寬頻網際網路服務品質保證(II)─子計畫七:寬頻網際網路規劃與容量管理(II)

行政院國家科學委員會專題研究計畫期末成果報告

總計劃名稱：寬頻網際網路之服務品質確保 II

子計劃名稱：寬頻網際網路規劃與容量管理 II (子計劃七)

計劃編號：NSC89-2219-E-002-001

執行期限：民國 88 年 08 月 01 日至民國 89 年 7 月 31 日

計劃主持人：林永松副教授

[email protected]

執行機構：國立台灣大學資訊管理學系

Abstr act

An essential issue in designing, operating and managing modern networks is to assure end-to-end (both unicast and multicast modes) Quality-of-Service (QoS) from users’ perspective, and in the meantime to optimize certain performance objectives from the system’s perspective. In this second year final report, two near-optimal QoS-based unicast and multicast routing algorithms, respectively, developed by mathematical programming techniques are proposed for broadband Internet.

Contr ibution and Discussion

1. QoS-based Routing

The routing problem in virtual circuit networks has been a traditional research topic in computer networks and has attracted even more attention since the emergence of the Asynchronous Transfer Mode (ATM) technology. To ensure user-perceived end-to-end QoS requirement (e.g. mean packet delay, packet delay jitter and packet lost probability) and achieve good system-level performance measure (e.g.

overall network utilization or average cross-network delay) are important to the user and the system operator. However, these two perspectives/ objectives may not be entirely agreeable with each other. This then places a major challenge to network managers and therefore calls for an integrated metho-dology to consider these two pers-pectives in a joint fashion.

PROBLEM FORMULATIONAND SOLUTION APPROACHES

The virtual circuit network is modeled as graph where the processors are depicted as nodes and the communication channels are depicted as arcs. We show the definition of the following notation.

V ={1,2,… ,N}, the set of nodes in the graph

L the set of communication links in the communication network

W the set of Origin-Destination (O-D) pairs in the network

w

γ

(packets/sec): the arrival rate of new traffic for each O-D pair w

∈

W, which is modeled as a Poisson process for illustration purpose

Cl (packets/sec), the capacity of each link

l

∈

L

Pw a given set of simple directed paths

O-D pair w

xp a routing decision variable which is 1 when path p

∈

Pwis used to transmit the

packets for O-D pair w and 0 otherwise

pl

δ the indicator function which is 1 if link

l is on path p and 0 otherwise

gl the aggregate flow over link l, which is

∑ ∑

∈Pw ∈ p wW pl w p

x

γ

δ

Dw

the maximum allowable end-to-end

delay for O-D pair

w

∑

∑

∈ ∈

−

=

L l l l l W w w IP

g

C

g

Z

γ

1

min

"

(IP”)

subject to:

∑ ∑

∈ ∈

≤

−

L l w l l pl p P p

D

g

C

x

w

δ

W

w

∈ ∀

(1.1)

g

l

=

∑ ∑

∈Pw ∈ p wW pl w p

x

γ

δ

≤

C

l ∀

l

∈

L

(1.2)

∑

∈

=

w P p p

x

1

∀

w

∈

W

(1.3)

x

p

= 0 or 1

∀

p

∈

P

w

,

w

∈

W

.

(1.4)

Constraint (1.1) requires that the end-to-end packet delay should be no larger than

Dw for each O-D pair. Constraint (1.2)

requires that the aggregate flow on each link should not exceed the link capacity. Constraints (1.3) and (1.4) require that the all the traffic for each O-D pair should be transmitted over exactly one path. The above formulation is a nonlinear multicommodity flow problem, since each O-D pair transmits different type of traffic over the network. And it is easy to show that (IP”) is a nonconvex programming problem by verifying the Hessian of

∑ ∑

∈L ∈

−

l l l pl p P p

C

g

x

w

δ

with respect to xp.

For the purpose of applying Lagrangean relaxation method, we transform the above problem formulation (IP”) into an equivalent formulation (IP). In (IP), two auxiliary variables are introduced:

y

_wl is defined as

∑

∈Pw

p

pl p

x

δ

and fl denotes the

estimate of the aggregate flow.

∑

∑

∈ ∈

−

=

L l l l l W w w IP

f

C

f

Z

γ

1

min

(IP) subject to:

∑

∈

≤

−

L l w l l wl

_D

f

C

y

W

w

∈ ∀ (2.1)

∑

∈

=

w P p p

x

1

∀

w

∈

W

(2.2) xp= 0 or 1

∀

p

∈

P

w

,

w

∈

W

(2.3)

∑

∈Pw p pl p

x

δ

≤

y

_wl

∀

w

∈

W

,

l

∈

L

(2.4) wl

y

=0 or 1

∀

w

∈

W

,

l

∈

L

(2.5) l l

f

g

≤

∀

l

∈

L

(2.6) l l

C

f

≤

≤

0

∀

l

∈

L

. (2.7)

Redundant constraints associated with these auxiliary variables from (2.4) to (2.7) are added. Note that Constraints (2.4) and (2.6) should be equalities, and it is clear that the equality should hold at the optimal point. By introducing these auxiliary variables, the Lagrangean relaxation problem can be decomposed into independent and easily solvable subproblems.

The algorithm development is based upon Lagrangean relaxation. We dualize Constraints (2.1), (2.4) and (2.6) to obtain the following Lagrangean relaxation problem (LR).

∑

∑

∑

∈ ∈ ∈ + − = W w w L l l l l W w w D t f C f v u t Z γ 1 min[ ) , , ( ) ( w L l l l wl D f C y − −

∑

∈ +

∑∑

∑

−

∈ ∈ ∈ p Pw pl p W w l L wl

x

v

(

δ

wl

y

) +

∑

∈

−

L l l l l

g

f

u

(

)]

(LR) subject to:

∑

∈

=

w P p p

x

1

∀

w

∈

W

(3.1) xp= 0 or 1

∀

p

∈

P

w

,

w

∈

W

(3.2) wl

y

=0 or 1

∀

w

∈

W

,

l

∈

L

(3.3) l l

C

f

≤

≤

0

∀

l

∈

L

. (3.4)

We can decompose (LR) into two independent subproblems. Subproblem 1: for xp min

∑ ∑

∑

∈ ∈ ∈

+

W w l L pl p w l wl P p

x

u

v

w

δ

γ )

(

(SUB1) subject to (3.1) and (3.2).

Subproblem 2: for ywl and fl

min

−

−

+

−

∑

∑ ∑

∈ ∈ ∈ l l W w wl w L l l l l W w w

C

f

y

t

f

C

f

γ

1

(

[

∑

∈ − W w wl wly v

∑

∈ − W w w w l l f D t u ) ] (SUB2) subject to (3.3) and (3.4).

(SUB1) can be further decomposed into

W

independent shortest path problem with nonnegative arc weights. It can be easily solved by the Dijkstra’s algorithm. The

∑

∈

−

W w w w

t

D

tem in the objective function of (SUB2) can be dropped first and added back to the objective value since it will not affect

the optimal solution to (SUB2). Then

(SUB2) can be decomposed into

L

independent subproblems. For each link

L

l

∈ min l l W w wl w l l l W w w

C

f

y

t

f

C

f

−

+

−

∑

∑

∈ ∈

*

1

[

γ

−

u

_l

f

_l

−

∑

∈W w wl wly v ] (SUB2.1) subject to: wl

y

=0 or 1 ∀

w

∈

W

and

0

≤

f

_l

≤

C

_l.

Problem (SUB2.1) is a complicated problem due to the coupling of ywl and fl. On

the other hand, the

l l l W w w

C

f

f

−

∑

∈

*

1

γ

term

in the objective function of (SUB2.1) is a nonnegative and monotonically increasing function with respect to fl, and it will not

affect the optimal value of the following terms in the (SUB2.1). Therefore, the algorithm developed in [4] can be used to solve (SUB2.1). Hence, the algorithm to solve (SUB2.1) is as follows:

Step 1. Solve (

−

=

0

−

wl l l w

_v

f

C

t

) for each

O-D pair w, call them the break points of fl.

Step 2. Sorting these break points and denoted as f 1

l, f2l, .. , f nl

Step 3. At each interval,

f

_li

≤

f

l

≤

f

li+1,

ywl(fl) is 1 if

−

≤

0

−

wl l l w

_v

f

C

t

and is 0 otherwise.

let al be wl( l) W w wy f t

∑

∈ and bl be ) ( _l wl W w wly f v

∑

∈

, then the local

minimal is either at the boundary

point,

f

_lior

f

_li+1, or at point * l

f

=

C

_l -l l

u

a

.

Step 5. The global minimum point can be found by comparing these local minimum points.

According to the algorithms proposed above, we could successfully solve the Lagrangean relaxation problem optimally.

2. Multicast Routing

The rapid progress in World Wide Web and high bandwidth network technology have given rise to the new multicast traffic applications, e.g., video conferencing and electronic newspaper services. Here, we try to find a single minimum cost spanning tree to carry all the traffic for the multiple multicast groups rooted at the same source from the lower bound and upper bound approaches at the same time. This approach has good scalability nature when the number of member groups is becoming larger as compared to separate spanning tree for each multicast group.

PROBLEM FORMULATIONAND SOLUTION APPROACHES

The definition of the notation for revised multicast routing problem is shown below.

al cost associated with link l

T the set of all spanning trees rooted at

the source node

rg traffic requirement of multicast group g

Pgd the set of paths that destination d of

multicast group g may use

G the set of all multicast groups rooted at the common source node

hg the minimum number of hops to the

farthest destination node in multicast group g

Dg the set of destinations of multicast

group g

δpl the indicator function which is 1 if link

l is on path p and 0 otherwise

σtl the indicator function which is 1 if link

l is on tree t and 0 otherwise

And the decision variables for the revised multicast routing problem are denoted as follows.

ygl 1 if link l is on the subtree adopted by

multicast group g and 0 otherwise

xgpd 1 if path p is selected for group g

destined for destination d and 0 otherwise

zt 1 if spanning tree t is selected to be

shared by all the multicast groups and 0 otherwise 2 IP Z = min

∑∑

∈L ∈ l l gl G g g

y

a

r

(IP2) subject to: 1 or 0 = gl y

∀

g

∈

G

,

l

∈

L

(4.1)

}

,

max{

∑

∈

≥

L l g g gl

h

D

y

∀

g

∈

G

(4.2) 1 or 0 = t z ∀

t

∈

T

(4.3)

∑

∈

=

T t t

z

1

(4.4)

∑

∈

≤

gd P p gl pl gdp

y

x

δ

∀

g

∈

G

,

d

∈

D

_g

,

l

∈

L

(4.5)

∑

∈

≤

T t t tl gl

z

y

σ

∀

g

∈

G

,

l

∈

L

(4.6)

∑

∈

=

gd P p gdp

x

1

∀

g

∈

G

,

d

∈

D

_g (4.7) 1 or 0 = gdp x ∀

g

∈

G

,

d

∈

D

_g,

p

∈

P

_gd.(4.8)

The objective function in IP2 is to minimize the routing cost for all the multicast groups. Constraints (4.1) and (4.2) require that the number of links on the multicast subtree adopted by the multicast group g be at least the maximum of hg and

the cardinality of Dg. The hg and the

cardinality of Dg are the legitimate lower

bounds of the number of links on the multicast subtree adopted by the multicast group g. Apparently, Constraint (4.2) is a redundant constraint. From the computational experiments, the error gaps between the upper bound and the lower bound can be tighter after introducing Constraint (4.2).

Constraints (4.3) and (4.4) require that exactly one single shared tree be adopted by all multicast groups. Constraint (4.5) requires that if one path is selected for group

g destined for destination d, it must also be on the subtree adopted by multicast group g. Constraint (4.6) requires that the subtree adopted by any multicast group must be a subset of the shared spanning tree. This spanning tree is selected to be shared by all multicast groups. Constraints (4.7) and (4.8) require that exactly only one path be selected for any group g destined for its destinationd.

In (IP2), Constraints (3.5) and (3.6) are relaxed, which leads to the following formulation (LR2). ) , (u v ZD = min

∑∑

∈L ∈ l l gl G g g

y

a

r

+

∑ ∑∑ ∑

∈G ∈ ∈ ∈ g d D lL pP pl gdp gdl g gd x u ( δ −y_gl) + _gl G g l L gl

y

v

∑∑

∈ ∈

(

∑

∈

−

T t t tl

z )

σ

(LR2) subject to: 1 or 0 = gl y

∀

g

∈

G

,

l

∈

L

(4.1)

}

,

max{

∑

∈

≥

L l g g gl

h

D

y

∀

g

∈

G

(4.2) 1 or 0 = t z ∀

t

∈

T

(4.3)

∑

∈

=

T t t

z

1

(4.4)

∑

∈

=

gd P p gdp

x

1

∀

g

∈

G

,

d

∈

D

_g (4.5) 1 or 0 = gdp x ∀

g

∈

G

,

d

∈

D

_g,

p

∈

P

_gd. (4.6) We can decompose (LR2) into three independent subproblems. Subproblem 3: for zt min

∑∑

∑

∈ ∈ ∈

−

T t t tl gl G g l L

z

v

σ

(SUB3) subject to (4.3) and (4.4). Subproblem 4: for ygl min

∑∑

∈L ∈ l l gl G g g

y

a

r

−

∑ ∑ ∑

∈G ∈ ∈ g d D gl L l gdl g

y

u

+ _gl G g l L gl

y

v

∑∑

∈ ∈ (SUB4) subject to (4.1) and (4.2). Subproblem 5: for xgdp min

∑ ∑ ∑

∑

∈ ∈ ∈ g ∈ p Pgd pl gdp gdl G g d D l L

x

u

δ

(SUB5) subject to (4.5) and (4.6).

(SUB3) can be easily solved by the minimum weight arborescences algorithm which can be found in [1, 2, 3]. (SUB4) can be decomposed into |G| independent subproblems. For each multicast group g,

min

∑

∑

∈ ∈

−

+

L l d D gl gdl gl l g g

y

u

v

a

r

)

(

(SUB4-1) subject to: 1 or 0 = gl y

∀

l

∈

L

(5.1)

}

,

max{

∑

∈

≥

L l g g gl

h

D

y

. (5.2)

The algorithm to solve (SUB4-1) is stated as follows:

Step 1. Compute max{hg,|Dg|} for multicast

group g.

Step 2. Compute the number of negative

coefficient

∑

∈ − + g D d gdl gl l ga v u r for

all links on multicast group g. Step 3. If the number of negative coefficient

is greater than max{hg,|Dg|} for

multicast group g, then assign the corresponding negative coefficient of ygl to 1 and 0 otherwise.

Step 4. If the number of negative coefficient is no greater than max{hg,|Dg|} for

multicast group g, then assign the corresponding negative coefficient of ygl to 1. Then, assign [max{hg,|Dg|}

−the number of negative coefficient of ygl] numbers of smallest positive

coefficient of ygl to 1 and 0

otherwise.

(SUB5) can be further decomposed into |G||Dg| independent shortest path problem

with nonnegative arc weights. It can be easily solved by the Dijkstra’s algorithm.

According to the algorithms proposed above, the Steiner tree problem no longer exists in this Lagrangean relaxation problem. And we could successfully solve the Lagrangean relaxation problem optimally.

By applying the optimization-based solution approaches that we propose, we successfully develop two effective and efficient algorithms to solve the QoS-based unicast and multicast routing problems. From an observation of the computational results, the proposed algorithms calculate solutions which are within a few percent of an optimal solution in minutes of CPU time for test networks of tens of nodes.

References:

[1] P. A. Humblet, “ A Distributed Algorithm for Minimum Weight Directed Spanning Trees,” IEEE Trans. on Comm., Vol. COM-31, No. 6, pp. 756-762, June 1983.

[2] R. E. Tarjan, “Finding Optimum Branchings,” Networks, Vol. 7, pp. 25-35, 1977.

[3] Y. J. Chu and T. H. Liu, “On the Shortest Arborescence of a Directed Graph,”Sci. Sinica, Vol. 14, pp. 1396-1400, 1965. [4] K. T. Cheng and F. Y. S. Lin, “Minimax

End-to-End Delay Routing and Capacity Assignment for Virtual Circuit Networks,” Proc. IEEE Globecom, pp. 2134-2138, 1995.