conﬁguration of users and servers. m m=1 sponds to the set of backlogged users B = B(t). If some user ∈ A cluster C(t) is deﬁned as an ordered pair C(t) = j I (t) with positive service rate is eligible to get service m ∈ (I(t), S(t)), where I(t) and S(t) are non-empty subset of from server k in another cluster, π = 1 and k S (t), then j,k l users and non-empty subset of servers, respectively. Sup- it follows that rCm(t) > rCl(t). pose that all users are partitioned into M disjoint subsets ≤ ≤ I (t), 1 m M and all servers are partitioned into the m ≤ ≤ III. FQ ALGORITHM same number M disjoint subsets S (t), 1 m M. A m ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Deﬁnition

text: A

text: rate

text: allocation

text: mechanism

text: is

text: said

text: to

text: be

text: memoryless

text: if

text: the

text: service

text: rate

text: of

text: each

text: user

text: at

text: time

text: t

text: is

text: 𝑓𝑙𝑜𝑤

text: 𝑎

text: independent

text: of

text: its

text: service

text: rate

text: for

text: all

text: times

text: prior

text: to

text: t,

text: Deﬁnition

text: fairness)

text: A

text: rate

text: allocation

text: matrix

text: R(t)

text: is

text: said

text: to

text: satisfy

text: Constrained

text: Multi-user

text: Multi-server

text: Max-

text: 𝑓𝑙𝑜𝑤

text: 𝑏

text: 𝑆1

text: 𝑀𝑏𝑝𝑠

text: Min

text: fairness,

text: when

text: increasing

text: the

text: normalized

text: service

text: rate

text: of

text: any

text: user

text: results

text: in

text: decreasing

text: the

text: normalized

text: service

text: rate

text: of

text: some

text: other

text: user(s)

text: with

text: smaller

text: or

text: equal

text: normalized

text: 𝑓𝑙𝑜𝑤

text: 𝑐

text: 𝑆

text: 𝑀𝑏𝑝𝑠

text: service

text: rate.

text: It

text: should

text: be

text: mentioned

text: that

text: fairness

text: is

text: just

text: the

text: general

text: deﬁnition

text: of

text: max-min

text: fairness

text: which

text: has

text: been

text: applied

text: 𝑓𝑙𝑜𝑤

text: 𝑑

text: 𝑆

text: 𝑀𝑏𝑝𝑠

text: to

text: the

text: rate

text: allocation

text: in

text: the

text: constrained

text: multi-user

text: multi-

text: server

text: queuing

text: system

text: Since

text: fair

text: rate

text: allocation

text: is

text: deﬁned

text: in

text: terms

text: of

text: instantaneous

text: rates,

text: it

text: is

text: obviously

text: Fig.

text: A

text: sample

text: queuing

text: system,

text: the

text: weight

text: of

text: all

text: users

text: are

text: assumed

text: to

text: memoryless.

text: b{e

text: e}qu{al.

text: }Here

text: there

text: are

text: two

text: clusters,

text: a,b,c

text: d

text: The

text: normalized

text: service

text: rate

text: of

text: users

text: in

text: cluster

text: is

text: Theorem

text: Consider

text: a

text: ﬂuid

text: ﬂow

text: system

text: conforming

text: to

text: Mbps,

text: and

text: the

text: normalized

text: service

text: rate

text: of

text: user

text: d

text: in

text: cluster

text: is

text: the

text: system

text: model

text: of

text: Fig.

text: A

text: rate

text: allocation

text: matrix

text: R(t)

text: Mbps.

text: satisﬁes

text: fairness,

text: if

text: and

text: only

text: if

text: statement

text: holds

text: for

text: any

text: backlogged

text: user

text: j:

text: clustering

text: consisting

text: of

text: M

text: clusters

text: is

text: deﬁned

text: as

text: C

text: (t)

text: M

text: m

text: where

text: C

text: (t)

text: (I

text: (t),

text: S

text: (t)).

text: if

text: r

text: (t)

text: and

text: π

text: rj(t)

text: ri(t)

text: Deﬁnitmion

text: Supmpose

text: thme

text: rate

text: allocation

text: matrix

text: R(t)

text: satis-

text: i,k

text: j,k

text: φj

text: φi

text: ﬁes

text: fairness

text: when

text: B(t)

text: B

text: is

text: the

text: set

text: of

text: backlogged

text: users

text: at

text: time

text: t.

text: Clustering

text: C

text: (t)

text: M

text: is

text: said

text: to

text: be

text: a

text: Fairness

text: m

text: Theorem

text: that

text: is

text: a

text: concise

text: statement

text: of

text: Theorem

text: in

text: Oriented

text: Clustering

text: (FOC)

text: in

text: conjunction

text: with

text: B

text: if:

text: offers

text: a

text: necessary

text: and

text: sufﬁcient

text: condition

text: on

text: R(t)

text: to

text: be

text: a)

text: For

text: any

text: cluster

text: C

text: (t),

text: any

text: user

text: i

text: I

text: (t)

text: has

text: the

text: same

text: m

text: m

text: fair

text: in

text: conjunction

text: with

text: fairness

text: deﬁnition.

text: Since

text: the

text: normalized

text: service

text: rate,

text: ri(t)/ϕi

text: rCm(t).

text: We

text: deﬁne

text: proof

text: follows

text: the

text: same

text: line

text: of

text: arguments

text: as

text: what

text: provided

text: for

text: rCm(t)

text: as

text: the

text: cluster

text: service

text: rate.

text: Theorem

text: in

text: it

text: is

text: omitted

text: here.

text: To

text: get

text: a

text: better

text: intuition

text: b)

text: For

text: any

text: two

text: distinct

text: clusters

text: C

text: (t)

text: and

text: C

text: (t),

text: m

text: l,

text: m

text: l

text: on

text: the

text: concept

text: of

text: fairness,

text: we

text: consider

text: an

text: example.

text: rCm(t)

text: rCl(t).

text: Example

text: Consider

text: the

text: queuing

text: system

text: shown

text: in

text: Fig.

text: c)

text: Server

text: k

text: belongs

text: to

text: cluster

text: Cm(t)

text: with∈rCm(t)

text: if

text: Assume

text: trafﬁc

text: stream

text: of

text: users

text: to

text: be

text: ﬂuid

text: ﬂow.

text: According

text: to

text: and

text: only

text: if

text: r

text: (t)

text: for

text: some

text: user

text: i

text: I

text: (t).

text: i,k

text: m

text: fairness

text: deﬁnition,

text: the

text: fair

text: service

text: rate

text: of

text: user

text: d

text: equals

text: Lemma

text: For

text: given

text: set

text: of

text: backlogged

text: users

text: and

text: service

text: to

text: Mbps,

text: while

text: the

text: fair

text: service

text: rate

text: of

text: users

text: a,

text: b

text: and

text: c

text: rates

text: of

text: servers,

text: there

text: exists

text: a

text: unique

text: FOC.

text: equals

text: to

text: Mbps.

text: To

text: achieve

text: such

text: fair

text: service

text: rates,

text: the

text: For

text: a

text: given

text: set

text: of

text: backlogged

text: users

text: at

text: time

text: t,

text: B(t)

text: B,

text: total

text: service

text: rate

text: of

text: server

text: should

text: be

text: allocated

text: to

text: user

text: d.

text: it

text: may

text: be

text: possible

text: that

text: more

text: than

text: one

text: rate

text: allocation

text: matrices

text: However,

text: there

text: exist

text: several

text: ways

text: for

text: allocating

text: the

text: service

text: satisfying

text: fairness

text: exist.

text: However,

text: Lemma

text: implies

text: rate

text: of

text: server

text: and

text: server

text: to

text: users

text: a,

text: b

text: and

text: c,

text: while

text: each

text: that

text: all

text: of

text: them

text: result

text: in

text: the

text: same

text: unique

text: FOC.

text: The

text: proof

text: of

text: of

text: them

text: gets

text: a

text: service

text: rate

text: of

text: Mbps.

text: Lemma

text: can

text: be

text: found

text: in

text: the

text: appendix.

text: A

text: similar

text: clustering

text: The

text: reader

text: can

text: verify

text: that

text: in

text: this

text: example

text: there

text: are

text: more

text: method

text: referred

text: to

text: as

text: rate

text: clustering

text: is

text: presented

text: in

text: In-

text: than

text: one

text: rate

text: allocation

text: matrices

text: satisfying

text: fairness.

text: deed,

text: the

text: FOC

text: satisﬁes

text: the

text: rate

text: clustering

text: property.

text: However,

text: However,

text: all

text: of

text: such

text: matrices

text: share

text: some

text: common

text: properties.

text: the

text: rate

text: clustering

text: does

text: not

text: result

text: in

text: a

text: unique

text: clustering

text: for

text: a

text: First,

text: in

text: all

text: cases

text: users

text: a,

text: b

text: and

text: c

text: get

text: the

text: same

text: service

text: rate.

text: given

text: set

text: of

text: backlogged

text: users.

text: Second,

text: in

text: all

text: of

text: them

text: users

text: with

text: higher

text: service

text: rate

text: are

text: not

text: Fact

text: For

text: given

text: service

text: rate

text: of

text: servers,

text: FOC

text: at

text: time

text: t

text: given

text: service

text: on

text: the

text: server

text: allocated

text: to

text: the

text: user

text: with

text: lower

text: could

text: be

text: viewed

text: as

text: a

text: function

text: of

text: the

text: set

text: of

text: backlogged

text: users

text: service

text: rate.

text: Based

text: on

text: these

text: observations,

text: the

text: set

text: of

text: users

text: and

text: B

text: B(t

text: i.e.,

text: {C

text: (B

text: )}M

text: servers

text: could

text: be

text: classiﬁed

text: into

text: two

text: clusters

text: (Fig.

text: satisfying

text: m

text: It

text: is

text: straightforward

text: to

text: show

text: that

text: for

text: any

text: two

text: clusters

text: m,

text: l

text: the

text: following

text: conditions.

text: I)

text: Users

text: in

text: each

text: cluster

text: have

text: the

text: such

text: that

text: rCm

text: rCl,

text: any

text: user

text: i

text: in

text: Cl

text: can

text: not

text: be

text: eligible

text: for

text: same

text: fair

text: service

text: rates.

text: II)

text: All

text: users

text: in

text: a

text: cluster

text: are

text: only

text: getting

text: service

text: from

text: servers

text: k

text: in

text: cluster

text: C

text: (i.e.,

text: π

text: m

text: i,k

text: given

text: service

text: by

text: servers

text: in

text: that

text: cluster.

text: In

text: the

text: following,

text: the

text: This

text: is

text: formally

text: stated

text: in

text: the

text: following

text: corollary.

text: concept

text: of

text: clustering

text: is

text: deﬁned

text: in

text: general

text: for

text: any

text: arbitrary

text: Corollary

text: Let

text: C

text: (t)

text: M

text: be

text: the

text: FOC

text: which

text: corre-

text: conﬁguration

text: of

text: users

text: and

text: servers.

text: m

text: sponds

text: to

text: the

text: set

text: of

text: backlogged

text: users

text: B

text: B(t).

text: If

text: some

text: user

text: A

text: cluster

text: C(t)

text: is

text: deﬁned

text: as

text: an

text: ordered

text: pair

text: C(t)

text: j

text: I

text: (t)

text: with

text: positive

text: service

text: rate

text: is

text: eligible

text: to

text: get

text: service

text: m

text: (I(t),

text: S(t)),

text: where

text: I(t)

text: and

text: S(t)

text: are

text: non-empty

text: subset

text: of

text: from

text: server

text: k

text: in

text: another

text: cluster,

text: π

text: and

text: k

text: S

text: (t),

text: then

text: j,k

text: l

text: users

text: and

text: non-empty

text: subset

text: of

text: servers,

text: respectively.

text: Sup-

text: it

text: follows

text: that

text: rCm(t)

text: rCl(t).

text: pose

text: that

text: all

text: users

text: are

text: partitioned

text: into

text: M

text: disjoint

text: subsets

text: I

text: (t),

text: m

text: M

text: and

text: all

text: servers

text: are

text: partitioned

text: into

text: the

text: m

text: III.

text: FQ

text: ALGORITHM

text: same

text: number

text: M

text: disjoint

text: subsets

text: S

text: (t),

text: m

text: M.

text: A

text: m

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: ##LTLine##

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text: ##LTLine##

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text: 5 TABLE I TABLE II THE PSEUDO CODE OF CM4FQ ALGORITHM THE SUBROUTINES OF CM4FQ ALGORITHM CM4FQ Algorithm • ActivateServers(i) • Initialization V 0 = 0; ≤ { | ∞} (cid:54) ∅ For (i = 1; i N; i + +) If ( k V k < = ) { | ∞} Fi = Di = 0; V 0 = max V k V k < + δ; (7) ≤ ≤ ∞ For (k = 1; k K; k + +) For (l = 1; l K & π = 1 & V l = ; l + +) i,l V k = Dk = 0; V l = V 0; (8) • En-queuing Module: On arrival of packet p of user i • UpdateV(i,k) If (Empty(Queuei)) //updating work level of servers Activate-Servers(i) Vˆk = V k; { { | }} ≤ Fi = max Fi,max V k πi,k = 1 ; (4) For (l = 1; l K & πi,l = 1; l + +) ∞ Enqueue(i,p); V l = ; (9) //Enqueue packet p to queue of user i and update B If (Bl (cid:54)= ∅) { | ∈ } V l = min Fj j Bl (10) • De-queuing Module(server k) //regulating the gap among work level of servers dk = 0; While(Bk (cid:54)= ∅) If ({l | V l < Vˆk} (cid:54)= ∅) i∗ ← argmini{Fi | i ∈ Bk}; (5) dk = min{V l | Vˆk ≤ V l} p = Dequeue(Head(Queuei∗)); −max{V l | V l < Vˆk} − δ; (11) //Dequeue head of queue of user i and update B If (dk > 0) Lp = Size(p); For (j = 1; j ≤ N & Fj ≥ Vˆk; j + +) Fi∗ = Fi∗ +∗ Lp/ϕi∗; (6) Fj = Fj − dk; (12) UpdateV (i ,k); Service(k,p); //Give service to packet p on server k Dj = Dj≤+ dk; ≥ (13) For (l = 1; l K & V l Vˆk; l + +) − V l = V l dk; (14) Dl = Dl + dk; (15) greater than or equal to V k, for any server k that is eligible to give service to user i, ((4) in Table I). To provide a better understanding of the algorithm, we con- sider some illustrative examples. In these examples, we assume served that all existing packets in the queue of users a and that UpdateV (i∗, k) subroutine is restricted to updates in (9) b at t = 0 are given service during interval [0, t0), where and (10) and other updates in it are not implemented. The t0 = 10 msec. In fact, the last existing packet in the queue of resultant algorithm will be referred to as reduced CM4FQ user b is chosen by server 1 at 9.6 msec. Hence, Fa and Fb algorithm. remain constant during (9.6, 20) msec. On the other hand, Fc The ﬁrst example describes the role of ActivateServers(·) increases with an average rate of 1 Mbps. Hence, Fc is greater subroutine. The second example examines the role of (4) than Fa and Fb at time t1 (as shown in Fig. 4). by which Fi is updated for a returning customer i. The At time 9.6 msec that the set of backlogged u∞sers connected third example, reveals the shortage of the reduced CM4fQ to server 1, B1 becomes empty, V·1 is set to (as stated in algorithms to achieve fairness and shows the necessity for 9). The role of ActivateServers( ) subroutine is to reset the additional updates (i.e. (11)-(15)) in UpdateV (i, k) subroutine work level of this server to a proper ﬁnite value as soon as of Table II. it becomes backlogged. Given that user a and user b become Example 3.1: Consider again the queuing system shown in backlogged at t+, V 1 will be updated to V 2 + δ ((8) and (8) 1 · Fig. 1. Assume that input trafﬁc stream of users comprises of in ActivateServers( ) subroutine), where δ is assumed to be packets of ﬁxed length L = 1000 bits. User c is assumed to δ = 2L. be continuously backlogged since t = 0. It is assumed that 13 It can be observed that by updating V 1 in this way and by packets already exist in the queue of each user a and b at time choosing δ = 2L, no packet of user a or user b is serviced on t = 0. It is assumed that no packet arrives at these queues server 2 from time t1 onwards (Fig. 4). In contrast, if such an during [0, t ), till t = 20 msec. Users a and b are assumed to update were not performed or if δ were chosen equal to Lmax 1 1 become backlogged at t+ and remains backlogged afterwards. (or some smaller amount), it would be possible for users a 1 Applying the reduced CM4FQ algorithm, it can be ob- and b to have some packets serviced on server 2. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TABLE

text: I

text: TABLE

text: II

text: THE

label: caption

text: PSEUDO

text: CODE

text: OF

text: ALGORITHM

text: THE

text: SUBROUTINES

text: OF

text: ALGORITHM

label: table

text: Algorithm

label: list

text: ActivateServers(i)

text: Initialization

text: V

label: equation

text: For

text: (i

text: i

text: N;

text: i

text: If

text: k

text: V

text: k

text: Fi

text: Di

text: V

text: max

text: V

text: k

text: V

text: k

text: δ;

text: For

text: (k

text: k

text: K;

text: k

text: For

text: (l

text: l

text: K

text: π

text: V

text: l

text: l

text: i,l

text: V

text: k

text: Dk

text: V

text: l

text: V

text: En-queuing

text: Module:

text: On

text: arrival

text: of

text: packet

text: p

text: of

text: user

text: i

text: UpdateV(i,k)

text: If

text: (Empty(Queuei))

text: //updating

text: work

text: level

text: of

text: servers

text: Activate-Servers(i)

text: Vˆk

text: V

text: k;

text: Fi

text: max

text: Fi,max

text: V

text: k

text: πi,k

text: For

text: (l

text: l

text: K

text: πi,l

text: l

text: Enqueue(i,p);

text: V

text: l

text: //Enqueue

text: packet

text: p

text: to

text: queue

text: of

text: user

text: i

text: and

text: update

text: B

text: If

text: (Bl

text: V

text: l

text: min

text: Fj

text: j

text: Bl

text: De-queuing

text: Module(server

text: k)

text: //regulating

text: the

text: gap

text: among

text: work

text: level

text: of

text: servers

text: dk

text: While(Bk

text: If

text: ({l

text: V

text: l

text: Vˆk}

text: i∗

text: argmini{Fi

text: i

text: Bk};

text: dk

text: min{V

text: l

text: Vˆk

text: V

text: l}

text: p

text: Dequeue(Head(Queuei∗));

text: −max{V

text: l

text: V

text: l

text: Vˆk}

text: δ;

text: //Dequeue

text: head

text: of

text: queue

text: of

text: user

text: i

text: and

text: update

text: B

text: If

text: (dk

text: Lp

text: Size(p);

text: For

text: (j

text: j

text: N

text: Fj

text: Vˆk;

text: j

text: Fi∗

text: Fi∗

text: Lp/ϕi∗;

text: Fj

text: Fj

text: dk;

text: UpdateV

text: (i

text: ,k);

text: Service(k,p);

text: //Give

text: service

text: to

text: packet

text: p

text: on

text: server

text: k

text: Dj

text: Dj≤+

text: dk;

text: For

text: (l

text: l

text: K

text: V

text: l

text: Vˆk;

text: l

text: V

text: l

text: V

text: l

text: dk;

text: Dl

text: Dl

text: dk;

text: greater

text: than

text: or

text: equal

text: to

text: V

text: k,

text: for

text: any

text: server

text: k

text: that

text: is

text: eligible

text: to

text: give

text: service

text: to

text: user

text: i,

text: in

text: Table

text: I).

text: To

text: provide

text: a

text: better

text: understanding

text: of

text: the

text: algorithm,

text: we

text: con-

text: sider

text: some

text: illustrative

text: examples.

text: In

text: these

text: examples,

text: we

text: assume

text: served

text: that

text: all

text: existing

text: packets

text: in

text: the

text: queue

text: of

text: users

text: a

text: and

text: that

text: UpdateV

text: (i∗,

text: k)

text: subroutine

text: is

text: restricted

text: to

text: updates

text: in

text: b

text: at

text: t

text: are

text: given

text: service

text: during

text: interval

text: where

text: and

text: and

text: other

text: updates

text: in

text: it

text: are

text: not

text: implemented.

text: The

text: msec.

text: In

text: fact,

text: the

text: last

text: existing

text: packet

text: in

text: the

text: queue

text: of

text: resultant

text: algorithm

text: will

text: be

text: referred

text: to

text: as

text: reduced

text: user

text: b

text: is

text: chosen

text: by

text: server

text: at

text: msec.

text: Hence,

text: Fa

text: and

text: Fb

text: algorithm.

text: remain

text: constant

text: during

text: msec.

text: On

text: the

text: other

text: hand,

text: Fc

text: The

text: ﬁrst

text: example

text: describes

text: the

text: role

text: of

text: ActivateServers(·)

text: increases

text: with

text: an

text: average

text: rate

text: of

text: Mbps.

text: Hence,

text: Fc

text: is

text: greater

text: subroutine.

text: The

text: second

text: example

text: examines

text: the

text: role

text: of

text: than

text: Fa

text: and

text: Fb

text: at

text: time

text: (as

text: shown

text: in

text: Fig.

text: by

text: which

text: Fi

text: is

text: updated

text: for

text: a

text: returning

text: customer

text: i.

text: The

text: At

text: time

text: msec

text: that

text: the

text: set

text: of

text: backlogged

text: u∞sers

text: connected

text: third

text: example,

text: reveals

text: the

text: shortage

text: of

text: the

text: reduced

text: to

text: server

text: becomes

text: empty,

text: is

text: set

text: to

text: (as

text: stated

text: in

text: algorithms

text: to

text: achieve

text: fairness

text: and

text: shows

text: the

text: necessity

text: for

text: The

text: role

text: of

text: ActivateServers(

text: subroutine

text: is

text: to

text: reset

text: the

text: additional

text: updates

text: (i.e.

text: in

text: UpdateV

text: (i,

text: k)

text: subroutine

text: work

text: level

text: of

text: this

text: server

text: to

text: a

text: proper

text: ﬁnite

text: value

text: as

text: soon

text: as

text: of

text: Table

text: II.

text: it

text: becomes

text: backlogged.

text: Given

text: that

text: user

text: a

text: and

text: user

text: b

text: become

text: Example

text: Consider

text: again

text: the

text: queuing

text: system

text: shown

text: in

text: backlogged

text: at

text: t+,

text: V

text: will

text: be

text: updated

text: to

text: V

text: δ

text: and

text: Fig.

text: Assume

text: that

text: input

text: trafﬁc

text: stream

text: of

text: users

text: comprises

text: of

text: in

text: ActivateServers(

text: subroutine),

text: where

text: δ

text: is

text: assumed

text: to

text: be

text: packets

text: of

text: ﬁxed

text: length

text: L

text: bits.

text: User

text: c

text: is

text: assumed

text: to

text: δ

text: be

text: continuously

text: backlogged

text: since

text: t

text: It

text: is

text: assumed

text: that

text: It

text: can

text: be

text: observed

text: that

text: by

text: updating

text: V

text: in

text: this

text: way

text: and

text: by

text: packets

text: already

text: exist

text: in

text: the

text: queue

text: of

text: each

text: user

text: a

text: and

text: b

text: at

text: time

text: choosing

text: δ

text: no

text: packet

text: of

text: user

text: a

text: or

text: user

text: b

text: is

text: serviced

text: on

text: t

text: It

text: is

text: assumed

text: that

text: no

text: packet

text: arrives

text: at

text: these

text: queues

text: server

text: from

text: time

text: onwards

text: (Fig.

text: In

text: contrast,

text: if

text: such

text: an

text: during

text: t

text: till

text: t

text: msec.

text: Users

text: a

text: and

text: b

text: are

text: assumed

text: to

text: update

text: were

text: not

text: performed

text: or

text: if

text: δ

text: were

text: chosen

text: equal

text: to

text: Lmax

text: become

text: backlogged

text: at

text: t+

text: and

text: remains

text: backlogged

text: afterwards.

text: (or

text: some

text: smaller

text: amount),

text: it

text: would

text: be

text: possible

text: for

text: users

text: a

text: Applying

text: the

text: reduced

text: algorithm,

text: it

text: can

text: be

text: ob-

text: and

text: b

text: to

text: have

text: some

text: packets

text: serviced

text: on

text: server

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 9 Deﬁnition 10: Assume that the set of backlogged users at Theorem 3: Consider the same set of conditions as in The- ∈ ∈ time t is B = B(t ). Consider an arbitrary cluster C = orem 2. Given that server k S becomes free at t (t , t ), 0 0 0 { } 0 1 (I, S) of the FOC which corresponds to B , C (B ) M . it follows that: 0 m 0 m=1 Minimum work level function (V C(t )), and maximum work 0 − ≤ level function (Vˆ C(t0)) corresponding to cluster C are deﬁned V k(t+) V C(t) (KC + 1)λC. (26) as: (cid:44) { | ∈ } For the proof, refer to the appendix. Intuitively, Theorem V C(t0) min V k(t0) k S (20) 3 implies that the maximum difference among work level of (cid:44) { | ∈ } Vˆ C(t0) max V k(t0) k S (21) servers in cluster C is limited to (KC + 1)λC. The following lemma is built upon this theorem. ∀ ∈ An important question is that how far can Vˆ C(t) be from Lemma 4: For Dk(τ0, τ1) = Dl(τ0, τ1), k, l S, to be established under conditions of Theorem 2, it sufﬁces to V C(t) for an arbitrary cluster C. To ﬁnd an answer to ≥ this question, we will consider a special situation. Speciﬁ- choose δ (KC + 1)λC. ∈ cally, consider an arbitrary interval [t , t ) during which the Proof: Consider any time t (t0, t1) at which some user set of backlogged users does not ch0ang1e, i.e., B˜(t , t ) = (server) in clust∈er C receives bonus. Speciﬁcally,∈Consider an Bthe(t0F,Ot1C) =corBre.spCoonndsiindgertoanthaerbistertaroyfcbluasctkelrogCge=d (u0Is,eSr1s) Bof. afrrebei.trAartytthiimseintstan(tt,∗0s,ot1m)eatuwsehrici∗h wsoimll ebesecrvheorsekn acScobredcionmg etos Assume that an initial condition, V k(t ) = V C(t ) holds for (5) and UpdateV (i , k) subroutine will be executed. all servers k ∈ S. Furthermore, assum0e that in th0e execution According to (26), unless server k has the m∈inimum work of the algorithm during interval [t , t ) cluster C is isolated level at −time t, t≤here exists at least one server l S such that from the rest of network, i.e., duri0ng1this interval no packet V k(t+) V l(t) (KC +1)λC. Hence, dk in (11) will be less of users i ∈/ I is chosen for service by servers k ∈ S, and no than or equal to 0, provided that V k(t) > V C(t). In this case packet of users i ∈ I is chosen for service by servers k ∈/ S. no user (server) in cluster C receives bonus. dk is positive only The following results are derived under such assumptions. when V k(t) = V C(t). In this case, all users (servers) in cluster Theorem 2: Consider an arbitrary interval [t , t ), for which C receive the same amount of bonus (because Fi(t) and V l(t) B˜(t , t ) = B(t , t ) = B. Consider an arbitr0ary1cluster C = is greater than or equal to V C(t) for all users and⊆all servers (I, S0) o1f the FO0C w1hich corresponds to the set of backlogged in cluster C). Hence, for any subinter∀val [τ∈0, τ1) [t0, t1) it users B. Assume that in the execution of the algorithm during follows that Dk(τ0, τ1) = Dl(τ0, τ1), k, l S. interval [t , t ) cluster C is isolated from the rest of network. Corollary 2: Consider an interval [t0, t1) for which Let V k(t 0) =1 V C(t ), for all servers k ∈ S. B˜(t0, t1) = B(t0, t1) = B. Consider a special case that Assum0e that δ is 0chosen arbitrarily large that Dk(τ , τ ) = the FOC corresponding to B include one cluster C = (I, S) Dl(τ , τ ) for any servers k, l ∈ S and for any sub0inte1rval which consists of all backlogged users and all K servers. ∈Let [τ , τ0) 1of the interval [t , t ). Given that server k ∈ S V k(t0) = V C(t0), for all servers. In order for any user i I be0com1 es free at t ∈ (t , t )0, it1 can be shown that: to get the same amo≥unt of bonus during interval [t0, t1), it 0 1 sufﬁces to choose δ (K + 1)λ , where λ is deﬁned as: 0 0 ≤ − wk(t0, t) rC(t t0) + KCλC (22) λ (cid:44) Lmax . (27) 0 min∀ ϕ i i where, wk(t , t) is deﬁned as: ≥ 0 By choosing δ (K + 1)λ , we can make sure that under 0 conditions of Theorem 2 and for any possible conﬁguration of (cid:44) wk(t0, t) V k(t0, t) + Dk(t0, t), (23) the FOC and in any cluster C = (I, S), all users receive the same amount of bonus. KC is the number of servers in cluster C, and λC is deﬁned as: C. Choosing the Control Parameter δ In this section, we discuss how to choose the parameter δ. λC (cid:44) Lmax . (24) As it stems from Example 3.3, it is desired to choose δ as min ∈ ϕ small as possible. On the other hand, according to Corollary i I i ≥ ∈ ∈ 2 it is desired to choose δ (K + 1)λ0. Hence, we propose Furthermore, for server k S and for any t (t0, t1), it to choose δ = (K + 1)λ0. can be shown that: Proposition: The control parameter, δ in CM4FQ algo- rithm could be chosen as: ≥ − − wk(t , t) rC(t t ) KCλC. (25) 0 0 δ = (K + 1)λ (28) 0 For the proof, refer to the appendix. ##LTLine## ##LTLine##

text: Deﬁnition

text: Assume

text: that

text: the

text: set

text: of

text: backlogged

text: users

text: at

text: Theorem

text: Consider

text: the

text: same

text: set

text: of

text: conditions

text: as

text: in

text: The-

text: time

text: t

text: is

text: B

text: B(t

text: Consider

text: an

text: arbitrary

text: cluster

text: C

text: orem

text: Given

text: that

text: server

text: k

text: S

text: becomes

text: free

text: at

text: t

text: (t

text: t

text: (I,

text: S)

text: of

text: the

text: FOC

text: which

text: corresponds

text: to

text: B

text: C

text: (B

text: M

text: it

text: follows

text: that:

text: m

text: Minimum

text: work

text: level

text: function

text: (V

text: C(t

text: and

text: maximum

text: work

text: level

text: function

text: (Vˆ

text: corresponding

text: to

text: cluster

text: C

text: are

text: deﬁned

text: V

text: k(t+)

text: V

text: C(t)

text: (KC

text: as:

text: For

text: the

text: proof,

text: refer

text: to

text: the

text: appendix.

text: Intuitively,

text: Theorem

text: V

text: min

text: V

text: k

text: S

text: implies

text: that

text: the

text: maximum

text: difference

text: among

text: work

text: level

text: of

text: Vˆ

text: max

text: V

text: k

text: S

text: servers

text: in

text: cluster

text: C

text: is

text: limited

text: to

text: (KC

text: The

text: following

text: lemma

text: is

text: built

text: upon

text: this

text: theorem.

text: An

text: important

text: question

text: is

text: that

text: how

text: far

text: can

text: Vˆ

text: C(t)

text: be

text: from

text: Lemma

text: For

text: k,

text: l

text: S,

text: to

text: be

text: established

text: under

text: conditions

text: of

text: Theorem

text: it

text: sufﬁces

text: to

text: V

text: C(t)

text: for

text: an

text: arbitrary

text: cluster

text: C.

text: To

text: ﬁnd

text: an

text: answer

text: to

text: this

text: question,

text: we

text: will

text: consider

text: a

text: special

text: situation.

text: Speciﬁ-

text: choose

text: δ

text: (KC

text: cally,

text: consider

text: an

text: arbitrary

text: interval

text: [t

text: t

text: during

text: which

text: the

text: Proof:

text: Consider

text: any

text: time

text: t

text: at

text: which

text: some

text: user

text: set

text: of

text: backlogged

text: users

text: does

text: not

text: i.e.,

text: B˜(t

text: t

text: (server)

text: in

text: clust∈er

text: C

text: receives

text: bonus.

text: Speciﬁcally,∈Consider

text: an

text: Bof.

text: wsoimll

text: ebesecrvheorsekn

text: acScobredcionmg

text: etos

text: Assume

text: that

text: an

text: initial

text: condition,

text: V

text: k(t

text: V

text: C(t

text: holds

text: for

text: and

text: UpdateV

text: (i

text: k)

text: subroutine

text: will

text: be

text: executed.

text: all

text: servers

text: k

text: S.

text: Furthermore,

text: that

text: in

text: execution

text: According

text: to

text: unless

text: server

text: k

text: has

text: the

text: m∈inimum

text: work

text: of

text: the

text: algorithm

text: during

text: interval

text: [t

text: t

text: cluster

text: C

text: is

text: isolated

text: level

text: at

text: −time

text: t,

text: t≤here

text: exists

text: at

text: least

text: one

text: server

text: l

text: S

text: such

text: that

text: from

text: the

text: rest

text: of

text: network,

text: i.e.,

text: interval

text: no

text: packet

text: V

text: k(t+)

text: V

text: l(t)

text: (KC

text: Hence,

text: dk

text: in

text: will

text: be

text: less

text: of

text: users

text: i

text: I

text: is

text: chosen

text: for

text: service

text: by

text: servers

text: k

text: S,

text: and

text: no

text: than

text: or

text: equal

text: to

text: provided

text: that

text: V

text: k(t)

text: V

text: C(t).

text: In

text: this

text: case

text: packet

text: of

text: users

text: i

text: I

text: is

text: chosen

text: for

text: service

text: by

text: servers

text: k

text: S.

text: no

text: user

text: (server)

text: in

text: cluster

text: C

text: receives

text: bonus.

text: dk

text: is

text: positive

text: only

text: The

text: following

text: results

text: are

text: derived

text: under

text: such

text: assumptions.

text: when

text: V

text: k(t)

text: V

text: C(t).

text: In

text: this

text: case,

text: all

text: users

text: (servers)

text: in

text: cluster

text: Theorem

text: Consider

text: an

text: arbitrary

text: interval

text: [t

text: t

text: for

text: which

text: C

text: receive

text: the

text: same

text: amount

text: of

text: bonus

text: (because

text: Fi(t)

text: and

text: V

text: l(t)

text: B˜(t

text: t

text: B(t

text: t

text: B.

text: Consider

text: an

text: C

text: is

text: greater

text: than

text: or

text: equal

text: to

text: V

text: C(t)

text: for

text: all

text: users

text: and⊆all

text: servers

text: (I,

text: the

text: corresponds

text: to

text: the

text: set

text: of

text: backlogged

text: in

text: cluster

text: C).

text: Hence,

text: for

text: any

text: subinter∀val

text: it

text: users

text: B.

text: Assume

text: that

text: in

text: the

text: execution

text: of

text: the

text: algorithm

text: during

text: follows

text: that

text: k,

text: l

text: S.

text: interval

text: [t

text: t

text: cluster

text: C

text: is

text: isolated

text: from

text: the

text: rest

text: of

text: network.

text: Corollary

text: Consider

text: an

text: interval

text: for

text: which

text: Let

text: V

text: k(t

text: V

text: C(t

text: for

text: all

text: servers

text: k

text: S.

text: B.

text: Consider

text: a

text: special

text: case

text: that

text: that

text: δ

text: is

text: arbitrarily

text: large

text: that

text: Dk(τ

text: τ

text: the

text: FOC

text: corresponding

text: to

text: B

text: include

text: one

text: cluster

text: C

text: (I,

text: S)

text: Dl(τ

text: τ

text: for

text: any

text: servers

text: k,

text: l

text: S

text: and

text: for

text: any

text: which

text: consists

text: of

text: all

text: backlogged

text: users

text: and

text: all

text: K

text: servers.

text: ∈Let

text: [τ

text: the

text: interval

text: [t

text: t

text: Given

text: that

text: server

text: k

text: S

text: V

text: V

text: for

text: all

text: servers.

text: In

text: order

text: for

text: any

text: user

text: i

text: I

text: es

text: free

text: at

text: t

text: (t

text: t

text: can

text: be

text: shown

text: that:

text: to

text: get

text: the

text: same

text: amo≥unt

text: of

text: bonus

text: during

text: interval

text: it

text: sufﬁces

text: to

text: choose

text: δ

text: (K

text: where

text: λ

text: is

text: deﬁned

text: as:

text: t)

text: rC(t

text: KCλC

text: λ

text: Lmax

text: min∀

text: ϕ

text: i

text: i

text: where,

text: wk(t

text: t)

text: is

text: deﬁned

text: as:

text: By

text: choosing

text: δ

text: (K

text: we

text: can

text: make

text: sure

text: that

text: under

text: conditions

text: of

text: Theorem

text: and

text: for

text: any

text: possible

text: conﬁguration

text: of

text: t)

text: V

text: t)

text: t),

text: the

text: FOC

text: and

text: in

text: any

text: cluster

text: C

text: (I,

text: S),

text: all

text: users

text: receive

text: the

text: same

text: amount

text: of

text: bonus.

text: KC

text: is

text: the

text: number

text: of

text: servers

text: in

text: cluster

text: C,

text: and

text: λC

text: is

text: deﬁned

text: as:

text: C.

text: Choosing

text: the

text: Control

text: Parameter

text: δ

text: In

text: this

text: section,

text: we

text: discuss

text: how

text: to

text: choose

text: the

text: parameter

text: δ.

text: λC

text: Lmax

text: As

text: it

text: stems

text: from

text: Example

text: it

text: is

text: desired

text: to

text: choose

text: δ

text: as

text: min

text: ϕ

text: small

text: as

text: possible.

text: On

text: the

text: other

text: hand,

text: according

text: to

text: Corollary

text: i

text: I

text: i

text: it

text: is

text: desired

text: to

text: choose

text: δ

text: (K

text: Hence,

text: we

text: propose

text: Furthermore,

text: for

text: server

text: k

text: S

text: and

text: for

text: any

text: t

text: it

text: to

text: choose

text: δ

text: (K

text: can

text: be

text: shown

text: that:

text: Proposition:

text: The

text: control

text: parameter,

text: δ

text: in

text: algo-

text: rithm

text: could

text: be

text: chosen

text: as:

text: wk(t

text: t)

text: rC(t

text: t

text: KCλC.

text: δ

text: (K

text: For

text: the

text: proof,

text: refer

text: to

text: the

text: appendix.

text: ##LTLine##

text: ##LTLine##