Multi-Server Queues with Time-Varying Arrival Rates

62 Pages·2010·1.58 MB·English

Multi-Server Queues with Time-Varying Arrival Rates
WardWhitt
(jointworkwithdoctoralstudentYunanLiu)
ColumbiaUniversity,http://www.columbia.edu/∼ww2040
INFORMSMarkovLecture,November8,2010
Time-Varying Arrival Rates
arrivalsperhourtoamedium-sizedfinancial-servicescallcenter
The Model
M /GI/s + GI
t t
time-varyingarrivalrateλ(t)
alargetime-varyingnumberofserverss(t)
customerabandonment(the+GI)
non-exponentialdistributions
service-timecdfG(x) = P(S ≤ x), patience-timecdfF(x) = P(A ≤ x)
unlimitedwaitingroomandtheFCFSservicediscipline
(modelparametersinred)
This talk has two parts.
Part 1. offered-load approximations
RecentrelatedworkhasbeendonebydiscussantGalitYom-Tov,
jointlywithherthesisadvisorAvishaiMandelbaum.
Part 2. deterministic fluid models
RecentrelatedworkhasbeendonebydiscussantBillMassey,
jointlywithRobertHampshire. SeetheirtutorialonTuesday.
Thediscussantsarecollaborators;e.g.,seeFeldman,Z.,Mandelbaum,A.,
Massey,W.A.&WW,StaffingofTime-VaryingQueuestoAchieve
Time-StablePerformance,ManagementScience54(2008)324-338.
One Unifying Idea:
Exploit Associated
Infinite-Server(IS)Models
M /GI/∞
t
First Topic
Offered-Load Approximations
To Set Staffing Levels
For capacity planning, specify capacity by seeing how
much would be used if there were an unlimited supply,
allowing for uncertainty.
The first Idea: Offered-Load (OL) Approximations
Howmanyserverswouldbeusediftherewereanunlimitedsupply?
ForM/GI/s +GI,lookattheassociatedM/GI/∞model.
t t t
LetX(t)bethenumberofbusyserversattimetinM/GI/∞.
t
d
X(t)= Poisson(m(t))≈ Normal(m(t),m(t)),
wheretheexpectednumberm(t) ≡ E[X(t)]iscalled“the”offeredload
andcanbeexpressedas
(cid:90) t
m(t) ≡ E[X(t)] = λ(u)P(S > t−u)du = E[λ(t−S )]E[S],
e
−∞
(cid:90) x
P(S ≤ x) ≡ (1/E[S]) P(S > u)du, x ≥ 0.
e
0
Formoreonm(t),seeEick,Massey&WW(1993a,b).
Implication: Square Root Staffing (SRS)
IfX(t) ≈ Normal(m(t),m(t)),then
(cid:32) (cid:33)
s(t)−m(t)
P(W(t) > 0) = P(X(t) ≥ s(t)) ≈ P N(0,1) ≥
(cid:112)
m(t)
= P(N(0,1) ≥ β) ≡ 1−Φ(β).
Hence,theOLapproximationsupportsthe
SquareRootStaffing(SRS)formula: Giventhetargetτ ≡ P(W > 0),
chooseβ suchthat1−Φ(β) = τ. Thenlet
(cid:112)
s(t) = m(t)+β m(t).
TheSRSisalsosupportedbyMSHTlimits.
A Markovian Example (used throughout the talk)
M /M/s + M with sinusoidal arrival rate
t t
λ(t) = 100 + 20 · sin(t)
G¯(x) = e−µx, µ = 1; F¯(x) = e−θx, θ = 0.5
The Example
Fromarrivalratetoofferedloadtostaffing
140
130
120
110
100
90
80
70
arrival rate lambda(t) = 100 + 20sin(t)
offered load m(t) = 100 + 10(sin(t) − cos(t))
60 s(t) = m(t) + 2*sqrt(m(t))
500 5 10 15 20
time
Student Version of MATLAB