UNM Load Balancing in Distributed Computing Over Wireless LAN: Effects of Network Delay S. Dhakal, M.M. Hayat, M. Elyas, J. Ghanem, C.T. Abdallah Department of Electrical and Computer Engineering University of New Mexico Albuquerque, NM 87131-0001, USA
The Load Balancing UNM Group at UNM This work is supported by the National Science Foundation under Information Technology Research (ITR) grants No. ANI-0312611 and ANI-0312182.
Goals UNM • Determine the statistics and model for load-transfer delay in wireless network in the context of distributed computing. • Determine the analytical solution to optimal load balancing (LB) in a two-node distributed system. • Understand the interplay between intensity of LB and delay and their effects on the workload completion time • Verify LB performance experimentally as well as Monte-Carlo simulation.
Overview UNM • Description of the LB policy. • Network delay characterization. • Regeneration-based queuing model for one-shot LB. • Results: analytical, experimental and MC simulation. • Conclusion and References.
Schematic Description of Distributed UNM System New Tasks Arrival N3 Tasks Served Communication Channel N2 N1 In this work, we present the initial value problem.
Description of the LB Policy UNM • There exists an exchange of information about load-states among all nodes (with a communication delay) • At the LB instant , viz., at t = t b Load LB intensity Partitioning ∈ Excess K [ 0 , 1 ] Load = ∑ p 1 kl ≠ l k Excess load at node l Load to be transferred to respective node (with load transport delay) System average load Q l (t b )
LB Policy UNM Communication delay: j to l ∈ Mathematically, at t = t b , K [ 0 , 1 ] ⎧ ⎛ ⎞ n − − − η − η ∑ ⎜ 1 ⎟ Kp Q ( t ) n Q ( t ) u ( t ) ⎪ ⎝ ⎠ kl l j lj lj = ⎪ j 1 ⎪ ⎛ ⎞ n = − − − η − η ≠ η ≤ ∑ ⎜ ⎟ ⎨ 1 L ( t ) . u Q ( t ) n Q ( t ) u ( t ) , k l , t , ⎝ ⎠ kl l j lj lj lj ⎪ = j 1 ⎪ ⎛ − ⎞ 1 ⎜ ⎟ ⎪ ( ) 1 , Kp Q t otherwise kl l ⎝ ⎠ ⎩ n where, l p ll = 0 , for n = 2 , p kl = 1 and ∀ k l ≠ ⎧ ⎛ ⎞ − η 1 Q ( t ) ⎜ − ⎟ ⎪ k lk 1 ⎜ ⎟ ⎪ − − η ∑ ⎝ ⎠ 2 ( ) n Q t = ≠ η ≤ ≠ ⎨ i l i li p , k l , t kl lk ⎪ 1 , otherwise ⎪ ⎩ − n 1
One-shot Load Balancing UNM • Each node sends to other nodes its queue length information at time t = 0. • All the nodes execute load balancing at a common instant t b with a common gain K. Remarks: Processing Speed • t b should be large enough so that Q 1 (0) each node is informed of the initial load state, but should be small enough such that much time T C1 N 1 is not wasted waiting for Transfer Delay communication. N 2 T C2 Q 2 (0) t b • K should be large enough to tackle the variability in the processor speed but should be t small enough such that transfer of t = 0 load does not take too long. Communication Delay Node 2 idle
One-shot LB over Wireless LAN UNM A. Queuing model for one-shot LB (to be described later) has been developed to accommodate: • Randomness of communication delay • Randomness of transfer delay • Load-size dependence of transfer delay • Variability in the processor speed among nodes B. Delays in wireless networks exhibit heavy random fluctuations. C. The wireless testbed is the most-suitable platform to validate the predictive ability of the analytical model. D. Delay probing experiments are conducted over the wireless LAN to estimate the channel statistics, which is later integrated in the model to develop optimal LB policy.
Delay Probing Experiments UNM Setup (the same setup is used for LB experiments) •Testbed: Two 1 GHz Transmeta processor machines communicating over ECE wireless LAN (802.11 b access point). •Each task is one row with fixed number of elements, where each element is generated uniformly and independently between 10 B to 100 B. ⎛ ⎞ x x ... x One task ⎜ ⎟ 11 12 1 n ⎜ ⎟ x x ... x 21 22 2 n ⎜ ⎟ m tasks � ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ x x ... x m 1 m 2 mn
Delay Probing Experiments UNM • In this experiment, tasks are independent, and size of each task is distributed as ~ U( [1KB, 10KB]). Load is a collection of tasks. • Task execution time is the time to multiply a row by a static matrix. The randomness in processing speed is introduced due to task size. • Both nodes use TCP to transfer load of different size between them. • Load transfer delay is calculated from the application’s layer perspective, i.e., this is the delay in transferring the entire load. • Communication delay is delay in transmitting a small time- stamped UDP packet carrying information about the number of tasks.
Empirical pdf of Delay UNM Communication Transfer delay delay (per task) • Average transfer delay per task is approximated as ~ exp(1/.35) • Communication delay is approximately as ~ exp(1/.089)
Empirical estimate of transfer delay UNM Fitting the parametric model: + β 1 1 /( Ld ) e θ = − d − β min 1 /( Ld ) 1 e With, d min = 0.162 s, d = 0.15, β = 0.085 Average transfer delay grows monotonically with the load size after some initial floor. • Each load (batch of tasks) was sent 25 times and the average delay was taken. • Traffic in the network changed during the time of experiments.
Empirical estimate of transfer UNM delay • Previous experiments are modified to obtain time- averaged estimate of transfer delay. Transfer delay (per task) • Transfer delay per task can be approximated with exponential pdf. • Time-averaged transfer delay grows monotonically as in the parametric model.
Regeneration-principle based queuing- UNM model to calculate the expected value of overall completion time Setup : Two-node system (The model has a trivial extension to n > 2 with little increase in algebraic complexity.) m n : Initial tasks at node 1 : Initial tasks at node 2 t : LB instant K : Balancing gain b [ ] µ = ( 0 , 0 ) 0 , 0 0 , 0 T ( t ) ( t ) E T ( t ) : Overall completion time, m , n b m , n b m , n b µ ( 0 , 0 ) ( t ) Goal : Calculate m , n b where, (0,0) is the knowledge state of the system at time t = 0 possible knowledge states : (0,0),(0,1),(1,0),(1,1)
Example of Regeneration Technique in UNM Context of Gambling • A gambler starts with an initial fortune $x • He can bid a dollar at a time and wins with probability p. • P(x) := P{ the gambler hits 20 | initial fortune =x} • Regeneration Event: Outcome of the first hand. P(x) = p*P(x+1) + (1-p)*P(x-1), with P(0) = 0, P(20) = 1
Regeneration Events and Knowledge States UNM • Regeneration event: The first event to occur among: • the completion of a task by any node. • the arrival of communication sent by any node. • Upon the occurrence of a regeneration event: • the stochastic dynamics of the new queues remain unchanged. • the new queues have different set of initial conditions, viz., • different load distribution if event is task completion • different knowledge state if event is communication arrival • Knowledge states: ( k 1 , k 2 ) where, k i ∈ {0,1} • (0,0) is the knowledge state at t =0, where each node does not know the load-state information of the other node. • (0,0) transits to (0,1) if node 2 receives communication from node 1.
Regeneration random variable: τ UNM W = waiting time for executing one task at node 1, W ~ exp (- λ D1 ) X = waiting time for executing one task at node 2, X ~ exp (- λ D2 ) Y = arrival of communication from node 1 to node 2, Y ~ exp (- λ 21 ) Z = arrival of communication from node 2 to node 1, Z ~ exp (- λ 12 ) τ = min( W , X , Y , Z ) = λ − λ t f ( t ) e u ( t ), τ λ = λ + λ + λ + λ where 1 2 21 12
Regeneration Equation: (0,0) case UNM ∞ µ = µ + ( 0 , 0 ) ∫ ( 0 , 0 ) ( t ) f ( s )[ ( 0 ) t ] ds τ m , n b m , n b t b λ t b + µ − + ∫ ( 0 , 0 ) D 1 f ( s ).[ ( t s ) s ]. ds τ − λ 1 , m n b 0 λ t b + µ − + ∫ ( 0 , 0 ) D 2 f ( s ).[ ( t s ) s ]. ds τ − λ m , n 1 b 0 λ t b + µ − + ∫ ( 0 , 1 ) 21 f ( s ).[ ( t s ) s ]. ds τ λ m , n b 0 λ t + b µ − + ∫ ( 1 , 0 ) 12 f ( s ).[ ( t s ) s ]. ds τ λ m , n b 0
Regeneration Equation UNM ∞ µ = µ + ( k , k ) ∫ ( k , k ) ( t ) f ( s )[ ( 0 ) t ] ds 1 2 1 2 τ m , n b m , n b t b λ t b + µ − + ∫ ( k , k ) D 1 f ( s ).[ ( t s ) s ]. ds 1 2 τ − λ 1 , m n b 0 λ t b + µ − + ∫ ( k , k ) D 2 f ( s ).[ ( t s ) s ]. ds 1 2 τ − λ m , n 1 b 0 λ t b + µ − + ∫ ( k , 1 ) 21 f ( s ).[ ( t s ) s ]. ds 1 τ λ m , n b 0 λ t + b µ − + ∫ ( 1 , k ) 12 f ( s ).[ ( t s ) s ]. ds 2 τ λ m , n b 0
Recommend
More recommend
