Contents Telektronikk - Telenor

222
and
(25)
(26)
The parameters ck are the factorial
moments of the feedback distribution fj ,
and h (1) the moments of the background
processing time distribution Ψ(t) with
respect to t = 0.
The average sojourn time for a zero-service-time
task passing the server k times
is
where
and
ξ (1)
k
ξ (2)
k =
¯tvk
(27)
(28)
(29)
The variance of the sojourn time may be
found as
where
�
d
= −
ds ξk(s,
�
1)
s=0
= h (1) 1 − � c1 + λh (1)�k
1 − c1 − λh (1)
� 2 d
ds2 ξk(s,
�
1)
s=0
= c3 +2c1λh (1) + λ 2 h (2)
h (1)
(1 − c1 − λh (1) ) 2
��
1+
�
c1 + λh (1)� k−1 �
ξ (1)
k
�
(1)
−2kh c1 + λh (1)� k−1 �
�
=(1−ρ) h (1) λbk
ak +
1 − c1 − λh (1)
�
ak = 1
2 λc2 +2c1λh (1) + λ2h (2)
�
1 − c1 − λh (1)�2 ξk (1)
k
bk = c1h (1) ξ (1)
k−1
+ 1
�
h(2) 1+λξ
2 (1)
k−1
·
�
+ λξ(1)
k
σ 2 vk =(1−ρ) �
h (1) λqk
pk +2akbk +
1 − c1 − λh (1)
�
− (¯tvk )2
(30)
pk =
λ
� 1 − c1 − λh (1)� 2 ξ (1)
∞ = lim
k→∞ ξ(1)
k =
� c2 +2c1λh (1) + λ 2 h (2)
+
2
ξ (2)
k
�� c2 +2c1λh (1) + λ 2 h (2)�2
2 � 1 − c1 − λh (1)�
+ c3 +3c2λh (1) +3c1λ 2 h (2) + λ 3 h (3)
�
ξ (1)
k
� 2 �
and
qk = 1
3 h(3)
�� 1+λξ (1)
�2 k−1
+λξ (1)
k
(31)
(32)
The parameters given by the formulae
(23) – (32) all form monotonously increasing
sequences (with respect to k).
This leads us to conclude that a function
with a total processing time distribution
with mean value t - p and variance σ2 p will
have a response time distribution W(t)
with parameters t - w and σ 2 w fulfilling:
¯tw1 ≤ ¯tw < ¯tw∞ and σ
(33)
2 w1 ≤ σ2 w