2 2 queuing theory

8/2/2019 2 2 Queuing Theory

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Queueing Theory

2008


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Queueing theory definitions

(Bose) the basic phenomenon of queueing arises

whenever a shared facility needs to be accessed for

service by a large number of jobs or customers.

(Kleinrock) We study the phenomena ofstanding,

waiting, and serving, and we call this study Queueing

Theory." "Any system in which arrivals place demandsupon a finite capacity resource may be termed a

queueing system.

(Mathworld) The study of the waiting times, lengths,

and other properties ofqueues.


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Applications of Queueing Theory

Telecommunications

Determining the sequence of computer operations

Predicting computer performance

One of the key modeling techniques for computer

systems / networks in general

Vast literature on queuing theory

Nicely suited for network analysis

Traffic control

Airport traffic, airline ticket sales Layout of manufacturing systems

Health services (eg. control of hospital bed

assignments)


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Queuing theory for studying networks

View network as collections of queues FIFO data-structures

Queuing theory provides probabilistic

analysis of these queues

Examples:

Average length (buffer)

Average waiting time

Probability queue is at a certain length

Probability a packet will be lost


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Use Queuing models to Describe the behavior of queuing systems

Evaluate system performance

Model Queuing System

Queuing SystemQueue Server

Customers


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Time

Time

Arrivalevent

Delay

Beginservice

Begin

service

Arrival

eventDelay

Activity

Activity

End

service

Endservice

Customern+1

Customern

Interarrival


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Characteristics of queuing systems

Kendall Notation 1/2/3(/4/5/6)

1. Arrival Distribution

2. Service Distribution

3.

Number of servers4. Total storage (including servers)

(infinite if not specified)

5. Population Size

(infinite if not specified)

6. Service Discipline

(FCFS/FIFO)


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Distributions

M: stands for "Markovian / Poisson" ,implying exponential distribution for

service times orinter-arrival times.

D: Deterministic (e.g. fixed constant)

Ek: Erlang with parameterk

Hk: Hyperexponential with param. k

G: General (anything)


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Poisson process & exponential distribution

Inter-arrival time t (time between arrivals) ina Poisson process follows exponential

distribution with parameter (mean)

(t)

1)(

)Pr(

=

=

tE

ett

fT(t)

t

1)( =TE


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Examples

M/M/1: Poisson arrivals and exponential service,

1 server, infinite capacity and population,FCFS (FIFO)

the simplest realistic queue M/M/m/m Same, but

m servers, m storage (including servers) Ex: telephone


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Analysis of M/M/1 queue

Given: : Arrival rate (mean) of customers (jobs)

(packets on input link)

: Service rate (mean) of the server(output link)

Solve:

L: average number in queuing system Lq average number in the queue ~ 1

W: average waiting time in whole system

Wq average waiting time in the queue ~ 1/


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M/M/1 queue model

1WqW

LLq


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Derivation

0 12 k-1 k k+1...

Po P1 Pk-1 Pk

Pk Pk+1P1 P2

P0=P1

P1=P2

P k = P k+1

since all probability sum to one

Pk =kP0

k=

k

P0 =

kP0k=0

=1


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Solving W, Wq and Lq

For stability, mean arrival rate must be less than

mean service rate

Utility factor < 1=

2

,

1 11

,(1 ) (1 )

(1 )

q

q

n

n

L L

W W

P

=

= =

= =

=


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Response Time vs. Arrivals

=1W

Waiting vs. Utilizatio

0

0.05

0. 1

0.15

0. 2

0.25

0 0.2 0.4 0.6 0.8 1 1.2

( % )

W(sec

)


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Example

On a network router, measurements show the packets arrive at a mean rate of 125

packets per second (pps)

the router takes about 2 millisecs to

forward a packet

Assuming an M/M/1 model

What is the probability of buffer overflow if

the router had only 13 buffers How many buffers are needed to keep packet

loss below one packet per million?


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Example Arrival rate = 125 pps Service rate = 1/0.002 = 500 pps Router util ization = / = 0.25 Prob. of n packets in router = Mean number of packets in router =

nn )25.0(75.0)1( =

33.057.0

25.0

1

==


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Example Probability of buffer overflow:= P(more than 13 packets in router)

= 13 = 0.2513 = 1.49x10 -8= 15 packets per bil l ion packets

To limit the probability of loss to lessthan 10 -6:

=9.96

610 n

( ) ( )25.0log/10log 6>n

2 2 queuing theory

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