A DYNAMIC PROGRAMMING MODEL FOR NETWORK

SERVICE SCHEDULING

Jesuk Ko

Department of Industrial and Information Engineering, Gwangju University

592-1 Jinwol-dong, Nam-gu, Gwangju 503-703, Korea

Keywords: Video on demand (VOD), Resource allocation, Dynamic programming.

Abstract: Video on Demand (VOD) is one of the most promising services in Broadband Integrated Services Digital

Network (B-ISDN) for the next generation. VOD can be classified into two types of services: Near VOD

(NVOD) and Interactive VOD (IVOD). For either service, some video servers should be installed at some

nodes in the tree structured VOD network, so that each node with a video server stores video programs and

distributes stored programs to customers. Given a tree-structured VOD network and the total number of

programs being served in the network, the resource allocation problem in a VOD network providing a

mixture of IVOD and NVOD services is to determine where to install video servers for IVOD service and

both IVOD and NVOD services. In this study we develop an efficient dynamic programming algorithm for

solving the problem. We also implement the algorithm based on a service policy assumed in this paper.

1 INTRODUCTION

The emergence of B-ISDN (Broadband-Integrated

Service Digital Network) and the advance of several

technologies such as ATM (Asynchronous Transfer

Mode) technology, image compression

/ retrieval

technology, and multimedia storage

/ transmission

technology make it possible to provide customers

with high bandwidth interactive services such as

video on demand (VOD), home shopping, video

conferencing, etc. VOD seems to be an especially

attractive service for the next generation. These

VOD services can be classified into two types:

interactive VOD (IVOD) and near VOD (NVOD)

(Petit et al, 1998; IGI Consulting, 2000).

IVOD is a real-time service that provides a

customer with a requested program for the customer

to control it. However, IVOD requires expensive and

highly developed Video Server

(VS)s and storage

media to support the real-time service, and incurs a

large amount of program storage and transmission

costs due to point-to-point connections on demand.

Consequently, NVOD service should be utilized

from the economical VOD service point of view

(Gelman and Halfin, 1999; Sincoskie, 1997).

NVOD distributes periodically some programs on

several channels for each program so that customers

can begin to watch their requested programs from

scratch after waiting an acceptable amount of time.

Customers who do not want to wait for the NVOD

service can switch the request to IVOD service.

NVOD service is not a real-time service and does

not depend on customers’ requests, but requires

relatively cheaper VS and storage media than those

for IVOD service. Moreover, NVOD service

requires a relatively small amount of program

storage and incurs lower transmission costs

compared with IVOD service because one channel

can be allocated to several customers simultaneously.

In this paper we consider the resource allocation

problem in a VOD network providing a mixture of

IVOD and NVOD services (RAPINVOD). The

RAPINVOD problem is to determine where to

install video servers for IVOD service and, by

considering customers demands, which programs

should be stored at each video server for both IVOD

and NVOD services so as to minimize the sum of

operating costs. There might be several costs related

to the operation of the mixed IVOD and NVOD

services, but we just consider three kinds of costs for

each service: a program transmission cost, a

program storage cost, and a VS installation cost.

To the best of our knowledge, the problem

RAPINVOD has yet to be carefully analyzed by

researchers. Hodge et al. (1994, 1998) and Ishihara

et al. (1996) have proposed only a service policy for

the mixture of IVOD and NVOD services such that

some popular programs are distributed through

Ko J. (2006).

A DYNAMIC PROGRAMMING MODEL FOR NETWORK SERVICE SCHEDULING.

In Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics, pages 47-53

DOI: 10.5220/0001199000470053

 SciTePress

NVOD service since the total cost will be increased

if all the programs are distributed only through

IVOD service. In particular, Hodge et al. (1994)

analyzed technologies and costs required for IVOD

and NVOD services. Kim et al. (1996) proposed a

dynamic programming algorithm for the resource

allocation problem in a VOD network providing

only IVOD service (RAPIVOD).

In this paper we propose a service policy for

providing NVOD service and also develop a

dynamic programming algorithm for solving

RAPINVOD under this policy by extending the

dynamic programming algorithm proposed earlier by

Kim et al. (1996).

This paper is organized as follows. In Section 2,

we first describe VOD network architecture and

several assumptions and then introduce the concepts

of program vision probability and mean service

demand. We also define the rate of lost service

request for an NVOD program. Section 3 addresses

a dynamic programming algorithm for the

RAPIVOD problem. In Section 4, we propose an

extension of the dynamic programming algorithm,

given in Section 3, so as to provide a solution for the

RAPINVOD problem. Section 5 concludes the paper.

2 THE TARGET PROBLEM

In this study, we consider two kinds of directed and

tree-structured VOD networks, one providing only

IVOD service and the other providing a mixture of

IVOD and NVOD services. We assume that these

networks consist of N interconnected central offices

(COs) represented by nodes which are labeled in the

Breadth First Search (BFS) order. It is assumed that

at most one VS for IVOD service can be installed

for each CO and exactly one VS for NVOD service

can be installed at the root node of the network. We

assume that the program warehouse containing the

programs to be provided is located at the root node

of the network. The program warehouse provides

some programs which are initially stored at the

program storage of a video server (VS) in a CO on

schedule whenever customers request those

programs. We also assume that each customer is

connected to exactly one leaf node (CO) in the

network by a dedicated link so that the transmission

cost from the leaf node to the customer can be

ignored. Each CO corresponding to a non-leaf node

not only transfers IVOD programs from the CO to

the immediately linked COs (i.e., its successors), but

also copies NVOD programs distributed from a VS

for NVOD service and multi-broadcasts those to its

successors.

Let

[

]

1,iP be the set of nodes on the path from

node i to the root node 1,

i.e.,

{

}

1,,,,1],[

⋅⋅⋅=== PDiPDiiiP

, where PD

the predecessor of node n for each

Nn ,,3,2

⋅

Then it is assumed that a customer connected to a

leaf node i can receive the requested IVOD program

from a VS on the path

[

]

1,iP . Therefore, all of the

IVOD programs requested by customers connected

to the leaf node i should be stored at some VS on the

path

[

]

1,iP . We assume that the unit storage cost for

every program is identical for all COs and the link

capacity between two consecutive COs is unlimited.

Let J be the total number of programs being

provided in the network. It is assumed that all of the

programs are sorted in decreasing order of

customers’ preference and an IVOD program with

higher preference is stored at a VS closer to

customers in order to reduce the transmission cost.

Moreover it is assumed that a more highly preferred

NVOD program has a higher priority to be stored at

the root node since more customers will be served

on each channel for an NVOD program.

NVOD program j is distributed on m

channels

from the root node where we assume that m

≥ m

< j so that the rate of lost service requests for

NVOD programs can be reduced (Ishihara et al,

1996). The service provider then needs to determine

the number of channels for each NVOD program.

2.1 Program Vision Probability and

Mean Service Demand

We assume in this paper that the demand for each

program is determined by customers preference

which is sorted in a decreasing order, although it

varies with several factors such as service time,

service type (IVOD or NVOD service), and

customers location, etc. Giovanni et al. (1994)

defined the vision probability of program j as

follows:

J2,3,...,j

−

⎟

⎠

⎞

⎜

⎝

⎛

−

⎟

⎠

⎞

⎜

⎝

⎛

−

, 1

∑

P (1)

where

D is the ratio between the (j-1)-th and j-

th program vision probabilities.

Note that

PPP ≥

⋅

≥≥

and thus 1≥

D . In

this paper we also use equation (1) as the definition

of the program vision probability. It is assumed that

the same program requested by customers connected

ICINCO 2006 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

to all leaf nodes in the network has the same

program vision probability.

We now define the mean service demand at node

n to be the mean traffic volume occurring during one

unit of time of the busiest period. The mean traffic

volume is the product of three values: the number of

customers connected to the node n, the probability

that customers will request the service during the

busiest period, and the mean service time. More

precisely speaking, let

()

EVT ,=

be a directed and

tree structured VOD network and

()

[]

{}

1, | qPnVqnT ∈∈= be the complete subtree of

rooted at node n, where

{}

NV ,,2,1

⋅

⋅⋅= and

{}

NiiPDE

,,3,2 | ),( ⋅⋅⋅== are the set of nodes and

the set of links (arcs), respectively. For convenience,

we denote an arc

()

iPD

, as just arc i since there is

point-to-point correspondence between E.

Let

L be the set of successors of node n(i.e.,

{

}

nPDVqL

=∈= |

). If n is a leaf node (i.e.,

∅=

L ), then the mean service demand

R at node

n can be determined by the following value: the

mean traffic volume at node n

÷ (the unit service

time). For nodes other than leaf nodes (i.e., n such

that

∅≠

L ),

∑

∈

RR where

})({ ∅=∈

L|nTq=W

2.2 Rate of Lost Service Requests

for an NVOD Program

NVOD service distributes programs on several

channels periodically. For instance, if a program

with the service duration of two hours is distributed

on five channels, then the program can be distributed

repeatedly through NVOD service per every 0.4

hours, i.e., 24 minutes. As mentioned earlier,

customers may feel that the waiting time is too long

and cancel the request.

We define the rate of lost service requests for an

NVOD program to be the probability that a customer

who requested the NVOD program cancels the

request. An NVOD program with vision probability

is distributed on m

number of channels. Then, if

(

)

mV is the time interval between the starts of two

consecutive distributions of this NVOD program, i.e.,

the maximum amount of time that a customer should

await the program can be obtained by

()

= , j = 1, 2, 3, …, J (2)

where

(

)

∞<<

mV0

and

is the service

duration of program.

Now, let the random variable T be the time that a

customer waits for the requested NVOD program,

with

()

tf its probability density function. Then the

probability that a customer will wait for more than t

hours is

() ()

dxxftTP

∫

∞

=> (3)

Therefore,

(

)

)(

mVP

, the probability that a

customer will wait the requested NVOD program

with vision probability P

, can be calculated by

()

)(

dttTP

mVP

∫

(4)

Consequently, the rate of lost service requests

for an NVOD program with vision probability P

denoted by

(

)

)(

mVP , is

(

)

(

)

)(1)(

jfjf

mVPmVP −= (5)

For example, if T is exponentially distributed

with parameter

, i.e., if

() ( )

ttf exp

−= with

∞

t0 and 0>

, then

()

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

⋅

−−

⋅

mVP

δτ

exp1

)( (6)

Here, the parameter

is the mean queuing rate

that a customer will receive an NVOD service.

Since NVOD service is usually cheaper than

IVOD service and each customer also makes a

decision to wait or not to await the requested NVOD

program, we assume the following: (i) if a program

is distributed through NVOD service, then a

customer who requested the program wants to

receive NVOD service rather than IVOD service, (ii)

a proportion of

of the customers who request

NVOD programs and choose not to wait for it

request IVOD instead. Consequently, a proportion of

−

of such customers clear their requests and

choose neither service.

3 DYNAMIC PROGRAMMING

FOR RAPIVOD

This paper considers three kinds of costs for IVOD

service: a program transmission cost, a program

storage cost, and a VS installation cost. Then the

resource allocation problem in a VOD network

providing only IVOD service (RAPIVOD) is to

decide where we should install VSs, which and how

many programs should be stored at each VS, so that

all the demands are satisfied with the minimum total

A DYNAMIC PROGRAMMING MODEL FOR NETWORK SERVICE SCHEDULING

cost. We propose a dynamic programming algorithm

for solving RAPIVOD in this section.

Although a complete enumeration of all the

possible solutions might be used to find the optimal

solution for the RAPIVOD problem, this would be

very inefficient if no computationally infeasible

when the number of COs and programs increase,

since the size of the solution space grows

exponentially. Moreover, cost functions are non-

linear in general and thus it is necessary to find an

efficient solution technique for this kind of problem.

Let

)(kTC

be the cost of the program transmission

on arc n

()

nPD

, when k programs are stored on

T(n). The transmission cost of the remaining (J-k)

programs, which will have lower program vision

probabilities, on arc n depends upon their mean

service demands. Therefore,

)(kTC

can be

expressed as follows:

()

⎪

⎩

⎪

⎨

⎧

≠

⎟

⎠

⎞

⎜

⎝

⎛

∑

otherwise,0

1if ,,

)(

n ,PRDCg

kTC

jnnt

where

C is the unit transmission cost of an

IVOD program and

D is the distance between

node n and its predecessor

PD .

For example, if

∅≠

L for 1

≠

n and

),,(

cbag is defined by

(

)

cba

×× with 0>

then

)(kTC

is expressed by

()

jnnt

PRDC

⎟

⎠

⎞

⎜

⎝

⎛

×××

∑

+= 1

where

is the parameter of the transmission cost.

The third quantity

()

∑

represents the total

amount of mean service demands on node n which is

equal to the total traffic volume on arc n during the

busiest period of time when k kinds of programs are

assumed to be stored on T(n).

Let

),(

xkSC be the cost function of the program

storage on node n when

x kinds of programs out of

k ones are stored at node n and the remaining

−

kinds of programs are stored on T(q) for all

Lq ∈ (i.e.,

xk − kinds of programs are stored at

some nodes on the path P[u, q] for each leaf node

)(qTu ∈ and all

Lq ∈ ). Note that programs

associated with the program vision probabilities

from the

)1( +−

xk -th through the k-th are stored at

node n because of our program storage policy

assumed in this paper. Here, we assume that the unit

program storage cost is the same for all programs.

Let

⎡

⎤

x be the smallest integer larger than or equal

to x. Then

),(

xkSC can be expressed as follows:

⎪

⎩

⎪

⎨

⎧

≠

⎟

⎠

⎞

⎜

⎝

⎛

⎥

⎤

⎢

⎡

∑

+−=

otherwise,0

0if ,

),(

x ,

xkSC

xkj

where

C is the unit storage cost of an IVOD

program and h is the number of multiple

accesses for an IVOD program.

For example, if

≠

x and ),(

bag is defined by

(

)

× with 0>

, then ),(

xkSC is expressed

xkj

⎟

⎠

⎞

⎜

⎝

⎛

⎥

⎤

⎢

⎡

∑

+−= 1

where

is the parameter of the storage cost. The

quantity

⎥

⎤

⎢

⎡

represents the number of

programs with the j-th program vision probability

stored at a VS located at node n.

If at least one program is stored at node n (i.e., if

≠

x ), then a VS should be installed in node n.

Let

),(

xkIC be the cost function of the installation

of a VS on node n under the same situation given for

),(

xkSC . Then ),(

xkIC can be expressed as

follows:

⎩

⎨

⎧

≠

otherwise 0,

0 if 1,

)( where ,))(,(),(

nnvn

xyxyCgxkIC

where

C is the installation cost of a VS for

IVOD service.

For example, if the function

),(

bag is defined

, then ),(

xkIC is expressed by )(

xyC × .

With these three cost functions, we now present

an efficient dynamic programming for solving

RAPIVOD. For a given node n, we assume that k

kinds of programs are stored on T(n) for k = 0, 1,

2, ..., J. Let f(n, k) be the minimum total cost related

to storing k kinds of programs on T(n). Suppose that

we have found f(q, k) for all

Lq ∈ and k = 0, 1,

2, ..., J. Then f(n, k) can be determined by the

following recursive formula:

⎪

⎩

⎪

⎨

⎧

∅≠+

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

−++

∑

∈

≤≤

otherwise. ,)(),(),(

if ,)(),(),(),(min

),(

kTCkkICkkSC

LkTCxkqfxkICxkSC

knf

nnn

It is important to notice that all the nodes in the

network are labeled in BFS order and our dynamic

ICINCO 2006 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

programming incorporates a bottom-up approach

which solves the restricted resource allocation

problem on T(n) by the node n in the reverse of BFS

order.

We now summarize the main idea of our dynamic

programming approach. We begin with the leaf node

N. If k kinds of programs are stored on T(N)={N},

then all of those programs should be stored at node

N itself and the (J-k) number of programs with the

lower program vision probabilities than the k-th (i.e.,

programs with the program vision probability from

(k+1)-th to the J-th) should be stored at some VSs

on the path

1],[

PDP . Therefore, to find the

minimum total cost f(N,k) for all k = 0, 1, 2, ..., J,

the cost of storing k kinds of programs at node N, the

video server installation cost at node N, and the

transmission cost of the J-k number of programs on

arc N should be evaluated by considering the service

demand for each program at node N. Consequently,

f(N,k) can be obtained by the sum of those costs, i.e.,

SC(k,k)+IC(k,k)+TC

(k), for all k = 0, 1, 2, ..., J.

Now, we consider the complete subtree T(N-1) of

rooted at node N-1. If the node N-1 is a leaf node,

then T(N-1)={N-1} and thus f(N-1,k) can be obtained

by the same argument for f(N,k), which is equal to

SC(k,k)+IC(k,k)+TC

N-1

(k), for all k = 0, 1, 2, ..., J.

Otherwise (i.e., if

∅

≠

−1N

L ), T(N-1) consists of

{N-1, N} where

−

= NPD

. Suppose that k kinds

of programs are stored on T(N-1) and

1−N

x kinds of

programs are stored at node N-1. Then, to find f(N-

1,k), it is enough to evaluate the cost of storing

1−N

kinds of programs at node N-1, the video server

installation cost at node N-1, and the transmission

cost of the (J-k) number of programs on arc N-1 for

each

k x

,,1,0

⋅⋅⋅=

−

, since

1−

−

xk kinds of

programs are stored at node N and we have already

found the minimum total cost f(N,

1−

−

xk ).

Therefore, f(N-1,k) can be obtained by

{}

)(),(),(),(min

1111

kTCxkNfxkICxkSC

NNNN

−−−−

≤≤

+−++

−

We continue the above procedure by visiting nodes

in the reverse of BFS order until we arrive at the root

node 1 of

, and finally find the optimal value

f(1,J) of RAPIVOD.

To find the optimal solution

x for n=1, 2, ..., N,

we first define the following value for each n=1,

2, ..., N and k=0, 1, 2, ..., J:

⎪

⎩

⎪

⎨

⎧

∅≠

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

−++

∑

∈

≤≤

otherwise ,

if ,),(),(),(argmin

),(

LxkqfxkICxkSC

knx

nnn

Then the optimal solution can be obtained by

),(

wJnxx

−= in the BFS order for all n = 2, 3, ...,

N, where

[]

∑

∈

PDPi

xw and

),(1

Jxx =

Note that the optimal solution holds the

information about the video server location and the

kinds of programs stored in the video server. In fact,

≠

x for some n, then a video server should be

installed at node n. Moreover, programs with the

program vision probability from

)1(

+−−

xwJ

-th

through

)( wJ

−

-th should be stored at node n, since

it is assumed that the program with the lower

program vision probability is stored at the farther

node from the customer. For example, if V={1,2},

J=7,

=x , and 4

=x , then programs with 7-th,

6-th, and 5-th program vision probabilities and

programs with 4-th, 3-rd, 2-nd, and 1-st program

vision probabilities should be stored at node 1 and

node 2, respectively.

4 EXTENSION TO THE MIXED

SERVICE OF IVOD AND NVOD

In this study we also consider three kinds of costs

for for both IVOD and NVOD services: a program

transmission cost, a program storage cost, and a VS

installation cost. The storage allocation problem in a

VOD network providing mixed IVOD and NVOD

(RAPINVOD) service is to decide where we should

install VSs for IVOD service, and which and how

many programs should be stored at each VS for both

IVOD and NVOD services, so that all the demands

are satisfied with the minimum total cost. Note that a

VS for NVOD service is assumed to be installed

only at the root node of the given network.

In this section, we propose a dynamic

programming for solving RAPINVOD. For mixed

IVOD and NVOD service, we first need to

determine an efficient rule for determining the

number of channels assigned to each NVOD

program. Note that we have assumed that impatient

customers unwilling to await the NVOD service will

receive IVOD service in the ratio of

, 10

≤

For this case, we might consider several possible

rules for determining the number of channels of each

NVOD program. Instead, we propose a rule which

assigns the number such that the expected number of

customers who cancel their NVOD service requests

do not exceed L, where L is a fixed number. In fact,

let m

be the number of channels for an NVOD

program with the j-th program vision probability.

Then

is determined by the minimum number of

A DYNAMIC PROGRAMMING MODEL FOR NETWORK SERVICE SCHEDULING

channels satisfying

(

)

(

)

LPRmVP

jjf

≤××

)( , where

(

)

)(

mVP is given in equation (5) and

represents the expected service demand for an

NVOD program with the j-th program vision

probability, since a VS for NVOD service can be

installed only at node 1. Note that the mean service

demand for an IVOD program with the j-th program

vision probability is

(

)

(

)

×××

jjf

PRmVP

)( . The

procedure for finding

can be described as

follows.

Procedure Find_

Step 1. (Initialization)

0←

m ;

Step 2.

1+←

;

Step 3.

(

)

(

)

)(1)(

jfjf

mVPmVP −← ;

Step 4. If

(

)

(

)

LPRmVP

jjf

>××

)( , then

go to Step 2.

Step 5.

mm ← ; Stop

Once we obtain the number of channels,

, for

all

Jj ,,2,1 ⋅⋅⋅= , we are able to decide which and

how many programs should be served through

IVOD and through NVOD, so as to minimize the

sum of the operating costs of IVOD and NVOD

services. Before we formulate the problem

RAPINVOD, we first introduce three kinds of cost

for NVOD service the transmission cost, the storage

cost, and the video server installation cost.

Let

NTC be the transmission cost for s kinds of

NVOD programs stored at a VS on node 1 to all leaf

nodes connected to customers by using

m number

of channels. For convenience, we set

=m . Then,

NTC can be expressed as follows:

()

jns

mDnctNTC

∑∑

⎟

⎠

⎞

⎜

⎝

⎛

××=

(7)

where nct is the transmission cost for an NVOD

program per unit distance and

′

is the parameter

for the transmission cost with

0≥

′

Let

NSC be the storage cost for s kinds of

NVOD programs at a VS on node 1. Then

NSC

can be expressed as follows:

′

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

⎥

⎤

⎢

⎡

×=

∑

ncsNSC

(8)

where ncs is the unit storage cost for an NVOD

program and

′

is the parameter for the storage cost

with

0≥

′

. The quantity

⎥

⎤

⎢

⎡

in (8) represents

the number of programs with the j-th program vision

probability stored at VS on node 1 for NVOD

service.

Let

NFC be the installation cost of a VS for

NVOD service on node 1 when s kinds of programs

are served by NVOD. Then, since a VS for NVOD

service should be installed at node 1 if at least one

program is served by NVOD,

NFC can be

expressed as follows:

yncvNFC

(9)

where ncv is the installation cost of a VS for

NVOD service and

y is defined by

⎩

⎨

⎧

≠

otherwise. 0,

0 if 1,

With the above three costs related to NVOD

service, the problem RAPINVOD can be formulated

as follows:

{

}

),1(min

JfNFCNSCNTC

ssss

≤≤

(10)

where

),1( Jf

is the minimum total cost for

providing IVOD programs with the program vision

probabilities rearranged by considering the rate of

lost service requests for s kinds of NVOD programs

and can be obtained by the dynamic programming

approach.

We now summarize the main idea of our dynamic

programming procedure for solving RAPINVOD.

Initially, the number of channels for each NVOD

program is obtained by applying the procedure

‘Find_m

’. We first evaluate the IVOD operating

cost corresponding to providing only IVOD service

by applying the dynamic programming technique

and then begin by allocating programs to the VS for

NVOD service in the decreasing order of program

vision probabilities and finding the total cost, i.e.,

the sum of the NVOD and IVOD operating costs. In

finding the IVOD operating cost, all the programs

for IVOD service should be rearranged in decreasing

order of program vision probabilities because all the

customers who cancel the requested NVOD

programs receive IVOD service in the ratio of

and

thus the program vision probabilities of programs for

IVOD service that are also allocated for NVOD

service should be changed. Once we determine the

maximum number,

s , of kinds of programs which

should be stored at the VS for NVOD service, the

locations of VSs for IVOD service and the kind and

number of IVOD programs stored at each VS can be

found simultaneously when

()

is evaluated

using the dynamic programming method. We now

describe our dynamic programming procedure for

solving equation (10) as follows:

Procedure Solve_RAPINVOD

Step 1. (Initialization)

Find_m

for all Jj ,,2,1 ⋅⋅⋅

;

∞

←

TCOST ; 0

←s ; 0←s ;

ICINCO 2006 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

Step 2.

sss

NFCNSCNTCNCOST ++← ;

Step 3. If

s = 0, then go to Step 6;

Step 4.

(

)

××← )(

sss

mVPPP ;

Step 5. Sort all the programs in descending order

of their updated program vision probabilities;

Step 6. Evaluate

),1( Jf

;

Step 7. If

()

TCOSTJfNCOST

<+ ,1 , then

()

JfNCOSTTCOST

,1+← ; ss ←

;

Step 8. If Js < , then 1+← ss and go to

Step 2; Otherwise, stop;

Note that the returned values TCOST and

from the procedure ‘Solve_RAPINVOD’ are the

optimal value and the optimal number of kinds of

programs for NVOD service, respectively. Moreover,

the total number of programs stored at a VS on node

1 for NVOD service is

∑

⎥

⎤

⎢

⎡

We now use the example to demonstrate our

algorithm. All the assumptions for IVOD service are

assumed to be the same as those used in Section 2,

except that the cost functions of the program

transmission and the program storage for IVOD

service as well as NVOD service are assumed to be

linear(i.e.,

1,,, =

′′

sstt

φφφφ

) here. We assume that

the program transmission cost, the program storage

cost, and the video server installation cost for IVOD

service are more expensive than those for NVOD

service. We also assume that the unit transmission

cost of an NVOD program is more expensive than

the unit storage cost of the program. Moreover, it is

assumed that all the customers unwilling to await the

NVOD service will receive IVOD service (i.e.,

) and the mean queuing time that a customer

will await the requested NVOD program is 20

minutes (i.e.,

05.0

). It is also assumed that

L = 710 and the service duration for all programs is

identically equal to 120 minutes (i.e.,

120

for

all

j = 1, 2, ..., J).

5 CONCLUSIONS

In this paper we have first introduced a dynamic

programming algorithm for optimally providing only

IVOD service (RAPIVOD). Then we have proposed

a procedure for determining the number of channels

assigned to each NVOD program under the

assumption that the mean number of customers who

cancel their requests for NVOD service is given.

Finally we have proposed an efficient dynamic

programming algorithm for optimally providing a

mix of IVOD and NVOD services (RAPINVOD) by

extending the key idea of the earlier dynamic

programming algorithm for solving RAPIVOD.

It is expected that our algorithms can be applied to

several optimization problems which arise in

resource allocation problems in networks that

provide various types of multimedia services.

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A DYNAMIC PROGRAMMING MODEL FOR NETWORK SERVICE SCHEDULING