Transcript Balancing Reduces Asymptotic Variance of Outputs

A bit on Queueing Theory:
M/M/1, M/G/1, GI/G/1
Yoni Nazarathy*
EURANDOM, Eindhoven University of Technology,
The Netherlands.
(As of Dec 1: Swinburne University of Technology, Melbourne)
Swinburne University Seminar, Melbourne,
July 29, 2010.
*Supported by NWO-VIDI Grant 639.072.072 of Erjen Lefeber
Outline
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The term: queueing theory
The single server queue
M/M/1, M/G/1, GI/G/1
Mean waiting time formulas
Derivation of the M/M/1 result
A glimpse at my queueing research
The Term: Queueing Theory
Queues
• Customers:
– Communication packets
– Production lots
– Customers at the ticket box
• Servers:
– Routers
– Production machines
– Tellers
• Queueing theory:
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–
–
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Quantifies waiting/congestion phenomena
Abstract models of reality
Mostly stochastic
Outputs:
• Performance evaluation (formulas, numbers, graphs)
• Design and control (decision: what to do)
Queueing Research
• 1909: Erlang – telephone lines
• Dedicated journal: Queueing Systems
• Other key journals:
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Stochastic Models
Applied Probability Journals (JAP/Advances)
Annals of Applied Probability
OR, ANOR, ORL, EJOR…
About 5 other applied probability journals
Books: Around 200 Teaching/Research
Active researchers: ~500
Researchers that “speak the language”: ~2000
Related terms: “Applied Probability”, “Stochastic Modeling”
Queueing Theory Applied in Practice
• Here and there…
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Practice motivates many new queueing problems
BUT: Queueing results not so often applied
Accurate data sometimes hard to obtain
Models are often too simple for very complex realities
• Simulation can do much more…
• …but say much less
• Insight gained from queueing theory is important
The Single Server Queue
The Single Server Queue
A 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
A 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
{tn  Tn  Tn1, n  1}  Inter-Arrivals times T0  0
{sn , n  1}  Service requirements
The sequence
(tn , sn ), n  1 Determines evolution of Q(t)
Wn 
The waiting time of customer n
Wn1  Max Wn  tn1  sn ,0
(tn , sn ), n  1
Wn
Q(t )
Performance Measures
Assume the sequence is stochastic and stationary
  E[tn ]
1
  E[sn ]
Stable when 
1


 Load

1
Some important performance measures:
P(Wn  x)
Little’s result:
W  lim E[Wn ]
n 
L  lim E Q(t ) 
t 
L  W  
We can quantify L (or W) under some further assumptions on
(tn , sn ), n  1
M/M/1, M/G/1, GI/G/1
Notation for Queues
• A/B/N/K
– A is the arrival process
– B is the service times
– N Is the number of servers
– K is the buffer capacity (default is infinity)
M/M/1, M/G/1, GI/G/1
(tn , sn ), n  1 :
Assumptions on
• M Poisson or exponential or memory-less
P  X  x    e
• G General
• GI Renewal process arrivals
x
0
 t
dt
Results for Mean Waiting Time
Mean Waiting Time
WM / M /1  
1

1 
2
1

c

s
WM / G /1   1
1  2
2
2 2
c


cs

1
a
WGI / G /1  
1 
2 2
ca 
2
Var (tn )
E  tn 
2
, cs 
2
Var (sn )
E  sn 
2
Derivation of the M/M/1 Result
A Markov Jump Process
Due to M/M (Exponential), at time t, Q(t) describes the state of the process

0


2
1



dP  Q(t )  j 
 (   ) P  Q(t )  j    P  Q(t )  j  1   P  Q(t )  j  1 , j  0,1, 2,...
dt
 j  lim P  Q(t )  j 
t 

j
, j  0  Stationary distribution

L   j j 1   0  Utilization
j 0
The Stationary Distribution

0
2
1
 j   j 1,





j 0


j
j  0,1, 2,...
1
Solution:
 j  (1   )  j
Performance measures:
L

1 
1  0  
lim P(Q(t )  j )   j
t 
My Research
During PhD
• Control and stability of Queueing Networks


• Queueing Output Processes

2
2
2 

c

c
1

s 
 K 
Var  DGI / G /1/ K ,  =1 (t )    a
 
lim

x 
t
 1 c 2  c 2  o (1) K  
s 
K
 3  a
During Post-doc
• BRAVO Effect
(Seminar tomorrow at Melbourne University)
• Sojourn Time Tail Asymptotics
• Methods of Control Theory Applied to Queues
• Stability of Queueing Networks
• Asymptotic scaling of stochastic systems
• Optical Packet Switching Applications
In future…
• Research area: Model selection and statistics
of queueing networks (from data)
• Engineering applications
• More on previous subjects
• Power supply networks
Thanks for Listening and
See you Dec 1, 2010