Side_1_360

θ+p-1(s) = Bg+,p-1(s+ λ+p-1– λ+p-1θ+p-1(s).

Combining the two last equations yields the

important relation:

s= σp-1(s) – λ+p-1(1 – Bg+,p-1(σp-1(s))).

Compared with the model without fragmentation

we see that the two results are similar in the way

the remaining service time of the lower priority

packets is introduced in the expression, however,

for the model with fragmentation the influence

from the lower priority traffic is limited to the

remaining service time of a single fragment.

A second observation we may mention is that

when the fragments are small the difference in

the service times of a packet and the correspond-

ing service times introduced by fragmentation

may be small. This can be seen from the LSTs of

the two variants. If we let ti= ibfwe may write

with dBp(ti) = Bp(ti) – Bp(ti–1).

This is a well known approximation of the

integral so when bfis

sufficiently small we will have

A major difference between the two models is

that the service of a low priority packet may be

interrupted after the completion of a fragment. It

will therefore also be of interest to find the wait-

ing time for the last fragment of a packet of pri-

ority class p. We denote this waiting time Wl,p.

We have Wl,p= Wf,p+ Dpwhere Dpconsists of

the service times of all the fragments from the

‘tagged’ packet of class pplus the delay cycles

generated from packets of the 1, 2, ..., p– 1

higher priority classes. The probability that a

‘tagged’ packet will consist of exactly ifragments

is given by so the probability that the last

fragment of a packet has exactly i fragments prior

in the queue hip(from that particular packet is:

and the

corresponding generating function is:

The LST of service time distribution of all these

fragments is then given as

As explained in [Takagi 1991] the corresponding

LST of Dpis obtained from Bpf*(s) by the rela-

tion:

Finally we get the LST for the waiting time of

the last fragment in a packet from priority class

pas:

.

4.3 Waiting Time Distribution for the

Highest Priority Traffic

This corresponds to the case p= 1 and in this

case we have σ 0 (s) = s, which gives the LST of

the waiting time as:

where

WM/G/1(s) is the LST of the waiting time distri-

bution in an M/G/1 queue with input rate λ 1 and

LST of the service time given as B 1 (s):

.

If we let w 1 (t) denote the density function for the

waiting time we get by inverting the equation

above:

where wM/G/1(t) is the density functions for the

M/G/1 queuing model and ~b 1 – (t) the DF of the

remaining service time for an arbitrary low pri-

ority packet.

In fact the latter formula may be explained as

follows: In the long run an arriving high priority

packet will find the server either idle or serving a

high priority packet with probability

and will ‘see’ the system as an M/G/1 queue, or

will find the server occupied with a low priority

1 − ρ^1

−

1 −ρ 1

w 1 (t)= 1 −

ρ 1 −

1 −ρ 1

⎛

⎝⎜

⎞

⎠⎟

wM/G/1(t)

+

ρ 1 −

1 −ρ 1

b ̃ 1 −(t)*wM/G/1(t)

WM/G/1(s)=

s( 1 −ρ 1 )

s−λ 1 +λ 1 B 1 (s)

W 1 *(s)=WM/G/1(s)1− ρ^1

−

1 −ρ 1

+ ρ^1

−

1 −ρ 1

⎛ B ̃ 1 −(s)

⎝⎜

⎞

⎠⎟

Wl*,p(s)=Wf*,p(s)D*p(s)

B∗g,p(s)=Gp(e−sbf)=

∑∞

i=1

e−stidBp(ti)

Bp∗(s)=

∫∞

t=0

e−stdBp(t)

Bg,p∗(s)≈Bp∗(s).

1

gp

igip

hpi=

1

gp

(i+1)gip+1 i=0, 1 ,...

Hp(z)=

1

gp

∑∞

i=0

(i+1)gpi+1zi=

1

gp

G′p(z).

Bf

∗

p(s)=

1

gp

G′p(Bf(s)) =

1

gp

G′p(e−sbf).

D∗p(s)=Bf

∗

p (σp−^1 (s)) =

1

gp

G′p

(

e−σp−^1 (s)bf

)

.