Parameter Estimation Problemsin Queueing and Related

Stochastic Models

Yoni NazarathySchool of Mathematics and Physics,

The University of Queensland.

Australian Statistical Conference, Adelaide,July 11, 2012

Talk Goal

• A taste of queueing theory• Parameter estimation problems in queues• Departure processes in queueing networks• Estimation through customer streams

Queues

• Customers: – Communication packets– Production lots– Customers at the ticket box, doctor or similar

• Servers: – Routers– Production machines– Tellers, etc…

This morning: Kayley Nazarathy aged 4, waited 55 minutes for a vaccination in QLD, she was reported by her mother as starting to be loud after 25 minutes saying, “when is it my turn, when is it my turn,….”

Queueing Theory• Overview:

– Quantifies waiting / congestion phenomena– Mostly stochastic– More than 10,000 papers, more than 100 books

• Types of research results:– Phenomena– Performance evaluation: Formulas, computational techniques,

asymptotic behavior…– Design and control

• Inference and estimation:• Less than 100 serious papers. 1st: “The Statistical Analysis of

Congestion”, D. R. Cox, 1955• Bib: “Parameter and State Estimation in Queues and Related

Stochastic Models: A Bibliography.” Y. N. and Philip K. Pollett, on-line

The Single Server Queue

The Single Server Queue

Buffer Server

0 1 2 3 4 5 6 …Number in System:

( )Q t Number in system at time t

( )Q t

t

The Single Server Queue

Buffer Server

0 1 2 3 4 5 6 …Number in System:

{ , 1}nT n Arrivals times

{ , 1}ns n Service requirements

1{ , 1}n n na T T n Inter-arrival times 0 0T

The sequence ( , ), 1n na s n Determines evolution of Q(t)

( )Q t Number in system at time t

( , ), 1n na s n

( )Q t

nW

nW The waiting time of customer n

1 1max ,0n n n nW W a s

Waiting Times

Performance MeasuresA “key” performance measure: ( )nP W x

limd

nn

W W

lim ( )d

tQ Q t

1[ ]nE a 1[ ]nE s

Often assume that the sequence is stochastic and stationary

Load

The core of queueing theory deals with the distributions

of W and/or Q under some assumptions on

Typically take i.id. with generic RVs denoted by A, S

If, , there are often limiting distributions:

Little’s result: If , [ ] [ ]E Q E W 1

( , ), 1n na s n

( , ), 1n na s n

( , ), 1n na s n

1

M/M/1, M/G/1, GI/G/1

Assumption types on A and S:

Notation for Queues• A/S/N/K– A is the arrival process– S represents the service time distributions– N is the number of servers– K is the buffer capacity (default is infinity)

M/M/1, M/G/1, GI/G/1

• M Poisson or exponential or memory-less• G General• GI Renewal process arrivals

Mean Stationary Waiting Time

1/ /1[ ]

1M ME W

21

/ /1

1[ ]

1 2s

M G

cE W

2 2 21

/ /1 2[ ]

1 2a s

GI G

c cE W

2

2

( )ns

n

Var sc

E s

2

2

( )na

n

Var ac

E a

Inference Interlude• Understanding the “congestion level” of a given situation

implies finding the distribution of W• Queueing theory tells us the distribution of W, based on

the distributions of A and S • To quantify the congestion level based on data we are

faced with two basic general options:– Perform inference for W directly (do not use queueing theory)– Perform inference for A and S and then use queueing theoryCox 1955:

“Such a prediction (i.e. using queueing theory) is of little value when we are merely interested in describing a particular situation, since it is usually no more difficult to measure (i)-(iv) (i.e. W) than to measure arrival or service times (i.e. A and S).

However our practical interest is usually in the effect of modifications designed to reduce congestion, and it is often difficult or impossible to find experimentally whether proposed changes are worth while.”

Illustration: Queueing Model for Load on the Swinburne Super Computer (Tuan Dinh, Lachlan Andrew, Y.N.)

Workload during first half of 2011

Interconnecting Queues:Queueing Networks

The Basic Model: Open Jackson NetworksJackson 1957, Goodman & Massey 1984

1

1

'

( ')

M

i i j j ij

p

P

I P

, ,P

ii

Traffic Equations (Stable Case):

i jp

1

M

1

1M

i jij

p p

Problem Data:

Assume: open, no “dead” nodes

1 11

lim ( ) ,..., ( ) 1jk

Mj j

M Mt

j j j

P X t k X t k

Product Form “Miracle”: If , i i

Customer Streams

Variance of Outputs

( )tVt o

t

Var ( )D t

Var ( )D T TV

* Stationary stable M/M/1, D(t) is PoissonProcess( ):

* Stationary M/M/1/1 with , D(t) is RenewalProcess(Erlang(2, )):

21 1 1

( )4 8 8

tVar D t t e

( )Var D t t V

4V

2 1 23V m c

m * In general, for renewal process with :

* The output process of most queueing systems is NOT renewal

2,m

Asymptotic Variance

Var ( )limt

VD t

t

Simple Examples:

Notes:

Balancing Reduces Asymptotic Variance of Outputs

Theorem (Y. N. , Weiss 2008): For the M/M/1/K queue with :

2

2 3 2

3 3( 1)

KV

K

Numerically tested Conjecture (Y. N. , 2011):For the GI/G/1/K queue with : 2 2

(1)3

a sK

c cV o

1

1

Theorem (Al Hanbali, Mandjes, Y. N. , Whitt 2010):For the GI/G/1 queue with ,under further conditions: 2 2 2

( ) 1a sV c c

1

Insight about the asymptotic variance is crucial for inference of customer streams

C D

One of the proofs tools: Markov Arrival Processes (MAPs)

* * 2 * 2 3 2Var ( ) 2( ) 2 2( ) 2 ( )r btD t D De t De O t e

0 0

1 1 1 1

1 1 1 1

( )

( )K K K K

K K

1

1

0 0

0 0

0

0K

K

* De *E[ ( )]D t t

0 0

1 1 1

1 1 1

0 ( )

0 ( )

0K K K

K

Generator Transitions without events Transitions with events

1( )e

, 0r b

Asymptotic Variance Rate

Birth-Death Process

Inference for MAPs

2000 Survey by Tobias Ryden, “Statistical estimation for Markov-modulated Poisson processes and Markovian arrival processes”

Proposition (stated loosely) (Y.N., Gideon Weiss): Many MAPs (those that are MMPPs) have equivalent processes that count all transitions of a CTMC (fully counting MAPs). The equivalence is in terms of the mean and variance function.

On-going work (with Sophie Hautphenne): Efficient methods (improving on EM for MAPs) for processes generated by all transitions of a CTMC.

Typical methods:

• MLE using EM (expectation maximization). The CTMC state is a “hidden variable”• Moments methods (typically for structured MAPS)

The idea: “fully counting MAPs” are easier than general MAPS and may approximate customer streams for network decomposition

Towards a Survey of Queuing Inference and Estimation Problems

Dichotomy for A/S/N/K Models

{Fixed, , ,prior}p np

{obs, unobs, part obs cust, part obs time}{stationary, non-stationary}

{constant,time varying, change point}

{fixed horizon, stopping times, other}

A S N K

na ns nW ( )Q t nL

[0, ]T

Bib: “Parameter and State Estimation in Queues and Related Stochastic Models: A Bibliography.” Y. N. and Philip K. Pollett, on-line

Closing Remarks

• Some processes are well modeled using queueuing models

• Using a white or gray box analysis for such systems is often better than a black box

• Estimation and inference in queues is NOT yet a highly developed field

• As is with other statistical models, there is not yet a definitive answer for model selection