On Analyzing the Capacity of WDM PONs
Jingjing Zhang and Nirwan Ansari
Advanced Networking laboratory
Department of Electrical and Computer Engineering
New Jersey Institute of Technology, Newark, NJ 07032, USA
Email: {jz58, nirwan.ansari}@njit.edu
Abstract—By taking advantage of the multiple wavelength
provisioning capability, WDM PON dramatically enlarges its
capacity as compared to TDM PON. The capacity of a WDM
PON system depends on many factors, such as the number and
capacities of wavelength channels, the network architecture, the
wavelength support of receivers, and the tuning ability of lasers.
This paper introduces the definition of achievable rate region to
describe the capacity of WDM PON and analyzes the achievable
rate region for a given network architecture from the perspective
of wavelength sharing. This can facilitate comparison of different
WDM PON architectures and help design an efficient access
control scheme.
I. I NTRODUCTION
Wavelength Division Multiplexing (WDM) Passive Optical
Network (PON) is a promising future-proof access network
technology to meet the rapidly increasing traffic demands
caused by the popularization of Internet and spouting of
bandwidth-demanding applications [1]–[3]. As compared to
Time Division Multiplexing (TDM) PON, WDM PON needs
to facilitate remote nodes, transmitters, and receivers with
multi-wavelength provision capability [4].
There are three major classes of light source generators,
depending on the wavelengths generation capability, namely,
wavelength-specified sources, wavelength-tunable sources, and
multi-wavelength sources [1]. A wavelength-specified source
emits only one specific wavelength, e.g., the common distributed feedback (DFB) / distributed Bragg reflector (DBR)
laser diode, or the vertical-cavity surface-emitting laser (VCSEL) diode. A multiple-wavelength source is able to generate
multiple WDM wavelengths simultaneously, including multifrequency laser, Gain-Coupled DFB LD Array, and ChirpedPulse WDM. Similar to multiple wavelength sources, a
wavelength-tunable source can generate multiple wavelengths.
However, it can only generate one wavelength at a time.
The receiver module consists of a photodetector (PD) and its
accompanying electronics for signal recovery. Common PDs
are positive-intrinsic-negative (PIN) and avalanche photodiode
(APD), which find different applications according to the
required sensitivity. Generally, one receiver detects data in one
wavelength at a time. So, the number of receivers determines
the number of simultaneous received wavelength channels. We
assume PDs can detect optical signal in any wavelength, and
the receiver is wavelength-inselective.
Formerly, Baik and Lee [5] analyzed the capacity of hybrid
WDM/SCMA-PON with the Reflective Semiconductor Optical
Amplifier (RSOA) and the (Fabry-Perot laser diode) FP-LDs
injected by external spectrum-sliced incoherent light, from
the physical layer performance point of view. In this paper,
we do not consider the physical layer performance of optical
devices, but analyze the network capacity from the perspective
of wavelength sharing, which constitutes another important
factor affecting the network capacity. Intuitively, the capacity
of WDM PON can be increased by increasing the number of
wavelengths, receivers, and lasers at OLT and ONUs, or by
broadening tuning ranges of lasers and receiving ranges of
receivers. However, this paper will show that increasing the
number of wavelengths, lasers, and receivers, and broadening
the lasers’ tuning range do not always increase the capacity.
The capacity of the network is determined by a combination of
all these factors. Simply by figuring out the bottleneck factor
and then giving the bottleneck factor more powerful capability
can increase the network capacity.
To better describe the network capacity, we introduce the
definition of achievable rate region as the set containing all
admissible traffic rates of a given WDM PON system. The
larger the volume of the achievable rate region, the larger the
capacity of the network. The paper makes further contributions
in the following two aspects:
• It provides a method to abstract any WDM PON system
into a tiered directed graph. Then, the analysis of the
WDM PON system is transformed into the analysis of
the tiered graph.
• It proposes a scheme to analyze tiered directed graph,
from which the achievable rate region for any given
WDM PON system can be derived.
With the knowledge of the achievable rate region, whether
an incoming traffic rate can be accommodated or not can be
easily decided. This will help design an efficient admission
control scheme in the MAC layer.
The rest of the paper is organized as follows. Section II
describes the network architecture we considered in this paper
and the system model. Section III discusses the scheme of
deriving the achievable rate region. We specially introduce definitions of two relations: “reach” relation and “block” relation
to facilitate derivation of the achievable rate region. Section
IV discusses the achievable rate region of one example of
WDM PON system. Section V presents concluding remarks.
II. S YSTEM M ODEL
This paper focuses on the upstream data transmission only.
The same idea can be applied to derive the achievable rate
Queue requests select lasers to generate the light signal
to carry the data traffic.
• Lasers select proper wavelengths to be tuned to.
• After the upstream signals arrive at the OLT side, signals
in certain wavelengths are received by receivers.
To describe the process, the abstracted graph consists of
four tiers of vertices, corresponding to queue requests at
ONUs, lasers at ONUs, wavelengths, and receivers at OLT,
respectively, as depicted in Fig. 1 (b). In the graph, vertices
in one tier only connect to vertices in its neighboring tiers.
The subgraph constituted by vertices in one tier and vertices
in its neighboring tier is a bipartite graph. The construction of
connecting edges proceeds as follows.
• One tier-one vertex connects to one tier-two vertex if and
only if the corresponding queue request in the first tier
can access that corresponding laser in the second tier.
Typically, one ONU has multiple queue requests which
can access all the lasers equipped at the ONU. So,
the bipartite subgraph constituted by queue requests and
lasers in the same ONU is fully connected.
• One tier-two vertex connects to one tier-three vertex if
and only if the corresponding laser in the second tier can
be tuned to that corresponding wavelength in the third
tier.
These edges are determined by the laser’s tuning ability.
If all the lasers are full-range tunable, the bipartite graph
is fully-connected.
• One tier-three vertex connects to one tier-four vertex if
and only if the wavelength in the third tier can be received
by the receiver in the fourth tier. The edges connecting
vertices in the two tiers depend on the receiver’s receiving
range. Usually, the receivers are wavelength-inselective,
implying that the subgraph constituted by the tier-three
vertices and the tier-four vertices are fully connected.
At each time, one laser schedules one queue request only;
one laser is tuned to one wavelength only; one wavelength is
tuned by one laser only, and one receiver can receive signals
in one wavelength only. So, if time is slotted in the system, the
upstream traffic transmission in an individual time slot can be
considered as a four-tuple matching problem in the four-tier
graph.
We next analyze the abstracted tiered graph to derive the
achievable rate region of the PON.
•
Fig. 1.
Graph abstraction of the data transmission process in WDM PON.
region for the downstream scenario as well.
There are two major architectures to realize multiwavelength provisioning in the upstream. One is to equip
each ONU with lasers responsible for its own upstream
traffic transmission [6]. Another one is to utilize lasers at
the OLT side to supply light for upstream data transmission
at ONUs. The unmodulated signal supplied at the OLT is
first transmitted down to ONUs, and then modulated and
reflected back by ONUs with Reflective Semiconductor Optical Amplifier (RSOA) combined with an electroabsorption
modulator (EAM) [7]. The latter architecture enables the
sharing of cost-intensive upstream light source generators. Its
deployment depends on the development and maturity of high
data rate RSOA products. In this paper, we focus on the former
architecture where each ONU is equipped with one laser for
its upstream traffic transmission as shown in Fig. 1 (a).
In this classes of PON architectures, ONUs usually employ
wavelength-fixed lasers or wavelength-tunable lasers for upstream data transmission. Multi-frequency lasers are currently
not favored owing to their high cost. Even wavelength-specific
lasers and wavelength-tunable lasers require significant capital
investment because of the large quantity of ONUs.
Generally, the upstream data transmission undergoes the
following processes: signals are first generated by lasers, then
modulated on certain wavelengths at ONUs, delivered by ODN
to OLT, then received by certain receivers, and processed at
OLT. The process of upstream transmission can be abstracted
by virtue of directed graphs, where the direction in the graph
conforms with the transmission direction of the light signal.
The upstream data transmission process proceeds as follows.
III. D ETERMINING T HE ACHIEVABLE R ATE R EGION FOR A
G IVEN PON SYSTEM
In this paper, we assume the data rate of each laser, the
capacity of each wavelength, and the receiving data rate of
each receiver all equal to C.
To better describe the capacity of the network, we define
the achievable rate region as the region containing traffic rates
which can be guaranteed by the network. Formal definitions of
the achievable rate and achievable rate region are given below.
Definition 1. Let R be the upstream traffic rate matrix, where
ri,j is the upstream traffic rate of queue j at ONU i. Rate
R is achievable if there exists a resource allocation scheme
which can allocate enough resources to guarantee rate R. The
resources include lasers, wavelengths, and receivers. For the
abstracted graph, the condition can be stated as that R is
achievable if there exists a rate assignment scheme satisfying:
1) for any vertex in the abstracted graph except those in the
first tier, the sum of rates on all incoming edges equals
to the rate of the vertex;
2) for any vertex in the abstracted graph except those in
the last tier, the rate of the vertex equals to the sum of
rates on all outgoing edges;
3) the rate of each vertex is no greater than C.
Achievable Rate Region, denoted as R, contains all the
achievable rates, R = {achievable rate RR}. The volume of R,
denoted as volR , is defined as volR = R dr1,1 dr1,2 ..., where
ri,j is the upstream traffic rate of queue j at ONU i.
Constraints 1) and 2) of the achievable rate state that there
exists a resource allocation scheme to guarantee successful
transmission of traffic with rate R. Constraint 3) assures that
wavelengths, lasers, and receivers are not overly exploited.
Fig. 2 (a) describes an example of the achievable rate and
non-achievable rate.
The next issue is to derive the achievable rate region for a
given network architecture. From the definition, we know that
checking whether rate R is achievable or not has to figure out
whether there exists a resource allocation scheme to guarantee
the traffic rate. It involves the resource allocation issue. To
determine the achievable rate region, we first introduce and
derive a few definitions and properties in the next section.
A. “Reach” Relation
We first introduce the definition of the “reach” relation,
which contains all constraints on the achievable rate.
Fig. 2.
Achievable rates, “reach” relations, and transitivity of “reach”
relations.
Definition 2. Assume set U contains vertices in tier i, and set
V contains vertices in tier j, j > i; if vertices in U can only
connect (directly or indirectly) to vertices in V , U is said to
reach V , denoted as U → V .
Property 1. Transitivity: Assume set U contains vertices in
tier i, set V contains vertices in tier i + 1, and set W contains
vertices in tier i + 2. If U → V and V → W , then U → W .
Fig. 2 (b) illustrates some examples of the “reach” relation
with respect to the graph shown in Fig. 1 (b). The “reach”
relation has the following effect on R. Note that the “reach”
relation is not only established between the vertex sets in two
neighboring tiers, but also between the vertex sets in any two
tiers.
Fig. 2 (c) describes one example of transitivity of the
“reach” relation. The vertex set in tier 1 reaches the vertex
set in tier 2, and the vertex set in tier 2 reaches the vertex set
in tier 3. Hence, the vertex set in tier 1 reaches the vertex set
in tier 3.
Theorem 1. The sum of rates of vertices in set U is no greater
than the sum of the maximum rates of vertices in set V if
U →V.
There are numerous sets exhibiting the “reach” relation.
Constraints exerted by some of the relations are naturally
satisfied, implying that they are redundant in delineating R.
To narrow down active constraints on R, we further introduce
the concept of the “block” relation.
A rate R is achievable only if it satisfies constraints exerted
by “reach” relations. In other words, constraints of “reach”
relations limit the achievable rate region of the network.
The “reach” relation exhibits the transitivity property, as
shown in Property 1. The transitivity property can transfer
constraints on vertices in higher tiers into those on vertices in
tier one, which denote the queue requests.
B. “Block” Relation
Definition 3. For any set V containing vertices in tier j, set
U containing vertices in tier i (j > i), and if U → V and
|U | > |V |, the sum of the maximum rates of vertices in V is
less than the maximum sum of rates of vertices in set U , i.e.,
|V | · C < |U | · C. V is said to block U , denoted as V 9 U .
If U → V and |U | ≤ |V |, the sum of rates of vertices in
U is less than the maximum sum of rates of vertices in set
V . The constraint exerted by U → V is naturally satisfied.
So, only “block” relations with |U | > |V | may exert active
constraints in limiting R. Fig. 3 (a) illustrates some examples
of “block” relations contained in the graph shown in Fig. 1
(b).
Similar to the “reach” relation, the “block” relation possesses the transitivity property as well, as described in Property
2.
Property 2. Transitivity: Assume set U contains vertices in
tier i, set V contains vertices in tier i + 1, and set W contains
vertices in tier i + 2. If W 9 V and V 9 U , then W 9 U .
However, constraints of some “block” relations are still
redundant constraints in limiting R. The following illustrate
two cases of redundant constraints exerted by some “block”
relations.
1
2
• Inclusion: assume V contains vertices in tier j, U , U , ...
k
containSvertices in tier i (j > i), and V 9 U , ∀k. Let
U = k U k , ∀k. Then, V 9 U . If the sum of rates
of vertices in U is less than the sum of maximum rates
of vertices in V , then, the sum of rates of vertices in
U k , ∀k, must be less than the sum of maximum rates
of vertices in V . The constraint exerted by the “block”
relation between U and V implies all constraints exerted
by “block” relations between U k and V . Constraints of
“block” relations between U k and V are redundant. Fig.
3 (b) shows an example. The three “block” relations V 9
U 1 , V 9 U 2 , and V 9 U , as shown in the left part of
Fig. 3 (b), is reducible to one “block” relation V 9 U
as shown in the right part of Fig. 3 (b).
1
2
• Independent: assume U and U contain vertices in tier
i, V 1 and V 2 contain vertices in tier i + 1, V 1 9
U 1 , and V 2 9 U 2 . Then, V 1 ∪ V 2 9 U 1 ∪ U 2 . If
U 1 ∩ U 2 = ∅ and V 1 ∩ V 2 = ∅, block relation between
V 1 and U 1 is independent from that between V 2 and U 2 .
If the constraints exerted by V 1 9 U 1 and V 2 9 U 2 are
satisfied, the constraint of V 1 ∪ V 2 9 U 1 ∪ U 2 is then
satisfied. So, V 1 ∪V 2 9 U 1 ∪U 2 is redundant in limiting
R. Fig. 3 (c) shows an example. The three “block”
relations V 1 9 U 1 , V 2 9 U 2 , and V 1 ∪ V 2 9 U 1 ∪ U 2 ,
as shown in the left part of Fig. 3 (c), is reducible to
two “block” relations V 1 9 U 1 and V 2 9 U 2 as shown
in the right part of Fig. 3 (c).
For the network example shown in Fig. 1 (b), Fig. 3
(d) shows all “block” relations after deleting redundant ones.
Hence, the achievable rate region R contains rates satisfying
the following constraints:

r1,1 + r1,2 + r1,3 + r1,4 ≤ 2C



r2,1 + r2,2 ≤ C
r3,1 + r3,2 ≤ C


 P
i,j ri,j ≤ 3C
In the above, we have shown that “reach” relations in a
graph determine the graph’s achievable rate region R. Then,
Fig. 3.
“block” relations and redundant constraints exerted by “block”
relations.
we reduce constraints exerted by “reach” relations into constraints exerted by “block” relations, and further provide some
rules of eliminating redundant “block” relations. The achievable rate region can be delineated based on the remaining
”block” relations.
IV. D ISCUSSIONS
In this section, we discuss the achievable rate regions of
WDM PON system with 8 ONUs.
First, assume the number of ONUs is 8, the number of
wavelengths is K, each ONU has only one queue, the laser
at each ONU can be tuned to all the K wavelengths, and
the number of receivers equals to the number of wavelengths.
Fig. 4 illustrates this WDM PON system. In the network, the
TABLE I
ACHIEVABLE RATE REGIONS FOR WDM PON S WITH EIGHT ONU S
Wavelength #
Admissible traffic rate R
Volume
1
ri ≤ C, ∀i
P8
r ≤C
i=1 i
C 8 /8!
4
ri ≤ C, ∀i
P8
r ≤ 4C
i=1 i
C 8 /2
7
ri ≤ C, ∀i
P8
r ≤ 7C
i=1 i
C 8 (1 − 1/8!)
8
ri ≤ C, ∀i
P8
r ≤ 8C
i=1 i
C8
to compare different network architectures, and is useful in
designing proper access control strategies.
ACKNOWLEDGMENT
This work was supported in part by the National Science
Foundation under Grant 0726549.
R EFERENCES
Fig. 4.
An example of WDM PON system
wavelength set blocks the laser set. Hence, the achievable rate
region R contains rates satisfying the following constraints:
½
ri,1 ≤ C
P
8
i=1 ri,1 ≤ KC
Table I shows the achievable rate region and its volume for
K = 1, 4, 7, 8, respectively. When K = 1, the volume of the
achievable rate region is C 8 /8!, which is the smallest among
the four cases; when K = 4, the volume of the achievable
rate region increases significantly to C 8 /2; when K = 7, the
volume of the achievable rate region equals to C 8 (1 − 1/8!),
less than twice of that with K = 4; when K = 8, the volume
of the achievable rate region achieves the maximum value
C 8 , and further increasing the number of wavelengths will
not enlarge the achievable rate region. It can be seen from
this example that when the number of wavelength is small,
increasing the number of wavelength will increase the volume
of the achievable rate region greatly, and the growth rate of
the achievable rate region declines with the increase of the
number of wavelengths.
V. C ONCLUSION
By utilizing WDM technology, WDM PONs dramatically
increase the capacity of broadband access. Its capacity depends
on the network architecture, the lasers’ tuning ability, the
receivers’ receiving range, the number of lasers, and the
number of receivers. In this paper, we have proposed to
describe the data transmission process in the PON network
by a directed tiered graph, where the direction aligns with
the direction of the light signal. We have also introduced
the concept of the achievable rate region to describe the
capacity of the WDM PON network. We have further defined
two relations, the “reach” and “block” relation, to derive the
achievable rate region for a given network. This can be applied
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