VARUN GUPTA Carnegie Mellon Google Research University of Chicago Booth School of Business With: Taayui Osogami (IBM Research-Toyo) 1
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Homogeneous servers 3
First-Come-First-Serve Buffer Homogeneous servers 4
Poisson(λ) First-Come-First-Serve Buffer Homogeneous servers λ = arrival rate 5
Poisson(λ) First-Come-First-Serve Buffer S i+2 S i+1 S i Homogeneous servers λ = arrival rate job sizes (S 1, S 2, ) i.i.d. samples from S load ρ λ E[S] 6
Poisson(λ) First-Come-First-Serve Buffer S i+2 S i+1 S i Homogeneous servers Waiting time (W) λ = arrival rate job sizes (S 1, S 2, ) i.i.d. samples from S load ρ λ E[S] GOAL : E[W M/G/ ] 7
λ S i+1 S i 1 2 ρ λ E[S] W =1 Case : S ~ Exponential (M/M/1) Analyze E[W M/M/1 ] via Marov chain (easy) Case: S ~ General (M/G/1) >1 Case : S ~ Exponential (M/M/) E[W M/M/ ] via Marov chain Case: S ~ General (M/G/) No exact analysis nown The Gold-standard approximation: Lee, Longton (1959) Sq. Coeff. of Variation (SCV) > 20 for computing worloads 8
Lee, Longton approximation: λ S i+1 S i 1 2 W Simple Exact for =1 Can not provision using this approximation! Asymptotically tight as ρ (thin Central Limit Thm.) 0.3 E[W M/G/ ] 0.2 0.1 0 9
λ S i+1 S i 1 2 W 2 moments not enough for E[W M/G/ ] Tighter bounds via higher moments of job size distribution 10
Lee, Longton approximation: λ S i+1 S i 1 2 W GOAL: Bounds on approximation ratio {G 2 moments} Lee-Longton Approximation E[W] 6 THEOREM: H 2 4 p 1-p Exp(µ 1 ) Exp(µ 2 ) 2 0 Increasing 3 rd moment (C 2 = 19, =10) [Dai, G., Harchol-Balter, Zwart] 11
{G 2 moments} THEOREM: If ρ < -1, Gap >= (C 2 +1) X COR.: No approx. for E[W M/G/ ] based on first two moments of job sizes can be accurate for all distributions when C 2 is large PROOF: Analyze limit distributions in D 2 mixture of 2 points Min 3 rd moment E[W M/G/ ] 3 rd moment 0 C 2 +1 1 1/² Approximations using higher moments? [Dai, G., Harchol-Balter, Zwart] 12
λ S i+1 S i 1 2 W 2 moments not enough for E[W M/G/ ] Tighter bounds via higher moments of job size distribution 13
λ S i+1 S i 1 2 W {G n moments}? tight bounds n moments? E[W] GOAL: Identify the extremal distributions with given moments RELAXED GOAL: Extremal distributions in some non-trivial asymptotic regime IDEA: Light-traffic asymptotics (λ 0) 14
RELAXATION: Identify the extremal distributions in light traffic Light traffic theorem for M/G/ [Burman Smith]: Probability of finding all servers busy S e1 1 i.i.d. copies of S e equilibrium excess of S pdf of S e : S e SUBGOAL: Extremal distributions for E[min{S e1,,s e }] s.t. E[S i ] = m i for i=1,..,n 15
λ S i+1 S i 1 2 W GOAL: Tight bounds on E[W M/G/ ] given n moments of S IDEA: Identify extremal distributions RELAXATION (Light Traffic): Extremal distributions for E[min{S e1,,s e }] s.t. E[S i ] = m i for i=1,..,n 16
GIVEN: Moment conditions on random variable X with support [0,B] E[X 0 ]=m 0 E[X 1 ]=m 1 E[X n ]=m n Principal Representations (p.r.) on [0,B] are distributions satisfying the moment conditions, and the following constraints on the support Lower p.r. Upper p.r. n even 0 B 0 B 1 + n/2 point masses 1 + n/2 point masses 17
GIVEN: Moment conditions on random variable X with support [0,B] Want to bound: E[g(X)] E[X 0 ]=m 0 E[X 1 ]=m 1 E[X n ]=m n 18
GIVEN: Moment conditions on random variable X with support [0,B] Want to bound: E[g(X)] E[X 0 ]=m 0 E[X 1 ]=m 1 E[X n ]=m n THEOREM [Marov-Krein]: If {x 0,,x n,g(x)} is a Tchebycheff-system on [0,B], then E[g(X)] is extremized by the unique lower and upper principal representations of the moment sequence {m 0,,m n }. 19
λ S i+1 S i 1 2 W GOAL: Tight bounds on E[W M/G/ ] given n moments of S IDEA: Identify extremal distributions RELAXATION (Light Traffic): Extremal distributions for E[min{S e1,,s e }] s.t. E[S i ] = m i for i=1,..,n THEOREM: For n = 2 or 3 RELAXATION 2: Restrict to Completely Monotone distributions (mixtures of Exponentials) (contains Weibull, Pareto, Gamma) THEOREM: For all n. 20
CONJECTURE: P.R.s are extremal for E[W M/G/ ] for all ρ, for all n, if moment constraints are integral. Given at least E[S], E[S 2 ] Not given E[S 2 ], even # of moment constraints in (0,2) 0 ρ 0 ρ 21
1.2 E[W M/G/ ] 0.8 Weibull 0.4 0 1 2 3 4 5 6 Number of moments 22
1.2 E[W M/G/ ] 0.8 Weibull Bounds via p.r. 0.4 0 1 2 3 4 5 6 Number of moments 23
1.2 E[W M/G/ ] 0.8 0.4 Weibull Bounds via p.r. Bounds via p.r. in CM class 0 1 2 3 4 5 6 Number of moments Approximation Schema: Refine lower bound via an additional odd moment, Upper bound via even moment until gap is acceptable 24
λ S i+1 S i 1 2 W 2 moments not enough for E[W M/G/ ] Tighter bounds via higher moments of job size distribution Many other hard queueing systems fit the approximation schema 25
Example 1: M/G/1 Round-robin queue Incomplete jobs Poisson(λ) arrivals Jobs served for q units at a time Need analysis to find q that balance overheads/performance THEOREM: Upper and lower p.r. extremize mean response time under λ 0, when S is a mixture of Exponentials. 26
Example 2: Systems with fluctuating load High Load Low Load High Load Low Load α T H Need analysis to tune sharing parameters α T L Beneficiary server Donor server THEOREM: Upper and lower p.r. extremize mean waiting time under α 0, when T H, T L are mixtures of Exponentials. 27
Example: Single server system S i+1 S i W i+1 = waiting time of S i+1 W i+1 = (W i, S i, A i+1 ) A i+1 time 28
Example: Single server system S i+1 S i W i+1 = waiting time of S i+1 W i+1 = (W i + S i - A i+1 ) + A i+1 time 29
Example: Single server system S i+1 S i W i+1 = waiting time of S i+1 d W = (W + S - A ) + A i+1 time Stationary behavior of a queueing system = d Fixed point of a stochastic recursive sequence of the form W = (W,S) Q: Given moments of S, under what conditions on f,, is E[f(W)] extremized by p.r.s? 30
λ 1 2 W All existing analytical approx for performance based on 2 moments, but 2 moments inadequate Provide evidence for tight n-moments based bounds via asymptotics for M/G/ and other queuing systems A new problem in analysis: Marov-Krein characterization of stochastic fixed point equations 31