Packet Queueing Delay in Resilient Packet Ring Network Nodes

408 Tsinghua Science and Technology, August 2004, 9(4): 406–409
For the second priority class, the queueing delay W2
has a similar expression as W1 except that an additional
queueing delay has to be counted due to higher priority
customers arriving while a lower priority customer is
waiting in the queue. W2 is obtained,
R+ ρ1W1
W2
= (3)
1−ρ−ρ
1 2
The derivation is similar for all priority classes (k>1),
and the formula for the waiting time in the scheduling
buffer is
R
Wk= , k > 1
1−ρ −L−ρ 1−ρ
−L−ρ ( )( )
1 k−1 1
k
(4)
Suppose that all nodes have the same transmission
pattern. The average number of transit buffers traversed
by each packet is then obtained,
α
N −1
= ∑ q ( j−1)
j=
2
j
(5)
where N is the number of the nodes on the ring, and qj
is the probability of node i transmitting packets to its jth
neighbor node.
Assume that traffic of all priorities has the same arrival
rate in the transmit buffer. The relationship between
the traffic arrival rates in the transit buffer and
in the transmit buffer is [9]
λTB = αλTx
(6)
We assume that all packets have the same service
time X . The relationship between the system utilization
and the arrival rate for each priority class in a node
is then
ρ1 = αρA
(7)
For a DQ ring, we identify five priority traffic
classes as shown in Fig. 3. Using Eqs. (6) and (7), the
system utilization of each priority class can be expressed
as:
ρ1 = αρA
(8)
ρ = ρ
(9)
2 A
3 = B
ρ ρ
4 C
5 ( B C)
(10)
ρ = ρ
(11)
ρ = α ρ + ρ
(12)
From Eqs. (1), (3), and (8)-(12), the queueing delays
for each priority class in different buffers are obtained:
DQ R
WPTQ
= (13)
1−
αρ
A
W
W
DQ
B
DQ
C
=
=
W
DQ
A
=
R
(1 −αρ ) 1 − 1 + α ρ
( ( ) )
A A
R
( )
( 1− ( 1+ α) ρA) 1− ( ( 1+
α) ρA + ρB)
R
( 1− ( ( 1+ α) ρA + ρB) ) 1− ( ( 1+
α) ρA + ρB + ρC)
W
DQ
STQ
=
( )
R
( 1− ( ( 1+
α) ρA + ρB + ρC)
)
1
( 1− ( ( 1+ α) ρA + ( 1+
α)( ρB + ρC)
) )
( 1 )( )
A B C
(14)
(15)
(16)
(17)
R = X + α ρ + ρ + ρ (18)
For the SQ ring, four priorities are identified as
shown in Fig. 2. Similar derivations to those used for
the DQ ring can be made. The equations for the system
utilization and queueing delay for each traffic class in
an SQ ring are obtained:
ρ = α ρ + ρ + ρ
(19)
W
SQ
A
W
=
W
SQ
PTQ
1
( )
A
B
C
ρ2 = ρA
(20)
ρ3 = ρB
(21)
ρ4 = ρC
(22)
R
=
1−
α ρ + ρ + ρ
( )
A B C
R
( )
( 1− α( ρA + ρB + ρC) ) 1−
( α( ρA + ρB + ρC) + ρA)
W
SQ
C
SQ
B
=
R
( 1−
( α( ρA + ρB + ρC) + ρA)
)
1
( 1− ( α( ρA + ρB + ρC) + ρA + ρB)
)
=
R
( 1−
( α( ρA + ρB + ρC) + ρA + ρB)
)
1
( 1− ( α + 1)(
ρA + ρB + ρC)
)
3 Simulation Results
(23)
(24)
(25)
(26)
In the normal operation state, the packets are routed on
a minimum hop basis in an RPR network. Therefore, it