Energy-Efficient Broadcast Scheduling. Speed-Controlled Transmission Channels

Transcription

1 for Speed-Controlled Transmission Channels Joint work with Christian Gunia from Freiburg University in ISAAC

2 Outline Problem Definition and Motivation 1 Problem Definition and Motivation 2 3 4

3 Outline Problem Definition and Motivation 1 Problem Definition and Motivation 2 3 4

4 Scenario A server that stores messages and clients that request messages. Communication done via wireless broadcasts. Speed of broadcasts can be chosen independently. For a given α 2 sending for one time unit at speed s consumes s α energy. Broadcasts are non-preemptive (e.g., sensor data).

5 Scenario A server that stores messages and clients that request messages. Communication done via wireless broadcasts. Speed of broadcasts can be chosen independently. For a given α 2 sending for one time unit at speed s consumes s α energy. Broadcasts are non-preemptive (e.g., sensor data).

6 Problem Definition Given Server stores a single message. n requests for that message. Each request has an individual release time and deadline. Wanted Schedule of broadcasts that answers each request and consumes least energy. Request is answered if the message is broadcasted completely at least once between its release time and deadline.

7 Problem Definition Given Server stores a single message. n requests for that message. Each request has an individual release time and deadline. Wanted Schedule of broadcasts that answers each request and consumes least energy. Request is answered if the message is broadcasted completely at least once between its release time and deadline.

8 A Simple Example

9 A Simple Example

10 A Simple Example

11 A Simple Example

12 Related Results Flowtime minimization of broadcasts [Bansal et al., 2006], [Edmonds et al., 2003] Transmission Energy used to reduce interference [Burkhart et al., 2004], [Moscibroda et al., 2005] Energy minimization of job scheduling [Yao et al., 1995], [Bansal et al., 2004], [Chan et al., 2007]

13 Outline Problem Definition and Motivation 1 Problem Definition and Motivation 2 3 4

14 Request (t, d) arrives at time t Channel idle? No Yes

15 Request (t, d) arrives at time t Start BC (t, f) Channel idle? No Yes f d

16 Request (t, d) arrives at time t Start BC (t, f) Yes Channel idle? No Half of request left? No Yes f d

17 Request (t, d) arrives at time t Start BC (t, f) Yes Channel idle? No t Half of request left? f No Yes f d

18 Request (t, d) arrives at time t Start BC (t, f) Yes Channel idle? No t Half of request left? f No Yes f d

19 Request (t, d) arrives at time t Start BC (t, f) Yes Channel idle? No t Half of f d request left? No Yes f d

20 Request (t, d) arrives at time t Start BC (t, f) Yes Channel idle? Yes No t Half of f d request left? No f t f d d f

21 Request (t, d) arrives at time t Start BC (t, f) Yes Channel idle? No Half of request left? No Abort current BC f min{f, d} Yes f d

22 Request (t, d) arrives at time t Put request on hold Start BC (t, f) Yes Channel idle? No Half of request left? No Abort current BC f min{f, d} Yes f d

23 Theorem Let E ON denote the energy consumption of algorithm ONLINE-SM and E OPT the optimal energy consumption. It holds that ( ) α 2 E ON 2 α E α 1 OPT. Theorem For requests of identical length it holds that E ON 2α ( ) 3 α α 1 E 2 OPT.

24 Theorem Let E ON denote the energy consumption of algorithm ONLINE-SM and E OPT the optimal energy consumption. It holds that ( ) α 2 E ON 2 α E α 1 OPT. Theorem For requests of identical length it holds that E ON 2α ( ) 3 α α 1 E 2 OPT.

25 Sketch of Proof Neglect cost of aborted broadcasts Factor 1 + 1/(α 1). Length of completed broadcast is at least half the length of initiating request.

26 Sketch of Proof Neglect cost of aborted broadcasts Factor 1 + 1/(α 1). Length of completed broadcast is at least half the length of initiating request. b

27 Sketch of Proof Neglect cost of aborted broadcasts Factor 1 + 1/(α 1). Length of completed broadcast is at least half the length of initiating request. b R

28 Sketch of Proof Neglect cost of aborted broadcasts Factor 1 + 1/(α 1). Length of completed broadcast is at least half the length of initiating request. b R E 1 E 2 E 3

29 Sketch of Proof E 1 : Every broadcast has deadline after f, thus, must be slow and cheap at this point. E 2 /E 3 : Every broadcast has length b /2, at most one broadcast is started after time s. b R E 1 E 2 E 3

30 Theorem The competitive ratio of every (randomized) online algorithm for single-message broadcasting is Ω((2 ε) α ) for any ε > 0. High Level Ideas of Proof Sequentially construct requests of length 2 i for i = 0, 1,...,n. Constructed such that OPT = O(1) 2 n α. Hence, each broadcast of algorithm has to be of length at least (1/2 + ε) r n. Show that some interval of the very last request is not used for broadcast: gap. Show that gap s relative length increases at constant rate.

31 Theorem The competitive ratio of every (randomized) online algorithm for single-message broadcasting is Ω((2 ε) α ) for any ε > 0. High Level Ideas of Proof Sequentially construct requests of length 2 i for i = 0, 1,...,n. Constructed such that OPT = O(1) 2 n α. Hence, each broadcast of algorithm has to be of length at least (1/2 + ε) r n. Show that some interval of the very last request is not used for broadcast: gap. Show that gap s relative length increases at constant rate.

32 How to Construct Such a Sequence? 2 i

33 How to Construct Such a Sequence? 2 i δ δ+ε 2 i 2 i

34 How to Construct Such a Sequence? 2 i δ δ+ε 2 i 2 i

35 How to Construct Such a Sequence? 2 i δ δ+ε 2 i 2 i δ+ε 2 i+1

36 How to Construct Such a Sequence? 2 i δ δ+ε 2 i 2 i δ+ε 2 i+1

37 δ δ+ε δ+ε 2 i 2 i 2 i+1 δ δ+ε δ+ε 2 i 2 i 2 i+1 δ+ε 2 i+1 (a) Front to front. (b) Front to back. ε δ ε δ 2 i 2 i 2 i 2 i δ+ε 2 i+1 δ+ε 2 i+1 (c) Back to front. (d) Back to back. δ+ε 2 i+1

38 has optimal asymptotic competitive ratio. Good in theory.... is extremely simple, thus, fast and easy to implement. Good in practice.

39 has optimal asymptotic competitive ratio. Good in theory.... is extremely simple, thus, fast and easy to implement. Good in practice.

40 Outline Problem Definition and Motivation 1 Problem Definition and Motivation 2 3 4

41 Algorithm ONLINE-MM can be (canonically) extended to multi-message version. Theorem Let E MM denote the energy consumption of algorithm ONLINE-MM for any sequence of requests with length varying between l and cl for some fixed c 1. It holds that E MM (4c 1) α E OPT.

42 Algorithm ONLINE-MM can be (canonically) extended to multi-message version. Theorem Let E MM denote the energy consumption of algorithm ONLINE-MM for any sequence of requests with length varying between l and cl for some fixed c 1. It holds that E MM (4c 1) α E OPT.

43 Speed Adaptation What if we allow the algorithm to adapt the speed of a running broadcast when new requests arrive? Theorem The competitive ratio of any (randomized) online algorithm for single-message broadcasting capable of speed adaptation is ω((γ ε) α ) ε > 0 and γ = ( )/( ) If requests have identical length the competitive ratio is Ω(1.09 α ).

44 Speed Adaptation What if we allow the algorithm to adapt the speed of a running broadcast when new requests arrive? Theorem The competitive ratio of any (randomized) online algorithm for single-message broadcasting capable of speed adaptation is ω((γ ε) α ) ε > 0 and γ = ( )/( ) If requests have identical length the competitive ratio is Ω(1.09 α ).

45 Outline Problem Definition and Motivation 1 Problem Definition and Motivation 2 3 4

46 Upper/lower bounds for the general multi-message case. Non-preemptive job scheduling. Improved algorithms using speed adaptation. s to preemptive broadcasting.

47 Upper/lower bounds for the general multi-message case. Non-preemptive job scheduling. Improved algorithms using speed adaptation. s to preemptive broadcasting.

48 Upper/lower bounds for the general multi-message case. Non-preemptive job scheduling. Improved algorithms using speed adaptation. s to preemptive broadcasting.

49 Upper/lower bounds for the general multi-message case. Non-preemptive job scheduling. Improved algorithms using speed adaptation. s to preemptive broadcasting.

50 Thanks for your attention.