Side_1_360

and with LTS.

We denote Wpthe waiting time for a packet of
priority class pand we denote the corresponding
Distribution Functions (DF) by Wp(t) = P(Wp t)

and with LST

4.1 The Unsaturated Case

We consider the unsaturated case ρ< 1, and we
define the higher priority intensity and load
(from the ρhighest priority classes) by

and

Further we also define the service time distribu-
tion of an arbitrary packet in one of the higher
priority classes denoted Bp+for 1, ..., p. The cor-
responding LST is given as the weighted sum

with mean and ithmoment given as

for i= 2, 3, ...

Similarly it is also efficient to define the service
time distribution of an arbitrary packet in one of
the lower priority classes p+ 1, ..., P, which we
denote Bp–. The corresponding LST is given as
the weighted sum

where the rate and correspond-

ing load Further the mean and

ithmoment are given as

and

for i = 2, 3, ...

We also define the remaining service time B~p–

(of the corresponding service time Bp–) and the

LST is given by

Based on the definitions above and by using the

results found in [Takagi 1991] we may write the

LST of the waiting time Wpon the following

compact form:

Wp*(s) = Wp+(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:

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

tion

Combining the two last equations yields the

important relation:

≤

̃bp(t)=^1
bp
P(Bp>t)=

1

bp

∫∞

τ=t

bp(τ)dτ

B ̃∗p(s)=^1 −B

∗p(s)

sbp

Wp∗(s)=

∫∞

t=0

e−stdWp(t).

λ+p=

∑p

k=1

λk ρ+p=

∑p

k=1

ρk.

Bp+(s)=

1

λ+p

∑p

k=1

λkB∗k(s)

b+p=^1

λ+p

∑p

k=1

λkbk=

ρ+p

λ+p

andb+p(i)=^1

λ+p

∑p

k=1

λkb(ki)

B−p(s)=

1

λ−p

∑P

k=p+1

λkBk∗(s)

λ−p=

∑P

k=p+1

λk

ρ−p=

∑P

k=p+1

ρk.

b−p=

1

λ−p

∑P

k=p+1

λkbk=

ρ−p

λ−p

b−

(i)

p =

1

λ−p

∑P

k=p+1

λkb(ki)

B ̃p−(s)=^1

ρ−p

∑P

k=p+1

ρkB ̃∗k(s).

Wp+(s)=WM/G/ 1 (s)

(

1 −

ρ−p

1 −ρ+p

+

ρ−p

1 −ρ+p

B ̃−p(s)

)

λ+p

(

=

∑p

k=1

λk

)

B+p(s)=

(

1

λ+p

∑p

k=1

λkBk∗(s)

)

:

WM/G/ 1 (s)=

s

(

1 −ρ+p

)

s−λ+p+λ+pB+p(s)

.

σp− 1 (s)=s+λ+p− 1 −λ+p− 1 θ+p− 1 (s)

θ+p− 1 (s)=Bp+− 1

(

s+λ+p− 1 −λ+p− 1 θ+p− 1 (s)

)

.

s=σp− 1 (s)−λ+p− 1 (1−B+p− 1 (σp− 1 (s))).