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: v f : value k f : size (no. packets) d f : period ∆ f : jitter s f : 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: v f : value k f : size (no. packets) d f : period ∆ f : jitter s f : slack arrival deadline s 2∆ � 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: v f : value k f : size (no. packets) d f : period ∆ f : jitter s f : slack arrival deadline s 2∆ f ’s packets arrive with period d f and jitter up to ± ∆ f slots, i.e., i -th one arrives in [ t ( f ) + ( i − 1) d f − ∆ f , t ( f ) + ( i − 1) d f + ∆ 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: v f : value k f : size (no. packets) d f : period ∆ f : jitter s f : slack arrival deadline s 2∆ f ’s packets arrive with period d f and jitter up to ± ∆ f slots, i.e., i -th one arrives in [ t ( f ) + ( i − 1) d f − ∆ f , t ( f ) + ( i − 1) d f + ∆ f ] s f : #steps to complete f since its last packet’s latest possible arrival; determines deadline D f = t ( f ) + ( k f − 1) d f + ∆ f + s 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: v f : value k f : size (no. packets) d f : period ∆ f : jitter s f : slack arrival deadline s 2∆ f ’s packets arrive with period d f and jitter up to ± ∆ f slots, i.e., i -th one arrives in [ t ( f ) + ( i − 1) d f − ∆ f , t ( f ) + ( i − 1) d f + ∆ f ] s f : #steps to complete f since its last packet’s latest possible arrival; determines deadline D f = t ( f ) + ( k f − 1) d f + ∆ f + s f 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 s f = d f for all f (nearly) uniform in Π: ∀ π ∈ Π π max π min ∈ 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 s f = d f for all f (nearly) uniform in Π: ∀ π ∈ Π π max π min ∈ O (1) Relations to classic job scheduling PPIs with d f = 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 s f = d f for all f (nearly) uniform in Π: ∀ π ∈ Π π max π min ∈ O (1) Relations to classic job scheduling PPIs with d f = 1 for all f : interval scheduling on single machine same with arbitrary slack: job scheduling on a single machine same with d f = 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 s f = d f for all f (nearly) uniform in Π: ∀ π ∈ Π π max π min ∈ O (1) Relations to classic job scheduling PPIs with d f = 1 for all f : interval scheduling on single machine same with arbitrary slack: job scheduling on a single machine same with d f = 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 v max v min · log k max k min · log d max d min ) 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 v max v min · log k max k min · log d max d min ) for general PPIs an O (1)-comp alg for PPIs nearly uniform in { v / k , d } would reduce log v max v min · log k max k min to log v max v min + log k max k min � 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 d f = 1; µ = min { v max k max v min , k min } not clear if it extends to PPIs with d f = 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 d f = 1; µ = min { v max k max v min , k min } not clear if it extends to PPIs with d f = d > 1 Slack Requirement for non-PPIs: s f ∈ Ω(∆ f ) uniform instance; large k and ∆, d ≥ 2∆ + s ; frames arrive at time 0. slack and deadline arrival possible arrival of last packet 2∆+ s ratio ≥ 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 � � �� log d max log v max v min + log k max implies O ratio for general instances d min k min mild assumptions: s f ∈ Ω(∆ f ) and d f ∈ Ω( s f ) 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 < d min ) Frame accepted or rejected upon arrival (may later be preempted by a frame ≥ twice its size) # active (accepted, neither preempted nor completed) frames ≤ d min (allows sending their packets within d min 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 < d min ) Frame accepted or rejected upon arrival (may later be preempted by a frame ≥ twice its size) # active (accepted, neither preempted nor completed) frames ≤ d min (allows sending their packets within d min steps of latest arrival) Charging ( ≈ interval scheduling): to chains of frames (via credit) preempted last/completed 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 < d min Reservations for last packets (cannot wait up to d min ) � 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 < d min Reservations for last packets (cannot wait up to d min ) In order not to delay other packaets: ◮ reduce # active ≤ d min / 2 ◮ have low slack frames Remain “active” d min steps after completion Charging of f : when OPT completes it, f charged to its “slack interval”, fully reserved by f ′ s.t. 2 k f ′ ≥ k f . � Lukasz Je˙ z (TU/e & UWr) Multipacket Frames With Deadlines SIROCCO, July 15, 2015 9 / 11
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