Contents Telektronikk - Telenor

34
Since Gw (t) is the survivor function, it
means the probability of the last previous
arrival still being in the system. If the
arrival interval distribution is given by
density f(t), then P{t < arrival interval ≤t
+ dt} = f(t) ⋅ dt. The product of the two
independent terms, G w (t) ⋅ f(t)dt, implies
that the next arrival happens in the interval
(t, t + dt), and that there is still one or
more customers in the system. By integration
over all t we obtain the implicit
equation for determination of σ:
� ∞
σ = e −µ(1−σ)·t · f(t)dt
0
(93)
It is immediately seen that if f(t) = λe –λt ,
then σ = λ/µ, and we get the M/M/1
solution as we should.
By analogy it is clear that the mean values
of waiting and sojourn times, which
are related to calls arriving, and queue
lengths as observed by arriving calls, are
given by formulas identical to those of
M/M/1, only that the mean load A is
replaced by the arrival related load σ.
Taken over time the mean values of number
in queue (Lq ) and number in system
(Ls ) are found by Little’s formula, based
on mean waiting time:
Lq = λ · W =
λ · s · σ
1 − σ
= A · σ
1 − σ
λ · s A
Ls = λ · (W + s) = = (94)
1 − σ 1 − σ
Example
In Table 5 is given an example set with
arrival interval distributions:
- Deterministic
- Erlang-2
- Exponential
- Hyperexponential (H2 ).
Table 5 Example with mean load A = 0.5 of GI/M/1 system
For the H2-distribution is chosen p = 0.9,
λ1 = 1.0 and λ2 = 1/11, like in the M/G/1
example previously, and two more cases:
p = 0.75/0.95, λ1 = 1.0/1.0 and λ2 =
0.2/0.0476. Arrival rate is chosen at
λ = 0.5 (consistent with the H2-examp les) and service time s = 1/µ = 1.0. Mean
load is A = λ/µ = 0.5, and service is
exponential.
The examples in the table indicate that
waiting time and queue length increase
with the form factor, but in a non-linear
way. This is different from the M/G/1
case, where there is strict proportionality.
16.4 The GI/GI/1 queue
When no specification is given for the
distributions of arrival and service processes,
other than independence within
each process and mutually, then there are
few general results available concerning
queue lengths and waiting time. A general
method is based on Lindley’s integral
equation method. The mathematics
are complicated, and it will not be discussed
any further. There are a lot of
approximations, most of them covering a
limited area with acceptable accuracy.
According to the Pollaczek/Khintchine
formula for M/GI/1 two moments of the
service distribution gives an exact solution.
The GI/M1 case is shown to be less
simple, and even though most approximations
for the GI/GI/1 queue apply only
two moments for both distributions, it is
clear that even the third moment of the
arrival distribution may be of importance.
According to [15] a reasonable approximation
for high loads (A→ 1) is the one
by Kingman [16], which is also an upper
limit given by
�
W ≈
(95)
A · s
2 · (1 − A) ·
�
(ca/A) 2 + c 2 s
Distribution Load at arrival (σσ) Waiting time (W) = Form factor (εε)
Queue length (L)/λλ
Deterministic 0.2032 0.255 1.0
Erlang-2 0.382 0.618 1.5
Exponential 0.5 1.0 2.0
Hyperexponential-2 0.642 1.79 3.5
" 0.74 2.85 6.5
" 0.804 4.11 11.5
where ca and cs are coefficient of variation
for arrivals and service respectively.
We immediately recognise the basic
terms giving proportionality with load
(A) and service time (s) and the denominator
term 2(1 – A). Next is the strong
dependence on second moments. The
formula approaches the correct value for
M/M/1 when A → 1. Otherwise it is not a
good approximation. An adjustment factor
has been added by Marchal [17]:
(1 + c 2
s )/(1/A2 + cs2 ) (96)
This gives the correct result for M/G/1
irrespective of load.
A better adjustment to Kingman’s formula
has been introduced by Kraemer
and Langenbach-Belz [18], using an
exponential factor containing the load as
well as the coefficients of variation. Distinct
factors are used for the two regions
ca ≤ 1 and ca ≥ 1. The best general approximation
over a wide range seems to be
that by Kimura [19]:
(97)
Here, σ is the mean load immediately
before an arrival, like the one in the
GI/M/1 case, given by an integral equation.
Thus the real distribution (not only
first and second moments) is taken into
account. Calculation is not straightforward
because of the implicit solution.
Another proposal [20], [21] is based on a
three moment approach for arrivals,
applying a three-parameter H2 distribution.
The three-moment match is much
more flexible than the two-moment
match, and it can be used to approximate
a general distribution of the type where
ca ≥ 1. The approximation utilises the
property of the H2 distribution that it can
model two extremes by proper choice of
parameters:
1 Batch Poisson arrivals with geometric
batch size
2 Pure Poisson arrivals.
Exact solutions are available for the
whole range of H2 /M/1 between the
extremes Mb /M/1 and M/M/1. In the general
service case the extreme points
Mb W ≈
/GI/1 and M/GI/1 have known exact
solutions. A heuristic interpolation between
those two points can then be
attempted for the GI case in analogy with
the known M case. A plot for the waiting
time as a function of the third moment is
shown in Figure 37.
σ · s · � c2 a + c2 �
s
(1 − σ) · (c2 a +1)