storage space estimation for videos using fading channels

Digital Signal Processing 19 (2009) 287–296

www.elsevier.com/locate/dsp

Storage space estimation for videos using fading channels

Satish Chand a,∗, Hari Om b

a Computer Engineering Division, Netaji Subhas Institute of Technology, Sector-3, Dwarka, New Delhi 110 078, Indiab Computer Science and Engineering Department, Indian School of Mines University, Dhanbad 826 004, India

Available online 13 February 2008

Abstract

In this paper, we propose a model for estimating the buffer storage for providing continuous delivery of the video data to users.The model incorporates the jitter delay, which is a switching delay between two consecutive video segments being transmittedthrough a logical channel. The jitter delay is assumed to be Rayleigh distributed as it characterizes the channel fading.© 2008 Elsevier Inc. All rights reserved.

Keywords: Jitter delay; Rayleigh distribution; Video channelization

1. Introduction

In multimedia applications, the videos require large storage space and high bandwidth for transmitting themthrough a channel. Therefore, it is very important to provide them to users at an affordable cost and at the sametime they should be of good quality. The quality is described in terms of quality of service (QoS), which plays an im-portant role in real life multimedia applications. The paper [1] defines what is meant by QoS in the context of packetnetworks and the basic packet QoS taxonomy using the simple analogy of the postal system. In [2], various studiesrelated to QoS in digital television broadcasting have been discussed that have led to the establishment of variousstandards and recommendations by organizations such as European Telecommunications Standards Institute (ETSI),International Telecommunication Union (ITU), International Standard Organization (ISO). The ISO defines QoS asa concept for specifying how “good” the offered services are, for example, networking services provided by networkmultimedia systems. These types of systems support four layered QoS: user QoS, application QoS, system QoS, andnetwork QoS. The user QoS is a quantitative measure of the multimedia object and it is called perceptual QoS alsobecause it describes perceptual quality of video [3,4]. The application QoS is defined in terms of media quality andrelationship among media [5,6]. The system QoS is given in terms of the requirements on communication services andoperating system services as outcome from the application QoS [7,8]. The network QoS is related to the requirementson low-level network services and is given in terms of the network load and network performance [9,10].

There are two types of QoS management: static and dynamic management [11]. The static QoS managementtechniques provide a guarantee for availability of resources whenever required, and a dynamic approach allocates anddeallocates resources during the lifetime of an application. In dynamic approach, the application is started with an

* Corresponding author.E-mail addresses: [email protected] (S. Chand), [email protected] (H. Om).

1051-2004/$ – see front matter © 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.dsp.2008.01.006

288 S. Chand, H. Om / Digital Signal Processing 19 (2009) 287–296

initial allocation of resources and in case not meeting its QoS requirements a resource manager attempts to allocatemore resources to that application until its QoS requirement is met. In [12], a video streaming system has beenproposed that adapts QoS control method for video streaming with high level of user satisfaction. The QoS affects thenumber of users desiring viewing the video services. If the quality is poor, a viewer may not be interested even if theservices are fairly cheap. In [13], a packet scheduling scheme has been discussed that ensures bandwidth as a QoSrequirement and optimizes revenue of the network service provider. The paper [14] discusses a scheme that maximizesthe profit of a network service provider by controlling the service-list and path-allocation under the constraint ofavailable network resources for multi-service-class networks, i.e., multiple levels of QoS. In [15], a Scalable VideoCoding (SVC) technique has been discussed that ensures QoS in multimedia communication. This scheme compressesa raw video into multiple bit-streams that is composed of a base bit-stream and enhancement bit-streams to supportmulti scalabilities such as SNR, temporal, and spatial. It can extract an appropriate bit-stream from original coded bit-stream without re-encoding to adapt a video to user environment. One of the important QoS parameters that affectsthe video services is jitter delay. It may be defined as the time delay occurred between two consecutive segments whileswitching of the segments takes place. In this paper, the jitter delay is taken into account for developing a model forthe buffer storage.

For utilizing bandwidth efficiently channelization is one of the important techniques. In channelization the physicalchannel is divided into logical channels allocating bandwidth using time-division multiplexing technique. Allocationof bandwidth to logical channels is called channelization. Uniform allocation of bandwidth among logical channels iscalled constant bandwidth channelization and non-uniform allocation is called variable bandwidth channelization. Fortransmitting the videos we divide them into segments so that only one segment of each video is transmitted through alogical channel. Since a logical channel transmits one segment of each video, there is switching of the segments thatcreates some delay, which is termed as jitter delay. The jitter delay is of random nature and affects the video servicesvery badly. The poor quality services may lead to drastic economic losses because the users may not be interested inviewing the poor quality services. In [16], it has been reported that non-uniform allocation of bandwidth to logicalchannels requires less buffer storage than the uniform allocation. It is well known that the Rayleigh distributioncharacterizes the channel fading. We assume jitter delay as a Rayleigh-distributed random variable. In this paper,based on variable bandwidth channelization and incorporating jitter delays as Rayleigh-distributed, we propose astochastic model for estimating the storage space for continuous delivery of the video data to users. The mean and thevariance are estimated. This study is important as the storage space directly affects the system cost and hence the videoservices. We believe that the study proposed in this paper has good applications in on-demand services such as cable-TV, news-on-demand, movie-on-demand, etc. Development of stochastic model for estimating the buffer storage isthe main contribution of this paper. The rest of the paper is organized as follows. Section 2 proposes a model for thebuffer storage incorporating the Rayleigh distributed jitter delays. Section 3 presents the results and in Section 4 weconclude the paper.

2. Modeling of storage space

We consider a video server with m videos, denoted by V1,V2, . . . , Vm, running on it. Let the bandwidth of aphysical channel that connects the users and the video server be BW units. We divide the physical channel into n

logical channels using the time-division multiplexing technique. Let the ith logical channel be utilized for a timeduration τi . We divide each of the videos into n segments, that is, equal to the number of logical channels so thatone segment of each video is transmitted through a logical channel. For simplicity, we assume that between any twoconsecutive segments transmitted through a logical channel the jitter delay is independent and identically distributedand thus can be denoted by a single random variable. It may be noted that, for variable bandwidth channelization,the utilization times of all logical channels (i.e. τi , i = 1,2, . . . , n) are not equal. So we need to calculate their actualutilization times. The actual utilization time of the ith logical channel is (τi − (m − 1)βi ), where βi denotes the jitterdelay occurring in the ith logical channel and τi its utilization time. Under the assumption of uniform buffer allocationto logical channels the bandwidth Bi of the ith logical channel is obtained by dividing the buffer space with its actualutilization time. Thus, we have

Bi =(

F

τi − (m − 1)βi

),

where F denotes the buffer space. Summing up all Bis gives bandwidth of the physical channel BW , i.e.,

S. Chand, H. Om / Digital Signal Processing 19 (2009) 287–296 289

BW =n∑

i=1

Bi =n∑

i=1

(F

τi − (m − 1)βi

).

It gives

F = BW∑ni=1

( 1τi−(m−1)βi

) .

It can be written as

F = BW

∏ni=1(τi − (m − 1)βi)∑n

j=1j �=i

∏ni=1(τi − (m − 1)βi)

.

In the above expression of F , βi is a random variable and τi and m are fixed quantities; thus (τi − (m − 1)βi) is arandom variable. The F being a function of n-random variables is a random variable. Its average value, denoted byFavg, is given by

Favg = BW

�2∫�1

· · ·�2n∫

�2n−1

∏ni=1(τi − (m − 1)βi)∑n

j=1∏n

i=1,i �=j

(τi − (m − 1)βi)f (β1 . . . βn) dβ1 · · · dβn,

n∑i=1

τi = T and τi > (m − 1)�2i , i = 1,2, . . . , n, (1)

where βi ∈ [�2i−1,�2i] and 0 < �2i−1 < �2i (i = 1,2, . . . , n), m signifies the number of videos running on thevideo server, and f (β1, β2, . . . , βn) is the joint probability density function of the random variables (jitter delays) βi ,i = 1,2, . . . , n.

The condition τi > (m − 1)�2i is required in order to have the storage space non-negative because the bufferstorage requirement can be at most zero but not negative. Since βi , i = 1,2, . . . , n, are independent random variables,the function f (β1, β2, . . . , βn) can be written as fβ1(β1), fβ2(β2), . . . , fβn(βn). We consider these random variablesas Rayleigh distributed to develop the model for estimating the storage space. The Rayleigh model basically describesthe fading effect of a communication channel.

2.1. Rayleigh distribution

The Rayleigh distribution of a random variable X is given by

fX(x) ={

xex2/2, for x � 0,

0, otherwise.(2)

Since the jitter delay βi takes values in the interval [�2i−1,�2i ], a normalizing factor, denoted by NRi, is needed and

it is given by

NRi=

�2i∫�2i−1

xe−x2/2 dx = (e−�2

2i−1 − e−�22i).

Thus, for βi the probability density function (2) is modified to

fβi(x) =

{1

NRixe−x2/2, for �2i−1 � x � �2i ,

0, otherwise.(3)

In (1), the storage space depends on n random variables, so it is also a random variable. We attempt to reformulate (1)as a function of single random variable. A function of single random variable is much easier to analyze than that ofmultiple random variables. The integrand in (1) contains terms of the form 1/(τi − (m − 1)x), where x is a randomvariable. Therefore, we define a new random variable Yi as gβi

(x), a function of βi as

Yi = gβi(x) = 1

, for �2i−1 � x � �2i . (4)
τi − (m − 1)x


Since jitter delay βi ∈ [�2i−1,�2i], the random variable Yi will assume values in the interval[1

τi − (m − 1)�2i−1,

1

τi − (m − 1)�2i

].

The characteristic function of a random variable χ(x), which is a function of a random variable X with probabilitydensity function fχ(x), is defined as the expected value of ejωx . Denoting the characteristic function of χ(x) asΦχ(ω), we have

Φχ(ω) = E{eiωx

} =∞∫

−∞fχ(x)eiωχ(x) dx.

Since Yi defined in (4) is a function of the random variable βi and the probability density function of βi defined in (3)assumes the values in the interval [�2i−1,�2i ], the characteristic function of Yi is given by

ΦYi(ω) =

�2i∫�2i−1

1

NRi

ejw

τi−(m−1)x xe−x2/2 dx. (5)

Carrying out integration in (5), we have

ΦYi(ω) = e

(jω

τi−(m−1)�2i−1−�2

2i−12 ) − e

(jω

τi−(m−1)�2i−�2

2i

2 ) − I

NRi

, (6)

where

I = jω(m − 1)

�2i∫�2i−1

e(

jωτi−(m−1)x

− x2

2 )

(τi − (m − 1)x)2dx. (7)

The integrand in (7) can be written as

e(

jωτi−(m−1)x

− x22 )

(τi − (m − 1)x)2= e

jωτi−(m−1)x e− x2

2

(τi − (m − 1)x)2=

∑∞p=0

1p!

( jωτi−(m−1)x

)p ∑∞q=0(−1)q 1

q!(

x2

2

)q

(τi − (m − 1)x)2. (8)

Using (8), we can write (7) in the following form:

I = (m − 1)

∞∑p=0

∞∑q=0

(−1)p+q (jω)p+1

p!q!2q

�2i∫�2i−1

x2q

(1

τi − (m − 1)x

)p+2

dx.

We write I as follows:

I = (m − 1)

∞∑p=0

∞∑q=0

(−1)q(jω)p+1

p!q!2q

�2i∫�2i−1

S(x)dx, (9)

where S(x) is given by

S(x) = x2q

(1

τi − (m − 1)x

)p+2

. (10)

For having non-negative storage, we have τi > (m − 1)�2i . It gives τi > (m − 1)x for all x ∈ [�2i−1,�2i]. Forτi > (m − 1)x or for (m − 1)x/τi < 1, the series

∑∞k=0((m − 1)x/τi)

k is convergent and its sum is given by1/(1 − (m − 1)x/τi).


Thus, we can write S(x) given in (10) in the following form:

S(x) = x2q

τp+2i

( ∞∑k=0

((m − 1)x

τi

)k)p+2

= x2q

τp+2i

(1 +

((m − 1)x

τi

)+

((m − 1)x

τi

)2

+(

(m − 1)x

τi

)3

+ · · ·)p+2

.

Simple manipulations give

S(x) = x2q

τp+2i

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 + (p + 2)

((m − 1)x

τi

)(1 +

((m − 1)x

τi

)+

((m − 1)x

τi

)2

+ · · ·)

+ (p + 2)(p + 1)

2!(

(m − 1)x

τi

)2(1 +

((m − 1)x

τi

)+

((m − 1)x

τi

)2

+ · · ·)2

+ (p + 2)(p + 1)p

3!(

(m − 1)x

τi

)3(1 +

((m − 1)x

τi

)+

((m − 1)x

τi

)2

+ · · ·)3

+ (p + 2)(p + 1)p(p − 1)

4!(

(m − 1)x

τi

)4(1 +

((m − 1)x

τi

)+ · · ·

)4

+ · · ·

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (11)

We may write S(x) in the following form:

S(x) = 1

τp+2i

(C0x

2q + C1x2q+1 + C2x

2q+2 + C3x2q+3 + · · ·) = 1

τp+2i

∑r�0

Crx2q+r , (12)

where Ci is the coefficient of x2q+i .The values of the coefficients Ci can be obtained by collecting the coefficients of the same powers of x from the

expressions occurring in different brackets of (11). For i = 0,1,2,3,4, Ci are given below:

C0 = 1, C1 = (p + 2)(m − 1)

τi

, C2 =(

(p + 2) + (p + 2)(p + 1)

2

)((m − 1)

τi

)2

,

C3 =(

(p + 2) + (p + 2)(p + 1)

1! + (p + 2)(p + 1)p

3!)(

(m − 1)

τi

)3

,

C4 =(

(p + 2) + 3(p + 2)(p + 1)

2! + (p + 2)(p + 1)p

2! + (p + 2)(p + 1)p(p − 1)

4!)(

m − 1

τi

)4

.

Substituting the value of S(x) from (12) in (9), we have

I = (m − 1)

∞∑p=0

∞∑q=0

(−1)q(jω)p+1

p!q!2qτp+2i

∑r�0

Cr

�2i∫�2i−1

x2q+r dx. (13)

Carrying out integration in (13), we have

I = (m − 1)

∞∑p=0

∞∑q=0

(−1)q(jω)p+1

p!q!2qτp+2i

∑r�0

Cr

(�

2q+r+12i − �

2q+r+12i−1

2q + r + 1

). (14)

Denote θ2i−1 = 1/(τi − (m − 1)�2i−1) and θ2i = 1/(τi − (m − 1)�2i ). We write (6), using (14), in the followingform:

ΦYi(ω) = 1

NRi

(A2i−1e

−jωθ2i−1 + A2ie−jωθ2i + (m − 1)

∞∑(iω)p+1Bp

), (15)

p=0


where A2i−1 = e−�22i−1/2, A2i = e−�2

2i /2, and

Bp =∞∑

q=0

(−1)q+1

p!q!2qτp+2i

(∑r�0

Cr

(�

2q+r+12i − �

2q+r+12i−1

2q + r + 1

)). (16)

There are n random variables (jitter delays) βi, i = 1,2, . . . , n. Corresponding to these jitter delays β1, β2, . . . , βn,we have n functions, denoted by Y1, Y2, . . . , Yn. These functions Yi , i = 1,2, . . . , n, are random variables and theyassume values in the intervals [θ2i−1, θ2i]. We define a new random variable Y as the sum of Y1, Y2, . . . , Yn, i.e.,

Y = Y1 + Y2 + · · · + Yn. (17)

The characteristic function of Y = Y1 + Y2 + · · · + Yn is given by the product of the characteristic functions ΦYi ,i = 1,2, . . . , n, i.e.,

ΦY (ω) =n∏

i=1

ΦYi(ω). (18)

Using (15) and (16), we write (18) as

ΦY (ω) =n∏

i=1

1

NRi

(A2i−1e

−jωθ2i−1 + A2ie−jωθ2i + (m − 1)

∞∑p=0

(iω)p+1Bp

). (19)

Denote Pi(jω) = A2i−1e−jωθ2i−1 + A2ie

−jωθ2i and Q(jω) = (m − 1)∑n

p=0(jω)p+1Bp .We can write (19) in the following form:

ΦY (ω) =n∏

i=1

1

NRi

n∏i=1

(Pi(jω) + Qi(jω)

). (20)

Now we simplify∏n

i=1(Pi(jω) + Qi(jω)). It can be written as

n∏i=1

(Pi(jω) + Qi(jω)

) =(

Pn(jω)

(n−1∏i=1

(Pi(jω) + Qi(jω)

))

+ Qn(jω)

(n−1∏i=1

(Pi(jω) + Qi(jω)

))).

For n = n − 1, n − 2, . . . ,2, we haven−1∏i=1

(Pi(jω) + Qi(jω)

) =(

Pn−1(jω)

(n−2∏i=1

(Pi(jω) + Qi(jω)

))

+ Qn−1(jω)

(n−2∏i=1

(Pi(jω) + Qi(jω)

))),

n−2∏i=1

(Pi(jω) + Qi(jω)

) =(

Pn−2(jω)

(n−3∏i=1

(Pi(jω) + Qi(jω)

))

+ Qn−2(jω)

(n−3∏i=1

(Pi(jω) + Qi(jω)

))),

...2∏

i=1

(Pi(jω) + Qi(jω)

) = (P2(jω)

(P1(jω) + Q1(jω)

) + Q2(jω)(P1(jω) + Q1(jω)

)).

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(21)

Using (21), we can write (20) in a compact form as


ΦY (ω) =(

n∏i=1

1

NRi

)(n∏

i=1

Pi(jω) + Q(jω)

n∑k=1

n∏i=1,i �=k

Pi(jω) + · · ·

+ (Q(jω)

)n−1n∑

k=1

Pi(jω) + (Q(jω)

)n

). (22)

In order to evaluate the product of the terms of the form Pi(jω), we use the following P-to-S Theorem.

2.2. P-to-S Theorem

Let gi(x) be a function of the form f2i−1(x)ea2i−1x + f2i (x)ea2i x , i = 1,2, . . . , n, where f2i−1(x) and f2i (x) arefunctions of an independent variable x, and a2i−1 and a2i are real or complex constants. The product of n such gi(x)

functions, denoted by G(x), is given by

G(x) =n∏

i=1

gix =2n∑

k=1

Fk(x)e(ak1 +ak2 +···+akn )x, (23)

where Fk(x) is given as Fk(x) = fk1(x)fk2(x) . . . fkn(x).

In (22), we have expressions of the form∏

i=1,...,n Pi(jω), i.e.,∏n

i=1(A2i−1e−jω2i−1 + A2ie

−jω2i ) for differentvalues of n and these types of expressions can be simplified using (23). Basically, the relation in (23) helps writinga product form in a sum form and, for a sum form, we can apply linear operators easily. A linear operator appliedto a sum of terms gives the same result as the sum of the terms obtained after applying the same linear operator toindividual terms.

Now we define a function hY (y) of the random variable Y , denoting it Z, as follows:

Z = hy(y) = 1

y, for (θk1 + θk2 + · · · + θkn) � y �

n∑i=1

θ2i . (24)

Since Y assumes values in [(θk1 + θk2 + · · · + θkn),∑n

i=1 θ2i], 1/Y , i.e., Z will assume values in the interval[1/

∑ni=1 θ2i ,1/(θk1 + θk2 + · · · + θkn)]. The probability density function of Z is given by the inverse characteristic

function. Thus, the probability density function of Z, which is a function of Y , can be obtained from the characteristicfunction ΦY (ω) of Y and it is given by

fZ(z) = 1

2π

∞∫−∞

ΦY (ω)eiωhY (y) dω, for1∑n

i=1 θ2i

� z � 1

θk1 + θk2 + · · · + θkn

. (25)

Substituting the value of ΦY (ω) from (22) and hY (y) = 1/y = z from (24) in (25), we have

fZ(z) = 1

2π

(n∏

i=1

1

NRi

) ∞∫−∞

(n∏

i=1

Pi(jω) + Q(jω)

n∑k=1

n∏i=1,i �=k

Pi(jω) + · · ·

+ (Q(jω)

)n−1n∑

k=1

Pi(jω) + (Q(jω)

)n

)ejω/z dω, for

1∑ni=1 θ2i

� z � 1

θk1 + θk2 + · · · + θkn

. (26)

It is the probability density function of Z, which signifies the storage space.

3. Results

The formulation in (26) specifies the buffer storage in probabilistic terms. The buffer storage is indeed a randomvariable because there are different parameters which are themselves of random nature. For example, the load in thenetwork, which depends on how many users are using the system and what are the application being used, what


the network conditions are, e.g., bandwidth, intermediate memory, etc. Thus, to model the buffer storage as randomvariable is logically justifiable. In fact, it is not just random variable, but a function of random variables. That iswhat we have tried to develop in this paper. Since the buffer storage is a random variable, it cannot be computeddeterministically; however, we can compute it in terms of its moments. The first two moments are sufficient to findits mean and variance. The average value (mean) of the storage space is defined by the first-order moment, denotedby M1, and it is given by

M1 =1/(θk1 + θk2 +···+ θkn )∫

1/∑n

i=1 θ2i

zfZ(z) dz. (27)

The second moment, denoted by M2, is given by

M2 =1/(θk1 + θk2 +···+ θkn )∫

1/∑n

i=1 θ2i

z2fZ(z) dz. (28)

The variance of Z, denoted by σ 2, is given by σ 2 = M2 − M21 .

We compute the mean of the storage space using a physical channel dividing it into 3, 5, 8, and 10 logical channels.The buffer storage is represented in terms of BW , the bandwidth of the physical channel. The numbers of videos m aretaken as 15 and the utilization time of the physical channel is taken in normalized form, i.e., T = 1. The jitter delayhas been discussed in [17,18]. The input parameters are given below.

For n = 3

τ1 = 0.26, τ2 = 0.40, τ3 = 0.34;�1 = 0.000008, �2 = 0.000024, �3 = 0.00001,

�4 = 0.000025, �5 = 0.000018, �6 = 0.00005.

For n = 5

τ1 = 0.10, τ2 = 0.18, τ3 = 0.30, τ4 = 0.22, τ5 = 0.20;�1 = 0.000010, �2 = 0.000020, �3 = 0.000014, �4 = 0.000028, �5 = 0.000018,

�6 = 0.000036, �7 = 0.000022, �8 = 0.000044, �9 = 0.000025, �10 = 0.000050.

For n = 8

τ1 = 0.13, τ2 = 0.14, τ3 = 0.20, τ4 = 0.17,

τ5 = 0.12, τ6 = 0.09, τ7 = 0.08, τ8 = 0.07;�1 = 0.000010, �2 = 0.000018, �3 = 0.000014, �4 = 0.000022,

�5 = 0.000018, �6 = 0.000030, �7 = 0.000022, �8 = 0.000035,

�9 = 0.000025, �10 = 0.000045, �11 = 0.000027, �12 = 0.000050,

�13 = 0.000029, �14 = 0.000035, �15 = 0.000031, �16 = 0.000042.

For n = 10

τ1 = 0.14, τ2 = 0.12, τ3 = 0.12, τ4 = 0.13, τ5 = 0.11,

τ6 = 0.10, τ7 = 0.09, τ8 = 0.08, τ9 = 0.07, τ10 = 0.04;�1 = 0.000001, �2 = 0.000015, �3 = 0.000014, �4 = 0.000020, �5 = 0.000016,

�6 = 0.000025, �7 = 0.000018, �8 = 0.000030, �9 = 0.000020, �10 = 0.000032,

�11 = 0.000022, �12 = 0.000034, �13 = 0.000024, �14 = 0.000035, �15 = 0.000026,

�16 = 0.000040, �17 = 0.000028, �18 = 0.000046, �19 = 0.000003, �20 = 0.000006.


Fig. 1. Buffer storage (in BW × 10−6) vs logical channels.

The average value and variance of the buffer storage Z, in terms of the bandwidth BW , for n = 3,5,8, and 10,respectively, have been obtained as 71.76E–006, 57.11E–006, 42.87E–006, 28.35E–006, and 7.72E–6, 2.21E–6,0.59E–6, 0.25E–6. Fig. 1 shows the bar graphs for the buffer storage (mean and variance) taking different numberof logical channels. It is evident from Fig. 1 that as the number of logical channels increases, the buffer storage re-quirement decreases. It can be analogized with a high capacity and low speed vehicle to a low capacity and high speedvehicle. For transferring a given commodity in a given time by a low capacity and high speed vehicle, it requires morenumber of rounds as compared to a high capacity and low speed vehicle.

4. Conclusion

In this paper, we have developed a stochastic model for estimating the buffer storage for transmitting the videodata through a physical channel by dividing it into logical channels and incorporating the jitter delays as Rayleigh-distributed. The development of this work required some published work, which regrettably is not sufficiently wellknown. Being a very new evolving area of research to the best of our knowledge there does not appear an alternativepublished work and hence comparison is not possible. This study has good applications in on-demand services likecable-TV, news-on-demand, movie-on-demand, etc.

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Satish Chand received the M.Sc. degree in mathematics from Indian Institute of Technology, Kanpur, India, the M.Tech.degree in computer science from Indian Institute of Technology, Kharagpur, India, and the Ph.D. degree in computer science fromJawaharlal Nehru University, New Delhi, India. Presently he is an Assistant Professor in Computer Engineering Division at NetajiSubhas Institute of Technology, Delhi, India. Areas of his research interest are multimedia broadcasting, networking, video-on-demand, cryptography, and image processing.

Hari Om received the M.Sc. degree in mathematics from Agra University, Agra, India, in 1997, the M.Tech. degree in computerscience and engineering from Kurukshetra University, Kurukshetra, India, in 2002, and the Ph.D. degree in computer science fromJawaharlal Nehru University, New Delhi, India, in 2007. Presently he is a Senior Lecturer in Computer Science and EngineeringDepartment, Indian School of Mines University, Dhanbad, India. Areas of his research interest are cryptography, data mining, and
video-on-demand.

storage space estimation for videos using fading channels

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