Improved Online Scheduling in Maximizing Throughput of Equal Length Jobs Nguyen Kim Thang (university Paris-Dauphine, France)

Improved Online Scheduling in Maximizing Throughput of

Equal Length Jobs

Nguyen Kim Thang

(university Paris-Dauphine, France)

Motivation

Enterprise: perishable product (electricity, ice-cream, ...).

Clients: single-minded, arrive online, different demands.

Goal: maximize the profit.

Profit maximization

$

$

$

Objective: maximize the total value.

Other applications

ATM network: online packages, typically of the same length.

Data broadcast: online broadcast pages

Model

Objective: maximize the total value of jobs completed on time.

Online Scheduling

Jobs: arrive at , processing time , deadline , value (weight) .^pi d_i ^ri w_i

Preemption is necessary

Preemption with restart: when a job is scheduled again, it must be executed from the beginning (e.g., data

broadcast).

Preemption with resume: when a job is scheduled again, the previously done work can be resumed (e.g., ATM

network) .

Competitive ratio

An algorithm ALG is -competitive if for any instance α I

(maximization problem) OP T (I)

ALG(I) ≤ α

Competitive ratio

An algorithm ALG is -competitive if for any instance α I

(maximization problem)

Measure the performance of an algorithm (worst-case analysis) What is a competitive ratio?

The price of an object (the problem):

Algorithm negotiation Adversary (lower bound) (upper bound)

OP T (I)

ALG(I) ≤ α

Contribution

unit weight

unbounded processing times equal processing

times

bounded processing times (by )k

general

α = 1 α = Θ(log k) ∞

≤ 4.24 ^{α = Θ(k/ log k)} ∞

3√ 3

2 ≤ α ≤ 5

Contribution

unit weight

unbounded processing times equal processing

times

bounded processing times (by )k

general

α = 1 α = Θ(log k) ∞

∞

Improved algorithms for both models of preemption Weights and correlation between jobs’ deadlines

≤ 4.24 ^{α = Θ(k/ log k)}

3√ 3

2 ≤ α ≤ 5

Settling the competitivity

i

w_j

j α · w_i

Methods: charging scheme, potential function, etc

Settling the competitivity

i

w_j

j α · w_i

Methods: charging scheme, potential function, etc

ALG

ADV

Starting point

Paradox: low weight,

imminent deadline

which jobs? higher weight, later deadline

Starting point

Paradox: low weight,

imminent deadline

which jobs? higher weight, later deadline : initial job length, : length of job at time ^p q_j(t) j t

A job is pending at time if j t t + q_j(t) ≤ d_j

Starting point

Paradox: low weight,

imminent deadline

which jobs? higher weight, later deadline

If no currently scheduled job, schedule the pending one with highest weight

A 5-competitive algorithm (preemption with restart)

If a new pending job arrive with weight at least twice that of the currently scheduled job, then schedule the new one (by interrupting the current job)

At any time

: initial job length, : length of job at time ^p q_j(t) j t

A job is pending at time if j t t + q_j(t) ≤ d_j

Observations

Some job would be delayed by new urgent jobs (even with low weight)

Treatment:

A job is urgent at time if i t ^di < t + q_i(t) + p

Correlation among jobs’ deadlines is ignored

Ensure no significant lost if new heavy jobs arrive.

Algorithm

At time , let be a new released job and the currently scheduled job, respectively. At any interruption, if then ^{t i, j}

Initially, set Q = ∅, α = 0, 1 < β < 3/2

α > 0 α := α + 1

Algorithm

At time , let be a new released job and the currently scheduled job, respectively. At any interruption, if then ^{t i, j}

Initially, set Q = ∅, α = 0, 1 < β < 3/2

If dow_i ≥ 2w_j, w_i ≥ 2^αw(Q) schedule ⁱ

set Q = ∅, α = 0

α > 0 α := α + 1

Algorithm

At time , let be a new released job and the currently scheduled job, respectively. At any interruption, if then ^{t i, j}

Initially, set Q = ∅, α = 0, 1 < β < 3/2

If dow_i ≥ 2w_j, w_i ≥ 2^αw(Q) schedule ⁱ

set Q = ∅, α = 0

If doα = 0, βw_j ≤ w_i ≤ 2w_j

urgent and

i d_j ≥ t + 2p

schedule job which is

arg max{w_! : d_! < t + 2p}

set Q = {j}, α = 1 α > 0 α := α + 1

Algorithm

At time , let be a new released job and the currently scheduled job, respectively. At any interruption, if then ^{t i, j}

Initially, set Q = ∅, α = 0, 1 < β < 3/2

If dow_i ≥ 2w_j, w_i ≥ 2^αw(Q) schedule ⁱ

set Q = ∅, α = 0

If doα = 0, βw_j ≤ w_i ≤ 2w_j

urgent and

i d_j ≥ t + 2p

schedule job which is

arg max{w_! : d_! < t + 2p}

set Q = {j}, α = 1

If is urgent doi

no job such that^!

S_j(t) + 2p ≤ d_! < t + 2p, w_! ≥ w_j

schedule ⁱ

w_i ≥ 2w_j + w_j^!

α > 0 α := α + 1

The charging scheme

j i

w_i w_j

phase of job i

f (i) i

2w_f(i)

Theorem: the algorithm is -competitive^{(2 + √5)} ALG

ADV

Theorem: there is a -competitive algorithm for model of preemption with resume^{(2 +}

√5)

Conclusion

Improved algorithms for both models of preemption

Open questions:

Settling the right competitive ratio 2.5 ≤ α ≤ 4.24

Interesting: not to reduce the gap but new methods.

Thank you!

Thank Kristoffer!