Peak load reduction for distributed backup scheduling


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1 Peak load reduction for distributed backup scheduling Peter van de Ven joint work with Angela Schörgendorfer (Google) and Bo Zhang (IBM Research)
2
3 My research interests are in the design and analysis of large (distributed) stochastic systems:  performance analysis; scheduling; stochastic optimization and control
4 Part I: Peak load reduction for distributed backup scheduling
5 load 8am 6pm Load is highly varying throughout the day  network costs are nonlinear in the hourly load  aim to reduce network costs by shifting peak load
6 load backups 8am 6pm  the load is comprised of different types of traffic  we focus on the backup traffic
7 load backups 8am 6pm Users (workstations) aim to do daily backups  the backup transmitted across the network to a central server  backup traffic can be shifted because of delaytolerance
8 load load backups backups 8am 6pm 8am 6pm current target Backups are initiated by the users, based on local information only  backup policy is characterized by hourly backup probabilities However  users are not always connected  the server may have some capacity constraint
9 load load backups backups 8am 6pm 8am 6pm current target Question 1: How much traffic can the network handle for given backup probabilities? Question 2: How to choose the backup probabilities to minimize the network costs while ensuring regular backups?
10 Model outline
11 N users network server Consider N users that generate data (bits) over time  user backup to a central server  access to server over a data network
12 A 1(t) B(t) i A 2(t) A 3(t) A 4(t) A 5(t) network server  time is slotted  user i generates A i (t) bits in slot t  data is added to the backlog B i (t)
13 A 1(t) B(t) i A 2(t) A 3(t) A 4(t) A 5(t) network C i (t) server  users can only do a backup when connected to the network  connectivity is represented by C i (t) {0, 1}, i = 1,..., N
14 A 1(t) B(t) i σ i (t) A 2(t) A 3(t) A 4(t) A 5(t) network C i (t) server Connected users may decide to do a backup (σ i (t) = 1) or not (σ i (t) = 0)
15 A 1(t) B(t) i σ i (t) A 2(t) A 3(t) A 4(t) A 5(t) network C i (t) server Connected users may decide to do a backup (σ i (t) = 1) or not (σ i (t) = 0) The backlog evolution and offered backup load are given by B i (t + 1) = (A i (t) + B i (t))(1 C i (t)σ i (t)), N H(t) = (A i (t) + B i (t))c i (t)σ i (t) i=1
16 User type & Backup probability Consider the number of days since backup W i (t) {0, 1,... } (user type): W i (t + 1) = W i (t) ( 1 σ i (t)c i (t) ) + 1 {η(t+1)=0}, with η(t) := t mod T the hour of the day
17 User type & Backup probability Consider the number of days since backup W i (t) {0, 1,... } (user type): W i (t + 1) = W i (t) ( 1 σ i (t)c i (t) ) + 1 {η(t+1)=0}, with η(t) := t mod T the hour of the day The backup decision is random and may depend on W i (t) and the hour of the day u(t) {0,..., T 1}: { 1 w.p. ν(u(t), Wi (t)) σ i (t) = 0 otherwise
18 Capacity & network cost minimization
19 Let W(t) = (W 1 (t),..., W N (t)) and consider the Markov process U := {(η(t), W(t)), t = 0, 1,... } For fixed ν, we can analyze U to obtain insights into the backup system
20 The stationary distribution π of the process U satisfies where π(u, w) = π(0, 1)k 1 (u, w), u = 0,..., T 1; w 1, π(u, 0) = π(0, 1)k 2 (u), u = 1,..., T 1, w 1 k 1 (u, w) := m(0, u 1, w) m(0, T 1, y), k 2 (u) := u v=1 y=1 and with normalizing constant [ π(0, 1) = y=1 k 1 (v 1, y)c(v 1)ν(v 1, y), T 1 w=1 u=0 T 1 k 1 (u, w) + u=1 k 2 (u)] 1.
21 Let u m(u, u, w) := (1 c(v)ν(v, w)), u, u = 0,..., T 1; w = 0, 1,..., v=u and define g(u, w) := 1 m(u, T 1, w)m(0, u 1, w + 1), u, u = 0,..., T 1; w = 0, 1,..., Proposition If α(u) := lim w w( g(u, w) w 1) exists and α(u) (0, ) for all u {0, 1,..., T 1}, then U is positive recurrent.
22 Lemma The expected backlog size of a typew user at hour u: S(u, w) = where T 1 u =0 = p(u, u, w)e π [ B(t) η(t) = u, W (t) = w, η(τ (t)) = u ]. [ E π B(t) η(t) = u, W (t) = w, η(τ (t)) = u ] p(u, u, w) = u 1 v=u +1 T 1 v=u +1 w =0 w =0 a(v), w = 0, a(v) + (w 1)â + u 1 v=0 a(v), w 1, π(u,w ) π(u,0) c(u )ν(u, w )m(u + 1, u 1, 0), w = 0, π(u,w ) π(0,1) c(u )ν(u, w )m(u + 1, T 1, 0), w 1.
23 load B(t) L(t) 8am 6pm We consider the sum of H(t) (backup load) and L(t) (extraneous load)  denote H(t; ν) to reflect the impact of ν
24 Network costs log10(kb) The data network is associated with a loaddependent hourly cost function g : R R We are interested in the average perday network costs (T = 24) G(ν) = T 1 u=0 E{g ( L(u) + H(u; ν) ) },
25 We approximate G(ν) G(ν) := T 1 u=0 g ( E [ L(u) ] + E π [ H(u; ν) ]), The E [ L(u) ] can be obtained from measurements. Exploiting user independence: E π [H(u; ν)] = NT π(u, w) ( S(u, w) + a(u) ) c(u)ν(u, w), w=0
26 We aim to minimize the network costs, while ensuring regular backups min ν G(ν) such that T w=w j π(0, w) γ j, j = 1, 2,.... This problem is nonconvex.
27 Implementation
28 We consider an IBM location with N = 5397 users 25 L u c u u u load L(u) connectivity c(u)
29 w = 1 w = 2 w = 3 w = 4 w 5 We truncate the state space at w = 5 and compute the ν(u, w)  the probability is high at night and low during working hours  users that have not done a backup in a while are more likely to schedule during peak hours
30 Comparison to original scheduling table Europe team s schedule: = Our schedule: =
31 Comparison to original scheduling table  Performance days without backup total load
32 Constrained system β Consider a constraint on the total amount of backup traffic: B i (t + 1) = (A i (t) + B i (t))(1 C i (t)σ i (t) min{1, β/b(t)}) Users are no longer independent  stability conditions and proofs are more involved  use mean field limit to compute longterm average behavior
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