Contents Telektronikk - Telenor

At the moment our test task enters the
server it will have waited a time t and
there will be j other tasks in the system.
The corresponding distribution becomes
V1 (s,z)
= Q(s,(q + pz)Ψ(s + λ(1 – z)),z)
= Q(s,ξ 1 (s,z),ξ0 (s,z)) (12)
Assume that our tagged task is a zeroservice-time
task and that it passes the
server k times. When it passes for the
k’th time it will have been in the system a
time t and there will be j other tasks in
the system. The corresponding distribution
will have the transform
Vk (s,z) = V1 (s,ξk–1 (s,z)) (k = 1, 2, ...)(13)
We are now in the position to rewrite
Uk (s,z) as
Uk (s,z) =
� �
k−1 �
Ψ(s + λ(1 − ξℓ(s, z))) Vk(s, z)
ℓ=0
(14)
Expressions for the moments of the
sojourn time distribution corresponding
to U k (s,1) may be found in terms of
moments of the sojourn times of zero service
time tasks, i.e.
� r d
dsr ξℓ(s,
�
1)
(15)
by taking the logarithm of equation (14)
and differentiating successively. Using
this procedure also makes it possible to
develop the formulas for the first two
moments (which still have a relatively
comprehensible structure) of the distribution
with transform
∞�
Φ(s) =q p k−1 Uk(s, 1)
k=1
s=0
(R(s) ≥ 0)
as dealt with in the paper of Takács.
(16)
Application to a deterministic
sequence of tasks
Feedback queuing has been used in processor
performance models, cfr. e.g. [2],
[3] and [4].
A processor in a telecommunication system
normally performs an extensive set of
different functions, each initiating a deterministic
sequence of individual process-
ing jobs. The processing times may be
deterministic or have specific distributions.
Such a processor may provide different
performance, in terms of e.g. response
times, for each of the various functions.
For each specific function the performance
will depend on
1 the whole set of functions offered and
their arrival intensities, i.e. on the
workload characteristics. This may be
modeled by means of a single server
queuing process with feedback, as in
the paper of Takács, but possibly with
a slightly more general distribution fi for the number of feedbacks which
may be generated as a result of the
processing of a job. The external
arrival intensity and the service time
distribution Ψ(t) will depend on the
mix of functions and their frequencies.
One may call this queuing process the
background process,
2 the processing requirement of the specific
function and the way it is handled
by the operating system in terms of a
sequence of processor jobs. This may
be modeled as a sequence of tagged
jobs with service times according to
different distributions. Some, if not
most, of the processing times will be
deterministic.
To derive the response time of a specific
function comprising a sequence of k
tagged jobs one has to make changes
corresponding to the introduction of different
service time distributions in the
formula
Uk (s,z) =
�
k�
�
Ψ(s + λ(1 − ξk−ℓ(s, z))) Vk(s, z)
ℓ=1
(17)
which is just a rearranged formula (14).
The corresponding response time / number
of left jobs distribution will have the
transform
Wk (s,z) =
�
k�
�
Ψℓ(s + λ(1 − ξk−ℓ(s, z))) Vk(s, z)
ℓ=1
(k = 1, 2, ...) (18)
where Ψ l is the Laplace-Stieltje transform
of the processing time distribution
for tagged job No. l.
If all the tagged jobs have deterministic
processing times this formula may be
written as
Wk
e (19)
(k = 1, 2, ...)
− �k ℓ=1 (s+λ(1−ξk−ℓ(s,z)))tℓ Vk(s, z)
Note that the definition of ξℓ(s, z) now is
slightly different from the previous
chapter, as we will have
( ℓ = 1, 2, ...) (20)
and ξ 0 (s,z) = z. f(z) is the z-transform of
the feedback distribution f i .
Stationarity of the process is assured
whenever the load
(22)
Of course, the Uk (s,z) discussed previously
may be obtained by assuming that
all tagged jobs have processing times
with the same distribution as the other
jobs, and that the feedback distribution
has z-transform f(z) = q + pz.
The form of the response time distribution
is conceptually convenient inasmuch
as it illustrates the impact of the partition
of work of each particular function on its
system response time.
The mean and variance of the response
times are easily calculated.
where
�
s; {tℓ} k
�
ℓ=1 ; z =
ξℓ(s, z) =f(ξℓ−1(s, z))
Ψ(s + λ(1 − ξℓ−1(s, z)))
λα
ρ =
(21)
1 − f
of the server is less than 1.
Also the definition of Vk (s,z) has to be
slightly changed. Eq. (13) is still valid,
but now starts with:
V1 (s,z) =
′ (1)
�
(1 − ρ) 1+ λ(1 − ξ1(s, z))
s + λ(ξ1(s, z) − z) ·
�
ξ1(s, z) − f(z)Ψ(λ(1 − ξ1(s, z)))
ξ1(s, z) − f(ξ1(s, z))Ψ(λ(1 − ξ1(s, z)))
¯twk = ¯tvk +
σ 2 wk = σ2 vk +
σ 2 Ψℓ
k�
ℓ=1
�
1+λξ (1)
�
k−ℓ · ¯tΨℓ
k�
ℓ=1
+ λξ(2)
k−ℓ ¯tΨℓ
�� 1+λξ (1)
�2 k−ℓ
�
(23)
(24)
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