Side_1_360

In the second example we have taken a slower
link with capacity 0.5 Mbit/s. In this case we see
that approximately one out of 100 high priority
packets will get a queuing delay greater than 80
ms. This delay lies in the range where it adds up
with other types of delay and may reach the limit
where QoS is not possible to maintain, for
instance for speech services.

If the packet length of the background traffic
increases, the distribution function of the queu-
ing delay will decrease very slowly, and with a
relatively high probability the high priority traf-
fic will experience unacceptable delays and
therefore fragmentation will be necessary in
order to maintain QoS.

To conclude the numerical examples it seems
that the DiffServ model with the EF traffic hav-
ing (non-preemptive) priority over the other
classes seems to give satisfactory performance
on access-links higher than 2 Mbit/s. For links
with lower bitrate the multiplexing disturbance
from lower priority traffic may be so high that it
will be difficult to maintain stable QoS. In this
case fragmentation of the lower priority traffic,
for instance by deploying ATM as a link proto-
col, will be an efficient alternative in order to
solve these multiplexing problems. This is how-
ever a critical limit since a broad part of the
access links will typically be ADSL links with
access rates in the range of 2 Mbit/s.

Further investigations could be done with more
realistic arrival time distributions, especially
with regard to the background traffic.

4 Some Methods for Calculat-

ing Delay Distributions in

Non-Preemptive Priority

Queues

In this section we will consider some methods to

calculate waiting time distributions for priority

queues. Most of the results for the M/G/1 queues

with priorities are given in terms of Laplace-

Stieltjes transforms(LSTs). To get the actual

distributions we then have to invert these trans-

forms.

We consider a non-preemptivequeuing system

with Ppriority classes where the priority order-

ing is according to the increasing numbers

indexed by p= 1, 2, ..., P.

Packets from class parrive according to a Poisson

process with rate λpand the service times (de-

noted Bpin each class) are all independent with

Distribution Functions (DF) Bp(t) = P(Bp t)

and LST We denote

mean of the service time bpand the ithmoments

bp(i), i= 2, 3, .... Further the load from class pis

given by ρp= λpbpand the total arrival rate and

load on the server are:

and.

Sometimes we will also need to consider the

remaining service time

~

Bp(from an arbitrary

time until the service is finished for a given pri-

ority class). Then the Probability Density Func-

tion (PDF) of this stochastic variable is given as:

ρ= ρp

p= 1

P

λ= λp ∑

p= 1

P

∑

0

-0.5

-1

-1.5

2

-2.5

-3

-3.5

Log[P(W>t)]

0.025 0.05 0.075 0.1

sec

0.125 0.15 0.175 0.02

0

-0.5

-1

-1.5

2

-2.5

-3

-3.5

Log[P(W>t)]

0.025 0.05 0.075 0.1

sec

0.125 0.15 0.175 0.02

Figure 4 The complementary waiting time distribution for priority traffic with ATM fragmentation (the lower curves) and without any
fragmentation (upper curves) for a link of capacity 0.5 Mbit/s, and high priority packet length of 200 bytes, low priority packet length
of 3000 bytes in the left figure and 6000 bytes in the right figure. The load is 0.2 and 0.5 for high priority and low priority traffic

≤

Bp∗(s)=

∫∞

t=0

e−stdBp(t).