Transcript Balancing Reduces Asymptotic Variance of Outputs

Parameter Estimation Problems
in Queueing and Related
Stochastic Models
Yoni Nazarathy
School 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
Server
Buffer
Number in
System:
Q(t ) 
0
1
2
3
4
5
6
…
Number in system at time t
Q(t )
t
The Single Server Queue
Server
Buffer
Number in
System:
Q(t ) 
0
1
2
3
4
5
6
…
Number in system at time t
{Tn , n  1}  Arrivals times
{an  Tn  Tn1, n  1}  Inter-arrival times
{sn , n  1}  Service requirements
The sequence
T0  0
(an , sn ), n  1 Determines evolution of Q(t)
Waiting Times
Wn 
The waiting time of customer n
Wn1  max Wn  an1  sn ,0
(an , sn ), n  1
Wn
Q(t )
Performance Measures
P(Wn  x)
A “key” performance measure:
Often assume that the sequence
  E[an ]
1
  E[sn ]
1
(an , sn ), n  1 is stochastic and stationary


 Load

If,   1 , there are often limiting distributions:
d
d
Q  lim Q(t )
W  lim Wn
n 
Little’s result: If
 1,
t 
E[Q]   E[W ]  
The core of queueing theory deals with the distributions
of W and/or Q under some assumptions on
Typically take i.id.
(an , sn ), n  1
(an , sn ), n  1
with generic RVs denoted by A, S
M/M/1, M/G/1, GI/G/1
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
Assumption types on A and S:
• M Poisson or exponential or memory-less
• G General
• GI Renewal process arrivals
Mean Stationary Waiting Time
E[WM / M /1 ]  
cs 
Var (sn )
ca 
Var (an )
2
2
E  sn 
2
E  an 
2
1

1 
2
1

c

1
s
E[WM / G /1 ]  
1  2
2
2 2
c


cs

1
a
E[WGI / G /1 ]  
1 
2 2
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 theory
Cox 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 Networks
Jackson 1957, Goodman & Massey 1984
Problem Data:
 , , P
1
Assume: open, no “dead” nodes
Traffic Equations (Stable Case):
M
i
i
pi j
M
p i  1   pi j
j 1
i   i    j p j i
j 1
    P '
  ( I  P ') 1
Product Form “Miracle”: If i  i ,
M
 j
lim P  X 1 (t )  k1 ,..., X M (t )  kM    1 
 
t 
j 1 
j
M
  j

 
 j



kj
Customer Streams
Variance of Outputs
Var  D(t )   Vt  o(t )
Asymptotic Variance
Var  D(t ) 
V  lim
t 
t
Var  D(T )   VT
t
Simple Examples:
* Stationary stable M/M/1, D(t) is PoissonProcess( ): Var  D(t )   t
* Stationary M/M/1/1 with    ,
1
1 1
D(t) is RenewalProcess(Erlang(2,  )): Var  D(t )   t   e 2 t
4
Notes:
2
1 2

3 m c
* In general, for renewal process with m,  : V 
m
2
* The output process of most queueing systems is NOT renewal
8 8
V 
V 
4
B
alancing
R
educes
A
Theorem (Y. N. , Weiss 2008):
For the M/M/1/K queue with 
symptotic
V
ariance of
O
utputs
2
3K  2
V   
2
3
3( K  1)
 1:
Theorem (Al Hanbali, Mandjes, Y. N. , Whitt 2010):
For the GI/G/1 queue with   1,
under further conditions:
 2
V   (ca  cs ) 1  
 
Numerically tested Conjecture (Y. N. , 2011):
For the GI/G/1/K queue with  1 :
2
2
ca  cs
V 
  oK (1)
3
2
2
Insight about the asymptotic variance is crucial for inference of customer streams
One of the proofs tools:
Markov Arrival Processes (MAPs)


0
(1  1 ) 1
 K 1






(K 1   K 1 ) K 1 

K
  K 
E[ D(t )]   * t

 0

 0






*   De
0
(1  1 ) 1
0






(K 1   K 1 ) K 1 

0
  K 

D
Transitions with events
Transitions without events
Generator
 0

 1







C

0

 1






0
0
0
 K 1
K






0

0 
(  e  )1
Var  D(t )     *  2( * ) 2  2  D  De  t  2( * ) 2  2    De  O(t 3r  2 e  bt )
r , b 0
Asymptotic Variance Rate
Inference for MAPs
2000 Survey by Tobias Ryden, “Statistical estimation for Markov-modulated Poisson
processes and Markovian arrival processes”
Typical methods:
•MLE using EM (expectation maximization). The CTMC state is a “hidden variable”
•Moments methods (typically for structured MAPS)
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.
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
 A S N
K
{Fixed, p , np ,prior}
{constant,time varying, change point}
an s n Wn Q(t ) Ln
{obs, unobs, part obs cust, part obs time}
{stationary, non-stationary}
[0, T ]
{fixed horizon, stopping times, other}
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