Methods for Moment Manipulation of Various Traffic
Sessions in Modern Small Cell Environment
Paramita Bhattacharya1, Samya Bhattacharya1,2
1
1
2
M/S Untamed Spectrum , Post Box 119 Espoo 02320, Finland ; School of Electronic and Electrical Engineering , University of
2
2
Leeds , LS2 9JT, United Kingdom
1
1
2
[email protected] ; [email protected] , [email protected]
ABSTRACT
In this paper, we exercise some important derivation involving moment manipulations to estimate various traffic sessions
in modern small cell environment. In the light of some previous work on cellular environment, we derive accurate
expressions for a proposed Semi-Markov model that accounts for the smaller population size of the users in modern small
cells. The small cells are generally powered by access points (WiFi/mini base stations) and are typically used as mobile
clouds, hotspots. We validate the analysis for small cell model with an existing cellular traffic model by increasing the
population size of the users to a large value (typical for a macro/micro-cellular environment). Regardless of the use of
communication technology and nature of user sessions (voice, video, data etc.), the generic analysis can provide a good
insight to the state-of-the-art traffic dimensioning for small cells.
CONCLUSION
We developed some traffic estimation techniques using various moment manipulations for Semi-Markov model
applicable in a small cell scenario. The analysis accounts for the limited population size of the users in modern small cells. It
is observed that the distributions of the traffic streams largely vary from the large population scenario in traditional cells.
These affect the quality of service (QoS) parameters such as congestions both for fresh session requests and hand-off
requests. The deviation between the small cell model and large population model increases with the user mobility while the
same decreases with the increase in population size. Thus, it can be concluded that as the cell size diminishes, small cell
model becomes more appropriate for dimensioning. Further, if the cell size increases and population size becomes
comparable with the large cell case, both small cell models and R-T models yield same results because population size shapes
the fresh traffic to be Poisson nature. Thus, for accurate traffic dimensioning in small cells, it is essential to adopt proper
modelling and estimation technique.
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Citation
Paramita Bhattacharya, Samya Bhattacharya, “Methods for Moment Manipulation of
Various Traffic Sessions in Modern Small Cell Environment,” International Journal of
Research in Wireless Systems (IJRWS), Vol. 2, No. 3, pp. 23–34, October, 2013
ISSN: 2320 - 3617
International Journal of Research in Wireless Systems (IJRWS), Volume 2, Issue 3, October (2013)
23
Methods for Moment Manipulation of Various Traffic
Traffic Sessions in Modern Small Cell Environment
I. INTRODUCTION
With the rapid growth of mobile users, deployment of
smaller cells powered by miniature base stations (BSs), such as
access points (APs)) or WiFi clouds have become increasingly
popular cost-effective
effective choice for the mobile operators around
the world. Supporting multimedia sessions in such cells
require accurate traffic dimensioning to maintain various
quality of service (QoS) requirements. In addition, to maintain
seamless connectivity on the move, these small cells require
better session hand-off
off management. Therefore, hand-off
hand
traffic plays an important role in state-of-the-art
state
traffic
analysis. The researchers in the past have extensively
extensivel studied
the above in the context of cellular networks. Early traffic
analyses for cellular networks were carried out with M/M/C
loss systems and single-moment
moment approach where fresh (i.e.
new) and hand-off
off calls were assumed to be Poisson arrival
processes [1]. In the context of micro-cellular
cellular systems where a
user undergoes frequent hand-offs,
offs, it was shown [4] that the
hand-off
off process does not remain Poisson distributed. As a
result, even if the fresh
sh call arrival process can be represented
as a Poisson distribution, the aggregated traffic stream is more
realistically represented by a General distribution. It was also
proposed [7] that in case of General distributed traffic,
traff the
two-moment
moment representation of traffic (using mean and
variance) is a better approach than a single-moment
single
representation. Rajaratnam and Takawira [7] in their studies
pointed out that the hand-off
off traffic is the main distinguishing
dis
features between the PSTN and mobile cellular networks by
analysing the traffic using General distributed hand-off
hand
traffic.
These analyses were applicable for densely populated macromacro
micro cells. With diminishing cell sizes and thus user
population
tion in a cell, fresh call arrivals, or a new session
request in the context of modern mobile environment, follow
Engset distribution [10] instead of Poisson distribution. The
corresponding effects on the traffic streams and user
us sessions
in small cell environment have not been thoroughly
investigated in the literature to the best of our knowledge. In
[11][12], authors suggested to follow the models developed in
[7] for a less populated area. But the use of the above models
leads to difficulties, when the fresh call arrival process is
Engset distributed. Firstly, the estimation of traffic offered to
the virtual cell in their model needs a Semi-Markov
Semi
analysis
with Engset distribution. Secondly, the suggested use of
Binomial-Poisson-Pascal (BPP) method in [7] yields inaccurate
estimates of the carried traffic in a cell from the traffic offered
to that cell when the peakedness
akedness factor (or peakedness) of the
offered traffic falls below 0.5 (common in presence of an
Engset distributed traffic stream) [13].. Also, the suggested use
of Girard's method [9] to estimate the
he carried traffic in a cell
from the traffic offered to that cell is limited to Pure Chance
Type-I (PCT-I)
I) traffic; making it inapplicable in the case of
Engset distributed fresh call arrival process i.e. Pure Chance
Type-II (PCT-II) traffic [10].. Thus, a thorough Semi
Semi-Markov
analysis for Engset distributed fresh call traffic and the
corresponding traffic estimates in modern mobile
environment merits investigation.
Therefore, our contributions in this paper are two
two-fold.
Firstly, we propose a Semi-Markov
Markov analysis for the Engset
distributed fresh traffic sessions and General distributed
handoff sessions in a small cell environment. Secondly, we
derive two moments (mean and variance) of fresh and
handoff traffic sessions and use tthose estimates to compute
corresponding congestion parameters in a small cell. In
Section II,, we derive various traffic estimates for the models
developed in [7] for micro-cellular
cellular environment. This forms
the background for the proposed work in Section III. Section IV
presents some numeric results related to the model
verification and validation. Finally, we conclude in Section
Error! Reference source not found.
found..
II. RELATED WORK
In this section we study the various traffic sessions related
to traditional cellular networks. In cellular networks, a channel
was considered to be a single call carrying resource [7]. Fixed
Channel Allocation (FCA) was considered where the number of
simultaneous ongoing calls or sessions is fixed in a cell. A
traffic stream was represented by two moments ((‫ ܯ‬ǡܸ)
denoting the corresponding mean and variance. The hand
hand-off
traffic was obtained when a fraction of users from a cell hand
handoff to an adjacent cell in their two two
two-cell model (Fig. 1).
Fig. 1. The two-cell model with a virtual
ual cell [7].
Initially, it was assumed that the originating cell receives no
hand-off
off traffic. Later, the restriction was removed through an
iterative process. To estimate the offered traffic to the target
cell, the carried traffic
affic estimate in the previous (originating)
cell is needed. It has been shown that the variance of the
carried traffic in the originating cell does not match with that
of the traffic offered to the target cell, though the
corresponding means match. This iis because the offered
traffic used to be stochastically different from the carried
traffic. Thus, a virtual (hypothetical) cell of infinite capacity has
been introduced between the two cells. The carried traffic in
the virtual cell (also known as freed car
carried traffic [14]) will
now be equivalent to the offered traffic in the target cell.
Later, the two-cell
cell model was generalised to a multi
multi-cell
International Journal of Research in Wireless Systems (IJRWS), Volume 2, Issue 3, October
ber (201
(2013)
24
Methods for Moment Manipulation of Various Traffic Sessions in Modern Small Cell Environment
model where the estimated hand-off traffic using the two-cell
model was aggregated with fresh call offered traffic to form a
combined traffic stream offered to a cell.
Using various moment manipulation techniques presented
in [13],[14], we in this section, present an alternative
approach to [7] to formulate and derive the moments of the
corresponding traffic streams in their two-cell model. These
form the background of our work in the next section involving
small cells. A session initiated by a mobile user may suffer
blocking at two stages - (a) at session initiation, known as
fresh blocking or fresh congestion, and (b) at the cell
boundary, which is called hand-off blocking or hand-off
congestion. Let us consider the ݅௧௛ cell ‫ܮ‬௜ (݅= 1, 2) in Fig. 1.
Let ‫ ܥ‬be the number of channels assigned to the cell; ߣ௜ = ߣ௙
be the mean arrival rate of fresh requests to the cell, which is
Poisson distributed. A mobile user vacates a channel in a cell
due to either session termination or session hand-off to the
other cell. Both processes are assumed to be Poisson
distributed with respective mean rate ߤ௧ and ߤ௛௜. Thus, the
session departure process from a cell is also Poisson
distributed with mean rate ߤௗ௜ = ߤ௧ + ߤ௛௜. It is imperative to
state that the channel holding time is Negative Exponential
distributed with mean 1/ߤௗ௜. It is noted that some studies
modelled the channel holding time as Gamma distribution
[15] or Hyper-Erlang distribution [16] or Lognormal
distribution [17] or a combination of constant and Negative
Exponential distribution [8]. However, considering the urban
scenario [18], the channel holding time is modelled as
Negative Exponential distribution in this work. Assuming that
the originating cell ‫ܮ‬ଵ initially receives no hand-off traffic, the
offered traffic to that cell is Poisson distributed (fresh requests
only) with mean ‫ܣ‬௙ = ߣ௙ /ߤௗଵ. The carried traffic (‫ ܯ‬஼భ ,ܸ஼భ ) in
the cell is obtained as below.
A. Expression for the Carried Traffic in ࡸ૚
Let there be ݇ ongoing sessions in ‫ܮ‬ଵ. Then the probability
(‫݌‬௞ ) that ‫ܮ‬ଵ is in state [13], is given by
‫ܣ‬௙௞
‫ܣ‬௙௞
݇!
‫݌‬௞ =
= ݇!
௜
‫ܣ‬
݁஼ ൫‫ܣ‬௙൯
∑஼௜ୀ଴ ௙
݅!
(1ܽ)
‫ܣ‬௙஼
‫݌‬஼ = ‫ !ܥ‬௜ = ‫ܤ‬൫‫ܣ‬௙, ‫ܥ‬൯(1ܾ)
‫ܣ‬
∑஼௜ୀ଴ ௙
݅!
஺೔
where݁஼ ൫‫ܣ‬௙൯= ∑஼௜ୀ଴ ೑ is the incomplete exponential
௜!
function and ‫݌‬஼ is the Erlang loss function. Since the carried
traffic has only the fresh component, it is sufficient to use
Probability Generating Function (PGF) [13] alone, which is
defined as G(z) = ∑ஶ௞ୀ଴ ‫݌‬௞ ‫ݖ‬௞ . Using Eqn. 1(a), we obtain
‫= )ݖ(ܩ‬
Therefore,
and
‫ܩ‬
ᇱ(‫)ݖ‬
=
‫ܩ‬′′(‫= )ݖ‬
∑஼௞ୀ଴ ‫ݖ‬௞
‫ܣ‬௙௞
݇!
‫ܣ‬௜
∑஼௜ୀ଴ ௙
݅!
ಲೖ
೑
ೖష భ
∑಴
ೖసభ ௞௭
ಲ೔
೑
∑಴
೔సబ ೔!
(2)
(3)
ೖ!
ಲೖ
೑
ೖష మ
∑಴
ೖసమ ௞(௞ିଵ)௭
ಲ೔
೑
∑಴
೔స బ ೔!
(4)
ೖ!
We express the mean (‫ ܯ‬஼భ ) of the carried traffic from [13] as
‫ ܯ‬஼భ = ‫ܩ‬ᇱ(1)
=
=
ಲೖ
೑
∑಴
ೖసభ ௞
[Using Eqn. (3)]
ೖ!
ಲ೔
೑
∑಴
೔సబ
೔!
ಲೖ
೑
∑಴
ೖసభ(ೖషభ)!
ಲ೔
೑
∑಴
೔సబ
೔!
ಲ೔
೑
಴షభ
∑೔సబ
೔!
௙
೔
ಲ೑
∑಴
೔సబ ೔!
಴
ಲ೔
೑ ಲ೑
∑಴
ି಴!
೔సబ ೔! ௙
ಲ೔
೑
∑಴
೔సబ ೔!
ಲ಴
೑
=‫ܣ‬
=‫ܣ‬
= ‫ܣ‬௙ ቌ1 −
಴!
ಲ೔
೑
∑಴
೔సబ
೔!
ቍ
= ‫ܣ‬௙ൣ1 − ‫ܤ‬൫‫ܣ‬௙, ‫ܥ‬൯൧[Using Eqn. (1b)]
(5)
We express the variance (ܸ஼భ ) of the carried traffic from [13]
as
ଶ
ܸ஼భ = ‫ܩ‬ᇱᇱ(1) + ‫ܩ‬ᇱ(1) − ൫‫ܩ‬ᇱ(1)൯
ଶ
= ‫ ܯ‬஼భ − ቂ൫‫ܩ‬ᇱ(1)൯ − ‫ܩ‬ᇱᇱ(1)ቃ[Using Eqn. (5)] (6)
Using Eqn. (1b), (3), (4), and (5), we get
ଶ
ቂ൫‫ܩ‬ᇱ(1)൯ − ‫ܩ‬ᇱᇱ(1)ቃ= ‫ܣ‬௙ଶ(1 − ‫݌‬஼ )ଶ −
‫ܣ‬௙௞
(݇ − 2)!
‫ܣ‬௜
∑஼௜ୀ଴ ௙
݅!
∑஼௞ୀଶ
‫ܣ‬௙௜
݅!
= ‫ܣ‬௙ଶ(1 − ‫݌‬௖)ଶ − ‫ܣ‬௙ଶ
௜
‫ܣ‬
∑஼௜ୀ଴ ௙
݅!
௜
‫ܣ‬஼
‫ܣ‬௙஼ିଵ
஼ ‫ܣ‬௙
∑௜ୀ଴ − ௙ −
݅!
‫ ܥ( !ܥ‬− 1)!
= ‫ܣ‬௙ଶ(1 − ‫݌‬௖)ଶ − ‫ܣ‬௙ଶ
‫ܣ‬௜
∑஼௜ୀ଴ ௙
݅!
∑஼ିଶ
௜ୀ଴
International Journal of Research in Wireless Systems (IJRWS), Volume 2, Issue 3, October (2013)
25
Methods for Moment Manipulation of Various Traffic Sessions in Modern Small Cell Environment
moment of the ‫ݏ‬௧௛ traffic stream, where ‫ = ݏ‬1 denotes the
overflow stream and ‫ = ݏ‬2 denotes the freed carried traffic
stream [14]. In our notation, the mean (‫ ܯ‬ଵ) of the overflow
traffic [14] is written as
‫ܣ‬௙஼
‫ܣ‬௙஼
‫ܥ‬
‫ !ܥ‬൪
= ‫ܣ‬௙ଶ(1 − ‫݌‬௖)ଶ − ‫ܣ‬௙ଶ ൦1 − ‫ !ܥ‬௜ −
‫ܣ‬
‫ܣ‬
‫ܣ‬௜
௙
௙
∑஼௜ୀ଴
∑஼௜ୀ଴ ௙
݅!
݅!
‫ܥ‬
=
‫ ݌‬቉[Using Eqn. (1b)]
‫ܣ‬௙ ஼
= ‫ܣ‬௙ଶ − 2‫ܣ‬௙ଶ‫݌‬஼ + ‫ܣ‬௙ଶ‫݌‬஼ଶ − ‫ܣ‬௙ଶ + ‫ܣ‬௙ଶ‫݌‬஼ + ‫ܣܥ‬௙‫݌‬஼
= ‫ܣܥ‬௙‫݌‬஼ − ‫ܣ‬௙ଶ‫݌‬஼ + ‫ܣ‬௙ଶ‫݌‬஼ଶ
= ‫ܣ‬௙‫݌‬஼ ൣ‫ ܥ‬− ‫ܣ‬௙(1 − ‫݌‬஼ )൧
= ‫ܣ‬௙‫݌‬஼ ൣ‫ ܥ‬− ‫ ܯ‬஼భ ൧
‫ܣ‬௙ଶ(1
− ‫݌‬௖)ଶ − ‫ܣ‬௙ଶ ቈ1 − ‫݌‬஼ −
஼
1
1
݈!
‫ܥ‬
=
=෍ ቀ ቁ
݈ ‫( ܯ‬௟ାଵ)
‫ ܯ‬ଵ ‫ ܯ‬ଵ,(ଵ)
௟ୀ଴
Therefore, ‫ ܯ‬ଵ =
=
= ‫ܣ‬௙‫ܤ‬൫‫ܣ‬௙, ‫ܥ‬൯(7)
=
Substituting Eqn. (7) in Eqn. (6), we obtain
ܸ஼భ = ‫ ܯ‬஼భ − ‫ܣ‬௙‫ܤ‬൫‫ܣ‬௙, ‫ܥ‬൯൫‫ ܥ‬− ‫ ܯ‬஼భ ൯(8)
‫( ܯ‬௞)
௞ିଵ
1
=
ෑ
ߤௗ (‫)ܣܧ‬
௝ୀଵ
݆‫ߤ݆( ∗ܣ‬ௗ )
݇ ∈ ‫(ܫ‬9)
1 − ‫ߤ݆( ∗ܣ‬ௗ )
where ߤௗ is the mean service rate (Poisson distributed), ‫ ܣܧ‬is
the mean inter-arrival time. If ‫ )ݏ( ∗ܣ‬denotes the L.S.T. of
‫)ݐ( ∗ܣ‬, then for Poisson distributed offered traffic, we have
‫= )ݏ( ∗ܣ‬
Further, from [14], we get
‫ = ܣܧ‬−‫( ∗ܣ‬0) = −
ߣ௙
‫ݏ‬+ ߣ௙
ଶอ
൫‫ݏ‬+ ߣ௙ ൯
௦ୀ଴
=
1
ߣ௙
Substituting ‫ ܣܧ‬from Eqn. (11) in Eqn. (9) we obtain
‫( ܯ‬௞) =
ଵ
ఓ೏
=
భ
ഊ೑
∏௞ିଵ
௝ୀଵ
ఒ೑
ఓ೏
ఒ
= ቀ ೑ቁ
where ‫= ܯ‬
ఒ೑
ఓ೏
ఓ೏
=‫ܯ‬
ഊ೑
ೕഋ ೏ శ ഊ೑
ఓ೏
(12)
is the first factorial moment of the Poisson
Let ‫ ܯ‬௦,(௞) (‫ = ݏ‬1, 2 and݇ = 1, 2, …) be the ݇௧௛ factorial
಴!
ಾ೔
೔!
∑಴
೗సబ
[Using Eqn. (1b)]
(13)
From [13], the variance (ܸଵ) of the overflow traffic can be
expressed in terms of the mean (‫ ܯ‬ଵ) and the second factorial
moment (‫ ܯ‬ଵ,(ଶ) ) of the overflow traffic as
ܸଵ = ‫ ܯ‬ଵ,(ଶ) − ‫ ܯ‬ଵ(‫ ܯ‬ଵ − 1)
(14)
The second factorial moment (‫ ܯ‬ଵ,(ଶ) ) of the overflow traffic
from [14] is written as
1
Therefore,
‫ ܯ‬ଵ,(ଶ)
ܸଵ + ‫ ܯ‬ଵ(‫ ܯ‬ଵ − 1) =
஼
‫݈( ܥ‬+ 1)!
=෍ ቀ ቁ
݈ ‫( ܯ‬௟ାଶ)
௟ୀ଴
1
ଵ
಴!(೗శభ)!
೗!(಴ష೗)!ಾ ೗శమ
∑಴
೗స బ
ଵ
಴!
ಾ ಴ష೗
∑಴ (௟ାଵ)(಴ష೗)!
ಾ ಴శమ ೗స బ
ಾ಴
ெ మ ಴!
∑಴
೗సబ(஼ାଵି௞)
=
(஼ାଵ) ∑಴
ೖసబ
=
[Using Eqn. (14)]
‫݈( ܥ‬+ 1)!
∑஼௟ୀ଴ ቀ ቁ
݈ ‫( ܯ‬௟ାଶ)
=
distributed offered traffic.
C. Expression for the Overflow Traffic in ࡸ૚
ಾ಴
಴!
ಾ ಴ష೗
∑಴
೗సబ(಴ష೗)!
ಾ಴
= ‫ ܯ(ܤ ܯ‬, ‫)ܥ‬
=
ఒ೑
௞
಴!ಾ ష೗
(಴ష೗)!
=
ೕഊ೑
ೕഋ ೏ శഊ೑
ଵି
∏௞ିଵ
௝ୀଵ
(11)
[Using Eqn. (12)]
∑಴
೗స బ
=‫ܯ‬
(10)
−ߣ௙
ଵ
஼ ೗!
∑಴
೗స బቀ௟ቁಾ ೗శ భ
ெ
=‫ܯ‬
B. Factorial Moments for the Offered Traffic in ࡸ૛
Let the inter-arrival time of the offered traffic be denoted
as ‫)ݐ(ܣ‬. From [14], the ݇௧௛ factorial moments ൫‫( ܯ‬௞) ൯ of the
offered traffic is expressed as
ଵ
೗!
஼
∑಴
೗సబቀ௟ቁಾ (೗
శభ)
ಾೖ
ೖ!
ಾ಴
ெ మ ಴!
ೕ
ಾೖ
భಾ
ିெ ∑಴ష
ೕసబ ೕ!
ೖ!
ெమ
ಾ಴
಴!
ಾೖ
∑಴
ೖసబ ೖ!
ಾೕ ಾ಴
∑಴
ೕస బ ೕ! ష ಴!
(஼ାଵ)ିெ ൦
ಾೖ
∑಴
ೖస బ ೖ!
International Journal of Research in Wireless Systems (IJRWS), Volume 2, Issue 3, October (2013)
൪
26
Methods for Moment Manipulation of Various Traffic Sessions in Modern Small Cell Environment
ெ మ஻(ெ ,஼)
=
Therefore,
஼ାଵିெ [ଵି஻(ெ ,஼)]
ெ మெ భ
=
஼ାଵିெ ାெ భ
௏భ
ெభ
+ (‫ ܯ‬ଵ − 1) =
=
[Using Eqn. (13)]
=
ெ
஼ାଵିெ ାெ భ
‫ܯ ܯ‬ଵ
ܸଵ = ‫ ܯ‬ଵ − ‫ ܯ‬ଵଶ +
‫ ܥ‬+ 1 − ‫ ܯ‬+ ‫ܯ‬ଵ
(15)
D. Expression for the Freed Carried Traffic in ࡸࢂ
From the conservation of flow, the mean (‫ ܯ‬௏ ) of the
freed carried traffic can be expressed as
‫ ܯ‬௏ = ‫ ܯ‬ଶ = ‫ ܯ‬− ‫ ܯ‬ଵ=[1 − ‫ ܯ(ܤ‬, ‫[ ])ܥ‬Using Eqn. (13)] (16)
which is equal to the mean carried traffic. The second
factorial moment (‫ ܯ‬ଶ,(ଶ) ) of the freed traffic from [14] is
written as
‫ ܯ‬ଶ,(ଶ) = ‫ ܯ‬ଶ
஼
‫( ܯ‬ଶ)
− ‫ ܯ‬ଵ[ܸଵ + ‫ ܯ‬ଵ(‫ ܯ‬ଵ − 1)]
‫( ܯ‬ଵ)
௟
݉ ‫( ܯ‬௠ )
݈!
‫ܥ‬
×෍ ቀ ቁ
෍ ቈ
+ 1቉(17)
݈ ‫( ܯ‬௟ାଵ)
‫( ܯ‬௠ ାଵ)
௟ୀଵ
௠ ୀଵ
Using the relation between ordinary, central and factorial
moments of a distribution [13], we rewrite Eqn. (17) as
ܸଶ + ‫ ܯ‬ଶ(‫ ܯ‬ଶ − 1) = ‫ ܯ‬ଶ
஼
× ∑஼௟ୀଵ ቀ ቁ
௟!
௟ ெ (೗శభ)
‫( ܯ‬ଶ)
− ‫ ܯ‬ଵ[ܸଵ + ‫ ܯ‬ଵ(‫ ܯ‬ଵ − 1)]
‫( ܯ‬ଵ)
௠ ெ (೘ )
∑஼௟ୀଵ ൤
ெ (೘ శ భ)
+ 1൨
= ‫ ܯ‬ଶ‫ ܯ‬− ‫ ܯ‬ଵ[ܸଵ + ‫ ܯ‬ଵ(‫ ܯ‬ଵ − 1)] ∑஼௟ୀଵ
= ‫ ܯ‬ଶ‫ ܯ‬− ‫ ܯ‬ଵ[ܸଵ + ‫ ܯ‬ଵ(‫ ܯ‬ଵ − 1)] ∑஼௟ୀଵ
= ‫ ܯ‬ଶ‫ ܯ‬− ‫ ܯ‬ଵ[ܸଵ + ‫ ܯ‬ଵ(‫ ܯ‬ଵ − 1)]
× ൤∑஼௟ୀଵ
஼!ெ ష(೗శభ) ௟
(஼ି௟)!
+ ∑஼௟ୀଵ
஼!ெ ష(೗శభ)
(஼ି௟)!
஼!ெ ష(೗శభ)
(஼ି௟)!
஼!ெ ష(೗శమ) ௟(௟ାଵ)
ଶ(஼ି௟)!
൨
௠
ெ
௟(௟ାଵ)
ଶெ
ቃ
஼!ெ ష(಴ష ೖశభ) ௟(஼ି௞)
஼ିଵ
= ‫ ܯ‬ଶ‫ ܯ‬− ‫ ܯ‬ଵ[ܸଵ + ‫ ܯ‬ଵ(‫ ܯ‬ଵ − 1)] ቎∑
௞ୀ଴
ᇣᇧᇧᇧᇧᇧᇧᇤ
ᇧᇧᇧᇧᇧᇧᇥ +
஼!ெ ష (಴షೖశ మ) (஼ି௞)(஼ି௞ାଵ)
஼ିଵ
∑
௞ୀ଴
቉
ᇣᇧᇧᇧᇧᇧᇧᇧᇧᇤ
ᇧᇧᇧᇧᇧᇧᇧᇧᇥ
ଶ௞!
ୀ஻(௦௔௬)
Now,
‫∑ = ܣ‬஼ିଵ
௞ୀ଴
=
஼!ெ ష(಴షೖశ భ) (஼ି௞)
= ‫∑ !ܥܥ‬஼ିଵ
௞ୀ଴
஼஼!
ெ ಴శ భ
௞!
ୀ஺(௦௔௬)
∑஼ିଵ
௞ୀ଴
௞!
஼!ெ ష(಴ష ೖశభ)
ெೖ
௞!
−
௞!
஼!
ெ಴
− ‫∑ !ܥ‬஼ିଵ
௞ୀଵ
∑஼ିଶ
௝ୀ଴
ெೕ
௝!
=
(18)
ቂ∑஼௞ୀ଴
ಾೖ
಴
஼ ∑ೖసబ ೖ!
ಾ಴
಴!
ெ
஼
ଵ
(௞ିଵ)!
௞!
൥1 −
ெభ
−
=1+
and,
஼
ெభ
ெభ
−
‫∑ = ܤ‬஼ିଵ
௞ୀ଴
=
=
=
=
=
஼!
ቃ−
ಾ಴
಴!
ಾೖ
∑಴
ೖసబ ೖ!
஼!
ெ಴
൩−
−
ଵ
ଵ
஻(ெ ,஼)
ቂ∑஼௝ୀ଴
ಾೕ
ೕ!
ಾ಴
಴!
∑಴
ೕసబ
ଵ
஻(ெ ,஼)
஼
+1+
ெ
ெೕ
௝!
−
൥1 −
ெ಴
஼!
ெ ಴ష భ
− (஼ିଵ)!ቃ
ಾ಴
಴!
ಾೖ
∑಴
ೖసబ ೖ!
−
ቂ1 − ‫ ܯ(ܤ‬, ‫ )ܥ‬−
஼
ெ
஼
ெ
ಾ಴
಴!
ಾೖ
∑಴
ೖసబ ೖ!
൩
‫ ܯ(ܤ‬, ‫)ܥ‬ቃ
[Using Eqn. (13)]
(19)
஼!ெ ష(಴ష ೖశమ) (஼ି௞)(஼ି௞ାଵ)
஼!
∑஼௞ୀ଴
ଶெ ಴శ మ
஼!
∑஼௞ୀ଴
ଶெ ಴శ మ
஼!
ெೖ
௞!
ெೖ
௞!
ଶ௞!
(‫ ܥ‬− ݇)(‫ ܥ‬− ݇ + 1)
[‫ܥ‬ଶ + ‫ ܥ‬− 2‫ ݇ܥ‬+ ݇(݇ − 1)]
ቂ(‫ܥ‬ଶ + ‫∑ )ܥ‬஼௞ୀ଴
ଶெ ಴శ మ
஼!
ቂ(‫ܥ‬ଶ + ‫∑ )ܥ‬஼௞ୀ଴
ଶெ ಴శ మ
ெೕ
௝!
ಾೖ
ೖ!
ಾ಴
ଶெ మ
಴!
ಾ಴
಴!
ଶ
ಾೖ
∑಴
ೖసబ ೖ!
ଵ
ெ಴
஻(ெ ,஼)
‫ ܯ‬ଶ ∑஼௝ୀ଴
=
−
[1 − ‫ ܯ(ܤ‬, ‫ ]ܥ‬−
∑಴
ೖసబ
ଵ
ଶெ మ஻(ெ ,஼)
ଶ
−‫ܯ‬ଶ
ெ಴
஼!
ெೖ
௞!
ெೖ
௞!
ெೖ
− 2‫∑ ܯܥ‬஼௝ୀ଴
ெ ಴ష భ
− ‫ ܯ‬ଶ (஼ିଵ)!ቃ
൥‫ܥ‬ଶ + ‫ ܥ‬− 2‫ ܯܥ‬+ 2‫ܯܥ‬
− ‫ܯܥ‬
ಾ಴
಴!
ಾೖ
∑಴
ೖసబ ೖ!
ெೖ
− 2‫∑ ܥ‬஼௞ୀଵ (௞ିଵ)! + ∑஼௞ୀଶ (௞ିଶ)!ቃ
൩
ெೕ
௝!
+ 2‫ܯܥ‬
ಾ಴
಴!
ಾೖ
∑಴
ೖసబ ೖ!
ெ಴
஼!
+
+‫ܯ‬ଶ−
[‫ܥ‬ଶ + ‫ ܥ‬− 2‫ ܯܥ‬+ 2‫ ܯ(ܤ ܯܥ‬, ‫ )ܥ‬+ ‫ ܯ‬ଶ −
‫ ܯ(ܤ ܯ‬, ‫ )ܥ‬− ‫ ܯ(ܤ ܯܥ‬, ‫])ܥ‬
=
ଵ
ଶெ ெ భ
[‫ܥ‬ଶ + ‫ ܥ‬− 2‫ ܯܥ‬+ ‫ ܯܥ‬ଵ + ‫ ܯ‬ଶ − ‫ ܯ ܯ‬ଵ]
[Using Eqn. (13)]
(20)
Substituting ‫ ܣ‬from Eqn. (19) and ‫ ܤ‬from Eqn. (20) in Eqn.
(18), we obtain
ܸଶ + ‫ ܯ‬ଶ(‫ ܯ‬ଶ − 1) = ‫ ܯ‬ଶ‫ ܯ‬− ‫ ܯ‬ଵ[ܸଵ + ‫ ܯ‬ଵ(‫ ܯ‬ଵ − 1)]
‫ܥ‬ଶ + ‫ ܥ‬− 2‫ ܯܥ‬+ ‫ ܯܥ‬ଵ + ‫ ܯ‬ଶ
൬
൰൨
ெభ
஻(ெ ,஼)
ଶெ ெ భ
−‫ ܯ ܯ‬ଵ
ଵ
= ‫ ܯ‬ଶ‫ ܯ‬− [ܸଵ + ‫ ܯ‬ଵ(‫ ܯ‬ଵ − 1)] ቂ‫ ܯ‬ଵ + ‫ ܥ‬− ‫ ܯ‬+ (‫ܥ‬ଶ + ‫ ܥ‬−
× ൤1 +
஼
−
ଵ
+
ଵ
2‫ ܯܥ‬+ ‫ ܯܥ‬ଵ + ‫ ܯ‬ଶ − ‫ ܯ ܯ‬ଵ)ቃ
= ‫ ܯ‬ଶ‫ ܯ‬−
ெ ష (಴షೖశభ)
ெೖ
ெ ஻(ெ ,஼)
஼
஼஻(ெ ,஼)
‫ܯ‬
∑௟௠ ୀଵ ቀ + 1ቁ
ቂ݈+
=
஼஼!
ெ ಴శ భ
= ‫ ܯ‬ଶ‫ ܯ‬−
= ‫ ܯ‬ଶ‫ ܯ‬−
Therefore,
௏భାெ భ(ெ భିଵ)
ଶெ
ெ భିெ భమା
ெభ
ଶெ
[Using Eqn. (13)]
[‫ܥ‬ଶ − ‫ ܯ‬ଶ + ‫ ܥ‬+ ‫ ܯܥ‬ଵ + ‫ ܯ ܯ‬ଵ]
ಾಾభ
ାெ భ(ெ భିଵ)
಴శ భషಾ శಾ భ
ଶ(஼ାଵିெ మ)
‫ܥ‬ଶ − ‫ ܯ‬ଶ + ‫ ܥ‬+ ‫ ܯܥ‬ଵ
൤
൨
ଶெ
+‫ ܯ ܯ‬ଵ
[Using Eqn. (15)]
× [‫ܥ‬ଶ + ‫ ܥ‬+ ‫ ܯܥ‬− ‫ ܯܥ‬ଶ + ‫ ܯ ܯ‬ଶ]
International Journal of Research in Wireless Systems (IJRWS), Volume 2, Issue 3, October (2013)
[Using Eqn. (16)]
27
Methods for Moment Manipulation of Various Traffic
Traffic Sessions in Modern Small Cell Environment
௏మ
ெమ
ൌ ͳെ ‫ܯ‬ଶ ൅ ‫ ܯ‬െ
= 1−
= 1−
ெభ
ଶெ మ
ଶெ మమ
ቂ
ெభ
−
ெ ஻(ெ ǡ஼)
஻(ெ ǡ஼)
ଶ[ଵି஻(ெ ǡ஼)]
=1−
ଶ[ଵି஻(ெ ǡ஼)]
=1−
=1−
ଶெ ெ మ
ெభ
ଶெ ൫ଵି஻(ெ ǡ஼)൯
=1−
=1−
‫ ܥ(ܥ‬൅ ͳ − ‫ ܯ‬ଶ)
൤
൨
ଶெ మ(஼ାଵିெ మ) ൅‫( ܯ‬
(‫ ܥ‬− ‫ ܯ‬ଶ)
஻(ெ ǡ஼)
ெభ
ெభ
ቂ‫ ܥ‬െ ʹ‫ ܯ‬ଶ +
Therefore, ܸ௛ ൌ ܲ௜ଶܸ௏ ൅ ܲ௜ଶ‫ ܯ‬௏ଶ ൅ ܲ௜‫ ܯ‬௏ െ ܲ௜ଶ‫ ܯ‬௏ െ ‫ ܯ‬௛ଶ
ൌ ܲ௜ଶ‫ ܯ‬௏ ൅ ‫ ܯ‬௛ଶ ൅ ‫ ܯ‬௛ െ ܲ௜ଶ‫ ܯ‬௏ − ‫ ܯ‬௛ଶ [Using Eqn. (26a)]
ൌ ‫ ܯ‬௛ ൅ ܲ௜ଶ(ܸ௏ െ ‫ ܯ‬௏ )
ெ
[Using Eqn. (26a)]
ൌ ‫ ܯ‬௛ ൅ ܲ௜ ೓ (ܸ௏ െ ‫ ܯ‬௏ )
ெ (஼ିெ మ)
൅ ‫ܥ‬൅
ଶெ మ(ெ మିெ )
ቂ
ܸ௛ ൅ ‫ ܯ‬௛ଶ ൌ ܲ௜ଶ(ܸ௏ ൅ ‫ ܯ‬௏ଶ) + (ܲ௜ െ ܲ௜ଶ)‫ ܯ‬௏
ቃ
஼ାଵିெ మ
൅ ‫ܥ‬൅
ெ ((஼ି
ିெ మ)
஼ାଵ
ଵିெ మ
ቃ
ெೇ
[Using Eqn.
Eqn (13)]
ெ (஼ିெ మ)
஼ାଵିெ మ
ቃ
ൌ ‫ ܯ‬௛ ቂͳ ൅
[Using Eqn.
Eqn (16)]
ቂ‫ ܥ‬െ ‫ ܯ‬ଶ െ ቀ‫ ܯ‬ଶ −
ெ ((஼ିெ
ெ మ)
஼ାଵିெ
ெమ
ቁቃ
‫ ܯ(ܤ‬ǡ‫)ܥ‬
‫ ܥ( ܯ‬െ ‫ ܯ‬ଶ)
ቈ‫ ܥ‬െ ‫ ܯ‬ଶ ൅ ‫ ܯ‬ଵ − ቆ
ቆ‫ ܯ‬−
ቇ቉
2[ͳ െ ‫ ܯ(ܤ‬ǡ‫])ܥ‬
‫ ܥ‬൅ ͳെ ‫ܯ‬ଶ
[Using Eqn.
Eqn (16)]
஻(ெ ǡ஼)
ଶ[ଵି஻(ெ ǡ஼)]
஻(ெ ǡ஼)
ଶ[ଵି஻(ெ ǡ஼)]
ெ ஼ାெ ିெ
ெ ெ మିெ ஼ାெ ெ మ
ቂ‫ ܥ‬െ ‫ ܯ‬ଶ ൅ ‫ ܯ‬ଵ െ ቀ
ቂ‫ ܥ‬െ ‫ ܯ‬ଶ ൅ ‫ ܯ‬ଵ െ ቀ
ெ
஼ା
ାଵିெ మ
஼ାଵିெ మ
ቁቃ
ቁ (21)
ቁቃ
In the present case, ‫ ܯ‬ൌ ‫ܣ‬௙,‫ ܯ‬ଵ ൌ ‫ܣ‬௙‫ܤ‬൫
൫‫ܣ‬௙ǡ‫ܥ‬൯, ‫ ܯ‬ଶ ൌ ‫ ܯ‬௏
and ܸଶ ൌ ܸ௏ . Substituting the above
ve quantities in Eqn. (21),
we obtain
ܸ௏ ൌ ‫ ܯ‬௏ ቈͳ െ
‫ܤ‬൫‫ܣ‬௙ǡ‫ܥ‬൯
൫‫ ܥ‬െ ‫ ܯ‬஼భ + ‫ܣ‬௙‫ܤ‬ሺ‫ܣ‬௙ǡ‫ܥ‬൯
ʹൣͳ െ ‫ܤ‬൫‫ܣ‬௙ǡ‫ܥ‬൯൧
‫ܣ‬௙
(22)
቉
−
‫ ܥ‬െ ‫ ܯ‬஼భ + 1
E. Expression for the Hand-off Offered Traffic
affic
From [13], the first two ordinary moments (ߙ
( ଵǡߙଶ) of the
aggregated offered traffic and the ݅௧௛ (݅ൌ 1, 2, …) individual
traffic stream (ߙଵ௜ǡߙଶ௜) are related as
ߙଵ௜ ൌ ܲ௜ߙଵ
(23a)
ߙଶ௜ ൌ ܲ௜ଶߙଵ
(23b)
where ܲ௜ is the probability of a session belonging to
t stream ݅.
In the present case, there are two streams, viz. (i) the stream
containing sessions that are completing within a cell and (ii)
the stream containing sessions that are handed off to another
cell. For the second stream (ܲ௜) is expressed as
ܲ௜ =
ఓ೓
ఓ೓ ାఓ೟
ൌ ߞ
(24)
The parameter ߞ is called the mobility [1]. Using the relation
between the ordinary moments, mean and variance of a
distribution [13], we write
ߙଵ ൌ ‫ ܯ‬௏ ƒ†ߙଶ ൌ ܸ௏ + ‫ ܯ‬௏ଶ
ߙଵ௜ ൌ ‫ ܯ‬௛ ƒ†ߙଶ௜ ൌ ܸ௛ + ‫ ܯ‬௛ଶ
Using Eqn. (23), (24) and (25), we obtain
‫ ܯ‬௛ ൌ ܲ௜‫ ܯ‬௏ =
ఓ೓
ఓ೓ ାఓ೟
‫ܯ‬௏
௏ೇ
ൌ ‫ ܯ‬௛ ቂͳ ൅ ܲ௜ ቀ
(25a)
(25b)
(26a)
ெೇ
ఓ೓
െ ͳቁቃ
௏ೇ
ቀ
െ ͳቁቃ
ቃ
ఓ೓ ାఓ೟ ெ ೇ
(26b)
III. CASE OF FINITE POPULATION SIZE IN SMALL CELL WIFI /
MOBILE CLOUD ENVIRONMENT
As mentioned earlier,
rlier, the fresh traffic in a small cell
environment follows Engset distribution [10]. Thus, cell ‫ܮ‬ଵ in
the two-cell
cell model receives Engset distributed fresh requests
in our case instead of Poisson distributed fresh requests in
case of [7]. Thus, the user population in ‫ܮ‬ଵ is now finite. Let ܰ
be the number of users in ‫ܮ‬ଵ, ߣ௜ ൌ ߣ௙ᇱ be the mean arrival
rate of fresh sessions per idle user and ߤௗ be the mean
departure rate from ‫ܮ‬ଵ. We develop a joint Semi
Semi-Markov
model to obtain the variance of the offered traffic to ‫ܮ‬ଶ. The
Semi-Markov
Markov model does not have a closed form solution and
needs complex manipulation using Probability Generating
Function (PGF) and Binomial Moment Generating Function
(BMGF).
A. Carried Traffic in ࡸ૚ for the Engset type arrival
The state transition diagram for the Engset distribution is
shown in Fig. 2. Let ‫݌‬௜ be the probability of ‫ܮ‬ଵin state ݅.
Fig. 2.. State transitions of the Engset distribution.
The cut equation in Fig. 2 yields
) ௙ᇱ‫݌‬௜ିଵˆ‘”Ͳ ൑ ݅൑ ‫ܥ‬
݅ߤௗ ‫݌‬௜ = (ܰ െ ݅൅ ͳ)ߣ
Š‡”‡ˆ‘”‡ǡ݅‫݌‬௜ =
ఒᇲ
೑
ఓ೏
(ܰ െ ݅+ 1)‫݌‬௜ିଵ
(27)
The expression for the mean ((‫ ܯ‬஼ ) of the carried traffic is
written as
‫ ܯ‬஼ = ∑஼௜ୀ଴ ݅‫݌‬௜
= ∑஼௜ୀ଴
ఒᇲ
೑
ఓ೏
= ∑஼ିଵ
௜ୀ଴
=
ఒᇲ
೑
ఓ೏
ఒᇲ
೑
ఓ೏
(ܰ െ ݅൅ ͳ)‫݌‬௜ିଵ
(ܰ െ ݅)‫݌‬௜
ܰ ∑஼ିଵ
௜ୀ଴ ‫݌‬௜ −
ఒᇲ
೑
ఓ೏
∑஼ିଵ
௜ୀ଴ ݅‫݌‬௜
International Journal of Research in Wireless Systems (IJRWS), Volume 2, Issue 3, October
ber (201
(2013)
28
Methods for Moment Manipulation of Various Traffic
Traffic Sessions in Modern Small Cell Environment
ఒᇲ
೑
=
ఓ೏
ఒᇲ
೑
=
ఓ೏
ܰ (ͳ െ ‫݌‬஼ ) −
ܰ (ͳ െ ‫݌‬஼ ) −
Š‡”‡ˆ‘”‡ǡ ‫ ܯ‬஼ ൬ͳ ൅
‫ܯ‬஼ =
ఒᇲ
೑
ఓ೏
ఒᇲ
೑
ఓ೏
ఒᇲ
೑
ఓ೏
(∑஼௜ୀ଴ ݅‫݌‬௜ െ ‫݌ܥ‬஼ ) since ∑஼௜ୀ଴ ‫݌‬௜ = 1
൫‫ ܯ‬஼ െ ‫݌ܥ‬஼ ൯
൰ൌ
ఒᇲ
೑
ఓ೏
[ܰ െ (ܰ െ ‫݌))ܥ‬஼ ]
ܰ ߣ௙ᇱ
ܰെ‫ܥ‬
൤ͳ െ
‫݌‬஼ ൨
ߣ௙ᇱ ൅ ߤௗ
ܰ
(28)
The expression for the variance (ܸ஼ ) of the carried traffic is
written as
ܸ஼ = ∑஼௜ୀ଴(݅െ ‫ ܯ‬஼ )ଶ‫݌‬௜
= ∑஼௜ୀ଴ ݅ଶ‫݌‬௜ െ ʹ‫ ܯ‬஼ ∑஼௜ୀ଴ ݅‫݌‬௜ ൅ ‫ ܯ‬஼ଶ ∑஼௜ୀ଴ ‫݌‬௜
= ∑஼௜ୀ଴ ݅ଶ‫݌‬௜ െ ʹ‫ ܯ‬஼ଶ ൅ ‫ ܯ‬஼ଶ
= ∑஼௜ୀ଴ ݅ଶ‫݌‬௜ െ ‫ ܯ‬஼ଶ
Now, let ܺ ൌ ∑஼௜ୀ଴ ݅ଶ‫݌‬௜.
(29)
Š‡”‡ˆ‘”‡ǡ ܺ ൌ ∑஼௜ୀ଴ ݅ή݅‫݌‬௜
‫ܩ‬௡ (‫∑ = )ݖ‬ஶ௠ ୀ଴ ݉ ‫݌‬௡ǡ௠ ‫ݖ‬௠
ఒᇲ
೑
Therefore, ‫ܩݖ‬௡ (‫= )ݖ‬
= ∑஼௜ୀ଴ ݅ (ܰ െ ݅൅ ͳ)‫݌‬௜ିଵ
=
=
ఓ೏
ᇲ
஼ିଵ ఒ೑
∑௜ୀ଴ ݅ (݅൅ ͳ)(ܰ െ ݅)‫݌‬௜
ఓ೏
ఒᇲ
೑
஼ିଵ[ܰ݅
∑௜ୀ଴
‫݌‬௜ െ ݅ଶ‫݌‬௜ ൅ ܰ ‫݌‬௜ െ
ఓ
೏
ఒᇲ
೑
݅‫݌‬௜]
ܰ (‫ ܯ‬஼ െ ‫݌ܥ‬஼ ) − ∑஼ିଵ
( െ ‫݌‬஼ )
௜ୀ଴ ݅‫݌‬௜ ൅ ܰ (ͳ
൨
( ஼ െ ‫݌ܥ‬஼ )
−(‫ܯ‬
ఒᇲ
ఒᇲ ܰ ൅ ܰ ‫ ܯ‬஼ − ‫ ܯ‬஼ ൅ ‫ܥ‬ଶ‫݌‬஼
Š‡”‡ˆ‘”‡ǡ൬ͳ ൅ ೑ ൰ܺ ൌ ೑ ൤
൨
ఓ೏
ఓ೏ െ ܰ‫݌ܥ‬஼ ൅ ‫݌ܥ‬஼ െ ܰ ‫݌‬஼
=
=
=
ఓ೏
ఒᇲ
೑
ఓ೏
ఒᇲ
೑
ఓ೏
∑஼ିଵ
௜ୀ଴ ൤
1)‫݌‬஼ ]
Hence,
ܸ஼ =
ߣ௙ᇱ
ఒᇲ
೑ାఓ೏
[ܰ ൅ ‫ ܯ‬஼ (ܰ െ ͳ) + (‫ ܥ‬െ ܰ ) (‫ ܥ‬൅
ߣ௙ᇱ
[ܰ ൅ ‫ ܯ‬஼ (ܰ െ ͳ)
൅ ߤௗ
+ (‫ ܥ‬െ ܰ )(‫ ܥ‬൅ ͳ)‫݌‬஼ ]]‫ ܯ‬஼ଶ
From Eqn. (32a), we obtain
(32b)
‫ܩ‬௡ᇱ(‫∑ = )ݖ‬ஶ௠ ୀଵ ݉ ‫݌‬௡ǡ௠ ‫ݖ‬௠ ିଵ
= ∑ஶ௠ ୀ଴(݉ ൅ ͳ)‫݌‬௡ǡ௠ ାଵ‫ݖ‬௠
(32c)
(32d)
Further, the Binomial Moment Generating Function (BMGF)
of the joint probability distribution is defined as
‫ܨ‬௡ (‫ )ݔ‬ൌ ‫ܩ‬௡ (‫ ݔ‬൅ ͳ)) = ∑ஶ௠ ୀ଴ ߚ௡ǡ௠ ‫ݔ‬௠
[ܰ ൅ ‫ ܯ‬஼ (ܰ െ ͳ) + (‫ ܥ‬െ ܰ )(‫ ܥ‬൅ ͳ)‫݌‬
) ஼]
Š‡”‡ˆ‘”‡ǡ ܺ ൌ
(32a)
∑ஶ௠ ୀଵ ‫݌‬௡௡ǡ௠ ିଵ‫ݖ‬௠
Therefore, ‫ܩݖ‬௡ᇱ(‫∑ = )ݖ‬ஶ௠ ୀଵ ݉ ‫݌‬௡ǡ௠ ‫ݖ‬௠
= ∑ஶ௠ ୀ଴ ݉ ‫݌‬௡ǡ௠ ‫ݖ‬௠
[ܰ ൅ ‫ ܯ‬஼ (ܰ െ ͳ) ൅ ‫݌ܥ‬஼ (‫ ܥ‬െ ܰ ) ൅ ‫݌‬஼ (‫ܥ‬
( െ ܰ )]
ఒᇲ
೑
Fig. 3. Semi-Markov
Markov model for the carried sessions in ‫ܮ‬ଵ and offered sessions
to ‫ܮ‬௏ .
(30)
(31)
As mentioned before, mean of the offered traffic to ‫ܮ‬௏ is
equal to the mean of the carried traffic in ‫ܮ‬ଵ. Therefore, we
need to find out variance of the traffic in ‫ܮ‬௏ .
B. Semi-Markov Analysis to
o estimate the variance of the
traffic in ࡸࢂ
The state space Markov diagram for the carried sessions
in ‫ܮ‬ଵ and ‫ܮ‬௏ is shown in Fig. 3. Let ‫݌‬௡ǡ௠ denotes the joint
probability distribution, when there are ݊ carried sessions in
cell ‫ܮ‬ଵ and ݉ sessions are carried to ‫ܮ‬௏ . The PGF of the joint
probability distribution is defined as
‫ݍ‬
™ Š‡”‡ߚ௡ǡ௠ = ∑௡௤ୀ௠ ቀ ቁ‫݌‬
ቁ
݉ ௤
(33a)
(33b)
݇
™ Š‡”‡‫݌‬௤ = ∑ஶ௞ୀ଴(−1)௞ି௤ ൬ ൰(33c)
‫ݍ‬
From the above Markov state diagram, the B
B-D equations in
terms of joint probability distribution are written as
൫ܰ ߣ௙ᇱ ൅ ݉ ߤௗଶ൯‫݌‬଴ǡ௠ ൌ ߤௗଵ‫݌‬ଵǡ௠ ିିଵ
( ൅ 1)ߤௗଶ‫݌‬଴ǡ௠ ାଵˆ‘”݊ ൌ Ͳ(34a)
+(݉
ൣ(ܰ െ ݊)ߣ௙ᇱ ൅ ݊ߤௗଵ ൅ ݉ ߤௗଶ൧‫݌‬௡௡ǡ௠ = (݉ ൅ ͳ)ߤௗଶ‫݌‬௡ǡ௠ ାଵ
( + 1)ߤௗଵ‫݌‬௡ାଵǡ௠ ିଵ
+(ܰ െ ݊ ൅ ͳ)ߣ௙ᇱ‫݌‬௡ିଵǡ௠ + (݊
ˆ‘”Ͳ ൏ ݊ ൏ ‫(ܥ‬34b)
(݉ ߤௗଶ ൅ ‫ߤܥ‬ௗଵ)‫݌‬஼ǡ௠ = (݉ ൅ 1
1)ߤௗଶ‫݌‬஼ǡ௠ ାଵ
+(ܰ െ ‫ ܥ‬+ 1)ߣ௙ᇱ‫݌‬஼ିଵǡ௠ ˆ‘”݊ ൌ ‫(ܥ‬34c)
For ݊ ൌ Ͳ,, we transform Eqn. (34a) in terms of PGF, using
Eqn. (32), as
International Journal of Research in Wireless Systems (IJRWS), Volume 2, Issue 3, October
ber (201
(2013)
29
Methods for Moment Manipulation of Various Traffic Sessions in Modern Small Cell Environment
ܰ ߣ௙ᇱ‫ܩ‬଴(‫ )ݖ‬+ ߤௗଶ‫ܩݖ‬଴ᇱ(‫ߤ = )ݖ‬ௗଵ‫ܩݖ‬ଵ(‫ )ݖ‬+ ߤௗଶ‫ܩ‬଴ᇱ(‫)ݖ‬
⟹ ܰ ߣ௙ᇱ‫ܩ‬଴(‫ )ݖ‬+ ߤௗଶ(‫ݖ‬− 1)‫ܩ‬଴ᇱ(‫ߤ = )ݖ‬ௗଵ‫ܩݖ‬ଵ(‫)ݖ‬
⟹ ܰ ߣ௙ᇱ‫ܩ‬଴(‫ ݔ‬+ 1) + ߤௗଶ(‫ܩ)ݔ‬଴ᇱ(‫ ݔ‬+ 1)
= ߤௗଵ(‫ ݔ‬+ 1)‫ܩ‬ଵ(‫ ݔ‬+ 1)
Similarly, for 0 < ݊ < ‫ܥ‬, Eqn. (34b) leads to
ஶ
(35a)
(ܰ − ݊)ߣ௙ᇱ‫ܩ‬௡ (‫ )ݖ‬+ ߤௗଵ݊‫ܩ‬௡ (‫ )ݖ‬+ ߤௗଶ‫ܩݖ‬௡ᇱ(‫)ݖ‬
= ߤௗଶ‫ܩ‬௡ᇱ(‫ )ݖ‬+ (ܰ − ݊ + 1)ߣ௙ᇱ‫ܩ‬௡ିଵ(‫)ݖ‬
+ (݊ + 1)ߤௗଵ‫ܩݖ‬௡ାଵ(‫)ݖ‬
+(ܰ − ݊ + 1)ߣ௙ᇱ‫ܩ‬௡ିଵ(‫ ݔ‬+ 1)
Similarly, for ݊ = ‫ܥ‬, Eqn. (34c) leads to
஼
ߤௗଶ‫ ݔ‬෍ ߚ஼,௠ ݉ ‫ݔ‬௠ ିଵ + ‫ߤܥ‬ௗଵ ෍ ߚ஼,௠ ‫ݔ‬௠
௠ ୀ଴
ஶ
௠ ୀ଴
= (ܰ − ݊ + 1)ߣ௙ᇱ ෍ ߚ஼ିଵ,௠ ‫ݔ‬௠
for݊ = ‫(ܥ‬37c)
௠ ୀ଴
ߤௗଶߚ଴,ଵ + ܰ ߣ௙ᇱߚ଴,ଵ = ߤௗଵߚଵ,଴ + ߤௗଵߚଵ,ଵ
⟹ ൫ܰ ߣ௙ᇱ + ߤௗଶ൯ߚ଴,ଵ = ߤௗଵߚଵ,ଵ + ߤௗଵ‫݌‬ଵᇱ
where
(35b)
ߤௗଶ‫ܩݖ‬஼ᇱ(‫)ݖ‬
+ ‫ߤܥ‬ௗଵ‫ܩ‬஼ (‫)ݖ‬
= ߤௗଶ‫ܩݖ‬஼ᇱ(‫ )ݖ‬+ (ܰ − ‫ ܥ‬+ 1)ߣ௙ᇱ‫ܩ‬஼ିଵ(‫)ݖ‬
⟹ ߤௗଶ(‫ݖ‬− 1)‫ܩ‬஼ᇱ(‫ )ݖ‬+ ‫ߤܥ‬ௗଵ‫ܩ‬஼ (‫ ܰ( = )ݖ‬− ‫ ܥ‬+ 1)ߣ௙ᇱ‫ܩ‬஼ିଵ(‫)ݖ‬
⟹ ߤௗଶ(‫ܩ)ݔ‬஼ᇱ(‫ ݔ‬+ 1) + ‫ߤܥ‬ௗଵ‫ܩ‬஼ (‫ ݔ‬+ 1)
= (ܰ − ‫ ܥ‬+ 1)ߣ௙ᇱ‫ܩ‬஼ିଵ(‫ ݔ‬+ 1)
for 0 < ݊ < ‫(ܥ‬37b)
For ݊ = 0, equating coefficients of ‫ ݔ‬from Eqn. (37a), we get
݊)ߣ௙ᇱ
⟹ ߤௗଶ(‫ݖ‬− 1)‫ܩ‬௡ᇱ(‫ )ݖ‬+ ൣ(ܰ −
+ ݊ߤௗଵ൧‫ܩ‬௡ (‫)ݖ‬
= (݊ + 1)ߤௗଵ‫ܩݖ‬௡ାଵ(‫)ݖ‬
+ (ܰ − ݊ + 1)ߣ௙ᇱ‫ܩ‬௡ିଵ(‫)ݖ‬
ᇱ(‫ݔ‬
⟹ ߤௗଶ‫ܩݔ‬௡ + 1) + ൣ(ܰ − ݊)ߣ௙ᇱ + ݊ߤௗଵ൧‫ܩ‬௡ (‫ ݔ‬+ 1)
= (݊ + 1)ߤௗଵ(‫ ݔ‬+ 1)‫ܩ‬௡ାଵ(‫ ݔ‬+ 1)
× ∑ஶ௠ ୀ଴ ߚ௡ିଵ,௠ ‫ݔ‬௠
(35c)
Transforming the set of Eqn. (35) in terms of BMGF using Eqn.
(33), we obtain
ߚ௡,଴
= ‫݌‬௡ᇱ =
ᇲ
(38a)
೙
ഊ೑
ே
ቀ ቁ൭
൱
௡ ഋ ೏భ
ᇲ
ೖ
ഊ೑
ே
∑಴
ೖసబቀ௞ ቁ൭ഋ ൱
೏భ
ᇣᇧᇧᇧᇧᇧᇤ
ᇧᇧᇧᇧᇧᇥ
౏౪౗౪౛ౌ౨౥ౘ౗ౘ౟ౢ౟౪౯౥౜౪౞౛
ు౤ౝ౩౛౪ౚ౟౩౪౨౟ౘ౫౪౟౥౤[భబ]
(38b)
For 0 < ݊ < ‫ܥ‬, equating coefficients of ‫ ݔ‬from Eqn. (37b), we
get
ߤௗଶߚ௡,ଵ + ൣ(ܰ − ݊)ߣ௙ᇱ + ݊ߤௗଵ൧ߚ௡,ଵ
= (݊ + 1)ߤௗଵ൫ߚ௡ାଵ,଴ + ߚ௡ାଵ,ଵ൯+ (ܰ − ݊ + 1)ߣ௙ᇱߚ௡ିଵ,ଵ
ൣ(ܰ − ݊)ߣ௙ᇱ + ߤௗଶ + ݊ߤௗଵ൧ߚ௡,ଵ
ᇱ
= (݊ + 1)ߤௗଵ൫ߚ௡ାଵ,ଵ + ‫݌‬௡ାଵ
൯+ (ܰ − ݊ + 1)ߣ௙ᇱߚ௡ିଵ,ଵ (38c)
For ݊ = ‫ܥ‬, equating coefficients of ‫ ݔ‬from Eqn. (37c), we get
ߤௗଶߚ஼,ଵ + ‫ߤܥ‬ௗଵߚ஼,ଵ = (ܰ − ‫ ܥ‬+ 1)ߣ௙ᇱߚ஼ିଵ,ଵ
⟹ (ߤௗଶ + ‫ߤܥ‬ௗଵ)ߚ஼,ଵ = (ܰ − ‫ ܥ‬+ 1)ߣ௙ᇱߚ஼ିଵ,ଵ
(38d)
ߤௗଶ2ߚ଴,ଶ + ܰ ߣ௙ᇱߚ଴,ଶ = ߤௗଵߚଵ,ଵ + ߤௗଵߚଵ,ଶ
(39a)
ܰ ߣ௙ᇱ‫ܨ‬଴(‫ )ݔ‬+ ߤௗଶ‫ܨݔ‬଴ᇱ(‫ߤ = )ݔ‬ௗଵ(‫ ݔ‬+ 1)‫ܨ‬ଵ(‫ )ݔ‬for݊ = 0 (36a)
ߤௗଶ‫ܨ‬௡ᇱ(‫)ݔ‬+ൣ(ܰ − ݊)ߣ௙ᇱ + ݊ߤௗଵ൧‫ܨ‬௡ (‫ ݊( = )ݔ‬+ 1)ߤௗଵ(‫ ݔ‬+ 1)
For ݊ = 0, equating coefficients of ‫ݔ‬ଶ from Eqn. (37a), we
obtain
ߤௗଶ‫ܨݔ‬஼ᇱ(‫ )ݔ‬+ ‫ߤܥ‬ௗଵ‫ܨ‬஼ (‫ ܰ( = )ݔ‬− ݊ + 1)ߣ௙ᇱ‫ܨ‬஼ିଵ(‫)ݔ‬
For 0 < ݊ < ‫ܥ‬, equating coefficients of ‫ݔ‬ଶ from Eqn. (37b),
we obtain
× ‫ܨ‬௡ାଵ(‫ )ݔ‬+ (ܰ − ݊ + 1)ߣ௙ᇱ‫ܨ‬௡ିଵ(‫ )ݔ‬for 0 < ݊ < ‫(ܥ‬36b)
for݊ = ‫(ܥ‬36c)
Expressing the set of Eqn. (36) in terms of ߚ௡,௠ , we obtain
ஶ
ஶ
௠ ୀ଴
௠ ୀ଴
ߤௗଶ‫ ݔ‬෍ ߚ଴,௠ ݉ ‫ݔ‬௠ ିଵ + ܰ ߣ௙ᇱ ෍ ߚ଴,௠ ‫ݔ‬௠
ஶ
ஶ
= ߤௗଵ(‫ ݔ‬+ 1) ෍ ߚଵ,௠ ‫ݔ‬௠
௠ ୀ଴
for݊ = 0 (37a)
ஶ
ߤௗଶ‫ ݔ‬෍ ߚ௡,௠ ݉ ‫ݔ‬௠ ିଵ + ൣ(ܰ − ݊)ߣ௙ᇱ + ݊ߤௗଵ൧෍ ߚ௡,௠ ‫ݔ‬௠
௠ ୀ଴
ஶ
௠ ୀ଴
= (݊ + 1)ߤௗଵ(‫ ݔ‬+ 1) ෍ ߚ௡ାଵ,௠ ‫ݔ‬௠ + (ܰ − ݊ + 1)ߣ௙ᇱ
௠ ୀ଴
ߤௗଶ2ߚ௡,ଶ + ൣ(ܰ − ݊)ߣ௙ᇱ + ݊ߤௗଵ൧ߚ௡,ଶ = (݊ + 1)ߤௗଵ
× ൫ߚ௡ାଵ,ଵ + ߚ௡ାଵ,ଶ൯+ (ܰ − ݊ + 1)ߣ௙ᇱߚ௡ିଵ,ଶ
(39b)
For ݊ = ‫ܥ‬, equating coefficients of ‫ݔ‬ଶ from Eqn. (37c), we
obtain
ߤௗଶ2ߚ஼,ଶ + ‫ߤܥ‬ௗଵߚ஼,ଶ = (ܰ − ‫ ܥ‬+ 1)ߣ௙ᇱߚ஼ିଵ,ଶ
Adding Eqn. (39a), (39b) and (39c), we obtain
(39c)
ߤௗଶ൫2ߚ଴,ଶ + 2ߚଵ,ଶ + ⋯ + 2ߚ஼,ଶ൯
+ߤௗଵ൫ߚଵ,ଶ + 2ߚଶ,ଶ + ⋯ + ‫ߚܥ‬஼,ଶ൯
+ ߣ௙ᇱൣܰ ߚ଴,ଶ + (ܰ − 1)ߚଵ,ଶ + ⋯
+ (ܰ − ‫ ܥ‬+ 1)ߚ஼ିଵ,ଶ]
International Journal of Research in Wireless Systems (IJRWS), Volume 2, Issue 3, October (2013)
30
Methods for Moment Manipulation of Various Traffic Sessions in Modern Small Cell Environment
= ߤௗଵ൫ߚଵ,ଵ + 2ߚଶ,ଵ + ⋯ + ‫ߚܥ‬஼,ଵ൯
+ߤௗଵ൫ߚଵ,ଶ + 2ߚଶ,ଶ + ⋯ + 2ߚ஼,ଶ൯
+ߣ௙ᇱൣܰ ߚ଴,ଶ + (ܰ − 1)ߚଵ,ଶ + ⋯ + (ܰ − ‫ ܥ‬+ 1)ߚ஼ିଵ,ଶ൧
஼
஼
௡ୀ଴
௡ୀ଴
⟹ ߤௗଶ2 ෍ ߚ௡,ଶ = ߤௗଵ ෍ ݊ߚ௡,ଵ
஼
⟹ 2 ෍ ߚ௡,ଶ =
௡ୀ଴
஼
ߤௗଵ
෍ ݊ߚ௡,ଵ
ߤௗଶ
௡ୀ଴
(40)
nd
Where ∑େ୬ୀ଴ β୬,ଶ is the complete 2 Binomial moments of
the offered traffic to ‫ܮ‬௏ . The terms β୬,ଵ are the partial
Binomial moments of the offered traffic to ‫ܮ‬௏ .
From [13], using Eqn. (32), we express the variance of the
offered traffic to L୚ as
ܸ௏ = ‫ܩ‬௡ᇱᇱ(1) + ‫ܩ‬௡ᇱ(1) − [‫ܩ‬௡ᇱ(1)]ଶ
= ‫ܩ‬௡ᇱᇱ(1) − ‫ ܯ‬௏ଶ + ‫ ܯ‬௏
(41)
Using Eqn. (32) and (33), we obtain
஼
‫ܩ‬௡ (‫ = )ݖ‬෍
஼
=෍
ஶ
ஶ
෍ ߚ௡,௠ (‫ݖ‬− 1)௠
௡ୀ଴ ௠ ୀ଴
௠
௡ୀ଴ ௠ ୀ଴ ௞ୀ଴
(−1)௠
+ ݉ ‫(ݖ‬−1)௠ ିଵ
=൥ ݉ ଶ
൩
+ ቀ ቁ‫( ݖ‬−1)௠ ିଶ + ⋯
2
஼
‫ܩ‬௡ᇱ(‫ = )ݖ‬෍
ஶ
0 + ݉ (−1)௠ ିଵ
෍ ߚ௡,௠ ൤
൨
+݉ (݉ − 1)‫(ݖ‬−1)௠ ିଶ + ⋯
௡ୀ଴ ௠ ୀ଴
஼
ஶ
‫ܩ‬௡ᇱᇱ(‫ = )ݖ‬෍
௡ୀ଴ ௠ ୀ଴
Now
= ݉ (݉ − 1)(݉ − 2)(݉ − 3) + ⋯ ]
For݉ = 0 … 2
஼
‫ܩ‬௡ᇱᇱ(1)
= ෍ 2ߚ௡,ଶ
‫ܩ‬௡ᇱᇱ(1)
= ෍ ൣ2ߚ௡,ଶ + [(−3 ∙ 2) + (3 ∙ 2)]ߚ௡,ଷ൧
‫ܩ‬௡ᇱᇱ(1)
2ߚ௡,ଶ + [(−3 ∙ 2 + 3 ∙ 2)]ߚ௡,ଶ
=෍ ቈ
቉
+(4 ∙ 3 − 3 ∙ 2 + 4 ∙ 3 ∙ 2)ߚ௡,ସ
௡ୀ଴
‫ = ݉ݎ݋ܨ‬0 … 3
஼
௡ୀ଴
‫ = ݉ݎ݋ܨ‬0 … 4
⋮
஼
௡ୀ଴
⋮
൨
(43)
= ෍ 2ߚ௡,ଶ
௡ୀ଴
Finally, from Eqn. (41), we obtain
஼
ܸ௏ = 2 ෍ ߚ௡,ଶ − ‫ ܯ‬௏ଶ + ‫ ܯ‬௏
=
௡ୀ଴
஼
ߤௗଵ
ߤௗଶ
෍ ݊ߚ௡,ଵ − ‫ ܯ‬௏ଶ + ‫ ܯ‬௏
௡ୀ଴
[From Eqn. (40)]
(44)
We solve the equation set (38) for ߚ௡,ଵ as follows. Consider a
‫ × ܥ‬1 matrix ܺ that contains the ‫ ܥ‬+ 1 solutions in terms of
ߚ௡,ଵ. Further, consider a ‫ ܥ × ܥ‬matrix ‫ ܣ‬and a ‫ × ܥ‬1 matrix ‫ܤ‬
such that ‫ܤ = ܺܣ‬. We obtain the elements of ‫ ܣ‬and ‫ ܤ‬from
Eqn. (38) as
‫ܣ‬௡,௠ =
−(ܰ − ݉ )ߣ௙ᇱ
for 0 ≤ ݊ ≤ ‫ܥ‬, ݉ = ݊ − 1
⎧
⎪(ܰ − ݊)ߣᇱ + ߤ + ݊ߤ
ௗଵ
ௗଶ for 0 ≤ ݊ ≤ ‫ܥ‬, ݉ = ݊
௙
(45a)
for 0 ≤ ݊ ≤ ‫ܥ‬, ݉ = ݊ + 1
−(݊ + 1)ߤௗଵ
⎨
⎪
for݊ = ‫ܥ‬, ݉ = ݊
ߤௗଵ + ‫ߤܥ‬ௗଶ
⎩
(݊ + 1)ߤௗଵ‫݌‬௡ାଵ
for 0 ≤ ݊ ≤ ‫ܥ‬
‫ܤ‬௡,௠ = ቄ
(45b)
0
otherwise
IV. SMALL CELL MODEL VALIDATION AND DISCUSSION
௠ ିଶ
0 + 0 + ݉ (݉ − 1)(−1)
෍ ߚ௡,௠ ൤
+݉ (݉ − 1)(݉ − 2)
஼
Once we compute ܺ, the variance ܸ௏ can be easily computed
from Eqn. (44). Finally, following the steps in Section (II -E),
mean and variance of the hand-off offered traffic to ‫ܮ‬ଶ can
easily be computed. Since, various types of congestions are
important QoS parameters in traffic dimensioning, we
compare various types of congestions [12] estimated using
our analysis for small cell model with that of [7] in the next
section.
݉
෍ ෍ ቀ ቁ‫ݖ‬௞ (−1)௠ ି௞
݇
Therefore,
‫ܩ‬௡ᇱᇱ(1)
(42)
We validate the proposed model by considering a special
case where it reduces to some already existing model. As
mentioned in the literature, when the population size
increases and theoretically tends to infinity, the Engset
distribution behaves as Poisson distribution. In our small cell
case, we observed that around ܰ ≥ 150, the corresponding
fresh traffic behaves as Poisson distribution. Thus, we choose
ᇱ
ܰ , ߣ௙
such that some equivalent fresh offered traffic (Poisson
distributed) is injected into the proposed model. In such a
condition, the congestions estimated using the proposed
small cell model are compared with (a) the model proposed
by Rajaratnam and Takawira (R-T Model) [7] and (b) the
model proposed by Foschini et al. (Foschini’s Model) [1].
Table I presents the estimated congestion (fresh and handoff) for the above models. In the first case (Table I-a), the
number of users in a small cell is varied from 150 to 430,
while other parameters such as user mobility, session
duration, session request rate, cell capacity (channels) are
International Journal of Research in Wireless Systems (IJRWS), Volume 2, Issue 3, October (2013)
31
Methods for Moment Manipulation of Various Traffic
Traffic Sessions in Modern Small Cell Environment
(a) Varying number of users in a small cell.
kept constant. In the second case (Table I-b)
b) the user mobility
in a small cell is varied from 0 (static) to 0:9 (highly mobile),
while other parameters such as number of users,
us
session
duration, session request rate, cell capacity (channels) are
kept constant. Table I shows that all models are in good
agreement. This validates the correctness of the small cell
model. We further observe that in Table I-a,
I both types of
congestions increase with the number of users, which is
expected. In Table I-b, we notice that the fresh congestion
decreases with mobility. This is expected because the
dropped hand-off
off requests allow greater chances
chan
of
acceptance to the fresh requests. However, hand-off
hand
congestion decreases with user mobility as well, which is
unusual. This happens because of the insensitivity of the
existing models towards population size. The existing models
treat hand-off traffic
ic as fresh traffic from the same cell of
large population size. Hence, the phenomena
nomena for fresh traffic
repeat for hand-off
off traffic. This reveals the key limitation of
the existing models making them inapplicable for small cells.
Figure 4 shows the percentage of deviation in congestion
estimates using the Small Cell Model with that of the R-T
R
Model. Since the variance of the fresh requests, initiated
from a finite population size, is smaller compared to that
initiated from a large population
opulation size, we observe that the R-T
R
Model largely over estimates fresh congestion. The hand-off
hand
congestion is also over estimated because a part of the fresh
session requests hand-offs
offs to the adjacent cell. We observe in
(b) Varying user mobility in a small cell.
Fig. 5. Percentage deviation in congestion (Small Cell Model versus R
R-T
Model).
Fig. 6(a) that the deviation decrease
decreases with the increase in
number of users. This can be attributed to the fact that with
increasing number of population the fresh request
distribution (Engset) tends more towards Poisson
distribution. Thus, the variance of the fresh offered traffic
increases and the
he difference with the case of large population
size decreases. Since the fresh request is directly affected by
the population size, the deviation is more in this case.
However, with the increase in user mobility in Fig. 7(b), the
deviation for both fresh requests and hand-off request
increase drastically. This can be explained as follows. In the
Small Cell Model, hand-off
off congestion increases with the
mobility, which is not the case for the R
R-T Model. Thus, the
estimates largely differs for the hand
hand-off congestion. In case
of fresh congestion, deviation occurs from two phenomena.
Firstly, the small population size in the Small Cell Model limits
the variance of the fresh session requests which, in turn,
International Journal of Research in Wireless Systems (IJRWS), Volume 2, Issue 3, October
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Methods for Moment Manipulation of Various Traffic Sessions in Modern Small Cell Environment
makes the fresh traffic more predictable. This reduces the
fresh congestion. Secondly, the increasing hand-off
congestion favours the fresh session requests and reduces
the fresh congestion. R-T Model is insensitive to these
phenomena.
[9] M. Rajaratnam and F. Takawira, “Nonclassical traffic
modeling and performance analysis of cellular mobile
networks with and without channel reservation,” IEEE
Trans. Veh. Technol., vol. 49, no. 3, pp. 817–834, May
2000.
ACKNOWLEDGMENT
[10] V. B. Iversen, Teletraffic Engineering Handbook, revised
ed. Building 345v, DK-2800 Kgs. Lyngby: COM Center,
Technical University of Denmark, May 2006, vol. ITU-D
SG 2/16 & ITC Draft 2001-06-20.
The authors acknowledge Prof. H. M. Gupta, Department
of Electrical Engineering, Indian Institute of Technology Delhi,
and Vivek Pal, Untamed Spectrum for their valuable
comments and suggestions.
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Paramita Bhattacharya received the B.Sc. degree in Science from University of Calcutta, India in 2002
and M.Sc. degree in Information Technology from Sikkim Manipal University, India in 2005. She
worked as a project assistant at the Department of Electrical Engineering, Indian Institute of
Technology Delhi in 2008. Currently, she is associated with M/S Untamed Spectrum Ltd., Finland.
Her research interests include mobile networks, smart grid communication, next generation small
cell and carrier WiFi networks.
International Journal of Research in Wireless Systems (IJRWS), Volume 2, Issue 3, October (2013)
33
Methods for Moment Manipulation of Various Traffic Sessions in Modern Small Cell Environment
Samya Bhattacharya received the B.E. degree in Electrical Engineering from Jadavpur University,
Kolkata, India in 2000 and M.Tech. degree in Computer Science and Engineering from Vellore
Institute of Technology (V.I.T. University), Vellore, India in 2002. From July 2002 to December 2002,
he was attached to the Advanced Computing and Microelectronics Unit (ACMU), Indian Statistical
Institute (I.S.I.), Kolkata, India as a project trainee. In 2008, he obtained the Ph.D. degree in Electrical
Engineering from Indian Institute of Technology (I.I.T.) Delhi, New Delhi, India. Currently, he is a
Research Fellow and Project Manager at the School of Electronic and Electrical Engineering,
University of Leeds, United Kingdom and also is an advisory board member of M/S Untamed
Spectrum Ltd., Finland.
Dr. Bhattacharya is a member of technical program committee for IEEE Sarnoff Symposium, IEEE
ICC 2010-14, IEEE GLOBECOM 2010-14, ICIT workshop and a number of conferences organised by the
IEEE Communication Society. His current research interests include performance and energy efficient traffic engineering,
network calculus, call admission control, mobility modeling and quality of service in next generation communication systems.
International Journal of Research in Wireless Systems (IJRWS), Volume 2, Issue 3, October (2013)
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