Scheduling Multipacket Frames With Frame Deadlines Lukasz Je z - - PowerPoint PPT Presentation

Scheduling Multipacket Frames With Frame Deadlines

• Lukasz Je˙

z Yishay Mansour Boaz Patt-Shamir

Eindhoven University of Technology & University of Wroc law Microsoft Research & Tel Aviv University

SIROCCO, July 15, 2015

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 1 / 11

Introduction

High-level overview

heterogeneous network flows through a single link data frames, consisting of packets packets of each frame roughly periodic maximize # frames completed by their deadlines

Motivating examples

VoIP video streaming; compression level determines size and period

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 2 / 11

Model

MAX-Objective: total value of completed frames

Frames arrive online; f arrives at t(f ) and reveals: vf : value kf : size (no. packets) df : period ∆f : jitter sf : slack

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 3 / 11

Model

MAX-Objective: total value of completed frames

Frames arrive online; f arrives at t(f ) and reveals: vf : value kf : size (no. packets) df : period ∆f : jitter sf : slack

arrival deadline 2∆ s

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 3 / 11

Model

MAX-Objective: total value of completed frames

Frames arrive online; f arrives at t(f ) and reveals: vf : value kf : size (no. packets) df : period ∆f : jitter sf : slack

arrival deadline 2∆ s

f ’s packets arrive with period df and jitter up to ±∆f slots, i.e., i-th one arrives in [t(f ) + (i − 1)df − ∆f , t(f ) + (i − 1)df + ∆f ]

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 3 / 11

Model

MAX-Objective: total value of completed frames

Frames arrive online; f arrives at t(f ) and reveals: vf : value kf : size (no. packets) df : period ∆f : jitter sf : slack

arrival deadline 2∆ s

f ’s packets arrive with period df and jitter up to ±∆f slots, i.e., i-th one arrives in [t(f ) + (i − 1)df − ∆f , t(f ) + (i − 1)df + ∆f ] sf : #steps to complete f since its last packet’s latest possible arrival; determines deadline Df = t(f ) + (kf − 1)df + ∆f + sf

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 3 / 11

Model

MAX-Objective: total value of completed frames

Frames arrive online; f arrives at t(f ) and reveals: vf : value kf : size (no. packets) df : period ∆f : jitter sf : slack

arrival deadline 2∆ s

f ’s packets arrive with period df and jitter up to ±∆f slots, i.e., i-th one arrives in [t(f ) + (i − 1)df − ∆f , t(f ) + (i − 1)df + ∆f ] sf : #steps to complete f since its last packet’s latest possible arrival; determines deadline Df = t(f ) + (kf − 1)df + ∆f + sf Can transmit one packet per time slot.

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 3 / 11

Special Cases and Relations to Job Scheduling

Restricted Instance Classes

Perfectly Periodic Instances (PPI): ∆f = 0 and sf = df for all f (nearly) uniform in Π: ∀ π ∈ Π πmax

πmin ∈ O(1)

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 4 / 11

Special Cases and Relations to Job Scheduling

Restricted Instance Classes

Perfectly Periodic Instances (PPI): ∆f = 0 and sf = df for all f (nearly) uniform in Π: ∀ π ∈ Π πmax

πmin ∈ O(1)

Relations to classic job scheduling

PPIs with df = 1 for all f : interval scheduling on single machine same with arbitrary slack: job scheduling on a single machine

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 4 / 11

Special Cases and Relations to Job Scheduling

Restricted Instance Classes

Perfectly Periodic Instances (PPI): ∆f = 0 and sf = df for all f (nearly) uniform in Π: ∀ π ∈ Π πmax

πmin ∈ O(1)

Relations to classic job scheduling

PPIs with df = 1 for all f : interval scheduling on single machine same with arbitrary slack: job scheduling on a single machine same with df = m for all f : akin to job scheduling on m machines

1: 2:

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 4 / 11

Special Cases and Relations to Job Scheduling

Restricted Instance Classes

Perfectly Periodic Instances (PPI): ∆f = 0 and sf = df for all f (nearly) uniform in Π: ∀ π ∈ Π πmax

πmin ∈ O(1)

Relations to classic job scheduling

PPIs with df = 1 for all f : interval scheduling on single machine same with arbitrary slack: job scheduling on a single machine same with df = m for all f : not quite job scheduling on m machines

1: 2:

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 4 / 11

Immediate Upper Bounds

Upper Bounds (Classify and Randomly Select)

PPIs nearly uniform in {v, k, d}: O(1)-comp easy extends to O(log vmax

vmin · log kmax kmin · log dmax dmin ) for general PPIs

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 5 / 11

Immediate Upper Bounds

Upper Bounds (Classify and Randomly Select)

PPIs nearly uniform in {v, k, d}: O(1)-comp easy extends to O(log vmax

vmin · log kmax kmin · log dmax dmin ) for general PPIs

an O(1)-comp alg for PPIs nearly uniform in {v/k, d} would reduce log vmax

vmin · log kmax kmin to log vmax vmin + log kmax kmin

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 5 / 11

Immediate Lower Bounds

Interval scheduling on identical machines (Azar & Gilon ’15)

Ω(log µ) LB for PPIs with all df = 1; µ = min{ vmax

vmin , kmax kmin }

not clear if it extends to PPIs with df = d > 1

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 6 / 11

Immediate Lower Bounds

Interval scheduling on identical machines (Azar & Gilon ’15)

Ω(log µ) LB for PPIs with all df = 1; µ = min{ vmax

vmin , kmax kmin }

not clear if it extends to PPIs with df = d > 1

Slack Requirement for non-PPIs: sf ∈ Ω(∆f )

uniform instance; large k and ∆, d ≥ 2∆ + s; frames arrive at time 0.

arrival possible arrival of last packet slack and deadline

ratio ≥

2∆+s 2∆/k+s , i.e., Ω(k) if s/∆ → 0.

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 6 / 11

Our results

O(1)-competitive algorithms for:

instances with nearly uniform periods and densities PPIs with uniform size and value

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 7 / 11

Our results

O(1)-competitive algorithms for:

instances with nearly uniform periods and densities PPIs with uniform size and value

Note on 1st result

implies O

• log dmax

dmin

• log vmax

vmin + log kmax kmin

• ratio for general instances

mild assumptions: sf ∈ Ω(∆f ) and df ∈ Ω(sf ) for all f .

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 7 / 11

Instances with nearly uniform periods and densities

Ignore frame values, focus on sizes; Charge packets when OPT sends them.

Algorithm Overview (ignoring frames of slack < dmin)

Frame accepted or rejected upon arrival (may later be preempted by a frame ≥ twice its size) # active (accepted, neither preempted nor completed) frames ≤ dmin (allows sending their packets within dmin steps of latest arrival)

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 8 / 11

Instances with nearly uniform periods and densities

Ignore frame values, focus on sizes; Charge packets when OPT sends them.

Algorithm Overview (ignoring frames of slack < dmin)

Frame accepted or rejected upon arrival (may later be preempted by a frame ≥ twice its size) # active (accepted, neither preempted nor completed) frames ≤ dmin (allows sending their packets within dmin steps of latest arrival) Charging (≈ interval scheduling): to chains of frames (via credit)

last/completed preempted extra cover by the chain

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 8 / 11

Instances with nearly uniform periods and densities (2)

Remainder: frames of slack < dmin

Reservations for last packets (cannot wait up to dmin)

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 9 / 11

Instances with nearly uniform periods and densities (2)

Remainder: frames of slack < dmin

Reservations for last packets (cannot wait up to dmin) In order not to delay other packaets:

◮ reduce # active ≤ dmin/2 ◮ have low slack frames Remain “active” dmin steps after completion

Charging of f : when OPT completes it, f charged to its “slack interval”, fully reserved by f ′ s.t. 2kf ′ ≥ kf .

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 9 / 11

Instances with nearly uniform periods and densities (2)

Remainder: frames of slack < dmin

Reservations for last packets (cannot wait up to dmin) In order not to delay other packaets:

◮ reduce # active ≤ dmin/2 ◮ have low slack frames Remain “active” dmin steps after completion

Charging of f : when OPT completes it, f charged to its “slack interval”, fully reserved by f ′ s.t. 2kf ′ ≥ kf . Maximal intervals in the union of slack intervals (for f of size ≥ 1, 2, . . .):

fully reserved charged ≤ 2∆f∗ ≥ sf∗ ∈ Ω(∆f∗)

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 9 / 11

PPIs with uniform sizes and values

Initial Remarks

LBs: none for PPIs; no non-preemptive busy alg is 1-comp

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 10 / 11

PPIs with uniform sizes and values

Initial Remarks

LBs: none for PPIs; no non-preemptive busy alg is 1-comp Give such ALG that is 17-comp:

◮ accept a frame if feasible with all accepted so far ◮ send packet of accepted frame with earliest deadline

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 10 / 11

PPIs with uniform sizes and values

Initial Remarks

LBs: none for PPIs; no non-preemptive busy alg is 1-comp Give such ALG that is 17-comp:

◮ accept a frame if feasible with all accepted so far ◮ send packet of accepted frame with earliest deadline

Analysis Outline

each accepted frame completed ⇒ charge packets (intervals) f rejected at t: by Hall’s theorem, ∃T ≥ Df s.t. ALG sends ≥ T−t

2

− k ≥ k(df

2 − 1) packets of deadlines ≤ T in [t, T).

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 10 / 11

PPIs with uniform sizes and values

Initial Remarks

LBs: none for PPIs; no non-preemptive busy alg is 1-comp Give such ALG that is 17-comp:

◮ accept a frame if feasible with all accepted so far ◮ send packet of accepted frame with earliest deadline

Analysis Outline

each accepted frame completed ⇒ charge packets (intervals) f rejected at t: by Hall’s theorem, ∃T ≥ Df s.t. ALG sends ≥ T−t

2

− k ≥ k(df

2 − 1) packets of deadlines ≤ T in [t, T).

Let I be the set of such intervals for all rejected frames of OPT intervals in I may overlap, but ∃ I′ ⊆ I s.t.

• I∈I′

I =

• I∈I

I and |

• I∈I′

I| =

• I∈I′

|I| ≥ 1 2

• I∈I

|I|

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 10 / 11

Conclusion

Thank You!

• Lukasz Je˙

z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 11 / 11