Side_1_360

4.2 Unsaturated Case with Batch

Arrivals

To avoid the negative effect from the lower pri-
ority traffic on the high priority traffic, fragmen-
tation of the low priority packets may be neces-
sary in some part of the network. By cutting the
long packets into a number of smaller pieces the
maximum waiting time due to lower priority
traffic is limited to the length of one fragment of
a packet. We may model the fragmentation of
packets by representing a packet arrival as an
arrival of a batch of fragments (from that partic-
ular packet).

We assume that the fragments are of constant
length of bf, and the corresponding LST is

Bf(s)=e-sbf. With this assumption we may find
the distribution of the number of fragments a
packet (from priority class p) consists of as:

gip= P((i– 1)bf≤Bp< ibf) = Bp(ibf) –
Bp((i–1)bf) for i= 1, 2, ... and further we let the
corresponding generating function be:

and the corresponding mean value is

The corresponding LST of the service time of
a whole packet of class p(if it were not inter-
rupted by fragments from higher priority pack-
ets) is then:

Bg,p* (s) = Gp(Bf(s)) = Gp(e-sbf)

and the corresponding mean value is gpbf. The
load from the packets of class pis then ρp=
λpgpbf. As for the case without fragmentation
we define

and , and we consider

the unsaturated case ρ< 1.

We also define the higher priority intensity and
load (from the phighest priority classes) by

and. Further we also

define the service time distribution of an arbi-
trary packet (that is not interrupted) in one of the
higher priority classes 1, ..., p, denoted Bg,p+. The
corresponding LST is given as the weighted sum

with mean

Similarly we also define the rate

and corresponding load We

also define the remaining service time of a frag-

ment B~fwhich is uniformly distributed over the

interval (0, bf) with LST:

The queuing system described above is a non-

preemptive queuing model with batch arrivals.

By using the results found in [Takagi 1991] we

may write the LST of the waiting time for the

first fragment in a packet Wf,pon the following

compact form:

Wf*,p(s) = Wf+,p(σp-1(s)) where

and where WM/G/1(s) is the LST of the waiting

time distribution in an M/G/1 queue with input

rate and LST of the service

time given as

Further the function σp–1(s) is defined through

the LST of the busy period distribution, θ+p-1(s),

generated by packets of class 1, 2, ..., p– 1 (as

for the system without batch arrivals):

σp-1(s) = s+ λ+p-1– λ+p-1θ+p-1(s)

where θ+p-1(s) is the unique solution of the equa-

tion

Gp(z)=

∑∞

i=1

gpizi=

∑∞

i=1

(Bp(ibf)−Bp((i−1)bf))zi;

gp=

∑∞

i=0

Bpc(ibf).

λ=

∑P

p=1

λp ρ=

∑P

p=1

ρp

λ+p=

∑p

k=1

λk ρ+p=

∑p

k=1

ρk

B+g,p(s)=

1

λ+p

∑p

k=1

λkB∗g,k(s)

b+g,p=

bf

λ+p

∑p

k=1

λkgk=

ρ+p

λ+p

.

λ−p=

∑p

k=p+1

λk

ρ−p=

∑P

k=p+1

ρk.

B ̃f(s)=^1 −e

−sbf

sbf

Wf,p+(s)=WM/G/ 1 (s)

(

1 −

ρ−p

1 −ρ+p

+

ρ−p

1 −ρ+p

B ̃f(s)

)

λ+p

(

=

∑p

k=1

λk

)

B+g,p(s)

(

=

1

λ+p

∑p

k=1

λkB∗g,k(s)

)

:

WM/G/ 1 (s)=

s(1−ρ+p)

s−λ+p+λ+pB+s,p(s)

.