Diffusion Limits for Shortest Remaining Processing Time ... · Di usion Limits for Shortest...

Diffusion Limits for Shortest Remaining Processing TimeQueues under Nonstandard Spatial Scaling

Amber Puha

California State University San Marcos

[email protected]

June 26, 2015

Amber Puha (CSUSM) SRPT Diffusion Limits June 26, 2015 1 / 36

GI/GI/SRPT Queue

Initial condition:# jobs in the system a time 0, each with a strictly positive remainingprocessing time

Arrivals:rate α ∈ (0,∞) delayed renewal process E (·) with interarrival timestandard deviation a ∈ (0,∞)

Processing Times:strictly positive, i.i.d. with distribution ν, finite positive mean β, andstandard deviation b ∈ (0,∞)

Traffic Intensty:ρ = αβ

Shortest Remaining Processing Time (SRPT) discipline:job with the shortest remaining processing time served at rate one,with preemption


GI/GI/SRPT Queue

Performance Processes

Queue Length Process: Q(·)Q(t) is the number of jobs in the system at time t.

Workload Process: W (·)W (t) the total time required to process all jobs in the system at timet, ignoring future arrivals.

Other Quantities of Interest

Sojourn Times:For a particular job, the sojourn time is the total time spent in the system.


Some Related Literature: Early Work

GI/GI/1/SRPTL. Schrage, 1968, (optimality)D.R. Smith, 1976, (optimality)

M/GI/1/SRPT or GI/M/1/SRPTL.E. Schrage and L.W. Miller, 1966, (mean sojourn time)R. Schassberger, 1990, (steady state analysis)R. Perera, 1993, (steady state analysis)

SurveyF. Schreiber, 1993, (survey)


Some Related Literature: More Recent

Computer Science/FairnessN. Bansal and M. Harchol-Balter, 2001.A. Wierman and M. Harchol-Balter, 2003.

Tail Behavior/Large DeviationsR. Nunez Queija, 2002.M. Nuyens and B. Zwart, 2006.

Optimality for Discrete DistributionsD. Down and R. Wu, 2006.

Fluid Limit Theorems (flln)D. Down, H.C. Gromoll, and A. Puha, 2009.

Steady State Mean Sojourn Time Asymptotics (ρ↗ 1)M. Lin, A. Wierman, and B. Zwart, 2011.


Outline

1 Measure Valued State Descriptor

2 Diffusion Limits for Weibull Processing Time Distributions withNonstandard Spatial Scaling

3 Diffusion Limits under Standard ScalingProcessing Time Distributions with Bounded SupportProcessing Time Distributions with Unbounded Support

4 Revisiting the Unbounded Support CasePredictions Through Fluid ModelsInvestigation Via Simulation

5 Generalization to Rapidly Varying Processing Time Distributions

6 Work in Progress for Regularly Varying Processing Time Distributions


State Descriptor for the Stochastic System

b •• • • • • •• •• • • • •• • • •• • •0 Z(t)

For each Borel set A ⊂ [0,∞),

〈1A,Z(t)〉 = # jobs in the system at time t

with remaining processing time in A.

Observe that

Q(t) ≡ 〈1,Z(t)〉 = # jobs in system at time t,

W (t) ≡ 〈χ,Z(t)〉 = immediate workload at time t,

where χ(x) = x , x ∈ [0,∞).


State Descriptor for the Stochastic System

b •• • • • • •• •• • • • •• • • •• • •0 Z(t)

Z(t) ∈M, the space of finite, nonnegative Borel measures on R+,endowed with the topology of weak convergence.

Z(·) ∈ D([0,∞),M), the Skorokhod space of r.c.l.l. paths takingvalues in M.

Z(0) ∈M0 ={ξ ∈M : 〈1{0}, ξ〉 = 0 and 〈χ, ξ〉 <∞

}.


Diffusion Limits for SRPT Queues

Sequence of GI/GI/1/SRPT Diffusion Scaled Queues

Index set: R, positive real numbers r increasing to infinity

Initial Conditions: P (Zr (0) ∈M0) = 1

Arrivals: rate αr ∈ (0,∞) delayed renewal processes E r (·) withinterarrival time standard deviation ar ∈ (0,∞)

Processing times: i.i.d. with distribution νr ∈M0, meanβr ∈ (0,∞), and standard deviation br ∈ (0,∞)

Traffic Intensties: ρr = αrβr

Measure Valued State Descriptors: Zr (·) ∈ D([0,∞),M)

Standard Diffusion Scaling: Zr (·) =Zr (r2·)

r,

Qr (·) = 〈1, Zr (·)〉 and W r (·) = 〈χ, Zr (·)〉.



Standard Heavy Traffic Conditions

Fix critical data (α, ν), i.e., ρ = 1. As r →∞,

αr → α and ar → a ∈ (0,∞),

νrw→ ν, βr → β, and br → b ∈ (0,∞), and

r (1− ρr )→ γ ∈ R.

Also, the sequences of interarrival and processing time distributionsrespectively satisfy a Lindeberg-Feller condition.



Workload Diffusion Condition

Assume that the standard heavy traffic conditions hold and

W r (0)⇒W ?0 ,

where W ?0 is a random variable. Then, by [IW ‘70],

W r (·)⇒W ?(·),

where W ?(·) is a semimartingale reflecting Brownian motion with drift−γ, variance α(a2 + b2) per unit time, and W ?(0) =d W ?

0 .


Diffusion Limits under Nonstandard Spatial Scaling

Theorem (P ‘14)

Let λ, κ > 0 and let ν satisfy 〈1(x ,∞), ν〉 = exp (−(λx)κ), x ∈ R+.Assume that

the standard and workload heavy traffic conditions hold,

νr = ν for all r ∈ R, and

the sequence of initial conditions satisfies some mild asymptoticconditions.

Then, as r →∞,κ√

ln(r)Qr (·)⇒ λW ?(·),where W ?(·) is a semimartingale reflecting Brownian motion with drift−γ, variance α(a2 + b2) per unit time, and W ?(0) =d W ?

0 .


Diffusion Limits under Nonstandard Spatial Scaling

Observations

An implication is that Qr (·)⇒ 0(·) while W r (·)⇒W ?(·) as r →∞.This has not been observed for other non-idling disciplines. Usually,Qr (·)⇒ cW ?(·) for some c ∈ (0,∞).

The spatial scaling required to obtain a nontrivial limit for the queuelength process is distribution dependent.

The corrective spatial scaling (which specifies order of magnitude)and the constant c are explicitly identified.

Acknowledgement

Simulations by R. Hunsperger and S. Malter, supported by ViaSat, wereinstrumental formulating the result.


Diffusion Limits: Bounded Support Case

Definition

Let x∗ = inf{x ∈ [0,∞) : 〈1[0,x], ν〉 = 1

}.

Bounded Support Heavy Traffic Assumptions

x∗ <∞,

for all x > x∗, limr→∞

r〈χ1(x ,∞), νr 〉 = 0, and

for all x > x∗, limr→∞〈χ21(x ,∞), ν

r 〉 = 0.


Diffusion Limits: Bounded Support Case

Theorem (Gromoll, Kruk, & P ‘11)

Assume that both the standard and bounded heavy traffic conditions hold.Also assume that the workload diffusion condition holds and that

Zr (0)⇒ W ?(0)

x∗δx∗ , as r →∞,

Then

Zr (·)⇒ W ?(·)x∗

δx∗ , as r →∞.

Remarks

Similar in spirit to results for static priority queues, where work pilesup in the lowest priority job class.

Suggests a result for the case of unbounded support.


Diffusion Limits: Unbounded Support Case

Theorem (Gromoll, Kruk & P ‘11)

Assume that the heavy traffic and workload diffusion conditions hold. Alsoassume that x∗ =∞ and

Zr (0)⇒ 0, as r →∞,

ThenZr (·)⇒ 0(·), as r →∞.

In particular,Qr (·)⇒ 0(·), as r →∞.

Observation

Note Z?(·) = 0(·) and so 〈χ,Z?(·)〉 6= W ?(·).


Predictions Through Fluid Models

Model Inputs

Data (α, ν)

α ∈ (0,∞) is the arrival rate of fluid

ν is a Borel probability measure on R+ by which the fluid is distributedas it enters the system such that

⟨1{0}, ν

⟩= 0 and 0 < β = 〈χ, ν〉 <∞.

Initial Conditions

ξ ∈M0 is a finite, nonnegative Borel measure on R+ that gives the initialdistribution of fluid such that

〈1{0}, ξ〉 = 0 and 〈χ, ξ〉 <∞.



Definition (Down, Gromoll, & P ‘09)

A Fluid Model Solution for the data (α, ν) and initial condition ξ ∈M0

is a function ζ : [0,∞)→M such that ζ(0) = ξ and

ζ(·) is right continuous,

〈χ, ζ(t)〉 = [〈χ, ξ〉+ (ρ− 1) t]+ for all t ≥ 0,

for all t ∈ [0,∞) and g ∈ C+b (R+)

⟨g1(l(t),∞), ξ + αtν

⟩≤ 〈g , ζ(t)〉 ≤ 〈g1[l(t),∞), ξ + αtν〉,

where l(t) = inf{x ∈ R+ : 〈1[0,x), ζ(t)〉 > 0}.

bζ(t)

l(t)

0 ξ + αtν|)(

0

Figure: ζ(t) is the zero measure below l(t) and ξ + αtν above l(t).



A Naive Guess

Maybe the work concentrates near the left edge l(·)? Then

l(r ·)Qr (·) ≈ W r (·).

What is known about l(·)?

Informally, l(t) can be thought of as the initial processing time of fluidbeing processed at time t. This interpretation might lead one to speculatethat l(·) is an inverse of a fluid analog of sojourn time s(·) as a function ofthe initial processing time.



A Fluid Analog for Sojourn Time

In an SRPT queue for an initial job with x units of work at time 0, thefluid analog of the sojourn time s should satisfy

s = 〈χ1[0,x], ξ〉+ αs〈χ1[0,x], ν〉.

For critical data (α, ν) such that x∗ =∞ and ξ ∈M0, let

s(x) =〈χ1[0,x], ξ〉

1− α〈χ1[0,x], ν〉=〈χ1[0,x], ξ〉α〈χ1(x ,∞), ν〉

, x ∈ [0,∞).

The Right Continuous Inverse of Sojourn Time

Then s−1 : [0,∞)→ R+ is given by

s−1(t) = inf{x ∈ [0,∞) : s(x) > t}, t ∈ [0,∞).



The Right Continuous Inverse of Sojourn Time

For critical data (α, ν) such that x∗ =∞ and ξ ∈M0, s−1 : [0,∞)→ R+

is the right continuous inverse of

s(x) =〈χ1[0,x], ξ〉α〈χ1(x ,∞), ν〉

, x ∈ [0,∞).

Lemma (Down, Gromoll, & P ‘09)

Let (α, ν) be critical data such that x∗ =∞ and ξ ∈M0. Suppose thatζ(·) is a fluid model solution for the data (α, ν) and initial condition ξ.Then

l(t) = s−1(t), for all t ∈ [0,∞).



The Naive Guess Reformulated

Under suitable asymptotic assumptions,

s−1(r ·)Qr (·) ≈ W r (·).

where s−1 : [0,∞)→ R+ is the right continuous inverse of

s(x) =〈χ1[0,x], ξ〉α〈χ1(x ,∞), ν〉

, x ∈ [0,∞).



Example: Weibull Processing Time Distributions

Let

ξ ∈M0 be such that ξ 6= 0,

ν be a Weibull distribution with rate parameter λ > 0 andshape parameter κ > 0,

α = 1/〈χ, ν〉.

Then, it can be shown that s−1(t) ∼κ√

ln(t)

λ.

Resulting Prediction

If ν is Weibull with rate parameter λ > 0 and shape parameter κ > 0, then

κ√

ln(r)Qr (·)⇒ λW ?(·).


Evidence via Simulation

Simulation Results

Two realizations with r2 = n = 107 for exponential λ=1 processing andinterarrival times. Here W r (·) and Qr (·) = ln(r)Qr (·) are respectivelyplotted in red and green. Simulations by R. Hunsperger ‘12 and S. Malter‘13, both supported by ViaSat.


Generalization to Rapidly Varying Tails

Rapidly Varying Function

A function g : R+ → R+ is said to be rapidly varying with index infinity iffor all c > 1

limx→∞

g(cx)

g(x)=∞.

Exponential functions are examples.

Slowly Varying Function

A function g : R+ → R+ is said to be slowly varying if for all c > 0

limx→∞

g(cx)

g(x)= 1.

Logarithmic functions are examples.



A Rapid Variation Condition

Let ν be Borel probability distribution such that

〈1{0}, ν〉 = 0 and 0 < 〈χ, ν〉 <∞.

Further assume that ν has no atoms and a strictly increasing cdf. Set

S(x) =1

〈χ1(x ,∞), ν〉.

Let S−1(·) denote the right continuous inverse of S(·). Further assumethat for all c > 0

limt→∞

(S−1(ct)

S−1(t)− 1

)ln(S−1(t)) = 0.



Theorem (P ‘14)

Let ν satisfy the rapid variation condition. Assume that

the standard and workload heavy traffic conditions hold,

νr = ν for all r ∈ R, and

the sequence of initial conditions satisfies some mild asymptoticconditions.

Then, as r →∞,(S−1(r)Qr (·), W r (·)

)⇒ (W ?(·),W ?(·)) ,

where W ?(·) is a semimartingale reflecting Brownian motion with drift−γ, variance α(a2 + b2) per unit time, and W ?(0) =d W ?

0 .



Simplistic Overview of Proof

For ε > 0 and r ∈ R, l rε and urε are defined such that the following hold:

1 l rε < S−1(r) < urε for each ε > 0 and r ∈ R;

2 limε↘0

limr→∞

l rεurε

= 1;

3 for each ε > 0, the conventionally scaled work and theunconventionally scaled queue mass associated with jobs withremaining processing times with below l rε or above urε tends to zero asr →∞.

b [ | ]

0

lrε S−1(r) urε

• • • •• •• • • • •• • • •• • •Zr(t)

Figure: Asymptotic concentration of the remaining processing times in [l rε , urε].



Simplistic Overview of Proof

More specifically, for ε > 0 and r ∈ R,

l rε = S−1(r(S−1(r)

)−2−ε)

and urε = S−1(r(S−1(r)

)2+ε).

Then

urεS−1(r)

=S−1

(r(S−1(r)

)2+ε)

S−1(r).

The rapid variation condition on S(x) (equivalently the slow variationcondition on S−1(r)) implies that the above ratio converges to one as rtends to infinity. Thus conditions 1 and 2 hold.


In Progress: Processing Times with Regularly Varying Tails

Regularly Varying Function

A function g : R+ → R+ is said to be regularly varying with indexγ ∈ (0,∞) if for all c > 0

limx→∞

g(cx)

g(x)= cγ .

Polynomials, rational functions and roots are examples.



Example: Pareto Type 1

Let

ν be Pareto Type 1 with pdf f (x) =k

xk+1, x > 1, some k > 2,

α = (k − 1)/k.

Then, for x > 1 and t > α,

S(x) =k

k − 1xk−1 and S−1(t) = k−1

√kt

k − 1.



Resulting Prediction

If ν has pdf f (x) =k

xk+1, x > 1, for some k > 2, then

Qr (·)⇒W ?(·),

where

Qr (·) = S−1(r)Qr (·) = k−1

√kr

k − 1Qr (·).



Simulation Results

Two realizations with r2 = n = 105 for Pareto processing times, withk =100, 20 respectively. Here W r (·) and Qr (·) are respectively plotted inred and green. Simulations by J. Mulvany ‘15, supported by the CSMSummer Scholars Program.



Simulation Results

Two realizations with r2 = n = 105 for Pareto processing times, withk =4, 2.5 respectively. Here W r (·) and Qr (·) are respectively plotted inred and green. Simulations by J. Mulvany ‘15, supported by the CSMSummer Scholars Program.



Consistency with Lin, Wierman, Zwart ‘11

They develop asymptotics for the expected sojourn time E [S ] of anM/G/1 SRPT queue in steady state as ρ↗ 1.

Little’s Law (which applies in a wide variety of contexts) says that insteady state

E [Q] = αE [S ] .

Recall the heavy traffic assumption r (1− ρr )→ γ.Replacing 1− ρ with 1/r in the asymptotic expressions for E [S ]derived in [LWZ ‘11] yields expressions of order

r

S−1(r).


Dear Ruth,Wishing you the Happiest of Birthdays!

From Nam, Christian, and Amber


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