Side_1_360

service times are negative exponentially dis-

tributed with mean service times μp-1for priority

class p. With these assumptions we get:

where we assume that μk μ 1 (1 – ρ 1 ) for

k= 2, ..., P.

The similar result when fragmentation is per-

formed is found to be:

4.3.2 Constant Service Times for the

Highest Priority Traffic

In this case we assume that the highest priority

traffic is constant with mean b 1 = (μ 1 -1). For the

M/D/1 queue the DF of the waiting time may be

written as [Roberts1996]:

WM/D/1(t) = q(t/b 1 ; ρ) with

Below we show that it is possible to express the

convolution with an exponential density in terms

of a sum of q(x; ρ)’s in the following way:

where

packet with probability and will wait for

the remaining service time for that low priority
packet already in service plus the waiting time
for an M/G/1 queue.

Of main interest is W 1 c(t) = P(W 1 > t) the Com-
plementary Distribution Function (CDF) of the

waiting time. By definition

and by integrating the relation above we get:

where WM/G/1(t) is the DF and WMc/G/1(t) is the
CDF of the corresponding M/G/1 queue and
~
b 1 – (t) PDF for the remaining service time for an
arbitrary low priority packet. More explicitly we
have the following expressions for

~

b 1 – (t):

where

Bkc(t) = P(Bk> t) is the CDF of service time of
packets from priority class k.

In the case where fragmentation is performed the
results for the waiting time for the highest prior-
ity traffic looks very similar. If we let Wfc,1(t) =
P(Wf,1> t) be the CDF of the waiting time for
the first fragment of a high priority packet we
find:

where WM/G/1(t) is the DF and WMc/G/1(t) is the
CDF of the corresponding M/G/1 queue, and

is the unit step function.

4.3.1 Exponentially Distributed Service
Times
To carry the results further we must assume
some specific distributions. In the first (and the
most straightforward) case we assume that the

W 1 c(t)=

∫∞

τ=t

w(τ)dτ

W 1 c(t)=

(

1 −

ρ− 1

1 −ρ 1

)

WM/G/c 1 (t)

+

ρ− 1

1 −ρ 1

(

1 − ̃b− 1 (t)∗WM/G/ 1 (t)

)

̃b− 1 (t)=^1

ρ− 1

∑P

k=2

λkBkc(t),

=

q(x;ρ)=(1−ρ)

∑
x

k=0

[ρ(k−x)]k

k!

e−ρ(k−x).

ρ− 1

1 −ρ 1

Wf,c 1 (t)=

(

1 −

ρ− 1

1 −ρ 1

)

WM/G/c 1 (t)+

ρ− 1
1 −ρ 1

(

1 −^1

bf

∫t

τ=H(t−bf)(t−bf)

WM/G/ 1 (τ)dτ

)

H(x)=

{

1forx> 0

0forx< 0

W 1 c(t)=

ρ 1

1 −ρ 1

(

1 −ρ+

∑P

k=2

ρk

1 −μμk^1 (1−ρ 1 )

)

e−μ^1 (1−ρ^1 )t

+

1

1 −ρ 1

(P

∑

k=2

ρk

(

1 −

ρ 1

1 −μμk^1 (1−ρ 1 )

)

e−μkt

)

Wf,c 1 (t)=

ρ 1

1 −ρ 1

(

1 −ρ− ρ

− 1

(1−ρ 1 )bfμ 1

)

e−μ^1 (1−ρ^1 )t+

ρ− 1

1 −ρ 1

(ρ

1 e−μ^1 (1−ρ^1 )H(t−bf)

(1−ρ 1 )μ 1 bf +H(bf−t)

(

1 −bt

f

))

.

F(t;μ, ρ)=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)

)

.

μk

∫t

x=0

e−μk(t−x)WM/D/ 1 (x)dx=F(t/b 1 ;μkb 1 ,ρ)