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

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

𝜏𝑠=first time of loss

Steady state distribution of loss?

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.

• Traffic intensity • possess exponential moments• has moments up to infinite order

Logarithmic Asymptotic• Applications in communications, call centers…• Many-server loss system (Ridder 2009, Blanchet, Glynn & L. 2009,

Blanchet & L. 2012):

where is the first time of loss• Similar for delay of many-server queue in the same regime• Can be further extended to non-homogeneous functional of system

status -> application in insurance modeling etc. (Blanchet & L. 2011)• Steady-state phenomena (Blanchet & L. 2012):

• Rate function depends on the initial configuration of the queue• Goal: construct asymptotically optimal importance sampler

Model Dynamics

Required service time at arrival

Arrival time𝑉=0

𝑉 1

𝑈 0

𝑉 2

𝑈 1

𝑠=4

Model Dynamics

Arrival time𝑉=0

𝑉 1

𝑈 0

𝑉 2

𝑈 1

𝑠=4

Model Dynamics

Arrival time𝑉=0

𝑉 1

𝑈 0

𝑉 2

𝑈 1

𝑠=4

Model Dynamics

Arrival time𝑉=0

𝑉 1

𝑈 0

𝑉 2

𝑈 1

𝑠=4

𝜏𝑠

Model Dynamics

Arrival time𝑉=0

𝑉 1

𝑈 0

𝑉 2

𝑈 1

𝑠=4

Markov Representation

Arrival time𝑉=0

𝑠=4

Arrival time𝑉=0

𝑠=4

Markov state: customers at time with residual service time >

Arrival time𝑉=0

𝑠=4

Arrival time𝑉=0

𝑠=4

Arrival time𝑉=0

𝑠=4

A Numerical Example, , Poisson arrival with rate Service time

Parameters/Assumptions:

Erlang’s loss formula =

Time to simulate 1000 time units

Time to obtain 1 loss

Number of arrivals in this time

1.63×10−10

1000×100=105

The next algorithm takes 10 seconds to simulate one loss event.

Theoretical Performance

Theorem (Blanchet and L. ‘12 & Blanchet, Glynn and L. ‘09):

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 .

Steady-State Loss Probability

• Suppose is a recurrent set of the system• Kac's formula:

Notations:– = expectation with initial state in steady-state

conditional on being in – = number of loss before reaching again– = time units to reach again

What is a good choice of set ?

• is a - ball around the fluid limit of , given by

i.e. decays slower than the standard deviation exhibited by the diffusion limit of

Brief Comments on Importance Sampling and Rare-Event Simulation

• Want to estimate where is a rare event• Importance sampling (IS) identity: Given a

suitable probability measure ,

• So IS estimator is

Brief Comments on Importance Sampling and Rare-Event Simulation

• If then IS gives zero variance:

• Moral: Good IS mimics the conditional distribution given the rare event!

• Use large deviations, but carefully (counter-examples in Glasserman and Kou ‘97)

• Asymptotic optimality: Relative error does not grow exponentially

What is a good choice of set ?

• Visited infinitely often• Large deviation behavior is unique starting from every point in :

where is any point in and

• Possess good property of return time:

for any

Construction of Importance Sampler

• For simplicity let us first concentrate on Poisson arrivals

• Intuition:

where is the first time to experience a loss

Construction of Importance Sampler

• Observation 1: is the same for -server and infinite-server system

Remarkably handy

≈max𝑡𝑃𝑟 (𝑄∞ (𝑡 )>𝑠)

Implication : Bias process to induce 𝑄∞ (𝑡 )>𝑠 for some 𝑡 , then reconstruct backwards

• Observation 2: ≈ 𝑃𝑟 (𝑄∞ (𝑡 )>𝑠)

Importance Sampling Procedure

Arrival time𝑉=0

𝑠=4

STEP 1: Sample a random time over INDEPENDENT of the system

Arrival time𝑉=0 𝒕

𝑠=4

STEP 2: Sample the path given

Arrival time𝑉=0

𝑠=4

Use Poisson point process representation

Arrival time𝑉=0

𝑠=4

1. First sample given

Arrival time𝑉=0

𝑠=4

1. First sample given 2. Given , the points in triangle are distributed

independently according to intensity

Arrival time𝑉=0

𝑠=4

1. First sample given 2. Given , the points in triangle are distributed

independently according to intensity

Arrival time𝑉=0

𝑠=4

The rest of points outside the triangle follow non-homogeneous spatial Poisson process

Arrival time𝑉=0

𝑠=4

STEP 3: Identify and continue the process with the original measure

𝜏𝑠

Arrival time𝑉=0

𝑠=4

Arrival time𝑉=0

𝑠=4

𝜏𝑠

Arrival time𝑉=0

𝑠=4Until time

𝜏𝑠

Arrival time𝑉=0

𝑠=4Until time

STEP 4: Output

Change-of-Measure

The measure in this IS scheme is given by

where is an independent r. v.

General Renewal Arrivals• Use exponential tilting to induce • Represent • Gartner-Ellis limit (Glynn ‘95)

• Suggest state-dependent exponential tilting of each interarrival and service times according to an optimal (Szechtman and Glynn ‘02):

Specifications for Importance Sampling Procedure

• Distribution of random horizon:

• Likelihood ratio:

Proof of Efficiency

• Goal: The second moment of the estimator satisfies

• The second moment is bounded approximately as

• Main arguments:

Proof of Efficiency

• If , the area of is • Poisson arrivals: use thinning property• General arrivals: Conditioned on arrivals times, probability of each customer lying on

the area is independent and

Lower Bound• Construct the optimal sample path• Use the more general Gartner-Ellis limit

• Sample path large deviations: possesses a good rate function :

Logarithmic Estimate of Return Time

• Bound the return time for many-server system in terms of the infinite-server queue

where max of all residual service times at • Bounded service time:

– consider blocks of where the service time is bounded in – Return time bounded by a geometric random variable

independent of • Unbounded service time:

– need to estimate the residual service time from previous block– Use Borell’s inequality to ensure significant probability of the

Gaussian diffusion limit to stay in central region

Other Extensions

• Insurance portfolio problem: same algorithm with exponential tilting

• Time-inhomogeneous arrivals: same algorithm • Markov modulation (on finite state-space):

Sample Markov state ahead, then apply the same algorithm

Rare-Event Simulation for Many-Server Queues

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