Side_1_360

where we assume that μ 2 μ 1 (1 – ρ 1 ); and

where is the unit step

function.

The corresponding result when the highest prior-
ity traffic has constant packet lengths is more
complex and includes the waiting time distribu-
tion for the M/D/1 queue as well as an integral
over this distribution. (The origin of the formu-
las is described in more depth in Section 4.) We
find:

where

is the waiting time distribution for the M/D/1
queue with service time set to unity and

is the convolution of the waiting time with an
exponential distribution.

The corresponding result when fragmentation is
performed is found to be:

where

is the integral over waiting time distribution and

may be obtained by taking the limit

3 DiffServ IP Multiplexing in

the Access Network?

We are especially interested in investigating the

waiting time distribution when the link capacity

is quite low. In the examples below we have

taken the link capacity to be either 0.5 or 2.0

Mbit/sec. The high priority packet length is

taken to be 200 bytes and fragmentation is based

on ATM cells, i.e. 53 bytes. The load from the

priority traffic is assumed to be limited to the

values 0.2 or 0.3 and further the load from the

lower priority traffic is taken to be either 0.5 or

0.6 giving the total load in the range 0.7 to 0.9.

In each of the figures below we have plotted four

curves for the cases described, where the two

lower correspond to the case where fragmenta-

tion is performed. The lowest of these is for the

case where the high priority traffic has constant

distributed packet lengths, and the higher is for

exponentially distributed high priority packet

lengths. The two highest curves (which are

nearly overlapping in all the examples below)

correspond to the case with no fragmentation

and exponentially distributed low priority traffic.

(The lowest of these nearly overlapping curves

corresponds to the case where the high priority

traffic has constant packet lengths, and the

higher corresponds to the case where the high

priority traffic has exponentially distributed

packet lengths.)

Figure 1 shows that the influence of the load is

rather moderate as long as the system is stable.

In this example the link capacity is 2 Mbit/s.

With the given parameters we observe that the

waiting time distribution is nearly exponential

for both cases (straight lines in the log-plot). We

observe for the background traffic with packet

length up to 1500 bytes only one out of 100 high

priority packets will experience more than 20 ms

queuing delay. So for this case there seems to be

no need for packet fragmentation. However, if

the packet length of the background traffic

increases the picture will be different and high

priority traffic will quite frequently be delayed

more than 20 ms. This is shown in Figure 2.

=

Wf,c 1 (t)= ρ^1
1 −ρ 1

(

1 −ρ− ρ^2

(1−ρ 1 )bfμ 1

)

e−μ^1 (1−ρ^1 )t

+ ρ^2

1 −ρ 1

(ρ

1 e−μ^1 (1−ρ^1 )H(t−bf)(t−bf)

(1−ρ 1 )μ 1 bf

+H(bf−t)

(

1 −bt

f

))

H(x)=

{

1forx> 0

0forx< 0

W 1 c(t)=1−

1 −ρ 1 −ρ 2

1 −ρ 1

q(t/b 1 ;ρ 1 )

−

ρ 2

1 −ρ 1

F(t/b 1 ;b 2 /b 1 ,ρ 1 )

q(x;ρ)=(1−ρ)

∑
x

k=0

[ρ(k−x)]k

k!

e−ρ(k−x)

F(t;μ, φ)=μ

∫t

x=0

e−μ(t−x)q(x;ρ)dx

=

μ

μ+ρ

∑
t

k=0

(

ρ

ρ+μ

)k

(

q(t−k;ρ)−(1−ρ)eμ(k−t)

)

W 1 c,f(t)=1−

1 −ρ 1 −ρ 2

1 −ρ 1

q(t/b 1 ;ρ 1 )

−

ρ 2

1 −ρ 1

b 1

bf

(G(t/b 1 ;ρ 1 )

−H(t−bf)G

(

t−bf

b 1

;ρ 1

))

G(t;ρ)=

∫t

x=0

q(x;ρ)dx=limμ→ 0

1

μ

F(t;μ, ρ)

=

1

ρ

∑
t

k=0

(q(t−k;ρ)−(1−ρ))

lim

μ→ 0

1

μ

F(t;μ, ρ).