Contents Telektronikk - Telenor

If j lies in busy period (for the periodic
source) we can find a k value that gives
U(k,j) > 0, i.e. gives negative exponent in
the z-transform, which implies qj = 0.
For some cases of special interest the
boundary equations (3.5) and (3.6) can
be solved explicitly. This is the case if
the period P contains only one single
busy/idle period (for the periodic source).
We choose the numbering of the time
axis such that Dj = 0; j = 1,...,K (busy
period), and we have qj = 0; j = K+1,...,P
(idle period). In this case the boundary
equations will be of “Vandermonde”
type:
K�
qj = P (1 − ρ)
j=1
K�
qj(ζk) j−1 =0; k =2,...,K
j=1
(3.7)
(3.8)
where ζk = rk /B(rk ).
These equations are identical with (2.3)
and (2.4) in section 2.1.1, so the polynomial
generating the qj ’s may be written
as:
K�
j=1
qjx j−1 �K = P (1 − ρ)
(x − ζk)
(3.9)
(1 − ζk)
The coefficients qj in this case are easily
found by the relation between the roots
and the coefficients in a polynomial.
The form of the queue-length distributions
can be found by inverting the ztransforms
(3.3):
Qk(i) =
P �+k
j=k+1
qjΦ
k=2
� K
k=2
(i + U(k,j), j – k) (3.10)
where U(k,j) is the unfinished work
entering the queuing system from the
periodic source given by (3.4) and Φ(i,j)
is given by
Φ(i, j) = 1 d
i!
i
dzi �
B(z) −j
1 − zKB(z) −P
�
⌊ i
K ⌋
−(lP +j)
bi−Kl l=0
�
=
z=0
where b i –k is the i-th coefficient in the
expansion of B(z) –k , i.e.
(3.11)
(3.12)
For Poisson (M), Bernoulli (Geo) and
generalised negative binomial (Nb) the
coefficients above are given in Appendix
A, (where the offered A is set to ρB ).
As for the multiserver model, the explicit
expressions for the queue length distributions
are not effective to perform numerical
calculations for large i. This is due to
the fact that the function Φ(i,j) defined
by (3.11) contains all the roots of
zK - B(z) P also with modulus less than the
unity, and hence this function will grow
as power of the inverse of these roots. So
when i becomes large the expression for
Qk (i) becomes numerically unstable.
As for the multiserver model one can
overcome this problem by cancelling
these roots directly as it is done in the ztransform
(3.3) (by the boundary equations
(3.5) and (3.6)). To do so we also
need the roots of zK - B(z) P outside the
unit circle. We denote {rK+1 , rK+2 , ...} as
these roots (which in principle can be
infinite if b(i) is infinite). We also
assume that all the roots are distinct. By
making partial expansions of the z-transforms
(3.3) we get:
where
with
and
b −k
i
1
=
qk(i) = �
d
i!
i
dzi � −k
B(z) �
l=K+1
ck(rl) = 1
F (z) =
Hk(z) =
rl
�
1
ck(rl)
z=0
(3.13)
(3.14)
(3.15)
(3.16)
In the general case the series for q k (i)
given by (3.13)–(3.16) will not be convergent
for i < K = P - D. For numerical
calculations it is therefore necessary to
use (3.10) for small i, (at least for i < K)
and then the partial expansion (3.13) for
large i (i ≥ K). The first term in the series
for q k (i) (given by (3.13)–(3.16)) is the
dominant part for large i, and it can be
rl
� i
lim (rl − z)Qk(z)
z→rl
= F (rl)Hk(rl)
1 − z
[K − PzB ′ (z)/B(z)]
P �+k
j=k+1
qjz −U(k,j) B(z) k−j
shown that this root (with the smallest
modulus outside the unit circle) is real.
3.2 Finite buffer case
The finite buffer case is analysed by
Heiss [7] using an iterative approach by
solving the governing equations numerically.
However, we take advantage of the
analysis in section 3.1 and we apply the
same method as for the infinite buffer
case.
In this case the governing equation takes
the form
Qk = min[M, max[Qk-1 + Bk + dk - 1,0]] (3.17)
where M is the buffer capacity.
By using (3.17) we derive the following
set of equations for the distribution of the
number of cells in slot k (=1,...,P):
qk(0) =
�
b(i1)qk−1(i2)
qk(j) =
j = 1, ..., M – 1
qk(M) =
�
(3.18)
where qk (i) = 0 for i < 0 and i > M.
Also for the finite buffer case we introduce
the generating functions
Q M M�
k (z) = qk(i)z i .
By using (3.18) and solving for Q k M (z);
k = 0,...,P-1 we get:
j=k+1
z -U(k,j) B(z) P+k-j (3.19)
where
i1+i2+dk≤1
�
i1+i2+dk=j+1
i1+i2+dk≥M+1
i=0
Q M z − 1
k (z) =
zK − B(z) P
�
P �+k
q M j − z M+1
∞�
q M j = δ0,dj b(0)qj−1(0)
s=1
b(i1)qk−1(i2);
b(i1)qk−1(i2)
1 − zs 1 − z gM �
j (s)
is the probability that the system is empty
at the end of slot j, and
g M j (s) =
�
i1+i2+dk=s+M+1
b(i1)qj−1(i2)
is the probability that exactly s cells are
lost during slot j. (Due to the periodicity
we obviously have
213