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.

1

2

3

4

1

1

4

3

Outputs

Internally Nonblocking

Switch

Losing packet

Winning packetInput Queues

cannot access output 2

because it is blocked by the

first packet

3

2

head-of-line (HOL) blockinghead-of-line (HOL) blocking

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

1

2

3

4

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 3

Output 2

Output 1

the fictitious output queues used for analysisthe fictitious output queues used for analysis

– How about small N?

* : the maximum throughput with input queueing

– Simulation Results with

Large N

N*

20.75

30.68

40.66

50.64

– 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 e

k

( 1)

0

( ) Pr[ ] i z p

i

A z A i Z e

Throughout of Input-Buffered Switch

i

1

i

2

2

i

1

i

i

1

2

3

time slot m

2imA 1 3i

mC

time slot m-1

e.g. Fictitious Queue i

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 –

Meaning of Saturation Throughput

p0 = p = throughput

For finite buffer size, if p0 > p* = 0.586 at least (p0 - p*)/ p0 fraction of packets are dropped.

Must keep p0 < p*

Input Queue

Output 1

Fictitious Queues

Output 2

Output N

Input Queue

Time spent in HOL are

independent for successive

packets when N is large

Service 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

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

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

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

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.

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