lnmb course asymptotic methods in queueing theorysem/asqt/lecture02032015.pdf · 2015-03-10 ·...

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LNMB Course

Asymptotic Methodsin Queueing Theory

Lecture 3, March 2, 2015

Rudesindo Núñez-Queija (CWI/UvA), Sem Borst (TU/e)

http://www.win.tue.nl/˜sem/AsQT/


Course overview

Four main topics

• Large deviations and tail asymptotics

– Introduction, large deviations, large-buffer asymptotics for light-tailed queues

– Many-sources asymptotics

– Large-buffer asymptotics for heavy-tailed queues, impact of schedul-ing discipline

• Fluid and diffusion limits

• Perturbation analysis and time scale separation

• Heavy-traffic approximations


Heavy-tailed versus light-tailed distributions

A random variable X has a heavy-tailed distribution, when its tail exhibits(slow) ‘subexponential’ (e.g. polynomial) decay:

P{X > x}eθx→∞ as x →∞

for all θ > 0A prominent example of a heavy-tailed distribution is the Pareto distribution

P{X > x} ∼ γ x−ν as x →∞

In contrast, tails of light-tailed distributions exhibit (fast) exponential decay:

P{X > x} ∼ αe−βx as x →∞

While light-tailed distributions typically used to be assumed in (queueing)applications, in recent years empirical findings have pointed to widespreadoccurrence of heavy-tailed distributions


Traffic characteristics

Internet traffic exhibits burstiness on wide range of time scalesManifests itself in long-range dependence & self-similarity


Traffic characteristics (cont’d)

Contrasts with traditional traffic assumptions(Poisson arrivals, finite-variance service requirements)


Heavy-tailed distributions

Burstiness is caused by extreme variability in traffic processes(activity periods, file sizes)

Measurements suggest that file sizes follow Pareto distribution

P{X > x} ∼ γ x−ν as x →∞,

with 1 < ν < 2 (ν ≈ 1.7)

• E{X} <∞: finite mean

• E{X2} = ∞: infinite variance!!

Queueing models provide fundamental insight into performance impact ofheavy-tailed traffic characteristics and potential role of service discipline inlimiting detrimental effect


Heavy-tailed distributions (cont’d)

• Function F(·) is long-tailed if

limx→∞

1− F(x − y)1− F(x)

= 1, for all y ∈ R

• Function F(·) is subexponential if

limx→∞

1− F2∗(x)1− F(x)

= 2,

where F2∗(·) is two-fold convolution of F(·) with itself, implying

P{X1 + · · · + Xn > x} ∼ nP{X1 > x} as x →∞


Heavy-tailed distributions (cont’d)

• Function F(·) is regularly varying of index −ν if

F(x) = 1−L(x)xν

, ν ≥ 0,

where L : R+→ R+ is a function of slow variation,i.e., limx→∞ L(ηx)/L(x) = 1, η > 1 (e.g. L(x) = log(x))

• F(·) is intermediately regularly varying if

limη↓1

lim infx→∞

1− F(ηx)1− F(x)

= 1


Large-buffer asymptotics for heavy-tailed queues

As observed in first lecture, when X has a heavy-tailed distribution

• the log moment generating function 3(θ) = ∞ for all θ > 0

• the convex conjugate 3∗(x) = 0 for all x > E{X}

Consequently, the upper bound

1n

logP{Sn ≥ x} ≤ −3∗(x) = 0

has no meaning, nor do the corresponding LDP results for the stationarybuffer content distribution P{Q > q} as q →∞

Indeed, the large-buffer asymptotics for heavy-tailed queues do not decayexponentially, and turn out to exhibit starkly different behavior

A simple argument shows that one cannot expect exponential decay of thebuffer content distribution for heavy-tailed arrival processes


Large-buffer asymptotics for heavy-tailed queues (cont’d)

Specifically, consider the simple lower bound

P{Q > q} = P{supt≥0{St − Ct} > q} ≥ P{S1 − C > q} = P{A > C + q}

When the random variable A has a heavy-tailed distribution, the probabilityP{A > C + q} cannot show exponential decay

Indeed, if P{A > C + q} decays exponentially, i.e., there are constants KC <

∞ and θ > 0 such that

P{A > C + q} ≤ KCe−θq,

then

limx→∞

eθxP{A > x} ≤ limx→∞

eθx KCe−θ(x−C)= KCeθC <∞,

contradicting the definition of a heavy-tailed distribution



The previous observations indicate that the buffer content distribution forheavy-tailed arrival processes will have a ‘heavier’ tail, i.e., there is a ‘greaterprobability’ for ‘large buffer levels’ to occur

As it turns out, one can in fact obtain ‘exact’ (rather than logarithmic)large-buffer asymptotics for heavy-tailed arrival processes!

In order to gain some initial insight, it is useful to sharpen the simple lowerbound P{A > q + C} for P{Q > q}

First of all, we can consider arrivals t slots ago, rather than just one slot ago:

P{Q > q} = P{supt≥0{St − Ct} > q} = P{∃t ≥ 0 : St − Ct > q}

= P{∃t ≥ 0 : A−t + St−1 − Ct > q} ≥ P{∃t ≥ 0 : A−t − Ct > q}



The above lower bound considers the event that the buffer content at time 0exceeds level q due to arrivals in just a single slot some t time units ago

The above lower bound thus ignores all arrivals in the intermediate period,and can be sharpened by imposing that these are ‘not far below average’:

P{Q > q} = P{supt≥0{St − Ct} > q} = P{∃t ≥ 0 : St − Ct > q}

= P{∃t ≥ 0 : A−t + St−1 − Ct > q}≥ P{∃t ≥ 0 : A−t − (C − E{A} + ε)(t − 1)− C > q, St−1 > (E{A} − ε)(t − 1)}= P{∃t ≥ 0 : A−t − (C − E{A} + ε)(t − 1)− C > q, ‘likely’ event for large t}≈ P{∃t ≥ 0 : A−t − (C − E{A} + ε)(t − 1)− C > q} for large q



Without proof we state that when A has a Pareto (as a typical example of aheavy-tailed) distribution, the latter probability behaves asymptotically as∑

t≥1

P{A−t − (C − E{A} + ε)(t − 1)− C > q}

which in turn behaves asymptotically as

1C − E{A} + ε

∞∑u=0

P{A > q + u} =E{A}

C − E{A} + εP{Ar > q}

Here Ar is a random variable commonly referred to as the ‘residual’, ‘excess’,or ‘overshoot’ of A, with the ‘integrated-tail’ distribution

P{Ar≥ k} =

∑∞

l=k P{A > l}E{A}

or P{Ar≥ x} =

∫∞

y=x P{A > y}dy

E{A},

depending on whether A is integer-valued or real-valued



Since this holds for arbitrarily small ε > 0, we obtain the asymptotic lowerbound for P{Q > q},

E{A}C − E{A}

P{Ar > q}

Remarkably enough, it turns out that the above lower bound is in fact asymp-totically ‘exact’,

P{Q > q} ∼E{A}

C − E{A}P{Ar > q} =

E{A}/C1− E{A}/C

P{Ar > q} as q →∞

This result is essentially Veraverbeke’s theorem for the maximum of a ran-dom walk with negative drift and heavy-tailed step sizes



Indirectly, the derivation of the above lower bound shows that the typicalway for a large buffer level to occur is the scenario that we considered:an amount of arrivals q + (C − E{A})t in a single slot t time units ago,with the buffer ‘near-empty’, followed by ‘average behavior’

Establishing the matching asymptotic upper bound is considerably harderthough, and involves a proof that all other potential scenarios for the buffercontent to reach a large level are highly implausible, do not contribute tothe probability asymptotically, and can be neglected

We will further examine this in the context of the G/G/1 queue and variousqueues with fluid input



Focus on ‘exact’ large-buffer asymptotics of workload V in steady state

Other issues

• ‘logarithmic’ many-sources asymptotics

• other performance measures (delays, queue lengths)

• impact of service discipline (priority mechanisms, PS, LPS, DPS, GPS,SRPT, LAS)


Instantaneous input

Source generates instantaneous traffic burstsaccording to renewal process (G/G/1 queue)

time

Workload

Interarrival times generally distributed with mean 1/λBurst size distribution B(x) = P{B < x} with mean β <∞Traffic intensity ρ := λβResidual burst size distribution Br(x) = 1

β

∫ xy=0(1− B(y))dy


Instantaneous input (cont’d)

Theorem [Cohen, Pakes]If Br(·) is subexponential, and ρ < 1, then

P{V > x} ∼ρ

1− ρP{Br > x} as x →∞

time

Workload

Disaster scenario:Due to SINGLE extremely large burst


Heuristic derivation

Suppose that burst of size x + y(1− ρ) or largerarrives at time −y < 0Then workload is x or larger at time 0

P{V > x} ≈ λ

∫∞

y=0P{B > x + y(1− ρ)}dy

=λ

1− ρ

∫∞

y=0P{B > x + y}dy

=λβ

1− ρ1β

∫∞

y=0P{B > x + y}dy

=ρ

1− ρP{Br > x}


Three proofs for M/G/1 queue

• Proof via sample-path upper and lower bounds (based on heuristicderivation)

• Direct proof

• Proof via Laplace-Stieltjes Transform and Bingham-Doney


1. Proof via sample-path upper and lower bounds

Lower bound: ‘easy’

P{V > x} ≥ρ

1− ρ + δ

∫∞

x(1+ε)

P{B > z}E{B}

dz

Use Law of Large Numbers to show that single big jump yields this lowerbound

Upper bound: ‘hard’

P{V > x} ≤ρ

1− ρ − δ

∫∞

x(1−ε)

P{B > z}E{B}

dz + o(x1−ν)

Include all other scenarios (like two big jumps) and show that they canasymptotically be neglected.


2. Direct proof

Pollaczek-Khinchine formula for LST of amount of work

E{e−sV} =

(1− ρ)ss − λ(1− E{e−sB})

=1− ρ

1− ρE{e−sBr}=

∞∑n=0

(1− ρ)ρn(E{e−sBr

}

)n

=

∞∑n=0

(E{e−sBr

}

)nP{N = n} = E{e−s

∑Nn=1 B

rn},

implies that V may be represented as V =st Br1+· · ·+Br

N, with N ∼ Geo(ρ),and hence

P{V > x} = (1− ρ)∞∑

n=0

ρnP{Br1 + · · · + Br

n > x}

∼ (1− ρ)∞∑

n=0

ρnnP{Br > x}

=ρ

1− ρP{Br > x} as x →∞


Some observations

Observation 1

P{Br1 + · · · + Br

n > x} ∼ nP{Br > x} as x →∞

holds for subexponential distributions

Crucial property: if sum is large, it is most likely due to single big term

Observation 2If P{B > x} ∼ x−νL(x) as x →∞, then

P{Br > x} =∫∞

x

P{B > y}E{B}

dy

∼1

(ν − 1)E{B}x1−νL(x) as x →∞

“one degree worse”


3. Proof via Laplace-Stieltjes Transform and Bingham-Doney

Key lemma for regularly varying distributions [Bingham & Doney]:If k < ν < k + 1, k ∈ N, then next two properties are equivalent:

P{X > x} ∼ x−νL(x) as x →∞

and

E{e−sX} −

k∑j=0

E{X j}(−s) j

j !∼ −0(1− ν)sνL(

1s) as s ↓ 0


3. Proof via Laplace-Stieltjes Transform and Bingham-Doney (cont’d)

If P{B > x} ∼ x−νL(x) as x →∞, 1 < ν < 2, then (Bingham-Doney):

1− E{e−sBr} = 1−

1− E{e−sB}

sE{B}

∼ −0(1− ν)E{B}

sν−1L(1/s) as s ↓ 0

Combine with Pollaczek-Khinchine formula

E{e−sV} =

(1− ρ)ss − λ(1− E{e−sB})

=1− ρ

1− ρE{e−sBr}=

1− ρ1− ρ + ρ(1− E{e−sBr

}),

to obtain, for s ↓ 0:

1− E{e−sV} ∼ −

ρ

1− ρ0(1− ν)E{B}

sν−1L(1/s)

Then apply Bingham-Doney once again


Instantaneous input (cont’d)

In contrast, for light-tailed burst size distribution,

time

Workload

Conspiracy scenario:Combination of MANY relatively large burstsand MANY relatively short interarrival times


Fluid input

Source generates fluid traffic according to On-Off process

time

Workload

Off-periods generally distributed with mean 1/λOn-period distribution A(x) = P{A < x} with mean α <∞Fraction On-time p = α/(α + 1/λ)While On, flow produces traffic at constant rate rTraffic intensity ρ := prResidual On-period distribution Ar(x) = 1

α

∫ xy=0(1− A(y))dy


Fluid input (cont’d)

Theorem [Jelenkovic & Lazar]If Ar(·) is subexponential, and ρ < 1 < r , then

P{V > x} ∼ (1− p)ρ

1− ρP{Ar > x/(r − 1)} as x →∞

time

Workload

Due to SINGLE extremely long On-period


Heuristic derivation

Suppose that On-period of length xr−1 + y 1−ρ

r−ρ or largerstarts at time −y − x

r−1 < 0Then workload is x or larger at time 0

P{V > x} ≈1

α + 1/λ

∫∞

y=0P{A >

xr − 1

+ y1− ρr − ρ

}dy

=1

α + 1/λr − ρ1− ρ

∫∞

y=0P{A >

xr − 1

+ y}dy

=α

α + 1/λr − ρ1− ρ

1α

∫∞

y=0P{A >

xr − 1

+ y}dy

= pr(1− p)

1− ρP{Ar >

xr − 1

}

= (1− p)ρ

1− ρP{Ar >

xr − 1

}


Fluid input (cont’d)

In contrast, for light-tailed On-period distribution,

time

Workload

Combination of MANY relatively long On-periodsand MANY relatively short Off-periods


Several On-Off sources

Now suppose there are N statistically identical, independent On-Off sources,with Nρ < 1 for stability

Previous results suggest that large workload typically occurs due to singleextremely long On-periodIf r + (N − 1)ρ > 1, then long On-period of just a single source is sufficientto cause persistent positive drift

Large workload is typically caused by extremely long On-period of just asingle source, while other N − 1 sources show roughly normal behavior


Several On-Off sources (cont’d)

TheoremIf A(·) is regularly varying, and Nρ < 1 < r + (N − 1)ρ, then

P{V > x} ∼ NP{V1−(N−1)ρ > x} as x →∞

Here V1−(N−1)ρ represents workload in system with just a single source andservice rate c = 1 − (N − 1)ρ, i.e., original rate reduced by average rate ofother N − 1 sources

P{Vc > x} ∼ (1− p)ρ

c − ρP{Ar > x/(r − c)} as x →∞



What if r + (N − 1)ρ < 1?Multiple sources must experience long simultaneous On-periods in orderfor workload to build up

Let M be such that (M − 1)r + (N − M + 1)ρ < 1 < Mr + (N − M)ρThus, M is minimum number of sources that must be On in order to causepersistent positive drift

Large workload is typically caused by extremely long simultaneous On-periods of exactly M sources, while other N − M sources show roughlynormal behavior



TheoremIf A(·) is regularly varying, and (M−1)r + (N −M+1)ρ < 1 < Mr + (N −M)ρ, then

P{V > x} ∼(

NM

)P{V1−(N−M)ρ{1,...,M} > x} as x →∞

Here Vc{1,...,M} represents workload in critical system with only M sources

and service rate c = 1− (N −M)ρ, i.e., original rate reduced by average rateof other N − M sources

P{Vc{1,...,M} > x} ∼ G M

(P{Ar > x/(Mr − c)}

)M as x →∞

G M is some constant determined by an integral expression, which capturesgeometric probabilistic structure of overlap of M simultaneous On-periods


Delay and impact of service discipline

So far we have focused on workload asymptotics

Workload asymptotics hold for any service discipline

For FCFS, waiting time has same distribution as workload so

P{WFC F S > x} ∼ρ

1− ρP{Br > x} as x →∞

and

P{SFC F S > x} = P{WFC F S + B > x} ∼ P{WFC F S > x} ∼ρ

1− ρP{Br > x}

However, for non-FCFS service disciplines there is no simple relationshipin general between workload and waiting or sojourn time


Delay and impact of service discipline (cont’d)

Some service disciplines of interest

• Processor Sharing (PS)

– total service capacity is shared fairly among all customers

– when there are n customers in system, each receives fraction 1/n oftotal service capacity

– provides idealization of Round-Robin (RR) scheduling

• Limited Processor Sharing (LPS)

– total service capacity shared among up to M customers

– when there are n customers in system, oldest min{M, n} of them eachreceive fraction 1/min{M, n} of total service capacity

– remaining n −min{M, n} = max{n − M, 0} customers are waiting




• Discriminatory Processor Sharing (DPS)

– there are K customer classes with weights w1, w2, . . . , wK

– when there are nk class-k customers, k = 1, . . . , K , each class-l cus-tomer receives fraction wl

w1n1+w2n2+···+wK nKof total service capacity

– provides abstraction for service differentiation mechanisms, e.g.,Weighted Round-Robin (WRR) scheduling

• Generalized Processor Sharing (GPS):

– there are K customer classes with weights w1, w2, . . . , wK

– when subset of non-empty classes is L ⊆ {1, 2, . . . , K }, each classl ∈ L receives fraction wl∑

k∈Lwknkof total service capacity (which may

then be shared among the class-l customers in various ways)

– provides abstraction for service differentiation mechanisms such asWRR, but allocation on per-class rather than per-customer basis




• Shortest Remaining Processing Time First (SRPT)

– assigns priority to customer with shortest remaining service time

– minimizes number of customers in system sample-path wise(and hence expected sojourn time because of Little’s law)

– requires advance knowledge of service times

• Least Attained Service First (LAS)

– assigns priority to customer with least amount of service received sofar

– does not require advance knowledge of service times

– minimizes number of customers in system in distribution when ser-vice time distribution has decreasing failure rate(and hence expected sojourn time because of Little’s law)



For broad class of ‘preemptive’ service disciplines (PS, DPS, SRPT, LAS), ifB(·) is regularly varying, then

P{S > x} ∼ P{B > (1− ρ)x} as x →∞

In particular, for regularly varying distributions, sojourn time has same tailindex ν as service time, rather than one degree worse ν − 1 as for FCFS

This agrees with fact that E{S} <∞ even when E{B2} = ∞ for these service

disciplines, and reflects some notion of tail optimality

Interpretation

• During sojourn time of customer with large service time, other cus-tomers will take away fraction ρ of server capacity

• Thus large service time (1− ρ)x will result in sojourn time x


References

S. Aalto, U. Ayesta, S.C. Borst, V. Misra, R. Núñez-Queija (2007). BeyondProcessor Sharing. Perf. Eval. Rev. 34 (4), 36–43.R. Agrawal, A.M. Makowski, Ph. Nain (1999). On a reduced load equiva-lence for fluid queues under subexponentiality. Queueing Systems 33, 5–41.V. Anantharam (1999). Scheduling and long-range dependence. QueueingSystems 33, 73–89.N.H. Bingham, R.A. Doney (1975). Asymptotic properties of supercriticalbranching processes I: The Galton-Watson process. Adv. Appl. Prob. 6, 711–731.N.H. Bingham, C. Goldie, J. Teugels (1987). Regular Variation. CambridgeUniversity Press.S.C. Borst, O.J. Boxma, R. Núñez-Queija, A.P. Zwart (2003). The impact ofthe service discipline on delay asymptotics. Perf. Eval. 54 (2), 175–206.S.C. Borst, R. Núñez-Queija, A.P. Zwart (2006). Sojourn time asymptoticsin processor-sharing queues. Queueing Systems 53, 31–51.


References (cont’d)

O.J. Boxma, A.P. Zwart (2007). Tails in scheduling. Perf. Eval. Rev. 34 (4),13–20.J.W. Cohen (1973). Some results on regular variation for distributions inqueueing and fluctuation theory. J. Appl. Prob. 10, 343–353.P. Embrechts, C. Klüppelberg, T. Mikosch (1997). Modelling Extremal Events.Springer, Berlin.P. Embrechts, N. Veraverbeke (1982). Estimates for the probability of ruinwith special emphasis on the possibility of large claims. Insurance: Math.Econ. 1, 55–72.F. Guillemin, Ph. Robert, A.P. Zwart (2003). Tail asymptotics for processor-sharing queues. Adv. Appl. Prob. 36 (2), 525–543.P.R. Jelenkovic, A.A. Lazar (1999). Asymptotic results for multiplexingsubexponential on-off processes. Adv. Appl. Prob. 31, 394–421.J.K. Nair, A. Wierman, A.P. Zwart (2010). Tail-robust scheduling via limitedProcessor Sharing. Perf. Eval. 67, 978–995.


References (cont’d)

R. Núñez-Queija (2000). Processor-Sharing Models for Integrated-ServicesNetworks. PhD Thesis Eindhoven University of Technology.A.G. Pakes (1975). On the tails of waiting-time distributions. J. Appl. Prob.12, 555–564.N. Veraverbeke (1977). Asymptotic behaviour of Wiener-Hopf factors of arandom walk. Stoch. Proc. Appl. 5, 27–37.A.P. Zwart (2001). Queueing Systems with Heavy Tails. PhD Thesis Eind-hoven University of Technology.A.P. Zwart, O.J. Boxma (2000). Sojourn time asymptotics in the M/G/1processor-sharing queue. Queueing Systems 35, 141–166.

lnmb course asymptotic methods in queueing theorysem/asqt/lecture02032015.pdf · 2015-03-10 ·...

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