Analysis of Input Queueing
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Analysis of Input QueueingAnalysis of Input Queueing• More complex system to analyze than
output queueing case. • In order to analyze it, we make a
simplifying assumption of "heavy load", i.e. all queues are always full.
• This is a worst-case assumption.
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2
3
4
1
1
4
3
Outputs
Internally Nonblocking
Switch
Losing packetWinning packet
Input Queues
cannot access output 2 because it is blocked by the first packet
3
2
head-of-line (HOL) blockinghead-of-line (HOL) blocking
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01 (1 )NpN
0 1 (1 ) 1 pNp eN for large N
For p=1, 0 = 0.632
1232
Pr[ carry a packet ] = p
0 = Pr[ carry a packet ]
Internally Nonblocking Switch:Internally Nonblocking Switch:Dropping packetsDropping packets
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1234
Outputs
Internally Nonblocking
Switch
(input, output)
Fictitious Output Queues formed by HOL packets
(1,2)(1,1)(2,3)(2,1)(3,2)(3,2)(4,4)(4,1)
(1,2)(3,2)(2,3)(4,4) Output 4
Output 3Output 2Output 1
the fictitious output queues used for analysisthe fictitious output queues used for analysis
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– How about small N? * : the maximum throughput with input queueing
– Simulation Results with Large N
N*
20.75
30.68
40.66
50.64
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– Consider a fictitious queue i = # packets at start of time slot m. = # packets arriving at start of time slot m.
–
– is Poisson and independent of
as N
–
imCimA
max(0, 1)i i im m mC C A
1imB
imA 1
imB
Pr[ ]!
ki pm
pA k ek
( 1)
0
( ) Pr[ ] i z p
i
A z A i Z e
Throughout of Input-Buffered Switch
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i1i22i
1ii123
time slot m
2imA 1 3i
mC
time slot m-1
e.g. Fictitious Queue i
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Pr[ 0] Pr[ 1] Pr[ 2]...C C z C
0
( ) Pr[ ]k
k
B z z B k
Pr[ 0] Pr[ 1] ...B z B
1 1(1 ) Pr[ 0] ( )z C z C z
1 p
–
1 1( ) [(1 )(1 ) ( )] ( )C z z p z C z A z
( )( ( )) ( 1) ( )(1 )C z z A z z A z p 2 '(1)(1 '(1)) "(1) 2 '(1)(1 )C A A A p
2p pp1
under saturation
–
2 4 2 0 2 2 0.586p p p –
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Meaning of Saturation Throughput
p0 = p = throughputFor finite buffer size, if p0 > p* = 0.586 at least (p0 - p*)/ p0 fraction of packets are dropped.
Must keep p0 < p*
Input Queue
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Output 1
Fictitious Queues
Output 2
Output N
Input Queue
Time spent in HOL are independent for successive packets when N is largeService times at different fictitious queues are independent
2N
HOL
1/N
1/N
1/N
Queuing scenario for the delay Queuing scenario for the delay analysis of the input-buffered switchanalysis of the input-buffered switch
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X0 X3 X2 X1 X0
Busy periodIdle
periodBusy periodY
t
U(t)
Arrivals here are considered as arrivals in intervals i-2
Arrivals here are considered as arrivals in intervals i-1
Xi-1 Xi
The busy periods and The busy periods and interpretations for delay analysis interpretations for delay analysis
of an input queueof an input queue
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mi =2 prior
arrivals
Arrival of the packet of focus. One simultaneous arrival to be served before the packet; L=1.
Departure of packet of focus.
Xi Xi+1
Ri
W
-- Packet arrival in interval i.
-- packet departure in interval i+1.
-- number of arrivals(n)
(1) (1) (2)
Illustration of the meanings of Illustration of the meanings of random variables used in the delay random variables used in the delay
analysis of an input queueanalysis of an input queue
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Three Selection PoliciesThree Selection Policies• Random Selection Policy
– If k packets are addressed to a particular output, one of the k packets is chosen at random, each selected with equal probability 1/k.
• Longest Queue Selection Policy– The controller sends the packet from the longest queue
• Fixed Priority Selection Policy– The N inputs have fixed priority levels and of the k
packets, the controller send the one with highest priority
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W_
p0
Different contention-resolution policies Different contention-resolution policies have different have different waiting time versus waiting time versus load relationships, but a common load relationships, but a common maximum load at which waiting time maximum load at which waiting time goes to infinity.goes to infinity.
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ConclusionConclusion• Mean queue length are always greater for
queueing on inputs than on outputs
• Output queues saturate only as the utilization approaches unity
• Input queues saturate at a utilization that depends on N, but is approximately 0.586 when N is large