Rare-Event Simulation for Many-Server Queues Henry Lam Department of Mathematics and Statistics, Boston University Joint work with J. Blanchet, X. Chen and P. W. Glynn
Feb 22, 2016
Rare-Event Simulation for Many-Server Queues
Henry LamDepartment of Mathematics and Statistics, Boston University
Joint work with J. Blanchet, X. Chen and P. W. Glynn
2
Many-Server Loss System
Efficient simulation for many-server queues
Server 1
Server 2
Server 3
Server
Server 4
Customer
Efficient simulation for many-server queues 3
Many-Server Loss System
Server 1
Server 2
Server 3
Server
Server 4
Customer
ππ =first time of loss
Steady state distribution of loss?
Efficient simulation for many-server queues 4
Many-Server Loss System: Quality-Driven Regime
Server 1
Server 2
Server 3
Server
Server 4
Customers arrive according to a renewal process with rate i.e. interarrival times are i.i.d. with mean
Service times are i.i.d.
β’ Traffic intensity β’ possess exponential momentsβ’ has moments up to infinite order
Efficient simulation for many-server queues 5
Logarithmic Asymptoticβ’ Applications in communications, call centersβ¦β’ Many-server loss system (Ridder 2009, Blanchet, Glynn & L. 2009,
Blanchet & L. 2012):
where is the first time of lossβ’ Similar for delay of many-server queue in the same regimeβ’ Can be further extended to non-homogeneous functional of system
status -> application in insurance modeling etc. (Blanchet & L. 2011)β’ Steady-state phenomena (Blanchet & L. 2012):
β’ Rate function depends on the initial configuration of the queueβ’ Goal: construct asymptotically optimal importance sampler
Efficient simulation for many-server queues 6
Model Dynamics
Required service time at arrival
Arrival timeπ=0
π
π΄2
π 1
π΄1
π 0
π 2
π 1
π =4
π‘
Efficient simulation for many-server queues 7
Model Dynamics
Required service time at arrival
Arrival timeπ=0
π
π΄2
π 1
π΄1
π 0
π 2
π 1
π =4
π‘
Efficient simulation for many-server queues 8
Model Dynamics
Required service time at arrival
Arrival timeπ=0
π
π΄2
π 1
π΄1
π 0
π 2
π 1
π =4
Efficient simulation for many-server queues 9
Model Dynamics
Required service time at arrival
Arrival timeπ=0
π
π΄2
π 1
π΄1
π 0
π 2
π 1
π =4
ππ
Efficient simulation for many-server queues 10
Model Dynamics
Required service time at arrival
Arrival timeπ=0
π
π΄2
π 1
π΄1
π 0
π 2
π 1
π =4
Efficient simulation for many-server queues 11
Markov Representation
Required service time at arrival
Arrival timeπ=0
π
π =4
π‘
Efficient simulation for many-server queues 12
Required service time at arrival
Arrival timeπ=0
π
π =4
π‘
Markov state: customers at time with residual service time >
Markov Representation
Efficient simulation for many-server queues 13
Markov Representation
Required service time at arrival
Arrival timeπ=0
π
π =4
π‘
Markov state: customers at time with residual service time >
Efficient simulation for many-server queues 14
Markov Representation
Required service time at arrival
Arrival timeπ=0
π
π =4
π‘
Markov state: customers at time with residual service time >
Efficient simulation for many-server queues 15
Markov Representation
Required service time at arrival
Arrival timeπ=0
π
π =4
π‘
Markov state: customers at time with residual service time >
Efficient simulation for many-server queues 16
A Numerical Example, , Poisson arrival with rate Service time
Parameters/Assumptions:
Erlangβs loss formula =
Time to simulate 1000 time units
Time to obtain 1 loss
Number of arrivals in this time
1.63Γ10β10
1000Γ100=105
The next algorithm takes 10 seconds to simulate one loss event.
Efficient simulation for many-server queues 17
Theoretical Performance
Theorem (Blanchet and L. β12 & Blanchet, Glynn and L. β09):
1. Under current assumptions, the loss probability satisfies
where , and is the large deviations rate of , i.e.
where is the number of customers in an infinite-server system.
2. The algorithm we propose is asymptotically optimal for where
and and is the rate function starting from any initial configuration .
Efficient simulation for many-server queues 18
Steady-State Loss Probability
β’ Suppose is a recurrent set of the systemβ’ Kac's formula:
Notations:β = expectation with initial state in steady-state
conditional on being in β = number of loss before reaching againβ = time units to reach again
Efficient simulation for many-server queues 19
What is a good choice of set ?
β’ is a - ball around the fluid limit of , given by
where
i.e. decays slower than the standard deviation exhibited by the diffusion limit of
Efficient simulation for many-server queues 20
Brief Comments on Importance Sampling and Rare-Event Simulation
β’ Want to estimate where is a rare eventβ’ Importance sampling (IS) identity: Given a
suitable probability measure ,
β’ So IS estimator is
Efficient simulation for many-server queues 21
Brief Comments on Importance Sampling and Rare-Event Simulation
β’ If then IS gives zero variance:
β’ Moral: Good IS mimics the conditional distribution given the rare event!
β’ Use large deviations, but carefully (counter-examples in Glasserman and Kou β97)
β’ Asymptotic optimality: Relative error does not grow exponentially
Efficient simulation for many-server queues 22
What is a good choice of set ?
β’ Visited infinitely oftenβ’ Large deviation behavior is unique starting from every point in :
where is any point in and
β’ Possess good property of return time:
for any
Efficient simulation for many-server queues 23
Construction of Importance Sampler
β’ For simplicity let us first concentrate on Poisson arrivals
β’ Intuition:
where is the first time to experience a loss
Efficient simulation for many-server queues 24
Construction of Importance Sampler
β’ Observation 1: is the same for -server and infinite-server system
Remarkably handy
βmaxπ‘ππ (πβ (π‘ )>π )
Implication : Bias process to induce πβ (π‘ )>π for some π‘ , then reconstruct backwards
β’ Observation 2: β ππ (πβ (π‘ )>π )
Efficient simulation for many-server queues 25
Importance Sampling Procedure
Required service time at arrival
Arrival timeπ=0
π =4
π
STEP 1: Sample a random time over INDEPENDENT of the system
Efficient simulation for many-server queues 26
Importance Sampling Procedure
Required service time at arrival
Arrival timeπ=0 π
π =4
STEP 2: Sample the path given
Efficient simulation for many-server queues 27
Importance Sampling Procedure
Required service time at arrival
Arrival timeπ=0
π
π
π =4
STEP 2: Sample the path given
Use Poisson point process representation
Efficient simulation for many-server queues 28
Importance Sampling Procedure
Required service time at arrival
Arrival timeπ=0
π
π
π =4
STEP 2: Sample the path given
1. First sample given
Efficient simulation for many-server queues 29
Importance Sampling Procedure
Required service time at arrival
Arrival timeπ=0
π
π
π =4
STEP 2: Sample the path given
1. First sample given 2. Given , the points in triangle are distributed
independently according to intensity
Efficient simulation for many-server queues 30
Importance Sampling Procedure
Required service time at arrival
Arrival timeπ=0
π
π
π =4
STEP 2: Sample the path given
1. First sample given 2. Given , the points in triangle are distributed
independently according to intensity
Efficient simulation for many-server queues 31
Importance Sampling Procedure
Required service time at arrival
Arrival timeπ=0
π
π
π =4
STEP 2: Sample the path given
The rest of points outside the triangle follow non-homogeneous spatial Poisson process
Efficient simulation for many-server queues 32
Importance Sampling Procedure
Required service time at arrival
Arrival timeπ=0
π
π
π =4
STEP 3: Identify and continue the process with the original measure
Efficient simulation for many-server queues 33
ππ
Importance Sampling Procedure
Required service time at arrival
Arrival timeπ=0
π
π
π =4
STEP 3: Identify and continue the process with the original measure
Efficient simulation for many-server queues 34
π
Importance Sampling Procedure
Required service time at arrival
Arrival timeπ=0
π
π =4
STEP 3: Identify and continue the process with the original measure
Efficient simulation for many-server queues 35
ππ
Importance Sampling Procedure
Required service time at arrival
Arrival timeπ=0
π
π =4Until time
STEP 3: Identify and continue the process with the original measure
Efficient simulation for many-server queues 36
ππ
Importance Sampling Procedure
Required service time at arrival
Arrival timeπ=0
π
π =4Until time
STEP 4: Output
Efficient simulation for many-server queues 37
Change-of-Measure
The measure in this IS scheme is given by
where is an independent r. v.
Efficient simulation for many-server queues 38
General Renewal Arrivalsβ’ Use exponential tilting to induce β’ Represent β’ Gartner-Ellis limit (Glynn β95)
β’ Suggest state-dependent exponential tilting of each interarrival and service times according to an optimal (Szechtman and Glynn β02):
Efficient simulation for many-server queues 39
Specifications for Importance Sampling Procedure
β’ Distribution of random horizon:
where
β’ Likelihood ratio:
where
Efficient simulation for many-server queues 40
Proof of Efficiency
β’ Goal: The second moment of the estimator satisfies
β’ The second moment is bounded approximately as
β’ Main arguments:
Efficient simulation for many-server queues 41
Proof of Efficiency
β’ If , the area of is β’ Poisson arrivals: use thinning propertyβ’ General arrivals: Conditioned on arrivals times, probability of each customer lying on
the area is independent and
Efficient simulation for many-server queues 42
Lower Boundβ’ Construct the optimal sample pathβ’ Use the more general Gartner-Ellis limit
where
β’ Sample path large deviations: possesses a good rate function :
Efficient simulation for many-server queues 43
Logarithmic Estimate of Return Time
β’ Bound the return time for many-server system in terms of the infinite-server queue
where max of all residual service times at β’ Bounded service time:
β consider blocks of where the service time is bounded in β Return time bounded by a geometric random variable
independent of β’ Unbounded service time:
β need to estimate the residual service time from previous blockβ Use Borellβs inequality to ensure significant probability of the
Gaussian diffusion limit to stay in central region
Efficient simulation for many-server queues 44
Other Extensions
β’ Insurance portfolio problem: same algorithm with exponential tilting
β’ Time-inhomogeneous arrivals: same algorithm β’ Markov modulation (on finite state-space):
Sample Markov state ahead, then apply the same algorithm