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QUEUING SYSTEM IN SYNCHRONOUS OPTICAL
NETWORK (SONET)
S. Maragathasundari
Associate professor, Department of Mathematics,
Kalasalingam Academy of Research and Education.
Krishnankoil, (India).
E-mail: maragatham01@gmail.com ORCID: https://orcid.org/0000-0003-1210-6411
P. Suthersan
Assistant professor, Department of Mathematics,
Kalasalingam Academy of Research and Education.
Krishnankoil, (India).
E-mail: suthersan.p@klu.ac.in ORCID: http://orcid.org/0000-0003-0712-0518
K. S. Dhanalakshmi
Assistant professor, Department of Electronics and Communication Engineering,
Kalasalingam Academy of Research and Education.
Krishnankoil, (India).
E-mail: k.s.dhanalakshmi@klu.ac.in ORCID: https://orcid.org/0000-0001-6285-3656
Recepción:
05/12/2019
Aceptación:
08/01/2020
Publicación:
23/03/2020
Citación sugerida:
Maragathasundari, S., Suthersan, P., y Dhanalakshmi, K. S. (2020). Queuing system in Synchronous
Optical Network (SONET). 3C Tecnología. Glosas de innovación aplicadas a la pyme. Edición Especial, Marzo
2020, 231-245. http://doi.org/10.17993/3ctecno.2020.specialissue4.231-245
Suggested citation:
Maragathasundari, S., Suthersan, P., & Dhanalakshmi, K. S. (2020). Queuing system in Synchronous
Optical Network (SONET). 3C Tecnología. Glosas de innovación aplicadas a la pyme. Edición Especial, Marzo
2020, 231-245. http://doi.org/10.17993/3ctecno.2020.specialissue4.231-245
232 http://doi.org/10.17993/3ctecno.2020.specialissue4.231-245
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 – 4143 Edición Especial Special Issue Marzo 2020
ABSTRACT
This paper researches an investigation on Queuing framework in Synchronous Optical
Network (SONET). Optical ber utilized in SONET can blame on various conditions that
are capricious, which is a fundamental dependability worry for power lattice interchanges.
Dierent transmission advances have been utilized in whole deal interchanges, for example,
optical ber, microwave, or satellite. Optical ber can blame on various ighty conditions
which make it a noteworthy danger to arrange unwavering quality. Phases of administration
in SONET, administration intrusion in this system are well explained. The Queuing issue
occurring in this system is very much tackled by strengthening variable methodology and
the comparing line execution measures are inferred. The purpose of issue emerged is very
much anticipated by this Queuing approach and the administration interference could be
limited or to a NIL base. Numerical delineation encourages the model to be defended to an
incredible extent. Graphical portrayal unmistakably claries the presentation proportions
of the Queuing framework in SONET.
KEYWORDS
Batch arrival, Optional First Stage, Compulsory Second Stage, Service Interruption.
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1. INTRODUCTION
SONET is utilized to change over electrical sign into optical sign so it can travel longer
separations. Synchronous Optical NET work (SONET) is a typical for optical televise
communications transport, which was created in the mid-1980s, and stays in boundless
use today. Contrasted with Ethernet cabling that traverses separations up to100 meters,
SONET ber ordinarily runs a lot further. Indeed, even short achieve connections range as
much as 2 kilometers (1.2 miles); intermediate and long achieve connections spread many
kilometers. Along these lines it is appropriate for whole deal transmission, for example,
the one in the power lattice correspondences. Wu, Kobrinski, Ghosal, and Lakshman
(1994) examined a few DCS design upgrade choices, including a parallel handling/cross-
interface DCS engineering, which may improve the administration rebuilding time. Boehm,
Ching, Grith, and Saal (1986) gave an account of the exercises in dierent benchmarks
associations, with accentuation on a synchronous system proposition which is as of now
being talked about in the T1 advisory group. Way, Smith, Johnson, and Izadpanah
(1992) tentatively conrmed the system idea and talked about various system applications
for bursty information trac and persistent voice/video trac. Blumenthal et al. (2003)
explored the sign handling procedures, Hac and Mutlu (1989) researched the B-ISDN
convention, guidelines utilized in the Broadband reference model. Lee, Sherali, Han, and
Kim (2000) dealt with a system plan issue emerging from the sending of synchronous
optical systems (SONET), a standard of transmission utilizing optical ber innovation.
Cosares, Deutsch, Saniee, and Wasem (1995) inspected SONET framework by the Bellcore
customer organizations has spared 10 to 30 percent in expenses and requests of greatness
in time. Chao, Shtirmer, and Smoot (1989) broke down the physical layer of the system
utilizes the synchronous optical system transmission design. Fundamental ideas are talked
about and reviewed by Jue, Yang, Kim, and Zhang (2009). Kang, Park, Shin, and Jeong
(1995) watched the normal for the system relying on the collected transmission limit of
the network. Maragathasundari and Balamurugan (2015) contemplated the presentation
examination of bunch landing line with two phases of administration. Maragathasundari
and Dhanalakshmi (2018) investigated versatile adhoc systems issue A Queuing approach.
Maragathasundari and Srinivasan (2012) made an investigation on M/G/1 input line
with three phase and dierent server get-away. Maragathasundari and Srinivasan (2015)
examined a Non-Markovian Multistage Batch entry line with breakdown and reneging. An
examination on the investigation of execution proportion of mass information line with N
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sort of extra discretionary administration, administration interference and deterministic
get-away were inspected by Maragathasundari and Sowmiah (2016).
1.1. ADVANTAGES OF SONET
1) Transmits data to large distances.
2) Low electromagnetic interference.
3) High data rates.
4) Large Bandwidth.
Graphic 1. SONET Network Elements.
1.2. SONET CONNECTIONS
1) Section: Portion of system interfacing two neighboring gadgets.
2) Line: Portion of system interfacing two neighboring multiplexers.
3) Path: End-to-end segment of the system.
*STS Multiplexer:
Performs multiplexing of sign.
*STS Demultiplexer:
Performs demultiplexing of sign.
Converts optical sign to electrical sign.
*Regenerator:
It is a repeater, that takes an optical sign and recovers (builds the quality) it.
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*Add/Drop Multiplexer:
It permits including sign originating from various sources into a given way or expelling a
sign.
1.3. SONET LAYERS
Data Link
Path Layer
Line Layer
Section Layer
Physical
Photonic Layer
Graphic 2. SONET Layers.
SONET includes four functional layers:
1) Path Layer:
a. It is in charge of the development of sign from its optical source to its optical goal.
b. STS Mux/Demux gives way layer capacities.
2) Line Layer:
a. It is in charge of the development of sign over a physical line.
b. STS Mux/Demux and Add/Drop Mux give Line layer capacities.
3) Section Layer:
a. It is in charge of the development of sign over a physical area.
b. Each gadget of system gives segment layer capacities.
4) Photonic Layer:
a. It relates to the physical layer of the OSI model.
b. It incorporates physical determinations for the optical ber channel (nearness of light
= 1 and nonappearance of light = 0).
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1.4. PERFORMANCE REQUIREMENTS
We assume the following to describe the queuing model of our study.
1) Batch arrival – Queue - We consider a solitary server line which will give two distinct
administrations, an Essential Service and Optional Service.
2) Essential administration – 2 phases: One of the benets of SONET is that it can pass
on gigantic payloads (more than 50 Mbps). To achieve this capacity, the STS SPE can
be sub-apportioned into more diminutive sections or structures, known as VTs (Virtual
tributaries)
*Optional First Stage: Except for connected sign, all data sources are at last changed
over to a base setup of a synchronous STS–1 signal (51.84 Mbps or higher). Lower-
speed information sources, for instance, DS–1s are rst piece or byte-multiplexed
into VTs. Several synchronous STS–1s are then multiplexed together in either a
singular or two-mastermind system to outline an electrical STS–N signal (N >= 1).
* Compulsory Second Stage: Any kind of organization, running from voice to quick
data and video, can be recognized by various types of organization connectors. An
organization connector maps the sign into the payload envelope of the STS–1 or
VT. New organizations and sign can be transported by including new organization
connectors at the edge of the SONET sort out.
3) Optional administration – Service Interruption happens – Optical Cable Failures are
considered here as Service Interruption during this Optional Service.
Three kinds of optical strands have been utilized in the whole deal transport of information.
• Buried ber optic links have a higher disappointment rate than the two overhead
links.
• Optical ground wire links and introduced overhead on posts or transmission
towers.
• All dielectric self-supporting cables introduced overhead on posts or transmission
towers.
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Two sorts of disappointments are being considered,
(a) “cable cut” disappointments which will inuence both the working strand and the
assurance strand, and
(b) “Strand disappointments” which will bomb just one strand inside the link.
4) Completion of Both administrations– Dissatised Customers (not ready to utilize the
multiplexing procedure successfully) can join the tail of the rst line to get a Feedback
administration.
2. MATHEMATICAL PORTRAYAL OF THE QUEUING MODEL
The arithmetical portrayal of the Queuing frame work has the option to be described by
the resulting proposition:
Customers meet up at the structure in clusters of variable size in a compound strategy
pursues Poisson conveyance. Let
be the rst order probability that a
batch of j customers arrives at the system during a short duration of time (t,t+dt) where
and and is the mean landing rate of the batches.
The administration time pursues general(arbitrary) circulation. First stage of essential
service follows distribution function as
and density function . Let be
the conditional density function. Hence, we have:
(a)
For second stage of essential service,
(b)
For optional service,
(c)
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Service interruption follows Poisson distribution with mean rate .
3. GOVERNING EQUATIONS OF THE MODEL
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
4. BOUNDARY CONDITIONS
The following boundary conditions are used to solve the above equations:
(10)
(11)
(12)
(13)
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5. QUEUE LENGTH DISTRIBUTION
Usage of Supplementary variable technique
We multiply (1) by z
n
and sum over n from 1 to and add it to (2).
We get,
(14)
Again integrating the above from 0 to n, we get
Again integrating (*) by parts with respect to x yields,
(15)
Multiplying both sides of the (*) by
and integrating over x, we get:
(16)
Applying the same concept for the second stage (optional) in essential service
,
optional service
, and repair process , we get,
i)
(17)
Also we have,
(18)
ii)
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(19)
(20)
(21)
(22)
Now
(10), using (9) and further using (18), (20), (22) we get,
(23)
Substituting (23) in (15), (17), (19), (21) we get,
(24)
(25)
(26)
(27)
Where,
6. PROBABILITY CAPACITY FUNCTION OF THE QUEUE LENGTH
Let J
q
(z) be the PGFof the queue length
Adding (24) to (27), we get,
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(28)
7. IDLE TIME AND UTILIZATION FACTOR
Idle time is determined using the condition:
(29)
Applying LH rule we get,
(30)
(31)
From the idle factor Q , the utilization rate is calculated.
To nd L
q
the length of the Queue and the Queue performance measures.
We have
(indeterminate form)
(32)
Here
where N(z) and D(z) are the numerator and denominator of (28).
(33)
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D”(1)=-λE(1)2{1+(-λE(I))[E(L
(e
1
)
)+E(L
(e
2
)
)][(1-L
k
(β))+(1-m)+rL
k
(β)]
-[(1-L
k
(β)]+ λE(I)E(H)(1-L
k
(β))+ βλE(I)L’
k
+ rL’
k
λE(I)]}
+ β{-(-λE(I))
2
(E(L
(e
1
)
2
)+ E(L
(e
2
)
2
)+2E(L
(e
1
)
)E(L
(e
2
)
))[1-L
k
(β)
+(1-m)+ rL
k
(β)]+(-λE(I))L’
k
(E(L
(e
1
)
)+ E(L
(e
2
)
))
-(λE(1))(E(L
(e
1
)
+E(L
(e
2
)
))[1-L
k
(β)+ E(H) λE(I)(1-L
k
(β))
-λE(I) L’
k
+rλE(I)L’
k
]
-[(λE(I))E(H)(1-L
k
(β))+ β L’
k
+ E(H) λE(I)(1-L
k
(β))
+E(H
2
)(-λE(I))
2
(1-L’
k
(β))+βE(I)E(H) L’
k
+β (-λE(I)) L’
k
-β(-λE(I))
2
L’
k
E(H)+ βE(L
k
2
)(λE(I))
2
+rE(L
k
2
)(-λE(1))
2
-
]}
(33)
Substituting (33) in (32) we obtain L
q
in closed form.
Further, all the other queue performance measures can be found using Little’s law
8. NUMERICAL ILLUSTRATION
Table 1. Effect of change of .
Q Ρ Lq L Wq W
0.6485 0.3515 8.5449 8.8964 2.1362 2.2241
0.6917 0.3083 12.552 12.8602 3.138 3.2151
0.7382 0.2618 16.556 16.8173 4.1389 4.2043
0.7787 0.2213 20.983 21.2043 5.2457 5.3011
0.8190 0.1810 25.237 25.4176 6.3092 6.3544
0% 20% 40% 60% 80% 100%
2
2.5
3
3.5
4
Q
ρ
Lq
L
Wq
W
0
2
4
6
8
10
12
0.6 0.8 1 1.3 1.5
Q
ρ
Lq
L
Wq
W
Graphic 3. Effect of change ofβ.
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From Table 1 and Figure 3, it is clear that if the probability in service interruption increases
it leads to an increase in all the performance measures. Since the service interruption gets
increased the idle time gets amplied and utilization factor is decreased.
Table 2. Effect of change of .
Q Ρ Lq L Wq W
0.6485 0.3515 8.5449 8.8964 2.1362 2.2241
0.645 0.355 8.6013 8.9563 2.1503 2.2399
0.6391 0.3609 8.6967 9.0576 2.1742 2.2644
0.6277 0.3723 8.9197 9.2077 2.2299 2.3019
0.5977 0.4023 9.7597 10.162 2.4399 2.5405
0% 20% 40% 60% 80% 100%
2
2.5
3
3.5
4
Q
ρ
Lq
L
Wq
W
0
2
4
6
8
10
12
0.6 0.8 1 1.3 1.5
Q
ρ
Lq
L
Wq
W
Graphic 4. Effect of change of r.
Table 2 indicates that, as the probability of repair rate gets increased, length of the queue
is increased. Since the repair rate increased utilization factor gets increased and idle time
gets decreased.
9. CONCLUSIONS
In this paper we have studied a batch arrival, two phases of essential administration and
optional administration, service interruption, feedback service. This paper clearly analyses
the steady state results and some queuing performance measures. Further this model can
be extended by adding the concept of delay time, reneging, long vacation, short vacation
etc.
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