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

• 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:– 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:

– 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…– 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 QueueBuffer Server

0 1 2 3 4 5 6 …Number in

System:

A Single Server Queue:

( )Q t Number in system at time t

( )Q t

t

The Single Server QueueBuffer Server

0 1 2 3 4 5 6 …Number in

System:

A Single Server Queue:

{ , 1}nT n Arrivals times

{ , 1}ns n Service requirements1{ , 1}n n nt T T n Inter-Arrivals times 0 0T

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

( )Q t Number in system at time t

( , ), 1n nt s n

( )Q t

nW

nW The waiting time of customer n

1 1 ,0n n n nW Max W t s

Performance Measures

Some important performance measures:

( )nP W x lim [ ]nnW E W

Little’s result: L W

lim ( )t

L E Q t

1[ ]nE t 1[ ]nE s

Assume the sequence is stochastic and stationary

Load

Stable when 1

We can quantify L (or W) under some further assumptions on

( , ), 1n nt s n

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

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

( , ), 1n nt s nAssumptions on :

0

xtP X x e dt

Results for Mean Waiting Time

Mean Waiting Time1

/ /1 1M MW

21

/ /11

1 2s

M GcW

2 2 21

/ /1 21 2a s

GI Gc cW

2 2

2 2

( ) ( ),n na s

n n

Var t Var sc cE t E s

Derivation of the M/M/1 Result

A Markov Jump Process

0 1

2

Due to M/M (Exponential), at time t, Q(t) describes the state of the process

( )( ) ( ) ( ) 1 ( ) 1 , 0,1, 2,...

dP Q t jP Q t j P Q t j P Q t j j

dt

lim ( )j tP Q t j

, 0j j Stationary distribution

0j

j

L j

01 Utilization

The Stationary Distribution

0 1

2

(1 ) jj

1, 0,1,2,...j j j

0

1jj

1L

01

Solution:

Performance measures:

lim ( ( ) ) j

tP Q t j

My Research

During PhD

• Control and stability of Queueing Networks

• Queueing Output Processes

2 2

/ /1/ , =1

2 2

21Var ( )lim

1 (1)3

a sGI G K

x

a s K

c c KD t

tc c o K

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