text: − Similarly, we deﬁne the amount of disagreement l as the number of blocks they do not agree on: | | | | l− = TA + TB − 2l+. r − − The potential Φ is now simply deﬁned as Φ = l+ l . Proofs Corollary 4.3. Each iteration of Algorithm 1 without a hash collision or error increases the po- tential Φ by at least one. Proof. If TA = TB then both parties continue computing Π from the same place and, since no error happens, both parties correctly add the next r communications of Π to their transcripts. This increases l+ and therefore also the overall potential Φ by one. (cid:54) If TA = TB and no hash collision happens then both p|art|ies r|eal|ize this discrepancy and also learn the correct length of−the other party’s transcript. If TA = TB then both parties backtrack one block which reduces l by two and thus increases the potential by two. Otherwise, the party with the longer transcript backtracks one block while the other party does not change its transcript. − This reduces l by one and increases the overall potential Φ by one. Corollary 4.4. Each iterations of Algorithm 1, regardless of the number of hash collisions and errors, decreases the potential Φ by at most three. Proof. No matter what is received during an iteration a party never removes more than one block from its transcript. Similarly, at most −one block is added to TA and TB. Overall in one iteration this changes l+ by at most by one and l at most by two. The overall potential Φ changes therefore at most by three in any iteration. Next, we argue that the number of iterations of Algorithm 1 with a hash collision is negligible. (cid:54) To be precise, we say an iteration suﬀers a hash collision if TA = TB but either hashSB(TA) = hashSB(TB) or hashSA(TA) = hashSA(TB). In particular, we do not count iterations as suﬀering a hash collision if the random hash functions sampled would reveal a discrepancy but this detection, e.g., in Line 12, is prevented by corruptions in the transmission of a hash value or seed. To prove the number of hash collisions to be small we crucially exploit the fact that the randomness used for hashing is sampled afresh in every iteration. In particular, it is sampled after everything that is hashed in this iteration is already irrevocably ﬁxed. This independence allows to use the collision resistance property of Lemma 4.1 which shows that any iteration suﬀers from a hash collision with probability at most p = 1/n5. A union bound over all iterations then shows that with high probability no hash collision happens at all: − Corollary 4.5. With probability 1 1/n4 no iteration in Algorithm 1 suﬀers from a hash collision. While the success probability guaranteed by Corollary 4.5 is already quite nice it will be im- portant for subsequent algorithms to realize that one can easily prove stronger guarantees. In particular, using the independence between iterations in combination with a standard tail bound proves that the probability of having a number of hash collisions of the same order of magnitude − − as the number of errors is at least 1 2 Θ((cid:15)n): 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Similarly,

text: we

text: deﬁne

text: the

text: amount

text: of

text: disagreement

text: l

text: as

text: the

text: number

text: of

text: blocks

text: they

text: do

text: not

text: agree

text: on:

text: l−

text: TA

text: TB

text: r

text: The

text: potential

text: Φ

text: is

text: now

text: simply

text: deﬁned

text: as

text: Φ

text: l+

text: l

text: Proofs

text: Corollary

text: Each

text: iteration

text: of

text: Algorithm

text: without

text: a

text: hash

text: collision

text: or

text: error

text: increases

text: the

text: po-

text: tential

text: Φ

text: by

text: at

text: least

text: one.

text: Proof.

text: If

text: TA

text: TB

text: then

text: both

text: parties

text: continue

text: computing

text: Π

text: from

text: the

text: same

text: place

text: and,

text: since

text: no

text: error

text: happens,

text: both

text: parties

text: correctly

text: add

text: the

text: next

text: r

text: communications

text: of

text: Π

text: to

text: their

text: transcripts.

text: This

text: increases

text: l+

text: and

text: therefore

text: also

text: the

text: overall

text: potential

text: Φ

text: by

text: one.

text: If

text: TA

text: TB

text: and

text: no

text: hash

text: collision

text: happens

text: then

text: both

text: p|art|ies

text: r|eal|ize

text: this

text: discrepancy

text: and

text: also

text: learn

text: the

text: correct

text: length

text: of−the

text: other

text: party’s

text: transcript.

text: If

text: TA

text: TB

text: then

text: both

text: parties

text: backtrack

text: one

text: block

text: which

text: reduces

text: l

text: by

text: two

text: and

text: thus

text: increases

text: the

text: potential

text: by

text: two.

text: Otherwise,

text: the

text: party

text: with

text: the

text: longer

text: transcript

text: backtracks

text: one

text: block

text: while

text: the

text: other

text: party

text: does

text: not

text: change

text: its

text: transcript.

text: This

text: reduces

text: l

text: by

text: one

text: and

text: increases

text: the

text: overall

text: potential

text: Φ

text: by

text: one.

text: Corollary

text: Each

text: iterations

text: of

text: Algorithm

text: regardless

text: of

text: the

text: number

text: of

text: hash

text: collisions

text: and

text: errors,

text: decreases

text: the

text: potential

text: Φ

text: by

text: at

text: most

text: three.

text: Proof.

text: No

text: matter

text: what

text: is

text: received

text: during

text: an

text: iteration

text: a

text: party

text: never

text: removes

text: more

text: than

text: one

text: block

text: from

text: its

text: transcript.

text: Similarly,

text: at

text: most

text: −one

text: block

text: is

text: added

text: to

text: TA

text: and

text: TB.

text: Overall

text: in

text: one

text: iteration

text: this

text: changes

text: l+

text: by

text: at

text: most

text: by

text: one

text: and

text: l

text: at

text: most

text: by

text: two.

text: The

text: overall

text: potential

text: Φ

text: changes

text: therefore

text: at

text: most

text: by

text: three

text: in

text: any

text: iteration.

text: Next,

text: we

text: argue

text: that

text: the

text: number

text: of

text: iterations

text: of

text: Algorithm

text: with

text: a

text: hash

text: collision

text: is

text: negligible.

text: To

text: be

text: precise,

text: we

text: say

text: an

text: iteration

text: suﬀers

text: a

text: hash

text: collision

text: if

text: TA

text: TB

text: but

text: either

text: hashSB(TA)

text: hashSB(TB)

text: or

text: hashSA(TA)

text: hashSA(TB).

text: In

text: particular,

text: we

text: do

text: not

text: count

text: iterations

text: as

text: suﬀering

text: a

text: hash

text: collision

text: if

text: the

text: random

text: hash

text: functions

text: sampled

text: would

text: reveal

text: a

text: discrepancy

text: but

text: this

text: detection,

text: e.g.,

text: in

text: Line

text: is

text: prevented

text: by

text: corruptions

text: in

text: the

text: transmission

text: of

text: a

text: hash

text: value

text: or

text: seed.

text: To

text: prove

text: the

text: number

text: of

text: hash

text: collisions

text: to

text: be

text: small

text: we

text: crucially

text: exploit

text: the

text: fact

text: that

text: the

text: randomness

text: used

text: for

text: hashing

text: is

text: sampled

text: afresh

text: in

text: every

text: iteration.

text: In

text: particular,

text: it

text: is

text: sampled

text: after

text: everything

text: that

text: is

text: hashed

text: in

text: this

text: iteration

text: is

text: already

text: irrevocably

text: ﬁxed.

text: This

text: independence

text: allows

text: to

text: use

text: the

text: collision

text: resistance

text: property

text: of

text: Lemma

text: which

text: shows

text: that

text: any

text: iteration

text: suﬀers

text: from

text: a

text: hash

text: collision

text: with

text: probability

text: at

text: most

text: p

text: A

text: union

text: bound

text: over

text: all

text: iterations

text: then

text: shows

text: that

text: with

text: high

text: probability

text: no

text: hash

text: collision

text: happens

text: at

text: all:

text: Corollary

text: With

text: probability

text: no

text: iteration

text: in

text: Algorithm

text: suﬀers

text: from

text: a

text: hash

text: collision.

text: While

text: the

text: success

text: probability

text: guaranteed

text: by

text: Corollary

text: is

text: already

text: quite

text: nice

text: it

text: will

text: be

text: im-

text: portant

text: for

text: subsequent

text: algorithms

text: to

text: realize

text: that

text: one

text: can

text: easily

text: prove

text: stronger

text: guarantees.

text: In

text: particular,

text: using

text: the

text: independence

text: between

text: iterations

text: in

text: combination

text: with

text: a

text: standard

text: tail

text: bound

text: proves

text: that

text: the

text: probability

text: of

text: having

text: a

text: number

text: of

text: hash

text: collisions

text: of

text: the

text: same

text: order

text: of

text: magnitude

text: as

text: the

text: number

text: of

text: errors

text: is

text: at

text: least

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Algorithm 3 captioncolorCoding Scheme for Oblivious Adversarial Channels ← 1: Π n-round protocol to be simulated + ﬁnal conﬁrmation steps ← (cid:6)(cid:112) (cid:7) 2: hash inner product hashfamily from Deﬁnition 6.1 with o = Θ(1) and s = Θ(n) ← ← ← (cid:100) (cid:101) ← ∅ 3: Initialize Parameters: rc ← Θ(1); r r(cid:15)c ; Rtotal n/r + 65n(cid:15) ; T 4: Reset Status: k, E, v1, v2 0 √ · − 5: R = Robust Randomness Exchange(l = Rtotal s, δ = 2 Θ(nr o)) (cid:46) Requires Θ(n (cid:15)) rounds (cid:106) (cid:107) 6: for R iterations do total ← ← (cid:98) (cid:99) ← | | ← − 7: k ← k + 1; k˜ 2 log2 k ; MP1 k˜r ˜kTr ; MP2 MP1 k˜r (cid:46) Veriﬁcation Phase 8: S s new preshared random bits from R 9: Send (hash (k), hash (T), hash (T[1, MP1]), hash (T[1, MP2])) (cid:48)S (cid:48) (cid:48) S (cid:48) S S 10: Receive (H , H , H , H ); k T MP←1 MP2 11: (H , H , H , H ) (hash (k), hash (T), hash (T[1, MP1]), hash (T[1, MP2])) k T MP1 MP2 S S S S (cid:54) (cid:48) 12: if H = H then k← k 13: E E + 1 14: else ∈ { (cid:48) (cid:48) } 15: if H H , H then MP1← MP1 MP2 16: v1 v1 + 1 ∈ { (cid:48) (cid:48) } 17: else if H H , H then ← MP2 MP1 MP2 18: v2 v2 + 1 (cid:48) 19: if k = 1 and H = H and E = 0 then (cid:46) Computation Phase T T 20: continue computation and transcript T for r steps ← 21: Reset Status: k, E, v1, v2 0 22: else 23: do r dummy communications ≥ 24: if 2E k then (cid:46) Transition Phase ← 25: Reset Status: k, E, v1, v2 0 ≥ · 26: else if k = k˜ and v1 0.4 k˜ then 27: rollback computation and transcript T to position MP1 ← 28: Reset Status: k, E, v1, v2 0 ≥ · 29: else if k = k˜ and v2 0.4 k˜ then 30: rollback computation and transcript T to position MP2 ← 31: Reset Status: k, E, v1, v2 0 32: else if k = k˜ then ← 33: v1, v2 0 34: Output the outcome of Π corresponding to transcript T − As b(cid:22)efore, we deﬁne the amount of agreement l+ and disagr(cid:23)eement l between the two paths (cid:8) (cid:9) computed at Alice and Bob as | | | | l+ = 1 max l(cid:48) ∈ [1, min{|TA|, |TB|}] s.th. TA[1, l(cid:48)] = TB[1, l(cid:48)] and l− = TA + TB − 2l+. r r 22 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Algorithm

label: paragraph

text: captioncolorCoding

text: Scheme

text: for

text: Oblivious

text: Adversarial

text: Channels

text: Π

text: n-round

text: protocol

text: to

text: be

text: simulated

text: ﬁnal

text: conﬁrmation

text: steps

text: hash

text: inner

text: product

text: hashfamily

text: from

text: Deﬁnition

text: with

text: o

text: and

text: s

text: Θ(n)

text: Initialize

text: Parameters:

text: rc

text: r

text: Rtotal

text: n/r

text: T

text: Reset

text: Status:

text: k,

text: E,

text: R

text: Robust

text: Randomness

text: Exchange(l

text: Rtotal

text: s,

text: δ

text: Θ(nr

text: o))

text: Requires

text: Θ(n

text: rounds

text: for

text: R

text: iterations

text: do

text: total

text: k

text: k

text: k˜

text: k

text: k˜r

text: ˜kTr

text: k˜r

text: Veriﬁcation

text: Phase

text: S

text: s

text: new

text: preshared

text: random

text: bits

text: from

text: R

text: Send

text: (hash

text: (k),

text: hash

text: (T),

text: hash

text: hash

text: S

text: S

text: S

text: Receive

text: (H

text: H

text: H

text: H

text: k

text: T

text: (H

text: H

text: H

text: H

text: (hash

text: (k),

text: hash

text: (T),

text: hash

text: hash

text: k

text: T

text: S

text: S

text: S

text: S

text: if

text: H

text: H

text: then

text: k←

text: k

text: E

text: E

text: else

text: if

text: H

text: H

text: H

text: then

text: else

text: if

text: H

text: H

text: H

text: then

text: if

text: k

text: and

text: H

text: H

text: and

text: E

text: then

text: Computation

text: Phase

text: T

text: T

text: continue

text: computation

text: and

text: transcript

text: T

text: for

text: r

text: steps

text: Reset

text: Status:

text: k,

text: E,

text: else

text: do

text: r

text: dummy

text: communications

text: if

text: k

text: then

text: Transition

text: Phase

text: Reset

text: Status:

text: k,

text: E,

text: else

text: if

text: k

text: k˜

text: and

text: k˜

text: then

text: rollback

text: computation

text: and

text: transcript

text: T

text: to

text: position

text: Reset

text: Status:

text: k,

text: E,

text: else

text: if

text: k

text: k˜

text: and

text: k˜

text: then

text: rollback

text: computation

text: and

text: transcript

text: T

text: to

text: position

text: Reset

text: Status:

text: k,

text: E,

text: else

text: if

text: k

text: k˜

text: then

text: Output

text: the

text: outcome

text: of

text: Π

text: corresponding

text: to

text: transcript

text: T

text: As

text: we

text: deﬁne

text: the

text: amount

text: of

text: agreement

text: l+

text: and

text: l

text: between

text: the

text: two

text: paths

text: computed

text: at

text: Alice

text: and

text: Bob

text: as

text: l+

text: max

text: min{|TA|,

text: |TB|}]

text: s.th.

text: and

text: l−

text: TA

text: TB

text: r

text: r

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: GAMA: close-pairs and mergers 13 1 < r 20kpc h sep < R V 500km s e sep la 0.1tiv e E l rro l l l r l l l l l l l l l 0.01 l l l l ll l ll l l l l l ll l l l ll l ll ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l − Figure 10. Fit to the 2D close-pair density distribution for rsep < 20kpc and vsep < 500kms 1 sample (PSr20v500) as observed, i.e. not true pair corrected using the mock catalogues (see Section ). The ﬁt is represented by the grey shaded 2D manifold. The colour of the binned data represents the estimated relative error (so redder points dominate the ﬁt more). Circles on the base of the 3D plot represent missing data. volumes should contain more pairs by chance alone, regard- dows investigated the mass is well bound within the mass less of other physical processes further enhancing this num- range explored in this work. Using the appropriate scaling ber. α is similar for the two smallest dynamical selections, factors to account for galaxies being in potentially more CP (cid:62) − where values of α 1 indicate most stars in close-pairs than one close-pair (so to avoid double counting the ac- CP M∗ are found within galaxies of stellar masses around . creting mass), we ﬁnd the PS sample has a comov- CP r20v5×00 − α is larger for the largest dynamical window, but this is ing accreting mass density of 0.038 106/( M(cid:12)/ h 3 Mpc3), CP × − barely signiﬁcant given the calculated errors, and perhaps PS , PS has 0.100 106/( M(cid:12)/ h 3 Mpc3) and r50v500 r20v500 × − not physically notable. PS has 0.300 106/( M(cid:12)/ h 3 Mpc3). This com- r100v1000 pares to a total comoving stellar mass density for all galax- Using the analytic parameterisations presented in Ta- × − ies of 0.651 106/( M(cid:12)/ h 3 Mpc3) using the GSMF mea- ble 2 we can extrapolate to stellar masses beyond the range ∼ (cid:54) M (cid:54) sured in Baldry et al. (2012). This means that, e.g. 6% 108 / M(cid:12) 1012 used to constrain the ﬁts. Of clear of all stellar mass is available for minor merger accretion interest is the implied accreting mass in mergers, which can on the shortest dynamical timescale investigated here (the be thought of as the total mass of sub-dominant compo- PS selection). nents in close-pairs (i.e. the mass of the smaller galaxy in r20v500 close-pairs). This is straightforward to calculate analytically These numbers simply reﬂect the sub-dominant mass given the ﬁts to the data, and in all three dynamical win- in close-pairs, which is not to say that all of this mass will (cid:13) c 2014 RAS, MNRAS 000, 1–?? ##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## ##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##

text: GAMA:

text: close-pairs

text: and

text: mergers

font: default

text: r

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text: R

text: V

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text: Figure

text: Fit

text: to

text: the

text: close-pair

text: density

text: distribution

text: for

text: rsep

text: and

text: vsep

text: sample

text: as

text: observed,

text: i.e.

text: not

text: true

text: pair

text: corrected

text: using

text: the

text: mock

text: catalogues

text: (see

text: Section

text: The

text: ﬁt

text: is

text: represented

text: by

text: the

text: grey

text: shaded

text: manifold.

text: The

text: colour

text: of

text: the

text: binned

text: data

text: represents

text: the

text: estimated

text: relative

text: error

text: (so

text: redder

text: points

text: dominate

text: the

text: ﬁt

text: more).

text: Circles

text: on

text: the

text: base

text: of

text: the

text: plot

text: represent

text: missing

text: data.

text: volumes

text: should

text: contain

text: more

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text: 18 A.S.G. Robotham et al. Look−back Time / Gyrs circle on Figure 15, and is discussed in detail below). The 0 1 2 3 4 5 6 7 8 grey shaded region in Figure 15 shows the full 1σ range of 0.12 1.8 l RBeolbl o0t6ham (This Work) 0.12 allowed ﬁts using our model parameterisation. 1.7 Bdeu nPdryo p0r9is 07 The data analysed here does not extend to high enough )m 0.1 l 1.6m dKea rRtaalvteepl e0 907 0.1 rsehdifsthidftoswntotuwrnitninesst,heanmdatjhoerrecfloorsee-pﬁatirfofrr,acatnioynh(igi.he. rtehde- gction ( 0.08 1.51.4 LPXiaunt t10o28na 08 0.08 p20re0f6e)r.reOdurAv(a1lu+e zfo)mr ethc(e1+pzo)wmero-ldaewl (pmre)seisntgerdeaitnerCthonanseltihcee a ± −Pair Fr 0.06 0.018 0.019 0.02 0.021 0.022 0.023 1.3 l 0.06 mtha=n 0th.4e1m0=.220.2ﬁ±gu0r.e2pinubXliusheetdailn. (L2i0n12eat)a.lM. (o2s0t08o)f,tbhuetdlaetsas e A used to constrain our ﬁt overlaps with that presented in Xu s Clo 04 04 et al. (2012a), with the exception that we have much better or 0. 0. constraints at low redshift through the addition of the three Maj GAMA data points. The ﬂatter slope is partly driven by the 2 2 0.0 0.0 lack of signiﬁcant evolution seen for the GAMA data. l ( ) g = A 1 + z m We also note that Xu et al. (2012a) ﬁnd a normalisation 0 m 0 ± A = 0.013 0.001, which is signiﬁcantly less than the value 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.9 1.1 we ﬁnd. Inspecting their Figure 6, it is clear there ﬁt is Redshift (z) being heavily dragged down at low redshift by the lowest-z Kartaltepe et al. (2007) data point. This data point also has Figure 15. Major merger close-pair fraction as a function of red- by far the most tension with the best global ﬁt, being more shift (bottom) or look-back time (top). The stellar mass range explored is limited to z < 0.2 in GAMA since we are still conser- than 10σ away from the preferred functional form. This is M∗ vatively complete to galaxies out to this redshift. This Figure likely to be the main origin of the much lower normalisation uses data taken from Bell et al. (2006); Bundy et al. (2009); de and steeper power-law ﬁt seen in Xu et al. (2012a). To test Propris et al. (2007); de Ravel et al. (2009); Kartaltepe et al. this we attempted a ﬁt without the GAMA data, and then (2007); Lin et al. (2008); Patton et al. (2008); Xu et al. (2012a) either included or discarded the lowest redshift Kartaltepe atos purseeseancteodmminoXnu20eth−al1. k(p20c1p2rao),jewctheedreserpesauralttsioanr,e5s0c0aklemd/ssovaes- et al. (2007) data point. ±Including the data po±int returns ﬁt parameters A = 0.013 0.001 and m = 2.26 0.08, but locity separation and major merger deﬁnition of a 3:1 mass ratio ± ± M∗ discarding it returns A = 0.020 0.005 and m = 1.55 0.21. close to . The solid black line shows the Bayesian expectation for our simple 2 parameter model, and the grey shaded region The ﬁt parameters we ﬁnd when including the lowest shows the 1-σ marginalised range of allowed ﬁts. The top-left in- redshift Kartaltepe et al. (2007) data point is entirely con- set panel shows the 50%, 1σ and 95% percentile range contours sistent with that found in Xu et al. (2012a), whilst the ﬁt for our preferred model using the posterior MCMC chains. when discarding it is in excellent agreement with our new parameterisation including the new GAMA data points, al- These values are largely consistent with the mixture of pub- beit with larger errors. This is expected given the poorer lished values covering the same range of redshift. We see statistical constraints oﬀered by the available data without very mild evidence for major close-pair fraction evolution the GAMA results. Collective±ly this suggests that t±he new (increase with redshift) over the range investigated, but the ﬁtting parameters (A = 0.021 0.001 and m = 1.53 0.08) results are consistent with the fraction remaining constant are good estimates of the true power-law model, and pre- between z = 0.05–0.2. vious estimates have been systematically biased by the low Combining these data together, it is useful to attempt redshift SDSS derived Kartaltepe et al. (2007) data point. to ﬁnd the optimal parameterisation of the evolution of All but the lowest data point in Kartaltepe et al. (2007) major close-pair fractions. Comparing the marginalised log- is assembled from COSMOS HST data, however the lowest likelihoods of a number of simple models (including Az +C, data point itself was derived from the close-pairs catalogue A(1+z)+C A(1+z)m, A(1+z)m+C and A(1+z)mec(1+z)), of Allam et al. (2004). This work utilised a very restrictive we ﬁnd we prefer a simple two parameter model of the type: deﬁnition of close-pair, requiring spatial proximity relative γ = A(1 + z)m, (8) to the physical size of the galaxies rather than simply a con- m stant angular separation criterion. For this reason it seems where γM is the major close-pair fraction, z is the observed likely that the derived estimate is much lower than the in- redshift, and A and m are parameters to be ﬁtted. Using trinsic value. standard Metropolis Markov Chain Monte Carlo (MCMC) sampling we can estimate the posterior of the likelihood Both the Lin et al. (2008) and Xu et al. (2012a) param- space, with the likelihood model based on the Gaussian den- eterisations are strongly rejected once we include our new sity of the data given the model. This returns the expecta- GAMA data (see the posterior contours in the top-left inset tion of the three parameters we wish to ﬁt, along with the panel of Figure 15). Bridge et al. (2010) also consider a com- standard deviations and covariance. The ‘most likely’ pa- pendium of data including the Kartaltepe et al. (2007) data ± rameterisation of the evolution of close-pair major mergers points, and they ﬁnd m = 2.83 0.29. However, when using ± ± ± is found to be: A = 0.021 0.001 and m = 1.53 0.08 where only their CFHTLS-Deep data they ﬁnd m = 2.33 0.72, we censor out the highest tension low-z Kartaltepe et al. which brings their result for the power law slope into statis- (2007) data point (this is indicated by a transparent grey tical agreement with the ﬁgure presented here. 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text: 2 IEEE COMMUNICATIONS LETTERS, VOL. , NO. , 2014 ∗ · · · where indicates complex conjugation. Among U R(u), (a) + (b) + + (p) terms are needed to compute (3) for 2 xy the signal with maximum CORR is selected for transmission. times, 4 times, and 16 times oversampling, respectively. PTS scheme with CORR is treated in [5], but in this letter, Table I shows the coefﬁcients of the equations to compute we treat SLM scheme with CORR since SLM scheme is a CORR metric when L times oversampling is used. Indices are general scheme. arranged in numerical order (1–16). For example, in (9), the − In [6], analytic BER of SLM scheme is described when coefﬁcients of index 1, index 3, and index 6 are 0.04, 0.13, nonlinear HPA is (cid:18)presented. (cid:19) (cid:32)(cid:114) (cid:33) and 0.39, respectively. To compute the correlati{on betwee}n sequences derived from (8) with different L = 1, 2, 4, 16 , we only consider the coefﬁcients of each equation. P = 4 1 − √1 Q 3− SDNR . (4) Now, we use the Pearson correlation coefﬁcient (r). r is b log M M M 1 a measure of linear correlation between two sequences (vari- 2 − ables) and is giving a value [ 1, 1], where 0 is no correlation − According to [6], for better BER peformance, SDNR should and 1(or 1) is total positive(or negative) correlation. ∗ be large. To do so, E[x(u)y (u)] should be large and it is Pearson correlation(cid:80)coefﬁcient between two sequences n n identical to CORR metric. Thus, CORR metric is the optimal (x, y) is deﬁned as(cid:112)(cid:80) (cid:80) in terms of BER performance. Thus, we focus on CORR − − (x x¯)(y y¯) metric among many metrics targeting the BER performance r = i −i i − (10) and describe the oversampling effect in SLM scheme with i(xi x¯)2 i(yi y¯)2 nonlinearity. where x¯ denotes the sample mean and x denotes the ith III. OVERSAMPLING EFFECT element of the sequence x. i Based on [7], oversampled signal can be achieved by linear Table II shows the Pearson correlation coefﬁcients between combination of Nyquist rate samples. The impulse response two sequences obtained from Table I when different L is used. of interpolator for the L times oversampling is deﬁned as We can regard when L = 16 is used as a continuous signal which can estimate the nonlinear effect of HPA well. When sin(πn/L) comparing L = 1 and L = 16, r is 0.6023, which has very low h [n] = (5) L πn/L correlation. On the other hand, the value of r when comparing L = 2 & L = 16 and L = 4 & L = 16 are 0.9133 and 0.9839, where we use the limited ﬁlter length (I) in practice. Then, respectively, which have very high correlation. Therefore, we the output of the interpol(cid:88)ator can be expressed as can expect that 2 times and 4 times oversampling can be used instead of using 16 times oversampling. (cid:98)(nL+LI)/L(cid:99) − IV. SIMULATION RESULTS x˜ [n ] = x [k]h [n Lk] (6) In this section, we derive the BER performance of SLM L L 1 L L k=(cid:100)(nL−LI)/L(cid:101) schemes with CORR metric when different L times oversam- pling are used. Also, nonlinear HPA is considered. There are where x [n ] is n th element of L times oversampled four BER curves in Fig. 2. Original indicates OFDM system L L L signal x. In this letter, we use I = 2 for analyzation. without SLM scheme. Let us consider 16 times oversampled signal because we can V. CONCLUSION regard it as a continuous signal. Then, the interpolation matrix · · · REFERENCES M = [M , M , , M ] can be written as (7) at the top of the page 3. F1or ex2ample, x˜15 [16m+1] = h [17]x [16(m−1)]+ [1] S. H. Mu¨ller, R. W. B¨auml, R. F. H. Fischer, and J. B. Hu¨ber, −16 16 −1 “OFDM with reduced peak-to-average power ratio by multiple signal h [1]x [16m]+h [ 15]x [16(m+1)]+h [ 31]x [16(m+ representation,” Ann. Telecommun., vol. 52, no.1–2, pp. 58–67, Feb. 16 1 ≤ 1≤6 − 1 16 1 2)], where 0 m N 1. 1997. [2] M. R. D. Rodrigues and I. J. Wassell, “IMD reduction with SLM We can only express 16 samples among 16N samples and PTS to improve the error-probability performance of nonlinearly without loss of ge(cid:88)nerality. Then, we can rewrite (3) as distorted OFDM signals,” IEEE Trans. Veh. Technol., vol. 55, pp. 537– 548, Mar. 2006. [3] E. Al-Dalakta, A. Al-Dweik, A. Hazmi, C. Tsimenidis, and B. Sharif, 15 R¯(u) = x˜[16m + s](u)y˜[16m + s]∗(u). (8) “Efﬁcient BER reduction technique for nonlinear OFDM transmission x˜y˜ using distortion prediction,” IEEE Trans. Veh. Technol., vol. 61, pp. s=0 2330–2336, Jun. 2012. [4] D. Park and H. Song, “A new PAPR reduction technique of OFDM There are 16 terms and we assign alphabets, (a) to (p), system with nonlinear high power ampliﬁer,” IEEE Trans. Consum. ∗ Electron., vol. 53, pp. 327–332, May 2007. to each terms, i.e., (a) = x˜[16m + 1](u)y˜[16m + 1] (u) and ∗ [5] E. Al-Dalakta, A. Al-Dweik, A. Hazmi, C. Tsimenidis, and B. Sharif, (p) = x˜[16m + 15](u)y˜[16m + 15] (u). For better understand- “PAPR reduction scheme using maximum cross correlation,” IEEE ing, let us consider only (i) term and write it concretely as (9) Commun. Lett., vol. 16, no. 12, pp. 2032–2035, Dec. 2012. [6] K. Bae, C. Shin, and E. J. Powers, “Performance analysis of OFDM by (6) and (7). With same way, we can compute all the other systems with selected mapping in the presence of nonlinearity,” IEEE terms. Trans. Wireless Commun., vol. 12, no. 5, pp. 2314–2322, MAY 2013. Now, let us consider L times oversampling for [7] C.-L. Wang, S.-J. Ku, and C.-J. Yang, ”A low-complexity PAPR estimation scheme for OFDM signals and its application to SLM-based computing CORR metric. We only need (a) term for PAPR reduction”, IEEE J. Sel. Topics Signal Process., vol. 4, no. 3, Nyquist rate sampling. (a) + (i), (a) + (e) + (i) + (m), and pp. 637–645, Jun. 2010. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: IEEE

text: COMMUNICATIONS

text: LETTERS,

text: VOL.

text: NO.

text: where

text: indicates

text: complex

text: conjugation.

text: Among

text: U

text: R(u),

text: (a)

text: (b)

text: (p)

text: terms

text: are

text: needed

text: to

text: compute

text: for

text: xy

text: the

text: signal

text: with

text: maximum

text: CORR

text: is

text: selected

text: for

text: transmission.

text: times,

text: times,

text: and

text: times

text: oversampling,

text: respectively.

text: PTS

text: scheme

text: with

text: CORR

text: is

text: treated

text: in

text: but

text: in

text: this

text: letter,

text: Table

text: I

text: shows

text: the

text: coefﬁcients

text: of

text: the

text: equations

text: to

text: compute

text: we

text: treat

text: SLM

text: scheme

text: with

text: CORR

text: since

text: SLM

text: scheme

text: is

text: a

text: CORR

text: metric

text: when

text: L

text: times

text: oversampling

text: is

text: used.

text: Indices

text: are

text: general

text: scheme.

text: arranged

text: in

text: numerical

text: order

text: For

text: example,

text: in

text: the

text: In

text: analytic

text: BER

text: of

text: SLM

text: scheme

text: is

text: described

text: when

text: coefﬁcients

text: of

text: index

text: index

text: and

text: index

text: are

text: nonlinear

text: HPA

text: is

text: and

text: respectively.

text: To

text: compute

text: the

text: correlati{on

text: betwee}n

text: sequences

text: derived

text: from

text: with

text: different

text: L

text: we

text: only

text: consider

text: the

text: coefﬁcients

text: of

text: each

text: equation.

text: P

text: √1

text: Q

text: 3−

text: SDNR

text: Now,

text: we

text: use

text: the

text: Pearson

text: correlation

text: coefﬁcient

text: (r).

text: r

text: is

text: b

text: log

text: M

text: M

text: M

text: a

text: measure

text: of

text: linear

text: correlation

text: between

text: two

text: sequences

text: (vari-

text: ables)

text: and

text: is

text: giving

text: a

text: value

text: where

text: is

text: no

text: correlation

text: According

text: to

text: for

text: better

text: BER

text: peformance,

text: SDNR

text: should

text: and

text: is

text: total

text: positive(or

text: negative)

text: correlation.

text: be

text: large.

text: To

text: do

text: so,

text: E[x(u)y

text: (u)]

text: should

text: be

text: large

text: and

text: it

text: is

text: Pearson

text: between

text: two

text: sequences

text: n

text: n

text: identical

text: to

text: CORR

text: metric.

text: Thus,

text: CORR

text: metric

text: is

text: the

text: optimal

text: (x,

text: y)

text: is

text: deﬁned

text: in

text: terms

text: of

text: BER

text: performance.

text: Thus,

text: we

text: focus

text: on

text: CORR

text: (x

text: x¯)(y

text: y¯)

text: metric

text: among

text: many

text: metrics

text: targeting

text: the

text: BER

text: performance

text: r

text: i

text: −i

text: i

text: and

text: describe

text: the

text: oversampling

text: effect

text: in

text: SLM

text: scheme

text: with

text: i(xi

text: i(yi

text: nonlinearity.

text: where

text: x¯

text: denotes

text: the

text: sample

text: mean

text: and

text: x

text: denotes

text: the

text: ith

text: III.

text: OVERSAMPLING

text: EFFECT

text: element

text: of

text: the

text: sequence

text: x.

text: i

text: Based

text: on

text: oversampled

text: signal

text: can

text: be

text: achieved

text: by

text: linear

text: Table

text: II

text: shows

text: the

text: Pearson

text: correlation

text: coefﬁcients

text: between

text: combination

text: of

text: Nyquist

text: rate

text: samples.

text: The

text: impulse

text: response

text: two

text: sequences

text: obtained

text: from

text: Table

text: I

text: when

text: different

text: L

text: is

text: used.

text: of

text: interpolator

text: for

text: the

text: L

text: times

text: oversampling

text: is

text: deﬁned

text: as

text: We

text: can

text: regard

text: when

text: L

text: is

text: used

text: as

text: a

text: continuous

text: signal

text: which

text: can

text: estimate

text: the

text: nonlinear

text: effect

text: of

text: HPA

text: well.

text: When

text: sin(πn/L)

text: comparing

text: L

text: and

text: L

text: r

text: is

text: which

text: has

text: very

text: low

text: h

text: [n]

text: L

text: πn/L

text: correlation.

text: On

text: the

text: other

text: hand,

text: the

text: value

text: of

text: r

text: when

text: comparing

text: L

text: L

text: and

text: L

text: L

text: are

text: and

text: where

text: we

text: use

text: the

text: limited

text: ﬁlter

text: length

text: (I)

text: in

text: practice.

text: Then,

text: respectively,

text: which

text: have

text: very

text: high

text: correlation.

text: Therefore,

text: we

text: the

text: output

text: of

text: the

text: can

text: be

text: expressed

text: as

text: can

text: expect

text: that

text: times

text: and

text: times

text: oversampling

text: can

text: be

text: used

text: instead

text: of

text: using

text: times

text: oversampling.

text: IV.

text: SIMULATION

text: RESULTS

text: x˜

text: [n

text: x

text: [k]h

text: [n

text: Lk]

text: In

text: this

text: section,

text: we

text: derive

text: the

text: BER

text: performance

text: of

text: SLM

text: L

text: L

text: L

text: L

text: schemes

text: with

text: CORR

text: metric

text: when

text: different

text: L

text: times

text: oversam-

text: pling

text: are

text: used.

text: Also,

text: nonlinear

text: HPA

text: is

text: considered.

text: There

text: are

text: where

text: x

text: [n

text: is

text: n

text: th

text: element

text: of

text: L

text: times

text: oversampled

text: four

text: BER

text: curves

text: in

text: Fig.

text: Original

text: indicates

text: OFDM

text: system

text: L

text: L

text: L

text: signal

text: x.

text: In

text: this

text: letter,

text: we

text: use

text: I

text: for

text: analyzation.

text: without

text: SLM

text: scheme.

text: Let

text: us

text: consider

text: times

text: oversampled

text: signal

text: because

text: we

text: can

text: V.

text: CONCLUSION

text: regard

text: it

text: as

text: a

text: continuous

text: signal.

text: Then,

text: the

text: interpolation

text: matrix

text: REFERENCES

text: M

text: [M

text: M

text: M

text: can

text: be

text: written

text: as

text: at

text: the

text: top

text: of

text: the

text: page

text: h

text: S.

text: H.

text: Mu¨ller,

text: R.

text: W.

text: B¨auml,

text: R.

text: F.

text: H.

text: Fischer,

text: and

text: J.

text: B.

text: Hu¨ber,

text: −1

text: “OFDM

text: with

text: reduced

text: peak-to-average

text: power

text: ratio

text: by

text: multiple

text: signal

text: h

text: representation,”

text: Ann.

text: Telecommun.,

text: vol.

text: pp.

text: Feb.

text: where

text: m

text: N

text: M.

text: R.

text: D.

text: Rodrigues

text: and

text: I.

text: J.

text: Wassell,

text: “IMD

text: reduction

text: with

text: SLM

text: We

text: can

text: only

text: express

text: samples

text: among

text: samples

text: and

text: PTS

text: to

text: improve

text: the

text: error-probability

text: performance

text: of

text: nonlinearly

text: without

text: loss

text: of

text: Then,

text: we

text: can

text: rewrite

text: as

text: distorted

text: OFDM

text: signals,”

text: IEEE

text: Trans.

text: Veh.

text: Technol.,

text: vol.

text: pp.

text: Mar.

text: E.

text: Al-Dalakta,

text: A.

text: Al-Dweik,

text: A.

text: Hazmi,

text: C.

text: Tsimenidis,

text: and

text: B.

text: Sharif,

text: R¯(u)

text: s]∗(u).

text: “Efﬁcient

text: BER

text: reduction

text: technique

text: for

text: nonlinear

text: OFDM

text: transmission

text: x˜y˜

text: using

text: distortion

text: prediction,”

text: IEEE

text: Trans.

text: Veh.

text: Technol.,

text: vol.

text: pp.

text: Jun.

text: D.

text: Park

text: and

text: H.

text: Song,

text: “A

text: new

text: PAPR

text: reduction

text: technique

text: of

text: OFDM

text: There

text: are

text: terms

text: and

text: we

text: assign

text: alphabets,

text: (a)

text: to

text: (p),

text: system

text: with

text: nonlinear

text: high

text: power

text: ampliﬁer,”

text: IEEE

text: Trans.

text: Consum.

text: Electron.,

text: vol.

text: pp.

text: May

text: to

text: each

text: terms,

text: i.e.,

text: (a)

text: (u)

text: and

text: E.

text: Al-Dalakta,

text: A.

text: Al-Dweik,

text: A.

text: Hazmi,

text: C.

text: Tsimenidis,

text: and

text: B.

text: Sharif,

text: (p)

text: (u).

text: For

text: better

text: understand-

text: “PAPR

text: reduction

text: scheme

text: using

text: maximum

text: cross

text: correlation,”

text: IEEE

text: ing,

text: let

text: us

text: consider

text: only

text: (i)

text: term

text: and

text: write

text: it

text: concretely

text: as

text: Commun.

text: Lett.,

text: vol.

text: no.

text: pp.

text: Dec.

text: K.

text: Bae,

text: C.

text: Shin,

text: and

text: E.

text: J.

text: Powers,

text: “Performance

text: analysis

text: of

text: OFDM

text: by

text: and

text: With

text: same

text: way,

text: we

text: can

text: compute

text: all

text: the

text: other

text: systems

text: with

text: selected

text: mapping

text: in

text: the

text: presence

text: of

text: nonlinearity,”

text: IEEE

text: terms.

text: Trans.

text: Wireless

text: Commun.,

text: vol.

text: no.

text: pp.

text: MAY

text: Now,

text: let

text: us

text: consider

text: L

text: times

text: oversampling

text: for

text: C.-L.

text: Wang,

text: S.-J.

text: Ku,

text: and

text: C.-J.

text: Yang,

text: ”A

text: low-complexity

text: PAPR

text: estimation

text: scheme

text: for

text: OFDM

text: signals

text: and

text: its

text: application

text: to

text: SLM-based

text: computing

text: CORR

text: metric.

text: We

text: only

text: need

text: (a)

text: term

text: for

text: PAPR

text: reduction”,

text: IEEE

text: J.

text: Sel.

text: Topics

text: Signal

text: Process.,

text: vol.

text: no.

text: Nyquist

text: rate

text: sampling.

text: (a)

text: (i),

text: (a)

text: (e)

text: (i)

text: (m),

text: and

text: pp.

text: Jun.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: d n a m e D Time Figure 2: Sample demand realization for 3 buyers with our trace-inspired model for 600 simulated secs. Each color is a buyer’s demand. We simulate truthful bidding for both bks and vmm, even though vmm is not a truthful mech- anism. For memoryless demand models, this means that vmm results in the optimal allocation, acting as a benchmark for the truthful mechanisms. 6.2 Eﬃciency From our analysis, we know that spq is eﬃcient for memoryless demand models, but may not be eﬃcient for other demand models. Here, we look at both cases, and look at the magnitude of this eﬀect in some scenarios. We also examine the drop in eﬃciency due to bid resampling in bks. Fig. 3 shows the social welfare (sum of buyer and seller utilities) for a scenario with three buyers. The ﬁrst two have per-KB values 10 and 4, and average demand 10 KBps each, simulating a high value and a medium value user. The last buyer has a per-KB value of only 1, but has a higher average demand–30 KBps. All buyers use the stochastic demand model described above, and we compare four diﬀerent mechanisms–bks and vmm, both with spq routing and truthful bids, and fq and fifo, with a ﬁxed price of 1. (We study the eﬀect of changing this ﬁxed price in the next experiment). As the seller capacity increases, the social welfare increases for all the mechanisms, until all the demand is satisﬁed. The demand models in this simulation are memoryless, so spq routing for vmm is optimal given that we insist on truthful bidding. The small diﬀerence between the performance of vmm and bks is the eﬃciency cost of the sampling necessary for the truthfulness provided by bks. spq routing for bks is much more eﬃcient than either of the non-prioritized ﬁxed price policies. The diﬀerence between fq and fifo is also interesting–with fifo, the allocation is proportional to demand, so the low value, high-demand buyer gets three-ﬁfth of the channel on average, reducing eﬃciency. Fig. 4 shows the eﬀect of reserve price on eﬃciency, in the same scenario, with seller capacity ﬁxed at 25 KBps. Raising the reserve price slightly to avoid allocating to the lowest value buyer improves eﬃciency for fifo and fq. Since vmm and bks naturally prioritize high value data, raising 17 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: d

text: n

text: a

text: m

text: e

text: D

text: Time

text: Figure

text: Sample

label: caption

text: demand

text: realization

text: for

text: buyers

text: with

text: our

text: trace-inspired

text: model

text: for

text: simulated

text: secs.

text: Each

text: color

text: is

text: a

text: buyer’s

text: demand.

text: We

text: simulate

text: truthful

text: bidding

text: for

text: both

text: bks

text: and

text: vmm,

text: even

text: though

text: vmm

text: is

text: not

text: a

text: truthful

text: mech-

text: anism.

text: For

text: memoryless

text: demand

text: models,

text: this

text: means

text: that

text: vmm

text: results

text: in

text: the

text: optimal

text: allocation,

text: acting

text: as

text: a

text: benchmark

text: for

text: the

text: truthful

text: mechanisms.

label: section

text: Eﬃciency

text: From

text: our

text: analysis,

text: we

text: know

text: that

text: spq

text: is

text: eﬃcient

text: for

text: memoryless

text: demand

text: models,

text: but

text: may

text: not

text: be

text: eﬃcient

text: for

text: other

text: demand

text: models.

text: Here,

text: we

text: look

text: at

text: both

text: cases,

text: and

text: look

text: at

text: the

text: magnitude

text: of

text: this

text: eﬀect

text: in

text: some

text: scenarios.

text: We

text: also

text: examine

text: the

text: drop

text: in

text: eﬃciency

text: due

text: to

text: bid

text: resampling

text: in

text: bks.

text: Fig.

text: shows

text: the

text: social

text: welfare

text: (sum

text: of

text: buyer

text: and

text: seller

text: utilities)

text: for

text: a

text: scenario

text: with

text: three

text: buyers.

text: The

text: ﬁrst

text: two

text: have

text: per-KB

text: values

text: and

text: and

text: average

text: demand

text: KBps

text: each,

text: simulating

text: a

text: high

text: value

text: and

text: a

text: medium

text: value

text: user.

text: The

text: last

text: buyer

text: has

text: a

text: per-KB

text: value

text: of

text: only

text: but

text: has

text: a

text: higher

text: average

text: KBps.

text: All

text: buyers

text: use

text: the

text: stochastic

text: demand

text: model

text: described

text: above,

text: and

text: we

text: compare

text: four

text: diﬀerent

text: mechanisms–bks

text: and

text: vmm,

text: both

text: with

text: spq

text: routing

text: and

text: truthful

text: bids,

text: and

text: fq

text: and

text: fifo,

text: with

text: a

text: ﬁxed

text: price

text: of

text: (We

text: study

text: the

text: eﬀect

text: of

text: changing

text: this

text: ﬁxed

text: price

text: in

text: the

text: next

text: experiment).

text: As

text: the

text: seller

text: capacity

text: increases,

text: the

text: social

text: welfare

text: increases

text: for

text: all

text: the

text: mechanisms,

text: until

text: all

text: the

text: demand

text: is

text: satisﬁed.

text: The

text: demand

text: models

text: in

text: this

text: simulation

text: are

text: memoryless,

text: so

text: spq

text: routing

text: for

text: vmm

text: is

text: optimal

text: given

text: that

text: we

text: insist

text: on

text: truthful

text: bidding.

text: The

text: small

text: diﬀerence

text: between

text: the

text: performance

text: of

text: vmm

text: and

text: bks

text: is

text: the

text: eﬃciency

text: cost

text: of

text: the

text: sampling

text: necessary

text: for

text: the

text: truthfulness

text: provided

text: by

text: bks.

text: spq

text: routing

text: for

text: bks

text: is

text: much

text: more

text: eﬃcient

text: than

text: either

text: of

text: the

text: non-prioritized

text: ﬁxed

text: price

text: policies.

text: The

text: diﬀerence

text: between

text: fq

text: and

text: fifo

text: is

text: also

text: interesting–with

text: fifo,

text: the

text: allocation

text: is

text: proportional

text: to

text: demand,

text: so

text: the

text: low

text: value,

text: high-demand

text: buyer

text: gets

text: three-ﬁfth

text: of

text: the

text: channel

text: on

text: average,

text: reducing

text: eﬃciency.

text: Fig.

text: shows

text: the

text: eﬀect

text: of

text: reserve

text: price

text: on

text: eﬃciency,

text: in

text: the

text: same

text: scenario,

text: with

text: seller

text: capacity

text: ﬁxed

text: at

text: KBps.

text: Raising

text: the

text: reserve

text: price

text: slightly

text: to

text: avoid

text: allocating

text: to

text: the

text: lowest

text: value

text: buyer

text: improves

text: eﬃciency

text: for

text: fifo

text: and

text: fq.

text: Since

text: vmm

text: and

text: bks

text: naturally

text: prioritize

text: high

text: value

text: data,

text: raising

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: Figure 1: Illustration of our model. Routing: We adopt a ﬂow model of traﬃc, with a single ﬂow modeling both upstream and downstream traﬃc for each buyer. The router’s link to the Internet is assumed to have ﬁxed ≥ capacity c > 0. Each buyer has bid b 0, and receives a priority based on this reported value. i We consider three schemes for network prioritization: 1. First-in First-out (fifo) routes data in a ﬁrst-in, ﬁrst-out order as it arrives at the router. Since we do not model the queue of a router directly, we model fifo as a within-epoch allocation proportional to within-epoch demand. 2. Fair queueing (fq) allocates bandwidth to all buyers evenly, dividing any unused capacity recursively: in each period, each of the n buyers is guaranteed to be able to consume c/n. Any un- used capacity is divided evenly among buyers with further demand, and this repeats until everyone is satisﬁed or capacity is exhausted. 3. Strict Priority Queueing (spq) ﬁrst allocates capacity to the buyer with the highest bid in each epoch, up to c or the demand of the buyer. Then, capacity is allocated to the buyer with the second highest bid, and so forth, as long as capacity is available. Ties are broken at random. Let c denote the network capacity available to buyer i in epoch t (this depends on her bid, as i,t well as the bid and network usage of others.) Transport: We assume that the local network between router and buyers is much faster than the router’s link to the Internet and neglect delays there. Demand: Let x (bytes) denote the cumulative amount of traﬃc (upload or download) i,t associated with buyer i up to and including time t. This may be smaller than the total capacity available to buyer i up to and including time t because it depends on the buyer’s demand. Let D (t, x) (bytes) denote the demand of buyer i during epoch t, where x is the cumulative i amount of traﬃc (upload or download) used so far; i.e., x = x − . Equivalently, this represents the i,t 1 maximum amount of network capacity that buyer i wants to consume during the epoch. D (t, x) i is a random variable, and we adopt d (t, x) to denote a speciﬁc realization. We consider demand i models D that satisfy the following condition: i 2 2Conceptually, we can think about the demand for all (t, x) values as being realized by “nature” when a buyer arrives, capturing the type of the buyer (even if it is unknown to the buyer.) 5 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine##

text: Figure

text: Illustration

text: of

text: our

text: model.

text: Routing:

text: We

text: adopt

text: a

text: ﬂow

text: model

text: of

text: traﬃc,

text: with

text: a

text: single

text: ﬂow

text: modeling

text: both

text: upstream

text: and

text: downstream

text: traﬃc

text: for

text: each

text: buyer.

text: The

text: router’s

text: link

text: to

text: the

text: Internet

text: is

text: assumed

text: to

text: have

text: ﬁxed

text: capacity

text: c

text: Each

text: buyer

text: has

text: bid

text: b

text: and

text: receives

text: a

text: priority

text: based

text: on

text: this

text: reported

text: value.

text: i

text: We

text: consider

text: three

text: schemes

text: for

text: network

text: prioritization:

text: First-in

text: First-out

text: (fifo)

text: routes

text: data

text: in

text: a

text: ﬁrst-in,

text: ﬁrst-out

text: order

text: as

text: it

text: arrives

text: at

text: the

text: router.

text: Since

text: we

text: do

text: not

text: model

text: the

text: queue

text: of

text: a

text: router

text: directly,

text: we

text: model

text: fifo

text: as

text: a

text: within-epoch

text: allocation

text: proportional

text: to

text: within-epoch

text: demand.

text: Fair

text: queueing

text: (fq)

text: allocates

text: bandwidth

text: to

text: all

text: buyers

text: evenly,

text: dividing

text: any

text: unused

text: capacity

text: recursively:

text: in

text: each

text: period,

text: each

text: of

text: the

text: n

text: buyers

text: is

text: guaranteed

text: to

text: be

text: able

text: to

text: consume

text: c/n.

text: Any

text: un-

text: used

text: capacity

text: is

text: divided

text: evenly

text: among

text: buyers

text: with

text: further

text: demand,

text: and

text: this

text: repeats

text: until

text: everyone

text: is

text: satisﬁed

text: or

text: capacity

text: is

text: exhausted.

text: Strict

text: Priority

text: Queueing

text: (spq)

text: ﬁrst

text: allocates

text: capacity

text: to

text: the

text: buyer

text: with

text: the

text: highest

text: bid

text: in

text: each

text: epoch,

text: up

text: to

text: c

text: or

text: the

text: demand

text: of

text: the

text: buyer.

text: Then,

text: capacity

text: is

text: allocated

text: to

text: the

text: buyer

text: with

text: the

text: second

text: highest

text: bid,

text: and

text: so

text: forth,

text: as

text: long

text: as

text: capacity

text: is

text: available.

text: Ties

text: are

text: broken

text: at

text: random.

text: Let

text: c

text: denote

text: the

text: network

text: capacity

text: available

text: to

text: buyer

text: i

text: in

text: epoch

text: t

text: (this

text: depends

text: on

text: her

text: bid,

text: as

text: i,t

text: well

text: as

text: the

text: bid

text: and

text: network

text: usage

text: of

text: others.)

text: Transport:

text: We

text: assume

text: that

text: the

text: local

text: network

text: between

text: router

text: and

text: buyers

text: is

text: much

text: faster

text: than

text: the

text: router’s

text: link

text: to

text: the

text: Internet

text: and

text: neglect

text: delays

text: there.

text: Demand:

text: Let

text: x

text: (bytes)

text: denote

text: the

text: cumulative

text: amount

text: of

text: traﬃc

text: (upload

text: or

text: download)

text: i,t

text: associated

text: with

text: buyer

text: i

text: up

text: to

text: and

text: including

text: time

text: t.

text: This

text: may

text: be

text: smaller

text: than

text: the

text: total

text: capacity

text: available

text: to

text: buyer

text: i

text: up

text: to

text: and

text: including

text: time

text: t

text: because

text: it

text: depends

text: on

text: the

text: buyer’s

text: demand.

text: Let

text: D

text: (t,

text: x)

text: (bytes)

text: denote

text: the

text: demand

text: of

text: buyer

text: i

text: during

text: epoch

text: t,

text: where

text: x

text: is

text: the

text: cumulative

text: i

text: amount

text: of

text: traﬃc

text: (upload

text: or

text: download)

text: used

text: so

text: far;

text: i.e.,

text: x

text: x

text: Equivalently,

text: this

text: represents

text: the

text: i,t

text: maximum

text: amount

text: of

text: network

text: capacity

text: that

text: buyer

text: i

text: wants

text: to

text: consume

text: during

text: the

text: epoch.

text: D

text: (t,

text: x)

text: i

text: is

text: a

text: random

text: variable,

text: and

text: we

text: adopt

text: d

text: (t,

text: x)

text: to

text: denote

text: a

text: speciﬁc

text: realization.

text: We

text: consider

text: demand

text: i

text: models

text: D

text: that

text: satisfy

text: the

text: following

text: condition:

text: i

text: we

text: can

text: think

text: about

text: the

text: demand

text: for

text: all

text: (t,

text: x)

text: values

text: as

text: being

text: realized

text: by

text: “nature”

text: when

text: a

text: buyer

text: arrives,

text: capturing

text: the

text: type

text: of

text: the

text: buyer

text: (even

text: if

text: it

text: is

text: unknown

text: to

text: the

text: buyer.)

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