Photon Netw Commun (2015) 30:167–177 169
Fig. 1 The component (δ, L)
represents the burst, where the
first element δ is the offset time,
while the second element L is
the length of the burst. There are
three classes of burst associated
with offset time 2, 4 and 6. The
first plot shows a sample
channel horizon path. The
second plot shows the
transformed MLSFM. τ
δ
2
7
and
τ
δ
1
5
are corresponding to the
blocking time B
1
(7) and B
2
(5)
the channel is occupied for this incoming burst-v (v ≤ k,
since H (t) = H (t−) + L
k
+ δ
k
>δ
k
>δ
v
). We are inter-
ested the following important question that, when some class
of burst is blocked, how long is it before t he channel is avail-
able again for this class of incoming burst?
We assume that at time t , H (t ) = w>δ
k
and at
this time epoch BCP-k (burst-k) arrives, it is easy to see
that BCP-k (burst-k) is blocked, we refer to this time inter-
val from the time epoch at which BCP-k (burst-k)arrives
to the time epoch after which the channel is available
again for the potential burst-k as the blocking time of
burst-k. It is easy to see that the blocking time of burst-
k is dependent on the level of the channel horizon H(t),
and we denote the blocking time of burst-k by B
k
(w)
given the level of horizon is w at the time epoch that
the burst-k arrives. As depicted in Fig. 1, the blocking
time B
k
(w) is also the time duration from the time epoch
that the burst-k arrives and is blocked to the time epoch
that the channel horizon first hits the level δ
k
from the
above.
2.1 The multi-layer stochastic fluid model for an OBS
node
We first introduce some necessary notations. For a matrix A,
we let A
ij
be the block matrix of A, and denote its (i, j)th
element by [A]
ij
. We also write I
n
and e
n
as the unit matrix
and unit column vector with dimension n, when not causing
ambiguity, we write I and e for short. We let 0 be a zero
matrix with an appropriate dimension.
Stochastic fluid models (SFMs) have been studied exten-
sively for a few decades, and they have been particularly
promising tools for studying telecommunication networks
[10–13], insurance theoretical models [14–17], and for appli-
cations in some other areas [18–20]. SFM is an input–output
system of fluid with a finite (or infinite) buffer, which consists
of the level process and the background process. The level
process has a continuous domain and it indicates the amount
of workload (or fluid) in the system, while the background
process is a homogeneous Markov process on the finite state
space. The s tate space of the background process determines
the rate of net flow of workload into the system, which is
called fluid rate. In the basic version of SFM, the fluid rates
are independent of the fluid level, however, in a more general
set of Markov fluid models, the fluid rates is a determinis-
tic piecewise-constant function of the fluid level, then this
process is referred to as multi-layer stochastic fluid model,
or in short MLSFM. Other authors also call it feedback sto-
chastic fluid model (FSFM) in [21] or feedback Markov fluid
queue (FMFQ) in [5,7].
It is well known that time-dependent performance metrics
of fluid queueing systems play a crucial role in identifying the
Quality of Service (QoS) requirements guaranteed for users,
e.g. [12,15–17,22,23]. In the present paper, the value of our
technique to approximate the blocking time of multi-class
OBS nodes is that a stochastic transformation method is pro-
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