The Drift Method for Heavy-Traffic Limits

Transcription

1 The Drift Method for Heavy-Traffic Limits S. T. Maguluri (GaTech), R. Srikant (UIUC) and W. Wang (UIUC) Part III: Based on Wang, Maguluri, Srikant and Ying (2017)

2 Part III: Allocating bandwidth to data transfer flows New things you will see: A different type of state-space collapse A weighted inner product Insensitivity result

3 Data file transfer A data file is transferred along a route consisting of links The file leaves the system after all the bits of the file are delivered Source! Switches Destination Servers

4 Data file transfer A data file is transferred along a route consisting of links The file leaves the system after all the bits of the file are delivered File transfer flows use the bandwidths of links Source! Switches Destination! Servers

5 Data file transfer A data file is transferred along a route consisting of links The file leaves the system after all the bits of the file are delivered File transfer flows use the bandwidths of links Goal: How should we allocate bandwidth to flows to minimize delay? Source! Switches Destination! Servers

6 Queueing model! One queue per route Each link is modeled as a processor!

7 Queueing model! 1 2 1Gbps 1Gbps 3 1Gbps 4 1Gbps One queue per route Each link is modeled as a processor! Gbps 0.5 Gbps 1Gbps 1Gbps 1Gbps 1Gbps

8 Queueing model! One queue per route Each link is modeled as a processor! 5 For each queue, the associated links all allocate the same rate to it, and the rate is equally shared by all the files in the queue Capacity constraint at each link?? Capacity constraint

9 Heavy-traffic regime arrival rates mean file sizes

10 Scaling of data transfer delay

11 Scaling of data transfer delay

12 Yes: under proportionally fair sharing with exponential file size distributions Diffusion approximation: Kang et al. (2009), Shah et al. (2014), Ye and Yao (2012), Ye and Yao (2016)

13 What about general file size distributions? File size distributions may not be exponential in practice Conjectured yes Vlasiou et al. (2014) (phase-type distributions): the diffusion limit is insensitive No interchange of limits

14 Our results

15 Intuition

16 Intuition (phase-type file sizes)

17 Proportionally fair sharing

18 Digression: Single queue in continuous time 1

19 Bound on backlog 1

20 Bound on Backlog ` 1

21 Back to proportionally fair sharing

22 Identifying the subspace for state space collapse: Exponential file sizes

23 Identifying the subspace for state space collapse: Exponential file sizes

24 Identifying the subspace for state space collapse: Exponential file sizes Links

25 Identifying the subspace for state space collapse: Exponential file sizes

26 Identifying the subspace for state space collapse: Exponential file sizes

27 Identifying the subspace for state space collapse: Phase-type file sizes

28 Identifying the subspace for state space collapse: Phase-type file sizes

29 Identifying the subspace for state space collapse: Phase-type file sizes

30 Identifying the subspace for state space collapse: Phase-type file sizes

31 Identifying the subspace for state space collapse: Phase-type file sizes

32 State-space collapse

33 State-space collapse

34 Drift conditions

35 Our drift conditions

36 Our drift conditions

37

38

39

40 function of the parameters of file size distributions

41 Backlog bounds

42 Backlog bounds (1) (2) (3)

43 Backlog bounds (1) (2) (3)! Δ # % & = 0 +!, -. =

44 Extensions Our state-space collapse result in steady state implies interchange-oflimits for the diffusion approximation in Vlasiou et al. (2014) We can also use the drift method to study weighted proportionally fair sharing and obtain asymptotically tight bounds on the weighted sum of queue lengths The stationary distribution of the diffusion limit is unknown (no skew symmetry)

45 Summary In this part, we demonstrated the drift method in the problem of bandwidth sharing, where we show asymptotically tight bounds on the backlog the state-space collapse result is of a multiplicative type the file sizes have phase-type distributions: weighted inner product needs to be constructed