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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