Using Robust Queueing to Expose the Impact of Dependence in …wy2225/papers/INFORMS2016.pdf ·...
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Industrial Engineering & Operations Research DepartmentColumbia University
Using Robust Queueing to Expose the Impact ofDependence in Single-Server Queues
Wei Youjoint work with Ward Whitt
Industrial Engineering & Operations Research DepartmentColumbia University
INFORMS 2016Nashville
Based on “Using Robust Queueing to Expose the Impact of Dependence in Single-ServerQueues” by Whitt and You, submitted to Operations Research, revised in October 2016.
Outline
1 Review of Robust Queueing
2 Review of Dependence in Queues
3 Robust Queueing with Dependence
4 Numerical Examples
Review of Robust Queueing
Review of Robust Queueing
A robust optimization approach proposed by C. Bandi, D.Bertsimas, and N. Youssef (2015)
I analyzed the steady-state mean waiting time in singleserver queue with general interarrival and servicedistributions
I extended to open queueing networks with possibleenhancement to Queueing Network Analyzer;
I replaced probabilistic laws by uncertainty sets;
I used deterministic optimization and regression analysis.
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Review of Robust Queueing
Review of Robust Queueing Theory
A general FCFS queue is considered in Bandi et. al. (2015)
I {(Ui, Vi)}i>1: interarrival times and service times;
I λ, µ: arrival rate and service rate.
Lindley recursion
Wn = (Wn−1 + Vn−1 − Un−1)+ = max06k6n
{Ssk − Sak} ,
where Ss0 ≡ 0, Sa0 ≡ 0 and
Ssk ≡n−1∑i=n−k
Vi, Sak :=
n−1∑i=n−k
Ui, 1 ≤ k ≤ n.
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Review of Robust Queueing
Review of Robust Queueing
The worst case waiting time in Robust Queueing Theory
W ∗n = supU∈Ua
supV∈Us
max06k6n
{Ssk − Sak}
Ua =
{(U1, . . . , Un)
∣∣∣∣Sak − k/λk1/2> −Γa, 0 6 k 6 n
},
Us =
{(V1, . . . , Vn)
∣∣∣∣Ssk − k/µk1/26 Γs, 0 6 k 6 n
}.
I robustness is controlled by parameters Γa,Γs;
I standard CLT suggest that Γa = baσa and Γs = bsσs.
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Review of Robust Queueing
Review of Robust Queueing
With an interchange of maximum, they reduce the problem to
W ∗n = max0≤k≤n
{mk + b√k}
≤ supx≥0{mx+ b
√x} =
b2
4|m|=
λb2
4(1− ρ),
where m = µ−1 − λ−1 < 0, ρ = λ/µ and b ≡ Γs + Γa > 0,
I Closed-form solution depends only on ρ,Γa and Γs.
I The solution takes similar form as classical heavy-trafficlimits.
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Review of Dependence in Queues
Impact of Dependence on Queues
Dependence structures are ubiquitous in queueing systems:
I departure process is non-renewal unless all processes arePoisson;
I superposition of different arrival streams is non-renewalunless all processes are Poisson.
The dependence can’t be ignored
I the dependence will have huge impact on the systemperformance measures;
I the level of impact will depend on the traffic intensity.
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Review of Dependence in Queues
A Queueing Model with Dependence
Last queue of 5 queues in series (tandem queues)
E10Queue 1
H2, ρ = 0.99
Queue 2
E10, ρ = 0.98
Queue 5
M
I Consider the steady-state mean workload at the last queue;I The variability of the external arrival and the service at the
first 4 queues are alternative between low (Erlangdistribution E10) high (hyper-exponential distribution H2);
I The external arrival rate is 1;I The service rates/traffic intensities, at the intermediate
queues are set in a decreasing manner so as to exposedifferent variability.
I The service time at the last queue is exponential withmean ρ, the traffic intensity.
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Review of Dependence in Queues
A Queueing Model with Dependence
Normalized Steady-state mean workload, 2(1− ρ)E[Wρ(∞)]/ρ
0 0.5 1 1.5 2 2.5 3
- log10
(1- )
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
No
rma
lize
d m
ea
n w
ork
loa
d 2
(1-
)E[Z
]/
I The level of impact on the mean workload will changesdrastically as a function of the traffic intensity;
I The complex curve of mean workload cannot be captured withthe Kingman bound or classical heavy-traffic limits.
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Robust Queueing with Dependence
Continuous-time workload process
I {(Ui, Vi)}: interarrival times and service times;I λ, µ: arrival rate and service rate;I A(t): arrival counting process associated with {Uk};I Y (t): total input of work defined by Y (t) ≡
∑A(t)k=1 Vk;
I X(t): net-input process defined by X(t) ≡ Y (t)− t;Apply the one-sided reflection mapping to X(t) to get thesteady-state workload at time 0 in the queue staring empty atthe remote past −∞:
Z ≡ X(0)− inf−∞≤t≤0
{X(t)}.
= sup0≤s≤∞
{X(0)−X(−s)} ≡ sup0≤s≤∞
{X0(s)}
I X0(s) is interpreted as the net-input over time [−s, 0].I With an abuse of notation, we omit the subscript in X0(s).
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Robust Queueing with Dependence
Continuous-time workload process
We now insert the traffic intensity ρ into the model.
I We start with a unit-rate arrival counting process A(t).
I Assume that Aρ(t) in the ρ-th model takes a simple form:
Aρ(t) = A(ρt).
I For Poisson process, this is equivalent to changing thearrival rate from 1 to ρ.
I The total input process and net-input process are
Yρ(t) = Y (ρt), and Xρ(t) = Y (ρt)− t.
I The steady-state workload is
Zρ = sup0≤s≤∞
{Yρ(s)− s} = sup0≤s≤∞
{Xρ(s)}.
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Robust Queueing with Dependence
Stochastic versus Robust Queues
Zρ = sup0≤s≤∞
{Xρ(s)}.
Stochastic Queue
I Xρ(s) ≡∑N(ρs)
k=1 Vk − s, where N(t) and {Vk} are stationarypoint process and stationary sequence separately.
Robust Queue
I Xρ lies in a suitable uncertainty set Uρ of total inputfunctions to be defined later.
I There is no distribution involved, we hence focus on thedeterministic worse-case scenario
Z∗ρ ≡ supXρ∈Uρ
sup0≤s≤∞
{Xρ(s)}.
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Robust Queueing with Dependence
Robust Queueing for continuous-time workload
Now, we define the uncertainty set for the net-input process.
Uρ ≡{Xρ : R+ → R
∣∣∣∣ Xρ(s) ≤ E[Xρ(s)] + b√
Var(Xρ(s)), s ∈ R+
}={Xρ : R+ → R
∣∣∣ Xρ(s) ≤ −(1− ρ)s+ b√ρsIw(ρs), s ∈ R+
},
where Iw(t) is the index of dispersion for work (IDW), i.e.,
Iw(t) ≡ Var(Y (t))
t.
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Robust Queueing with Dependence
Robust Queueing for continuous-time workload
RQ for workload
Z∗ρ = supXρ∈Uρ
sup0≤s≤∞
{Xρ(s)},where
Uρ ={Xρ : R→ R
∣∣∣ Xρ(s) ≤ −(1− ρ)s+ b√ρsIw(ρs)
}.
Lemma (Dimensionality reduction)
The infinite-dimensional RQ problem can be reduced toone-dimensional
Z∗ρ = sup0≤s≤∞
supXρ∈Uρ
{Xρ(s)}
= sup0≤s≤∞
{−(1− ρ)s+ b
√ρsIw(ρs)
}.
Furthermore, if ρ < 1 and Iw(t)/t→ 0 as t→∞, then Z∗ρ <∞.W. Whitt, W. You Robust Queueing with Dependence 14 / 22
Robust Queueing with Dependence
Robust Queueing for continuous-time workload
In summary, the RQ optimization for steady-state workloadprocess reduces to one dimensional optimization problem
Z∗ρ = sup0≤s≤∞
{−(1− ρ)s+ b
√ρsIw(ρs)
}I above specifies the RQ algorithm;
I in application, weI estimate Iw(x) from data;I create a finite grid and search for the approximated
optimum over the finite grid.
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Robust Queueing with Dependence
Analyzing the Robust Queueing with Dependence
Theorem (Closed-from RQ solution)
The worst-cast RQ workload Z∗ρ for the model with trafficintensity ρ is
Z∗ρ =b2
2
ρIw(x∗ρ)
2(1− ρ)
1−
(x∗ρIw(x∗ρ)
Iw(x∗ρ)
)2 ,
where x∗ρ satisfies the equation
x∗ρ =b2ρ2Iw(x∗ρ)
4(1− ρ)2
(1 +
x∗ρIw(x∗ρ)
Iw(x∗ρ)
)2
.
Moreover, the associated optimal solution s∗ρ to the RQ problemis related to x∗ρ by s∗ρ = ρ−1x∗ρ.
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Robust Queueing with Dependence
Analyzing the Robust Queueing with DependenceImplication I: The choice of parameter b in the uncertainty set.
How to choose parameter b?
Uρ ={Xρ : R→ R
∣∣∣ Xρ(s) ≤ −(1− ρ)s+ b√ρsIw(ρs)
},
Z∗ρ =b2
2
ρIw(x∗ρ)
2(1− ρ)
1−
(x∗ρIw(x∗ρ)
Iw(x∗ρ)
)2 .
I We choose b =√
2 so that RQ is exact for all M/GI/1models.
I This choice of b is independent of model detail and trafficintensity.
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Robust Queueing with Dependence
Analyzing the Robust Queueing with DependenceImplication II: Asymptotically correct in heavy-traffic limit and light-traffic limit.
Theorem (RQ correct in Heavy-traffic and light-traffic)
For G/G/1 model, our RQ yields the exact steady-state meanworkload in both light-traffic and heavy-traffic limits.
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Numerical Examples
Numerical Example: 5 queues in series
Last queue of 5 queues in series (tandem queues)
E10Queue 1
H2
Queue 2
E10
Queue 5
M
0 0.5 1 1.5 2 2.5 3
- log10
(1- )
0
1
2
3
4
5
Norm
aliz
ed m
ean w
ork
load 2
(1-
)E[Z
]/
10-2
100
102
104
106
time t
0
1
2
3
4
5
6
IDW
, I w
(t)
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Numerical Examples
Numerical Examples - 5 Queues in series
0 0.5 1 1.5 2 2.5 3
- log10
(1-ρ)
0
1
2
3
4
5
6
Norm
aliz
ed m
ean w
ork
load
Simulation
RQ
I RQ automatically “matches” IDW to the mean workloadfor all traffic intensities.
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Summary
We
I develop new version of RQ for continuous-time workloadprocess in G/G/1 model to capture dependence amonginterarrival times and service times;
I show that RQ for continuous-time workload that are exactfor M/GI/1 queue and asymptotically correct for G/G/1in both light and heavy traffic;
I conduct simulation study and observe good approximationeven with extremely complex dependence structure.
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References
I Key references:
[BBY15] C. Bandi, D. Bertsimas, and N. Youssef, Robust Queueing Theory,Operations Research 63 (2015), no. 3, 676-700.
[FW89] K. W. Fendick and W. Whitt, Dependence in Packet Queues, IEEETransactions on Communications 37 (1989), no. 11, 1173-1183.
I Other references:
[IW70] D.L. Iglehart and W. Whitt, Multiple Channel Queues in Heavy Traffic II:Sequences, Networks and Batches. Advanced Applied Probability 2 (1970),355-369.
[Loy62] R. M. Loynes, The Stability of A Queue with Non-independent Inter-arrivaland Service Times, Mathematical Proceedings of the CambridgePhilosophical Society 58 (1962), no. 03, 497-520.
[SW86] K. Sriram and W. Whitt, Characterizing Superposition Arrival Processes inPacket Multiplexers for Voice and Data, IEEE JOurnal on Selected Areason Communications 4 (1986), no. 6, 833-846.
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Analyzing the Robust Queueing with DependenceImplication III: Connection to Fenick and Whitt (1989).
I Fendick and Whitt (1989) observed that the IDW Iw(t) isintimately related to the scaled mean workload c2
Z(ρ);I they proposed a deterministic time transformation (DTT)
method with variability-fixed-point approximation (VFP).I The red part below also acts as a heuristic refinement to
there result, we call it RQ-derived DTT and VFP.I The RQ approach provided a variation of the DTT method
and the VFP approximation, i.e.,
Z∗ρ =ρIw(x∗ρ)
2(1− ρ)
1−
(x∗ρIw(x∗ρ)
Iw(x∗ρ)
)2 ,
x∗ρ =ρ2Iw(x∗ρ)
2(1− ρ)2
(1 +
x∗ρIw(x∗ρ)
Iw(x∗ρ)
)2
.
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