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On modeling optical burst switching networks with fiber
delay lines: a novel approach
A. Rajabia, A. Khonsari∗,b,a,A.Dadlani
a,b
aInstitute for Studies in Theoretical Physics and Mathematics, School of Computer
Science, Tehran, Iran
bDepartment of Electrical and Computer Engineering, University of Tehran, Tehran, Iran
Abstract
Optical Burst Switching (OBS) has been widely admitted as a viable can-
didate for the evolving all-optical network control framework of nowadays
Internet core to meet the explosive demand of bandwidth constraints in mul-
timedia and real-time applications. A number of analytical models have been
proposed to characterize the behavior of OBS networks. These models, how-
ever, do not imitate the actual features of the Fiber Delay Line (FDL) com-
ponents present in contemporary optical switches. Underscoring the precise
behavior of FDLs leads to misleading results regarding major performance
measures of the OBS networks. While FDLs behavior has been neglected in
many studies, as a crude approximation, a few researches characterize FDLs
as conventional buffers present in electronic switching systems. In this pa-
per, we provide an inclusive discussion on the features of FDLs, followed by
presenting a novel analytical model to realize the inimitable characteristics of
FDLs. The proposed network-level model is based on the results of queuing
systems with impatient customers and deterministic impatience time. The
viability of the proposed model is validated through extensive simulation
experiments.
Key words: Optical burst switching, fiber delay lines, queuing systems
with impatient customers, performance evaluation
∗Corresponding author. Tel.: +98-21-22287014; fax: +98-21-22828687.
Email addresses: alirajabi@ipm.ir (A. Rajabi), ak@ipm.ir (A. Khonsari),
a.dadlani@ipm.ir (A. Dadlani)
Preprint submitted to Computer Communications May 28, 2009
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1. Introduction
Optical Burst Switching (OBS) has been identified as the most promising
technology to be employed in the next generation optical Internet. Having
the potential in meeting the fast growing bandwidth demand imposed by
multimedia applications such as video conferencing, Internet telephony, and
digital audio is the reason for such a choice [1]. OBS integrates the benefits of
Optical Circuit Switching (OCS) [2] and Optical Packet Switching (OPS) [3]
in order to reduce the end-to-end network latency and the overhead involved
in packet header-processing. Moreover, employing Dense Wavelength Divi-
sion Multiplexing (DWDM) in OBS, which enables a transfer rate of more
than 1 Tb/s on a single optical fiber, has opened up stimulating challenges to
researchers and network engineers so as to seek new paradigms to efficiently
utilize the bandwidth inherent in DWDM [4].
As a powerful performance evaluation tool, analytical modeling is essen-
tial to provide a platform on which robust judgments can be made on the
practicability of proposals regarding various aspects of OBS. Myriad amount
of work has been dedicated to providing effective analytical models in order to
accurately predict the behavior of an OBS network [1][4][5][6][7][8][9][10][11].
In most of these studies, the Erlang loss system (M/M/c/c queuing system)
plays a central role in modeling each OBS switch [4][7]. To capture the
precise characteristics of the arrival process, switch models adopting a finite-
population queuing system with no waiting room have been lately proposed
[5][11]. In all these models however, the impact of Fiber Delay Lines (FDLs)
have been completely neglected to relax the analysis complexity of the model.
Some other studies have included FDLs by applying an M/M/c/k queuing
system [1][25]. However, the behavior of FDLs in these models is approxi-
mated by that of conventional electronic buffers. As a result, these models
fail to capture the unique characteristics of FDLs that distinguish them from
electronic buffers. In [6], a model considering some FDL properties has been
presented, yet the analytical model has been developed for a special case
where optical fibers carry only two wavelengths, and no closed-form solu-
tion has been derived for the general case. Due to the significance of FDLs
in enhancing performance of optical networks, presenting effective analytical
models taking all FDL features into account are integral to network engineers
and designers. This study has been primarily motivated by the lack of such
models in the existing literature.
To the best of our knowledge, this is the first study in which a compre-
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hensive discussion on FDL properties is presented. The switch architecture
considered in this paper is the same as in [1]. Furthermore, we discuss on
the following properties that distinguish FDLs from ordinary buffers present
in electronic routers:
•how an FDL of definite length can delay a burst for a limited amount
of time.
•how a single FDL can be shared by more than one burst at the same
time.
•how a burst can hold an FDL busy while being transmitted via a wave-
length.
Unlike most of the existing models that have been proposed at either link-
level or switch-level [1][2][3][11][13], we propose a novel network-level analyt-
ical model based on queuing systems with deterministic impatience [12] to
capture the aforementioned characteristics of FDLs. The first two character-
istics are completely covered in the model, while the impact of the third one
is studied in a special case where the average burst length is less than the
FDL length, both measured in units of time. The derivation of the model
is straightforward and tractable. The simulation section verifies that the
model presented in this paper results in much better predictions of system
dynamics, such as blocking probability and network latency, than ordinary
M/M/c/k queuing system.
The rest of the paper is organized as follows: in Section 2, we briefly
introduce the OBS framework and provide a comprehensive explanation on
FDL characteristics. Section 3 introduces the assumptions and notations
used throughout the model, which is derived in Section 4. Simulation results
are presented in Section 5 followed by concluding remarks in Section 6.
2. Preliminaries of an OBS framework
Switches in an optical network are broadly categorized into edge and core
switches [14]. Edge switches collect traffic from various upper layer users such
as ATM switches and IP routers, while core switches forward this traffic in
the optical domain. Based on their functionality, edge switches are further
divided into two types: ingress and egress switches [9]. Figure 1 illustrates
the overall view of an optical network. In what follows, we briefly describe
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Optical Burst Switching Network
LAN/
WAN 2
packets
LAN/
WAN 3
packets
LAN/
WAN 1
C3
C1
C2C4
I1E1
I2
burst
Figure 1: An overall view of an optical network. I1and I2are ingress switches,
and E1is an egress switch. C1,C2,C3,andC4are core switches.
the functionality of ingress, egress, and core switches. A more detailed ex-
planation on optical switches can be found in [26].
2.1. Ingress switches
An ingress switch is where packet aggregation and burst assembly takes
place. It comprises of an array of queues, where each queue is associated
with a specific egress switch reachable from the ingress switch. Based upon
the destination IP address contained in the packet header, every incoming
packet is directed to the queue corresponding to its destined egress switch.
Each queue undergoes alternative periods of aggregation and transmission.In
each aggregation period, IP packets belonging to a queue are aggregated into
bursts prior to transmission into the optical domain. As shown in Figure 1,
ingress switch I2assembles the incoming packets from LAN/WAN 2 into
bursts and sends them towards the destined egress switch E1via core switches
C2and C3.
Some of the most common burst assembly algorithms can be classified
into timer-based [9], threshold-based [15], and mixed timer/threshold-based
[16] algorithms. In the timer-based approach, a timer is set at the beginning
of every new assembly cycle, determining the transmission time of the burst
into the core network. After a fixed amount of time, all the packets that
arrived during that time period are assembled into a burst. In the threshold-
based approach, a threshold is specified to determine the generation and
transmission time of a burst into the optical network. The incoming packets
are stored in the prioritized queues in the ingress node, until the threshold
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condition is satisfied. Once the threshold is reached, a burst is created and
sent into the optical core. The timeout value for the timer-based schemes
should be set carefully. If the value is chosen to be too large, the packet
delay at the edge might become intolerable. On the other hand, if the value
is too small, too many small-sized bursts will be generated, resulting in con-
trol overhead. While timer-based schemes might result in undesirable burst
lengths, threshold-based assembly algorithms do not guarantee on the packet
assembly delay. A mixed timer/threshold-based algorithm may perform bet-
ter, especially with self-similar traffic, but may experience higher operational
complexity.
A signaling protocol is the procedure through which a control packet re-
serves resources for the corresponding data burst by guiding it through a
routing path. In one-way reservation [7], a control packet reserves resources
along the path for the corresponding data burst without any acknowledge-
ment from the destination node. On the contrary, in a two-way reservation, a
control packet collects link and topology information instead of reserving re-
sources for the data burst. The acknowledgement packet from the destination
node to the source node reserves resources for the corresponding data burst
while traversing along the reverse path. Since one-way reservation protocols
are more flexible, have lower latency, and are more efficient as compared to
two-way reservation protocols, they are mainly adopted in OBS networks.
It is important to notice that from a modeling point of view, both control
packets and data bursts streaming out of an ingress switch represent essen-
tially the same process and can be used interchangeably in the derivation
of the model, provided that the offset time between the control packet and
data burst transmission remains unchanged during switching. We adopt the
burst stream to explicate our model since it is more concrete to deal with a
burst stream than a control packet stream.
2.2. Egress switches
The functionality of an egress switch is simply the reverse replica of an
ingress switch. In other words, each burst entering an egress switch undergoes
an optical to electrical conversion and is then disassembled into packets.
These disassembled packets are then transmitted to the outside non-optical
network based on normal IP forwarding as illustrated by the egress switch
E1in Figure 1.
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2.3. Core switches
A core optical switch comprises of a number of input and output inter-
faces, a switching fabric, and a scheduler as shown in Figure 5. Each output
interface is connected to an optical link which may be augmented with a
number of FDLs. The optical link carries a maximum of wwavelengths.
FDLs are optical buffers that can temporarily delay an arriving burst if no
free wavelength on the outgoing fiber is available for its transmission. More
on FDLs is covered in subsection 2.5.
Switch scheduler is the central component of a core switch and is respon-
sible for resource reservation, i.e., assigning a wavelength and/or an FDL to
every burst. When a control packet reaches a core switch, it undergoes an
optical to electrical conversion so that it can be processed in the electronic
domain. After conversion, the destined egress switch, corresponding burst
length, and time of burst arrival are extracted from the control packet. Based
upon this information, the switch scheduler attempts to reserve the required
resources for the time when the burst arrives at the switch. This type of
reservation is referred to as delayed reservation (DR) [14].
In case of a successful reservation, the control packet is converted back
into the optical format and forwarded to the next core switch to establish the
rest of the path. Before forwarding the control packet, the burst arrival time
to the next switch is calculated, and the newly calculated value is placed in
the control packet so that the successive core switch has a precise estimation
of burst arrival time for its scheduling process. In case of reservation failure,
the control packet is simply dropped.
3. FDL versus electronic buffer
An optical link may be allocated a number of FDLs, which are optical
buffers used to temporarily delay an incoming burst if all wavelengths of the
output link are busy transmitting other bursts. Previous work shows that
even employing a limited number of FDLs greatly improves the performance
of the optical network [1]. This section emphasizes on the fundamental dif-
ferences between buffers and FDLs and their impact on the analytical model.
1. Concurrent Access to Single FDL (CASF): In a queuing system, as
a newly arrived job finds all servers busy, it waits inside a buffer, if any
available, for one of the servers to become free. While residing inside the
buffer, it does not share the space with any other incoming jobs. As a
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L
L/4
t0t0 + L/4 t0 + L/2 t0 + 3L/4
Time
B
0
B
1
Figure 2: The Concurrent Access to Single FDL (CASF) scenario. (a) At time
t0,B0enters the FDL . (b) At time t0+L/2, the head of B0reaches the middle
of the FDL. At the same time, B1enters the FDL. (c) At time t0+3L/4, both
B0and B1are traveling inside the FDL, which implies that an FDL can be
shared by more than one burst at the same time.
result, in a queuing system with Dbuffers, the maximum number of jobs
awaiting service at any given time is bounded to D. However, a single FDL
might be shared by more than one burst at the same time, as shown in
Figure 2. For the sake of simplicity, assume that only one wavelength and
FDL are available. In Figure 2(a), burst B0with length L/4(measured
in time units) enters the system at time t0and finds the wavelength busy,
but the FDL free. Therefore, it enters the FDL. In Figure 2(b), the head
of B0reaches the middle of the FDL at time t0+L/2. At the same time,
another burst B1enters the system, finding the wavelength busy and the
FDL free (since B0released the FDL L/4 time units earlier), as a result
of which B1enters the FDL. In Figure 2(c), both B0and B1travel inside
the FDL at time t0+3L/4, implying that:
On contrary to a buffer in a queuing system which can accommodate at
most one job at any given time, an FDL can be shared by more than one
burst at the same time.
An immediate result is that employing a queuing system with wservers
and Dbuffers to model an output interface with wwavelengths and D
FDLs might result in an overestimation of the blocking probability. The
model claims that any arriving job finding all wservice lines busy and D
buffers occupied is blocked, while an arriving burst to an output interface
might not necessarily be blocked even when all wavelengths are busy and
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each FDL holds at least one burst.
CASF occurs frequently when the average burst length is less than that
of the FDL, both measured in units of time. In this case, the head of a
burst can enter an FDL while some other bursts are still being held in it
(as in Figure 2(b)). Consequently, if the average burst length equals the
FDL length, the impact of CASF on the model diminishes in long term.
This is true especially in cases where the burst length distribution has a
small variance, meaning that it slightly oscillates around the mean value.
If the average burst length is less than the FDL length, the model can be
adjusted to include the impact of CASF.
2. Concurrent FDL and Wavelength Holding (CFWH): In a queuing
system, a job inside the system might be in either of the following two
states: being served on one of the servers or waiting in one of the buffers
for service release. On the other hand, a burst may be so long that even
after commencing service, it still blocks the FDL, forbidding other bursts
from entering it. Figure 3 illustrates the CFWH scenario. At time t0,
burst B0with length 4L/3 enters the system, finds the only wavelength
busy, and enters the only available FDL which is free. Ltime units later,
the head of the burst reaches the end of the FDL while still blocking the
FDL. By this time, the wavelength on the optical link is freed (otherwise
the burst would have never entered the FDL and had been dropped at the
L
4L/3
t0t0 + L
Time
B
0
Figure 3: The Concurrent FDL and Wavelength Holding (CFWH) scenario.
(a)Attimet0, the head of B0enters the FDL. (b) At time t0+L, the head of B0
reaches the FDL and is about to start service, while its tail still remains outside
the FDL, thus blocking the FDL and prohibiting other bursts from entering it.
This implies that a burst can hold an FDL busy even while being transmitted
over a wavelength.
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instant of its arrival), and B0starts its service. From this time until time
t0+4L/3, at which B0releases the FDL, no other burst can enter it. In
other words, B0holds the wavelength and blocks the FDL concurrently,
implying that:
While a buffer in a queuing system is freed as soon as the job residing in
it commences service, an FDL might be blocked by a burst even after its
service (transmission) begins.
In contrast to CASF, CFWH might result in an underestimation of the
actual blocking probability. Again, consider a queuing system with w
service lines and Dbuffers. An incoming job finding less than w+Djobs
in the system definitely joins the system and remains in it until served.
On the other hand, at an output interface with wwavelengths and D
FDLs, a newly arrived burst might be blocked even at times when the
total number of bursts at the interface is less than w+D, as some bursts
might both, hold a wavelength and block an FDL simultaneously.
CFWH occurs frequently if the average burst length is greater than that
of the FDL. Similar to CASF, if the average burst length equals the FDL
length, and the burst length distribution has a small variance, then the
effect of CFWH on the model becomes negligible in long term. It can even
be claimed that on satisfying these two conditions, CASF and CFWH
neutralize the estimation impacts of each other on the analytical model.
3. FDL Holding Time Constraint (FHTC): In a queuing system, a
buffer can accommodate a job for any arbitrary amount of time. On the
contrary, as the length of an FDL is limited to some constant value, it can
hold a burst for only a limited amount of time, proportional to its length,
by the end of which the burst must leave the FDL.
On reaching an output interface, a burst either starts its transmission
immediately over a free wavelength or enters an unblocked FDL if assured
that by the end of its transversal inside the FDL, a free wavelength would
be available for its transmission. Otherwise, it is dropped and considered
as lost. In other words, the burst either balks at entering the interface and
is dropped or enters the interface and is served. A similar concept exists
for the term balking in queuing systems. It means that on arrival, a job
decides whether to enter a system or not. But, once entering the system,
it remains there until served. The condition upon which the entrance
decision is made may differ from one balking system to another. In an
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output interface (as a balking system), the decision is based on the waiting
time until beginning of service (WTUBS) which must be less than or equal
to the FDL length.
In contrast to a balking system, in a reneging queuing system, a newly
arrived job always joins the queue, but waits only for a limited amount
of time for its service to begin. If the service does not start within this
limited period, the job leaves the system and is lost. In other words,
joining the reneging system does not necessarily mean receiving service.
The term deterministic impatience is used to describe a reneging queue
in which the WTUBS is bounded to some constant value.
Due to [17], if the entrance decision in a balking queue is based on the
WTUBS, a reneging queue with the same WTUBS gives the same blocking
probability and waiting time distribution of served jobs as the original
balking queue, given that there are infinitely many number of buffers
available.
We present a reneging-based model for an output interface which is balk-
ing in nature. However, the number of FDLs allocated to an optical link
is limited to some constant value (not infinity), and hence a reneging-based
model is not applicable to the balking output interface. On the other hand,
we argued earlier that a single physical FDL is equivalent to wvirtual FDLs,
where wis the number of wavelengths on an optical fiber and is usually
a large number. Consequently, even a few physical FDLs result in a large
number of virtual FDLs which may be considered as infinity. As a result, a
reneging queue is employed to model an output interface with the hope that
the number of virtual FDLs is “large enough” to be thought of as infinity
so that the queue acts as a “balking queue” in terms of blocking probability
and average waiting time of served jobs. In Section 5, numerical examples
are given to clarify what we mean by “large enough”.
4. Assumptions and notations
In this section, the notations and assumptions used in the derivation of
the analytical model are presented. Tables 1 and 2 summarize, respectively,
the notations and functions used in this paper.
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Tab l e 1: Notations used in the derivation of the model
Notation Meaning
IjAn arbitrary ingress switch (1 ≤j≤m).
I=I1,...,I
mThe set of all ingress switches.
EjAn arbitrary egress switch (1 ≤j≤n).
E=E1,...,E
nThe set of all egress switches.
Qij The ith queue in the AoQ of Ij(1 ≤i≤n, 1≤j≤m).
Tij Random variable denoting the length of an arbitrary aggregation
period of Qij in terms of time.
1/λij Known constant denoting the mathematical expectation of Tij , i.e.,
E[Tij ]=1/λij .
πji The probability that the next coming burst from Ijis destined for
Ei(1 ≤i≤n, 1≤j≤m).
tj
0Known constant denoting the offset time between the transmission
of a control packet and its corresponding data burst from Ij.
tCKnown constant denoting the switching time of an arbitrary core
switch C, i.e., the time taken by the head of a burst to cross the
switching fabric of C.
1/μ Known constant denoting the average burst length in terms of time.
lAn arbitrary link.
fAn arbitrary FDL.
PAn arbitrary path, i.e., a sequence of links connecting an ingress
switch to an egress switch.
dKnown constant denoting the number of physical FDLs allocated to
l.
xlKnown constant denoting the length of lin terms of length unit.
tl
pThe propagation time on l, i.e., the time taken by the head of a burst
to reach from one end of the link to the other end.
LKnown constant denoting the length of fin terms of time units.
PlThe set of all paths including las a link.
λlLong-run rate of burst arrival at l.
δlLong-run rate of burst departure from l.
λP
lLong-run rate of burst arrival at lthrough Pgiven that Pincludes
l.
δP
lLong-run rate of burst departure from lthrough Pgiven that P
includes l.
blThe blocking probability at l.
(continued on next page...)
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Tab l e 1: Notations used in the derivation of the model (cont inu ed )
Notation Meaning
qlThe average queuing delay at l.
bNThe blocking probability of the network, i.e., the probability that a
randomly chosen burst is blocked somewhere on its path from source
to destination.
ANLPThe average network latency of P, i.e., the amount of time taken
by a burst traveling on Puntil its tail is received at the destination
switch from the time its corresponding control packet is sent from the
source switch, given that the burst is not blocked at any intermediate
core switch.
ANL The average network latency of an arbitrary burst, given that it is
not blocked at any intermediate core switch.
Tab l e 2: Function definitions
Function Definition
prev(l, P ) Returns the link preceding lon P.Iflis the first link of P, it returns
NIL.
next(l, P ) Returns the link following lon P.Iflis the last link of P, it returns
NIL.
head(l) Returns the switch, one of the output interfaces of which is connected
to l.
tail(l) Returns the switch, one of the input interfaces of which is connected
to l.
ingress(P) Returns the ingress switch connected to the first link of P.
egress(P) Returns the egress switch connected to the last link of P.
cores(P) Returns the set of all core switches in the path between ingress(P)
and egress(P).
lastlink(P) Returns the last link of Pconnecting to an egress switch.
index(Ij) Returns j.
index(Ei) Returns i.
The assumptions made below have been widely used in the previous lit-
erature [1][6][7][8][9][10][11][13]:
1. Each egress switch in Eis reachable from every ingress switch in I.
2. The routing algorithm directs a burst through a path resulting in the
minimum number of hops from source to destination.
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3. Qij stores IP packets destined for Ei.
4. Tij is exponentially distributed with mean 1/λij.
5. Control packet processing time at every core switch is negligible and can
be ignored.
6. The wavelength of a randomly chosen burst is uniformly distributed over
wavailable wavelengths.
7. Burst length distribution is exponential with mean 1/μ. The exponential
distribution of burst length is adopted for mathematical tractability. Fur-
thermore, there exists some kind of dependency between the length of an
aggregation period and mean burst length. The longer the aggregation
period, the greater is the average burst length. This dependency is made
insignificant by setting the average burst length to a constant value of 1/μ.
However, simulation results show that if the λij values are chosen close to
each other, such an assumption does not greatly influence the model.
8. On entering an FDL, a burst can leave it as soon as a wavelength becomes
free. This is also a simplifying assumption, as it is stated in the preceding
section that a burst may leave an FDL only at exit points. If a burst is
traveling between two exit points and a wavelength becomes free during
this period, the burst must proceed until the next exit point before being
sent over the link. However, simulation results show that if the exit points
are chosen to be close to each other, the behavior of an FDL with finite
number of exit points can be approximated by that of an FDL which
allows a burst to leave at any arbitrary point in the FDL.
9. There is no blocking at any link connecting an ingress switch to the net-
work.
10. For any arbitrary link l, the arrival process at lapproximates to a Poisson
process with rate λl[19].
5. Analytical model
The model is derived in three steps. First, an arbitrary optical link is
considered, and its performance metrics, such as blocking probability and
average queuing delay, are obtained assuming that the arrival rate at the link
is known. Next, a recursive set of equations are presented to obtain the arrival
rate at an arbitrary link of the network. The blocking probability and average
latency of an entire network are then derived in the third step. Finally, this
section is concluded with a discussion on the impact of wavelength converters
on the performance of optical networks.
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5.1. Step I: Performance metrics of an arbitrary link
In this subsection, we obtain the performance metrics related to an ar-
bitrary optical link, say l.Ifhead(l)∈I,thenbl=0andql=0(ac-
cording to assumption 10). Therefore, throughout this step, we assume that
head(l)/∈I.
Considering wwavelengths on land dphysicalFDLs,aswellasassump-
tion 7 and Lemma 3, one may conclude that a multi-server finite-buffer queu-
ing system with Poisson arrival and exponential service is the best choice for
capturing the behavior of l. However, taking the FHTC property of FDLs
into consideration, we employ an M/M/w/N queue with deterministic im-
patience to model l,whereNis the total queue capacity (including service
lines and buffers). Deterministic impatience is adopted since the length of
an FDL is limited to constant L. The better performance of this queueing
system with regard to the ordinary M/M/w/N queue in terms of blocking
probability and average network latency will be shown in the next section.
To obtain the number of buffers, notice that there are dphysical FDLs,
each equivalent to wvirtual FDLs. Furthermore, a single virtual FDL can
accommodate on average Lμbursts at a given time. As a result, the total
number of buffers, denoted by D, can be obtained as:
D=dwLμ.(1)
By assuming L≥1/μ (according to assumption 8), we have already
eliminated the impact of CFWH on the model. Now, by defining the total
number of buffers as in (3), instead of simply defining it as dw, we actually
take the impact of CASF into account since a single virtual FDL no longer
corresponds to a single buffer, but rather to Lμbuffers. The total system
capacity can be obtained from:
N=w+D=w(1 + dLμ).(2)
M/M/w/N queue with deterministic impatience has been thoroughly
studied in [20, 21], and its system size distribution for the steady state has
been obtained to be:
p−1
0=
w
j=0
ρj
l
j!+ρw
l
w!
D
j=1 λj
l
j−1
i=0 (wμ +Cw+j−i),
pj=ρj
l
j!p0,1≤j≤w, (3)
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pw+j=ρw
l
w!
λj
l
j−1
i=0 (wμ +Cw+j−i)p0,1≤j≤D,
where pjis the probability of jjobs in queue, ρl=λl/μ,andCw+jis the
long-term rate of time-out (reneging) when there are w+jjobs in queue and
is obtained from:
Cw+j=Lj−1e−wμL
L
0tj−1e−wμtdt.(4)
To obtain the blocking probability at l, notice that there are two types of
blocking in an M/M/w/N queue with deterministic impatience. A job may
balk at entering the queue if it finds all wservers and Dbuffers occupied by
jobs, the probability of which is pN. It may also time-out after joining the
queue if the waiting time until beginning of the service exceeds its patience
time, which in this case is equal to L. To obtain the latter probability, notice
that there is no time-out when the number of jobs in queue is less than or
equal to w. Furthermore, the long-run rate of time-out for w+jjobs in
queue is Cw+j, and the probability of having w+jjobs is pw+j.Asaresult,
the total rate of time-out is given by:
D
j=1
Cw+jpw+j.(5)
Dividing by λlgives the probability of time-out. Consequently, the block-
ing probability at lis derived as:
bl=pN+1
λl
D
j=1
Cw+jpw+j.(6)
Next, we obtain the average queuing delay at l. For this purpose, we
require to know the average number of jobs, denoted by E[S], and the prob-
ability of time-out given that a job joins the queue, denoted by PT|J,which
are given respectively as:
E[S]=
w+D
j=1
jpj,(7)
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and
PT|J=1
λl(1 −pN)
D
j=1
Cw+jpw+j.(8)
Equation (11) can be interpreted as follows: (1 −pN) is the probability
of joining the queue which when multiplied by PT|Jgives the probability of
time-out derived earlier. Applying Little’s formula [22] yields:
E[S]=λl(1 −pN)LPT|J+(ql+1/μ)1−PT|J,(9)
where λl(1−pN)andLPT|J+(ql+1/μ)1−PT|Jare the effective arrival
rate [22] at l, and the average response time of an arbitrary job joining the
queue, respectively. By response time, we mean the time period from when
the job enters the queue to the time when its service completes and leaves
the queue. The response time consists of two parts: if the burst times-out, its
response time is L(the patience time before time-out), otherwise the response
time is equal to queuing delay plus the service time. The probabilities of the
two cases are given by PT|Jand 1 −PT|J, respectively. Re-arranging (11)
yields:
ql=E[S]
λl(1 −pN)1−PT|J−LPT|J
1−PT|J−1
μ,(10)
which gives a formula to obtain the average queuing delay at l.
5.2. Step II: Obtaining arrival rate at l
In the previous step, we obtained the blocking probability and average
queuing delay of an arbitrary link l, assuming that the arrival rate at lis λl.
In this step, we calculate λl.
λlis the superposition of the arrival rates of the individual paths in Plat
l, i.e.,
λl=
P∈Pl
λP
l.(11)
The arrival rate of a path Pat lequals the departure rate of Pfrom
the link preceding lon Pif such a link exists. If lis the first link of P
connecting an ingress switch to a core switch, then λP
lequals a fraction of
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the total traffic rate generated by ingress(P) and destined for egress(P).
Let ϕ=index(ingress(P)) and γ=index(egress(P)). Consequently, λP
lis
given by:
λP
l=δP
prec(l,P )head(l)/∈I
πϕγ λhead(l)otherwise ,(12)
where λhead(l)is as defined in Lemma 1. A formula for δP
l, i.e., the departure
rate from lon P, can be easily obtained as:
δP
l=λP
l(1 −bl),(13)
where blis as in (9). Recall that bl=0ifhead(l)∈I.Aslongasthe
optical network is cycle-free, (9) and (14-16) can be employed to recursively
calculate the arrival rate at every arbitrary link.
5.3. Step III: Obtaining Average Network Latency and Network Blocking
Probability
In this step, we derive the network blocking probability (bN)aswellas
the average network latency (ANL). To obtain bN,noticethat1−bNis
the probability of burst delivery to egress switches and can be defined as the
long-term ratio of burst delivery to egress switches to the long-term rate of
burst generation at ingress switches. Therefore,
bN=1−tail(l)∈Eδl
head(l)∈Iλl
,(14)
where δl=λl(1 −bl).
To obt ain AN L, first consider a single path, say P, connecting an ingress
switch to an egress switch. Five components contribute to the total latency
on Pnamely, the offset time between control packet and data burst trans-
mission, the propagation time of every link on P, the switching time of every
core switch visited by a burst traveling on P, the average queuing delay of
every link on P, and the interval from the time the head of a burst is re-
ceived at an egress switch to the time its tail is received (which equals 1/μ
on average). With ϕas defined earlier, we consequently have:
ANLP=tϕ
0+
l∈P
(tl
p+ql)+
C∈cores(P)
tC+1
μ,(15)
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where tl
p=xl/(speed of light).
ANL can be defined as a weighted summation of the average latencies of
every path between a pair of ingress and egress switches where the weight of
the latency of each path is the ratio of the burst delivery rate through that
path to the total rate of burst delivery to egress switches, i.e.,
ANL =
P
δP
lastlink(P)
tail(l)∈Eδl
ANLP.(16)
5.4. Wavelength Converters (WCs)
In the derivation of the model, we have implicitly assumed that core
switches benefit from the presence of wavelength converters to change the
wavelength of a burst if necessary. A finite-buffer multi-server queue em-
ployed to model a link with wwavelengths is valid only if a burst can be
transmitted via every available wavelength and buffered inside every avail-
able FDL, which implies that its wavelength can be converted as necessary.
In this subsection, we slightly tune the model to cover the case in which core
switches lack the presence of WCs. In the rest of this subsection, “γ-burst”
refers to a burst with wavelength γ.
As pointed out earlier, a single optical fiber with wwavelengths can be
viewed as wvirtual links, each capable of transmitting bursts of a specific
wavelength. Similarly, a physical FDL is equivalent to wvirtual FDLs each
capable of optically buffering bursts of a particular wavelength. In case of
no WC, a γ-burst can only be transmitted via the virtual link and buffered
inside the virtual FDLs corresponding to wavelength γ. Virtual links and
FDLs corresponding to wavelengths other than γare completely ignored by
the scheduler when it tries to reserve resources for a γ-burst. It follows that
when no WC is available, bursts with different wavelengths arriving at a link
do not compete for the available resources, but the contention rather exists
only among the bursts with the same wavelength. Consequently, the arrival
stream of bursts at a link could be decoupled into wnon-overlapping streams,
one for each wavelength, where each stream is allocated with a single virtual
link and dvirtual FDLs (since there are dphysical FDLs, the total number of
virtual FDLs allocated to each stream is d). Figure 10 illustrates the models
of an arbitrary link, say l, with and without WCs.
As can be observed in Figure 10(a), the single queue with wwavelengths
and dw FDLs is modified to wqueues, each associated with one wavelength
and dFDLs. Due to the uniform distribution of a burst (assumption 6)
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d
w
(a)
Ȝ
l
dw
FDLs
Wavelengths
d
w
Ȝ
l
Ȝ
l
/w
Ȝ
l
/w
FDLs Wavelengths
(b)
Figure 4: (a) The model of an optical link with wavelength converters (WCs).
There is a single stream of bursts with rate λl. Each arriving burst may utilize
every dw FDL and every wwavelength made available. (b) The model of an
optical link without WCs. There are wdistinguished stream of bursts, each
having a specific wavelength and rate λl/w. Each stream can utilize only one
wavelength and dFDLs allotted to the wavelength of that stream.
as well as the decomposition property of Poisson process [22], the stream
of bursts into each queue in Figure 10(b) is also Poisson with rate λl/w.
As the wqueues in Figure 10(b) are equivalent and independent of each
other, one may conclude that the blocking probability and average queuing
delay at lis the same as those of one of the queues in Figure 10(b) (an
M/M/1/N queuing system with deterministic impatience, arrival rate of
λl/w,andN=1+dLμ). Such a queue is a special case of the queue studied
in step I of the model derivation. Consequently, its blocking probability and
average queuing delay can be obtained from (9) and (13), respectively. Once
the performance metrics of lare obtained this way, one can proceed with
steps II and III as before.
We end this section by arguing on how the presence of WCs enhance the
performance of optical networks by reducing the blocking probability. It is
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well-known in queuing theory that a queuing system in which the buffers and
servers are shared between arrivals results in lesser blocking probability than
a queue in which buffers are divided between the servers, and each arrival is
assigned to one server and may utilize buffers exclusively allocated to that
server. A special case of the above assertion has been deeply studied in [18].
As an optical fiber equipped with WCs can be modeled with a queue of the
former type, it is justified why WCs reduce the blocking probability. In the
next section, we verify the validity of the claim made above.
6. Numerical results
We constructed a discrete event-driven simulator in the Ptolemy environ-
ment [23] to verify the accuracy of the proposed model. An optical network
consisting of 3 ingress, 8 core, and 3 egress switches was adopted as the case
study. The topology of the network is as shown in Figure 11. The average
burst length is taken to be 3 time units in all scenarios. All ingress switches
generate traffic with the same rate in every simulation experiment All opti-
cal fibers carry the same number of wavelengths and are assigned the same
number of FDLs. tl
pand tCare assumed to be negligible compared to other
parameters contributing to the total network latency. Each simulation sce-
nario was made to run until its steady state., i.e, until no further increase
in simulation time altered the collected statistics appreciably. On average,
8 batches of bursts, each consisting of 1,000,000 bursts generated by ingress
switches, was sufficient to reach the steady-state. Statistical data gathered
during the first batch was discarded to avoid distortion due to the initial
warm-up conditions.
In the remainder of this section, we consider three analysis namely, A1,
A2,andA3.InA1, the blocking probability and average queuing delay at an
optical fiber are obtained from (9) and (13), respectively, and the number of
FDLs assigned to each optical fiber is calculated from (4). Consequently, A1
takes both, CASF and FHTC into consideration. In A2, the blocking proba-
bility and average queuing delay are obtained from an ordinary M/M/w/N
queuing system, and the number of FDLs assigned to each fiber is dw.A2
ignores both, CASF and FHTC, yet it has been widely used to model an
optical fiber in the existing literature. A3isthesameasA1, except that the
number of FDLs assigned to each optical fiber is dw (rather than dwLμ),
i.e., A3takes FHTC into account; however, it ignores CASF. In what follows,
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Optical Burst Switching Network
LAN/
WAN 1 LAN/
WAN 4
H
G
F
C
E
A
B
D
E1
E3
LAN/
WAN 6
Egress
E
2
E
E
E
2
2
I1
I2
LAN/
WAN 2
LAN/
WAN 3
Ingress
I3
LAN/
WAN 5
Figure 5: The topology of the simulated optical network with 3 ingress
switches, 8 core switches, and 3 egress switches.
we show that A1(the approach suggested in this paper) results in better per-
formance predictions of system dynamics than A2and A3.
Let ζbe the performance measure of interest and ζ(.) denote the function
which returns the value of ζobtained from its argument (for example, if ζis
the blocking probability, ζ(A1) is the value of blocking probability obtained
from A1). The relative error of A1with respect to the value obtained from
simulation is defined as:
R(Ai)=|ζ(Ai)−ζ(S)|
ζ(S)×100,(17)
where ζ(S) is the value obtained from simulation and |x|is the absolute value
of x.R(Ai) is the function we use to measure the effectiveness of Ai.The
lower the value of R(Ai), the more effective is the analysis Ai.
6.1. The equivalence of balking and reneging queuing systems
Towards the end of Section 2, we argued that a reneging queuing system
cannot be employed to model an optical fiber which is balking in nature
unless the number of buffers is large enough to be considered as infinity.
To show that a finite-buffer reneging queue is a good approximation of a
balking queue, we simulated both. Figures 12 and 13 depict respectively, the
blocking probability and average queuing delay of both queues as a function
of the number of buffers for λ=2,1/μ =3,w=4,andL=3,where
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λ,μ,w,andLare the average arrival rate, average service rate, number of
servers, and the waiting time until beginning of service, respectively. As can
be observed, for D≥10, both balking and reneging queues result in the same
blocking probability and average queuing delay of served jobs. Moreover, for
D<10, the difference between Figures 12 and 13 is negligible. For example,
in the worst case (D= 6), the blocking probability of the reneging queue is
0.351, which differs in only 1.2% with respect to that of the balking queue
which is 0.3467. Figures 12 and 13 imply that for traffic intensity ρ≤1.5,
finite-buffer balking and reneging queues behave alike, and the reneging-
based model presented in this paper holds well.
6.2. Blocking probability and average network latency as a function of traffic
intensity (a comparison of M/M/w/N and M/M/w/N with determin-
istic impatience)
Figure 14 illustrates the blocking probability as a function of traffic gener-
ation rate for w=4,d=4,andL= 3. Diamonds and squares represent the
analysis results of A1and A2, respectively, while triangles denote the values
obtained from simulation. Since L=1/μ =3,A1and A3are essentially the
same, and hence only A1is considered. As can be observed, the simulation
results are in good agreement with those of A1. For example, in the worst
case when λ=2,R(A1) is only 7.74, while R(A2) is 100. The increase in
relative error is noticeable as one adopts A2instead of A1to model an optical
0.34
0.345
0.35
0.355
0.36
0.365
0.37
0.375
0.38
0.385
0.39
01234567891011121314151617181920
Number of Buffers
Blocking Probability
Balking
Reneging
Figure 6: Comparison of balking and reneging queuing systems: blocking
probability versus number of buffers.
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0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 1 2 3 4 5 6 7 8 9 1011121314151617181920
Number of Buffers
Average Queuing Dela
y
Balking
Reneging
Figure 7: Comparison of balking and reneging queuing systems: average queu-
ing delay of served jobs versus number of buffers.
fiber.
Figure 15 depicts the average network latency as a function of traffic
generation rate. Again, it can be seen that A1results in better predictions
than A2.Forλ= 2, the relative error of A1is approximately 8.5, while
that of A2is 19.17. Based on the values adopted for base offset time (= 10)
and average burst length (= 3), the curve related to A2implies that burst
traveling through the network experience no queuing delay at intermediate
switches which is in contradiction with the results obtained from simulation.
The difference between the values obtained from A1and simulation is due to
the assumptions made in the derivation of the model.
6.3. The role of number of wavelengths on blocking probability and average
network latency
Figures 16 and 17 depict respectively, the blocking probability and aver-
age network latency as a function of the number of wavelengths when λIj=2
(1 ≤j≤m), d=2,andL= 3. Although the number of physical FDLs (d)
is constant, by increasing the number of wavelengths, the number of virtual
FDLs increases as well. In Figure 16, for w=4,R(A1)is7.79andR(A2)is
77.55, while in Figure 17, these values are 7.8 and 13.72, respectively. Thus,
even in this case, A1results in better estimations in both scenarios.
It should be noticed that increasing the number of wavelengths has a
great impact on reducing the blocking probability and queuing delay. By
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.25 0.75 1.25 1.75
Traffic Rate
Blocking Probability
A1
A2
Simulation
Figure 8: Blocking probability as a function of traffic rate.
12
13
14
15
16
17
18
19
0.25 0.75 1.25 1.75 2.25
Traffic Rate
Average Network Latenc
y
A1
A2
Simulation
Figure 9: Average network latency as a function of traffic rate.
doubling the number of wavelengths from 4 to 8, the blocking probability
decreases by approximately 88 percent, and by tripling this number, the
blocking probability becomes zero. Network queuing delay decreases as well
by a factor of 69 percent as the number of wavelengths increases from 4 to
8. The network queuing delay is calculated by the difference between the
values in Figure 17 and the sum of the base offset time and the average burst
length (which is 13). The final point regarding Figures 16 and 17 is that
by increasing the number of wavelengths, the analysis results of A1and A2
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converge towards each other. This is due to the fact that by increasing the
wavelength count, incoming bursts are less likely to be buffered inside FDLs
and are rather transferred instantly via optical fibers. A1is essentially the
impatient version of A2. Notice that in this case, the number of buffers is
the same in both approaches since L=1/μ in both cases. As an incoming
burst often finds a free wavelength when the number of wavelengths is large,
it rarely becomes impatient. Thus, the difference between A1and A2fades
away as wincreases.
6.4. The role of number of FDLs on blocking probability and average network
latency
In Figure 18, the influence of the number of physical FDLs on the net-
work blocking probability is illustrated. In this case, λIj=2(1≤j≤m),
w=2,andL= 3. As shown in Figure 18, the difference between the
blocking probability values obtained from A1and A2initially increases with
increase in the number of FDLs (when the number of FDLs increases from
2 to 5) and then remains constant for the rest of the simulation (when the
number of FDLs increases from 6 to 11). In the case of three FDLs, R(A1)
is 5.3, while R(A2) is 33.9. However, with increase in number of FDLs from
3 to 10, R(A1) remains the same, while R(A2) rises to 100. The relative
error obtained for the latter case is high, which shows the appropriateness
of the adopted A1approach. It can be seen that with increase in the num-
ber of FDLs, the blocking probability remains constant, concluding that the
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1234567891011
Number of Wavelengths
Blocking Probability
A1
A2
Simulation
Figure 10: Blocking probability as a function of the number of wavelengths.
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12
13
14
15
16
17
18
1234567891011
Number of Wavelengths
Average Network Latenc
y
A1
A2
Simulation
Figure 11: Average network latency as a function of the number of wave-
lengths.
blocking probability is merely affected by the slight increase in the number
of physical FDLs. In the following two subsections, we shall show that for
better FDL utilization, one should consider increasing the FDL length or the
number of exit points on an FDL.
Figure 19 presents the average latency behavior of the optical network
with respect to the number of physical FDLs for the same set of parametric
values as in Figure 18. In order to show the independence between the average
network latency and the number of existing physical FDLs, we calculated the
average latency experienced by the entire network for various values of FDL
counts, ranging from 2 to 11. The resulting trends show that the network
latency is not affected by the FDL count throughout the simulation. The
better performance of A1is justified by calculating the relative errors for each
of the two methods. In the case of seven FDLs, the values obtained for R(A1)
and R(A2) are 12.04 and 24.69, respectively. Thus, it can be inferred that
based on the trends in Figures 18 and 19, the FDL count alleviates neither
the total network blocking probability nor the average network latency.
6.5. The role of FDL length on blocking probability and average network la-
tency
In Figures 20 and 21, we study the impact of CASF. For the sake of
clarity, analysis results of A1and A3are only considered. Figure 20 plots
the blocking probability against the FDL length for λIj=1(1≤j≤m),
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 2 3 4 5 6 7 8 9 10 11 12
Number of Physical FDLs
Blocking Probability
A1
A2
Simulation
Figure 12: Blocking probability as a function of the number of physical FDLs.
12
13
14
15
16
17
18
19
20
1 2 3 4 5 6 7 8 9 10 11 12
Number of Physical FDLs
Average Network Latenc
y
A1
A2
Simulation
Figure 13: Average network latency as a function of the number of physical
FDLs.
w=3,andd= 2. For small FDL lengths, A1and A3behave similarly.
However, as the FDL length increases from 3 to 12, A1estimates the blocking
probability more accurately than A3.Asanexample,forL=8,R(A1)is
7, while R(A3) is 24. The same conclusion can be made from Figure 21
which depicts the average network latency in terms of FDL length. At first
glance, it may seem that A3predicts the network latency better than A1.
However, simulations show that by increasing the FDL length from 3 to 20,
the average network latency increases monotonically. This is logical as with
27
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increase in FDL length, bursts can be delayed longer inside FDLs before being
transmitted via optical fibers, which in turn increases the queuing delay (and
subsequently the network latency). A1agrees with the simulation, i.e., it
increases monotonically as the FDL length increases. On the other hand, for
L≥12, A2does not change appreciably as Lincreases. Although not shown
in Figure 21 due to space limitations, for larger values of L,R(A1)<R(A3).
It can be noticed that as the length of an FDL grows, A3approaches to A2
since the latter is an asymptotic case of A3in which patience time tends
to infinity. Finally, notice that on contrary of increasing the number of
physical FDLs, which had little impact on enhancing the performance of
optical networks, increasing the FDL length can prove effective in improving
performance. By tripling the length of an FDL from 3 to 12, the blocking
probability diminishes by a factor of 37 percent. The FDL length can be
virtually increased by employing FDLs with feedback architecture [4].
7. Conclusion
Recent studies have convincingly demonstrated that FDLs in Internet op-
tical core switches play a critical role in enhancing network performance by
obtaining enough time in order to not overload a potential congested down-
stream switch and henceforth avoid contention. Not taking FDLs behavior
into account leads to the inaccuracy of the calculated performance measures
of the network model and thus leads to incongruity with a real network.
0.15
0.2
0.25
0.3
0.35
0.4
2 3 4 5 6 7 8 9 10 11 12 13
FDL Length
Blocking Probability
A3
A1
Simulation
Figure 14: Blocking probability as a function of FDL length.
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10
15
20
25
30
35
40
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
FDL Length
Average Network Latenc
y
A3
A1
Simulation
7
Figure 15: Average network latency as a function of FDL length.
Previous analytical results of OBS networks subject to crude investigation
of FDLs are inadequate to capture the realistic behavior of optical switches.
This paper has presented a comprehensive discussion of FDL features and
their difference with conventional electronic buffers. Then a new analytical
model for OBS networks based on queuing systems with deterministic im-
patience time has been proposed so as to capture the important features of
FDLs. By including wavelength converters (WCs) in the model, we also have
shown the invaluable role of WCs in further reducing the network blocking
probability and thus decreasing the average network latency of messages in
the network. Simulation results have revealed that the proposed model pro-
vides more accurate prediction of the performance measures compared to the
M/M/c/k model of the past studies. As a future direction of this study, we
aim at extending the model so as to support real-world multimedia traffic
distributions.
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