於預算控制下計算M/G/K/K 和 GI/M/K/K 的滿載機率

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於預算控制下計算 M/G/K/K 和 GI/M/K/K 的滿載機率(第

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Blocking Probability of M/G/K/K and GI/M/K/K under Budget

Constraints

 : NSC 98-2221-E-004-001

 : 98  08  01  100  07  31 

Abstract. We address the issue of bandwidth

al-location on end-to-end communication networks with

multi-class traffic, where bandwidth is determined

optimally under the budget and network constraints.

We derive the blocking probabilities with respect to

bandwidth, traffic demand and the available

num-ber of end-to-end paths based on Erlang loss formula

for all service classes. Depending upon the

block-ing probability, the project presents different

perfor-mance metrics, such as budget ratio, utilization level

and bandwidth elasticity of blocking. Monotonicity

and convexity of blocking probabilities with allocated

bandwidth, traffic demand and the number of

end-to-end paths are also discussed.

1

Introduction

For a communication network providing performance

guarantees, it has to reserve resources and exercise

call admission control [43]. Network users are

main-ly interested in obtaining good quality

connection-s whenever they place requeconnection-stconnection-s. It iconnection-s the network

providers’ mission to have an end-to-end path with

suitable bandwidth. Clearly, it is too costly for the

network providers to have a 100% guaranteed

avail-ability for all connections under the budget constraint

at any time. This is also not necessary since demand

for connections or bandwidth capacity varies over

time. Traffic flow fluctuates with time, and

connec-tions do not last forever but occur at random times

and vanish in the network once the corresponding

dig-ital document has been transferred completely. This

results in a random dynamic set of active

connection-s. Moreover, the bandwidth assigned to each

connec-tion would determine how long that connecconnec-tion will

stay active and thus impacts the evolution of the set

of active connections. The network chooses an

opti-mal sharing scheme for the different users under the

total budget to fulfill connection requirements. In

ad-dition, the risk (probability) of rejecting connection

requests due to lack of resources is supposedly kept

below a negotiated level.

In this work, we aim to analyze the relationship

among blocking probability, bandwidth, traffic

de-mand and the available number of end-to-end paths

on communication networks with service from ISPs,

where requests for connections represent

customer-s arriving at the customer-sycustomer-stem. Acustomer-s customer-soon acustomer-s requecustomer-stcustomer-s are

accepted by the system, the service begins. The

in-stalled bandwidth allocation is used to maintain a

guaranteed connection availability where the

block-ing probability is kept below certain negotiated

lev-els. Our intention is to analyze the sensitivity of

the blocking probability with respect to these

sys-tem parameters, where the parameters for the syssys-tem

change one at a time.

We derive the relationship between the blocking

probability and allocated bandwidth under the

bud-get constraint, which has received relatively little

at-tention in the literature. The blocking probability

of connections for each QoS class is formulated as a

function of allocated bandwidth, traffic demand and

the available number of end-to-end paths. Monotone

and convex properties of the blocking probability are

shown in both theoretical construction and

numeri-cal examples. The results of this work can be helpful

in the operational processes involved in the efficient

set-up and usage of a core network under the

bud-get constraint, e.g., network design and provisioning

purposes.

The closed-form expression of the blocking

proba-bility in terms of bandwidth can be used to

investi-gate the optimal buffer size in capacitated

commu-nication systems so that the blocking probability is

kept below a specific threshold [35]. One application

of the relationship between blocking probability and

bandwidth allocation may be referred to designing

network pricing mechanisms for sharing bandwidth in

terms of blocking/congestion costs, whose examples

were given by Yacoubi et al. [43] and Anderson et al.

[3], etc. Another application of this work is used to

consider the admission control in end-to-end

network-s under different bandwidth network-sharing policienetwork-s including

throughput maximization, max-min fairness,

propor-tional fairness and balanced fairness, etc. Interested

readers may refer to Egorova et al. [10], Bonald et

al. [7], Nilsson and Pi´oro [29], etc.

2

Problem definitions

Consider a directed network topology G = (N, A),

where N and A denote the set of nodes and the set

of links in the network respectively. All connections

are delivered through G from the source node o to the

destination node d. There are m different Quality of

Service (QoS) classes in this core network G [4], and

M = {1, 2, . . . , m} denotes an index set consisting of

m QoS classes.

We assume connections of class i occur at the

source node o in accordance with independent

Pois-son processes at rate λ

i

(t) at period t, but the

connec-tion volume to be transmitted has an arbitrary

distri-bution with mean σ

i

(t) [40]. At period t, we intend to

allocate the bandwidth under a limited budget B(t)

in order to provide each class with maximal possible

QoS. The number of virtual paths of class i ∈ M is

denoted by s

i

(t). Every virtual path of class i ∈ M

is allocated the same amount of bandwidth x

i

(t) at

period t.

For each class i ∈ M , the mean sojourn time

1/µ

i

(t) of connections on virtual paths

correspond-s to the packet trancorrespond-smicorrespond-scorrespond-sion time, and it icorrespond-s equal

to average connection volume divided by bandwidth,

i.e.,

1

µ

i

(t)

=

σ

i

(t)

x

i

(t)

.

(1)

Suppose that connections occupy the virtual paths

in the order they occur and that sojourn times are

identically distributed and mutually independent.

In this article, we investigate the relationships

be-tween performance measures of interest and model

parameters at period t, which is similar at other

pe-riods. To simplify the notation, we skip the notation

(t), and the derivation is conducted in general

for-mat. The following definitions are given and will be

used throughout the whole context of this article.

Definition 1 The traffic demand y

i

for class i ∈

M is defined as the product of the mean occurrence

rate λ

i

and the average connection volume σ

i

, i.e.,

y

i

= λ

i

σ

i

.

(2)

This communication system is analyzed as an

Er-lang loss model under assumptions of Poisson

arrival-s, general sojourn time, preset s

i

virtual paths with

identical bandwidth x

i

, and no waiting space [28],

[40]. For a traffic class i ∈ M , we derive the

steady-state occupancy probabilities of n (0 ≤ n ≤ s

i

)

con-nections, P

n

. The unique steady-state probability

exists for this stable system [40]. Hence, we have

P

n

=

P

0

n!

µ

λ

i

σ

i

x

i

¶

n

, n = 1, 2, . . . , s

i

,

where λ

i

is the mean occurrence rate of connections,

σ

i

is the average connection volume, x

i

is the

band-width allocation and s

i

is the preset number of virtual

paths. Solving for P

0

in the equation

P

si

n=0

P

n

= 1,

we can obtain P

0

and P

n

for n = 1, 2, . . . , s

i

. Thus,

the blocking probability of incoming connections is

formulated as

P (x

i

, s

i

, y

i

) =

(y

i

/x

i

)

si

s

i

!

"

_s i

X

n=0

(y

i

/x

i

)

n

n!

#

−1

,

(3)

2

where x

i

is the allocated bandwidth, s

i

is the preset

number of virtual paths in the off-line optimization,

and y

i

= λ

i

σ

i

is the traffic demand from on-line

traf-fic flow. Moreover, the expected path occupancy in

the steady state is

L(x

i

, s

i

, y

i

) =

si

X

n=1

(y

i

/x

i

)

n

(n − 1)!





si

X

j=0

(y

i

/x

i

)

j

j!





−1

. (4)

Note that L(x

i

, s

i

, y

i

) = (y

i

/x

i

)(1 − P (x

i

, s

i

, y

i

)).

The average throughput for class i ∈ M can be

de-termined by x

i

L(x

i

, s

i

, y

i

).

In real world cases, the numbers of connections (or

users) on networks are always huge, i.e., s

i

À 0. If

the traffic intensity ρ

i

= y

i

/s

i

x

i

< 1, equation (3)

can be rewritten as

P (x

i

, s

i

, y

i

) ≈

(y

i

/x

i

)

si

_e

−yi/xi

s

i

!

, as s

i

→ ∞.

(5)

Moreover, we can conclude that

L(x

i

, s

i

, y

i

) ≈

y

i

x

i

µ

1 −

(y

i

/x

i

)

si

_e

−yi/xi

s

i

!

¶

,

(6)

as s

i

→ ∞.

3

Network

management

schemes

Network managers may wish to maximize the

aver-age revenue of the system [26] when regulating the

bandwidth allocation ~x = (x

1

, . . . , x

m

) and the

num-ber of virtual paths ~s = (s

1

, . . . , s

m

). Given traffic

demand y

i

for class i ∈ M , network managers would

like to determine the values of ~x and ~s to optimal the

system. As far as QoS is concerned, bandwidth

al-location x

i

and blocking probability P (x

i

, s

i

, y

i

) are

the key elements of the network revenue

managemen-t scheme [7], [10], [12], [14], [17], emanagemen-tc. The operamanagemen-ting

costs can be determined by the type of traffic

trans-mitted (data, voice, video) and the QoS guaranteed

for such transfer (delay constraint, bandwidth

allo-cation and blocking probability, etc) [43]. When

de-signing a network revenue management scheme, one

can formulate an optimization model with the

follow-ing average revenue function for traffic class i ∈ M

[43]:

f

i

(x

i

, s

i

, y

i

) = c

ti

L(x

i

, s

i

, y

i

)+c

bi

λ

i

x

i

(1−P (x

i

, s

i

, y

i

)),

(7)

where users of class i ∈ M are charged the cost c

b i

for using per unit of bandwidth and users of class

i ∈ M are charged the cost c

t

i

per unit of time for the

sojourn time 1/µ

i

= σ

i

/x

i

on those virtual paths.

Note that c

b

i

and c

ti

can possibly be varied according

to the time of the day to serve with a congestion

control mechanism. The total revenue is obtained by

summing over (7) for all traffic classes.

Let Ω(~s, B, G)) be the feasible set consisting of the

network constraints under preset numbers of virtual

paths ~s = (s

1

, . . . , s

m

), limited budget B and network

topology G. A network optimization scheme can be

executed as follows [3], [9], [16], [38], [39], etc.

max

X

i∈M

w

i

f

i

(x

i

, s

i

, y

i

)

(8)

s.t.

~x ∈ Ω(~s, B, G),

(9)

where w

i

∈ (0, 1) is a fixed weight assigned to each

class i by network managers. Here, ~x = (x

1

, . . . , x

m

)

is the decision variable, and ~s = (s

1

, . . . , s

m

), B,

y

i

are parameters.

The goal is to determine the

bandwidth allocation ~x under negotiated QoS

lev-el so that the revenue earned by the network access

providers is maximized. The feasible set Ω(~s, B, G))

is bounded. This result follows since the bandwidth

allocated to each class i in (8), ∀i ∈ M , has a

up-per bound due to limited budget B. Moreover, the

feasible set Ω(~s, B, G)) decreases to an empty set if

||~s||

2

= (

P

m

i=1

s

2i

)

1/2

increases to a sufficiently large

number, where || · ||

2

denotes the well-known

Eu-clidean norm on the vector space R

m

_.

Given fixed network topology G and limited

bud-get B, we can determine the optimal solutions ~x

∗

₌

(x

∗

1

, . . . , x

∗m

) under preset numbers of virtual

path-s ~path-s = (path-s

1

, . . . , s

m

), where x

∗i

represents the optimal

bandwidth allocated to every virtual path of class

i ∈ M . Note that the optimal bandwidth allocation

~x

∗

_{(~s, B, G) is a function of ~s, B and G. Consequently,}

the maximal throughput of s

i

virtual paths of class i

is s

i

x

∗i

.

4

Monotonicity and convexity

of blocking probability

The monotonicity and convexity properties of the

blocking probability (3) are listed below.

Proposition 1 The blocking probability P (x

i

, s

i

, y

i

)

is a decreasing function of bandwidth x

i

, given s

i

≥ 1

and y

i

> 0 fixed.

Corollary 1 In the case of large s

i

À 1, if the

traf-fic intensity ρ

i

= y

i

/(s

i

x

i

) < 1 holds, the first

deriva-tive of blocking probability P (x

i

, s

i

, y

i

) with respect to

bandwidth x

i

is always negative, i.e.,

∂P (x

i

, s

i

, y

i

)

∂x

i

= (

y

i

x

i

− s

i

)

y

si i

e

−yi/xi

s

i

!x

sii+1

< 0.

(10)

Proposition 2 For each s

i

≥ 1 and y

i

> 0, there

exists a subset (or region) S of positive real numbers

such that the blocking probability P (x

i

, s

i

, y

i

) is

con-vex (concave) in bandwidth x

i

for all x

i

∈ ( /

∈) S.

It should be noted that, as s

i

→ ∞, the limit of

the sequence {s

i

+

5₂

−

p

s

2

i

+ 4s

i

+ 2 | s

i

∈ N} is

0.5, where N is the set of positive integers. As the

number of end-to-end paths s

i

is huge in real-world

communication systems, Proposition 2 implies that

P (x

i

, s

i

, y

i

) is convex in bandwidth x

i

if we have

0.5 < P (x

i

, s

i

, y

i

) ≤ 1. Otherwise, there exist two

inflection points x

∗

i

and x

∗∗i

when 0 ≤ P (x

i

, s

i

, y

i

) <

0.5.

Proposition 3 If the traffic intensity ρ

i

= y

i

/s

i

x

i

>

1 holds in the case of large s

i

À 1, the expected

path occupancy L(x

i

, s

i

, y

i

) is a decreasing function

of bandwidth x

i

, given y

i

> 0 fixed.

Given y

i

> 0 and s

i

≥ 1 fixed, there exists an

inflection point x

∗

i

such that for all x

i

≤ (≥)x

∗i

the

expected path occupancy L(x

i

, s

i

, y

i

) is concave

(con-vex) in bandwidth x

i

.

It can also be observed that the utilization level

U is a decreasing function of bandwidth x

i

for given

y

i

> 0 and s

i

≥ 1. This is because the utilization level

U equals to the expected path occupancy L(x

i

, s

i

, y

i

)

divided by s

i

. Meanwhile, there exists an inflection

point x

∗

i

such that for all x

i

≤ (≥)x

∗i

the utilization

level U is concave (convex) in bandwidth x

i

.

Proposition 4 The blocking probability P (x

i

, s

i

, y

i

)

is increasing in traffic demand y

i

, given x

i

> 0 and

s

i

≥ 1 fixed.

5

Elasticity

For each traffic class, we investigate the elasticity of

blocking probability with respect to bandwidth,

traf-fic demand and the number of virtual paths

individu-ally. Based on the investigation of elasticity, one can

develop distributed algorithms for network revenue

management that takes user’s elasticity into

consid-eration [44], [3], etc. By using the concept of

elastici-ty, we can define the bandwidth elasticity of blocking

ε

b

i

for class i ∈ M as follows.

Definition 2 The bandwidth elasticity of

block-ing is defined as

ε

bi

=

4P (x

i

, s

i

, y

i

)/P (x

i

, s

i

, y

i

)

4x

i

/x

i

,

(11)

where 4x

i

is the change in allocated bandwidth, and

4P (x

i

, s

i

, y

i

) is the change in blocking probability.

The elasticity ε

b

i

represents the percent change in

blocking probability in response to a percent change

in bandwidth. Similarly, the demand elasticity of

blocking ε

d

i

and the capacity elasticity of blocking

ε

c

i

for class i ∈ M are given below.

Definition 3 The demand elasticity of blocking

is defined as

ε

d i

=

4P (x

i

, s

i

, y

i

)/P (x

i

, s

i

, y

i

)

4y

i

/y

i

,

(12)

where 4y

i

is the change in the traffic demand.

Definition 4 The capacity elasticity of blocking

is defined as

ε

c

i

=

4P (x

i

, s

i

, y

i

)/P (x

i

, s

i

, y

i

)

4s

i

/s

i

,

(13)

where 4s

i

is the change in the number of virtual

path-s.

Proposition 5 shows the phenomenon that the

blocking probability will decrease as the allocated

bandwidth increase. Proposition 6 infers that the

blocking probability will increase as the traffic

de-mand increases. Proposition 7 concludes that the

blocking probability is decreasing as enlarging the

number of virtual paths. Due to the limit of pages,

the proofs of those propositions are skipped here.

Those phenomena can also be observed in the

nu-merical results.

Proposition 5 The bandwidth elasticity of blocking

ε

b

i

is nonpositive and decreasing as bandwidth x

i

in-creases.

Proposition 6 The demand elasticity of blocking ε

d i

is nonnegative as the traffic demand y

i

≥ 0.

Proposition 7 The capacity elasticity of blocking ε

c i

is nonpositive and decreasing as the number of virtual

paths s

i

increases.

6

Conclusions

We consider the bandwidth allocation problem on

communication networks, where the network is

mod-elled with multiple classes of traffic. This work

con-centrates on study of the blocking probability

proper-ty of connections in terms of the available number of

end-to-end paths and the allocated bandwidth under

the budget constraint. We have presented

importan-t relaimportan-tions among importan-the blocking probabiliimportan-ty, allocaimportan-ted

bandwidth, traffic demand and the number of

end-to-end paths.

The monotonicity and convexity relationships have

been analyzed among model parameters and

perfor-mance measures of interest, e.g., blocking

probabili-ties and expected path occupancy. We also presented

three elasticities to investigate the effect of varying

model parameters on the average revenue in analysis

of economic models. Those results are verified with

numerical examples of the blocking probability and

utilization level. One can use those monotone and

convex properties to investigate the marginal revenue

in capacitated communication systems so that the

blocking probability is kept below a specific

thresh-old. Future work will be conducted in the direction

of further investigation for the network revenue

man-agement schemes.

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5TH INTERNATIONAL CONFERENCE ON QUEUEING THEORY AND NETWORK APPLICATIONS 2010 1

Dropping Behavior of a Random Early

Detection Mechanism by a Queueing Model of

Batch Arrivals and Multiple Servers

Hsing Luh, Chia-Hung Wang, Chung-Min Lin

Department of Mathematical Sciences, National Chengchi University

No. 64, Sec. 2, ZhiNan Rd., Wen-Shan District, Taipei 11605, Taiwan

Abstract

With the emerging popularity of multimedia applications, quality of service on Internet becomes

an important issue. Random Early Detection (RED) is one of the most widely used adaptive queue

management mechanisms in the Internet. In this paper, we study an analytic model for RED queues

with batch arrival and multiple servers. A Markovian approach is presented in order to study different

quality metrics by RED parameters, such as batch size, dropping rate, etc. Numerical results obtained

by simulation and analytic methods illustrate those performance measures of the proposed RED queue,

such as average system/queue size, average sysetm/queue waiting time, average dropping probability.

Keywords: Random Early Detection (RED), Adaptive Queue Management, Dropping Rate

I. I

NTRODUCTION

The Random Early Detection Algorithm (RED) have been proposed to be mainly used in the

imple-mentation of Active Queue Management (AQM) in the Internet [2], [9], [10], [11]. Rather than explicitly

sending a congestion notification packet to the source, RED implicitly notifies the source of congestion by

dropping its packets. A main purpose of active queue management is to provide congestion information

for sources to set their rates [19]. The design of active queue management algorithms must answer three

questions, assuming packets are probabilistically marked [20]:

1) How is congestion measured?

2) How is the measure embedded in the probability function?

3) How does it feed back to users?

RED is a congestion avoidance mechanism that takes advantage of TCP’s congestion control mechanism

[10], [16] . On the arrival of each packet, the average queue size is calculated by using the Weighted

Moving Average (WMA). The computation of the average queue size is compared with the minimum

and the maximum threshold to establish the next action.

When it comes to Quality of Service, there are two separate approaches. The first is congestion

management, which is setting up queues to ensure that the higher priority traffic gets serviced in times

of congestion. The other is congestion avoidance, which works by dropping packets before congestion

on the link occurs. RED takes a proactive approach to congestion. Instead of waiting until the queue is

completely filled up, RED starts dropping packets with a non-zero dropping probability after the average

queue size exceeds a certain minimum threshold. A dropping probability ensures that RED randomly

drops packets from only a few flows, avoiding global synchronization. A packet drop is meant to signal

the TCP source to slow down. Responsive TCP flows slow down after packet loss by going into slow

start mode.

The RED active queue management algorithm allows network operators to simultaneously achieve high

throughput and low average delay [21]. However, the resulting average queue length is quite sensitive to

the level of congestion and to the RED parameter settings, and is therefore not predictable in advance.

Delay being a major component of the quality of service delivered to their customers, network operators

would naturally like to have a rough a priori estimate of the average delays in their congested routers;

to achieve such predictable average delays with RED would require constant tuning of the parameters to

adjust to current traffic conditions.

However, because data traffic is inherently bursty, routers are provisioned with fairly large buffers to

absorb this burstiness and maintain high link utilization [17], [16]. The downside of these large buffers

is that if traditional drop-tail buffer management is used, there will be high queueing delays at congested

routers. Thus, drop-tail buffer management forces network operators to choose between high utilization

(requiring large buffers), or low delay (requiring small buffers). The merits of RED have been greatly

debated over the last decades. The first paper detailing random early detection’s merits was the Floyd

and Jacobson paper of 1993 [8].

Chandra and Subramani [1] briefly survey of various congestion control algorithms. It seems that

at present there is no single algorithm that can resolve all of the problems of congestion control on

computer networks and the Internet. More research work is needed in this direction. QoS is of particular

concern for the continuous transmission of high-bandwidth video and multimedia information. This type

of transmitting the content is difficult in the present Internet and network with the drop tail.

Wang et al. [3] examine the bursty nature of per-stream packet drops by means of conditional statistics

with respect to dropped periods and the probability that the queueing system stays in the dropped period.

The queueing model with a RED scheme can be modeled as a M AP/M/1/K queue. The distributions

of various absorbing times in the two hypothesized Markov chains are derived to compute the average

durations of the dropped periods and the conditional per-stream multimedia packet dropping probability

encountered during a dropped period.

While many of these phenomena have been seen in controlled experiments, much active research still

involves the refinement and verification of these claims in more realistic networks [8], [17], [18], [11].

Some of Floyd and Jacobson’s claims have since been refuted under certain conditions, including the

consecutive packet drop claim [16], [19], [20].

RED has some problems to face. First, it is not a thoroughly understood scheme. Second, it has many

parameters, and consequently, it is hard to tune. One of RED’s main weaknesses is that the average queue

size varies with the level of congestion and with the parameter settings. Delay being a major component

of the quality of service delivered to their customers, network operators would naturally like to have

a rough a priori estimate of the average delays in their congested routers; to achieve such predictable

average delays with RED would require constant tuning of RED’s parameters to adjust to current traffic

conditions. The second weakness of RED is that the throughput is also sensitive to the traffic load and to

RED parameters. In particular, RED often does not perform well when the average queue becomes larger

than max

th

, resulting in significantly decreased throughput and increased dropping rates. Avoiding this

regime would again require constant tuning of the RED parameters.

Although RED shows better performance than its predecessor, DropTail, its performance is highly

sensitive to parameter settings. Under non-optimum parameter settings, the performance degrades and

quickly approaches that of DropTail gateways. As the network conditions change dynamically on which

the optimum parameter settings depend, the RED parameters need to be updated accordingly. Since the

interaction between RED and UDP is not well understood as analytical solutions cannot be obtained,

stochastic approximation based parameter approach is proposed as an alternative. In this paper, we present

a Markovian approach for adjusting RED parameters that makes use of direct measurements in the

network. The algorithm presented here is found to show better approximation as compared to simulation

that adaptively tunes a RED parameter.

1) An novel idea is mentioned in this paper. It gives a possible solution to handle the congestion

problem in the network executing congestion avoidance.

2) Both the detailed algorithm and simulation results are given in this paper. Thus, we have a better

view of this method and tell the effect of this algorithm from the experiment results.

II. M

ODEL

D

ESCRIPTION

Edge routers assign IP precedences to packets as they enter the network. By randomly dropping

packets prior to periods of high congestion, RED signals the packet source to decrease its transmission

rate. Assume that there are c servers (links) in the router with a buffer size of K packets. Let n be the

instantaneous system size and d(n − c) be a drop function of queue size n − c at a router. When a packet

arrives, the following events occur:

1) The average queue size is calculated.

2) If the average is less than the minimum queue threshold, the arriving packet is queued.

3) If the average is between the minimum queue threshold for that type of traffic and the maximum

threshold for the interface, the packet is either dropped or queued, depending on the packet dropping

probability for that type of traffic.

4) If the average queue size is greater than the maximum threshold, the packet is dropped.

We would use the active/passive model or called as AP control model to illustrate the packet arrival

behavior. When the system is at the active mode, there would be packets coming to the Queue; otherwise,

when the system is the passive mode, there would be no packets coming to the Queue. Assume the

transition probabilities are

P (active | active)

= α

P (passive | active)

= 1 − α

P (active | passive)

= 1 − β

P (passive | passive) = β

where α, β ≤ 1. Thus, the length of passive and passive modes (in terms of packets) is considered to be

geometrically distributed with parameter α and β, respectively.

By adopting a active/passive mode mechanism, we first derive a model of a RED router with a single

input stream of bursty traffic. The processing times of the packets in the router are assumed to be

constant. The packet model is the same as the one proposed in [6]. The arrival rate of messages is

constant, where there is one time unit with m packets. We assume that 1 ≤ c ≤ m ≤ K because RED

mechanism is used for congestion control on the bottleneck links. Note that this model does not really

match empirically derived models of UDP and other bursty traffic patterns [13], [15], [21]. However, it is

analytically tractable; furthermore, our purpose here is to compare the relative impact of RED on bursty

(active mode)and less bursty traffic (passive mode).

To model the burstiness, two modes for working of the network element are defined: the active mode

in which packets are admitted if the buffer occupancy is less than K, and the passive mode in which

arriving packets are pending. At the active mode, a message continues to arrive with a batch size m

even when the buffer occupancy is at or above K packets. In this case, all packets of that message are

discarded. But, the service continues until there is no packet in buffer, i.e., the work conservation law is

assumed.

The number of packets buffered in the queue defines a semi-Markov process. Since the stationary

distribution can be computed, the dropping probability is also resolvable. We denote by π this stationary

distribution of system size in average. The dropping probability of a packet in a Tail Drop router is given

by

Fig. 1. The Dropping Rate of RED Queue

The RED buffer management algorithm manages the queue in a more active manner by randomly

dropping packets with increasing probability as the average queue size increases. If the queue size exceeds

min

_th

, the Queue would start to drop the incoming packet with the drop rate as shown below. The dropping

rate would increase monotonically as the queue size increases. When the buffer size meets max

th

, the

dropping rate would become 100%. The service time of the server is constant. For 0 ≤ x ≤ K − c, the

dropping rate d(x) of RED queue is a function of current queue sizes x (shown in Fig. 1), which is

defined as follows:

d(x) =







0

if x < min

th

(

_maxx−minth

th− minth

) max

p

if min

th

≤ x < max

th

1

if x ≥ max

th

We define the system state vector as the number of packets in the system, where its first and second

components are denoted with respect to active and passive mode. Let π

n

= (π

n

(1), π

n

(2)) be the

stationary probability of having n packets in the system and the system is in mode j, where j = 1

for the active mode and j = 2 for the passive mode.

Theorem 2.1: Given max

th

= K − c, it is on the active mode, i.e., α = 1 and β = 0, if

min_K−cth

→ 1,

then the average queue size of AP model L

RED_q

→ K − c.

Theorem 2.2: Given max

th

= K − c, if

min_K−cth

→ 1, then L

REDq

→ L

q

, where L

q

is the average queue

size of D

[m]

/D/c/K systms under AP control.

L

q

=

K

X

n=c

(n − c)(π

n

(1) + π

n

(2)).

It is our purpose to obtain in this paper the dropping probability of a packet in a RED router:

P

_RED

= π

_K

d(K − 1) + π

_K−1

d(K − 2) + . . . + π

1

d(0).

III. P

ROBLEM

F

ORMULATION

Here, we would introduce our traffic analysis model designed for RED. A bursty source is modelled

by Interrupted Bernoulli Process (IBP). In an IBP, it consists of a geometrically distributed period during

which no arrivals occur, followed a geometrically distributed period during arrivals occur in a Bernoulli

fashion. Let us assume that the time axis is segmented into a contiguous sequence of time intervals of

fixed (constant) duration which correspond to the elementary unit of time in the system. We assume that

each message arrives one unit of time. The arrival process is defined by a batch of m packets entering

the network in each slot. The service process is defined by transmitting one packet in each slot as long

as there are packets available in the queue.

Let P

n

(`) be the probability of ` packets randomly selected from m packets to enter the queue during

a service while the current system size is n. Denote by b

i

the position of the ith packet that is selected

in the message length m.

For 0 ≤ n ≤ K and 1 ≤ ` ≤ m, we have

P

n

(`) =

X

1≤b1<b2<···<b`≤m

{

`

Y

i=1

[d(Θ

n

+ i − 1)]

bi−bi−1−1

[1 − d(Θ

n

+ i − 1)]}[d(Θ

n

+ `)]

m−b`

,

(1)

where Θ

n

= max{n − 2c, 0}. In the case of ` = 0,

P

n

(0) = [d(Θ

n

)]

m

for all n = 0, 1, . . . , K.

For example, when ` = 1, it produces

P

n

(1) =

m

X

i=1

[d(Θ

n

)]

i−1

[1 − d(Θ

n

)][d(Θ

n

+ 1)]

m−i

.

When ` = 2, it produces

P

n

(2) =

m

X

1≤i<j≤m

[d(Θ

n

)]

i−1

[1 − d(Θ

n

)][d(Θ

n

+ 1)]

j−i−1

[1 − d(Θ

n

+ 1)][d(Θ

n

+ 2)]

m−j

.

Assume c ≤ m ≤ K, we write down the transition probability matrix P as follows.

P =



















































A0 0 A01 A02 · · · Am0 0 · · · · · · · · · · · · · · · · · · · · · 0 A1₀ A1₁ A1₂ · · · A1_m 0 · · · · · · · · · · · · · · · · · · · · · 0 A2₀ A2₁ · · · A2_m−1 A2_m 0 · · · · · · · · · · · · · · · · · · · · · 0 ._. . ._. . ._. . ._. . ._. . ._. . · · · · · · · · · · · · · · · . . . Ac 0 Ac1 · · · Acm−1 Amc 0 · · · · · · · · · · · · · · · · · · · · · 0 0 Ac+1₀ Ac+1₁ · · · Ac+1_m−1 Ac+1_m 0 · · · · · · · · · · · · · · · · · · 0 0 0 Ac+2₀ Ac+2₁ · · · A_m−1c+2 Ac+2_m 0 · · · · · · · · · · · · · · · 0 ._. . ._. . . . . . . . ._. . ._. . ._. . ._. . ._. . ._. . . . . Ac+m+1₀ Ac+m+1₁ · · · Ac+m+1 m 0 · · · · · · · · · 0 Ac+m+2₀ Ac+m+2₁ · · · Ac+m+2 m 0 · · · · · · . . . ._. . ._. . ._. . ._. . ._. . ._. . . . . ._. . AK−m₀ · · · AK−m_m 0 · · · ._. . ._. . ._. . ._. . . . . AK−m+c₀ · · · · · · AK−m+c_m ._. . ._. . ._. . ._. . . . . AK₀ · · · AK_c



















































where

A

0₀

=

·

0

0

1 − β β

¸

,

(2)

A

n₀

=

·

αP

_n

(0) (1 − α)P

_n

(0)

1 − β

β

¸

,

(3)

and

A

n_`

=

·

αP

n

(`) (1 − α)P

n

(`)

0

0

¸

,

(4)

for all 0 ≤ n ≤ K and 1 ≤ ` ≤ m. The matrix P is finite because K is finite. If the steady-state

probability exists (under stable conditions), then we can solve πP = π, where π is the state vector with

active and passive modes.

Under assumption of c ≤ m ≤ K, we can solve π from the following balance equations.



























































































π

₀

A

0 0

+ π

1

A

10

+ π

2

A

20

+ · · · + π

c

A

c0

= π

0

π

0

A

01

+ π

1

A

11

+ π

2

A

21

+ · · · + π

c

A

c1

+ π

c+1

A

c+10

= π

1

π

0

A

02

+ π

1

A

12

+ π

2

A

22

+ · · · + π

c

A

c2

+ π

c+1

A

c+11

+ π

c+2

A

c+20

= π

2

π

0

A

03

+ π

1

A

31

+ π

2

A

23

+ · · · + π

c

A

c3

+ π

c+1

A

c+1₂

+ π

c+2

A

c+2₁

+ π

c+3

A

c+3₀

= π

3

· · · ·

π

0

A

0m

+ π

1

A

m1

+ π

2

A

2m

+ · · · + π

c

A

cm

+ π

c+1

A

c+1m−1

+ π

c+1

A

c+2m−2

+ · · · + π

c+m

A

c+m0

= π

m

π

c+1

A

c+1m

+ π

c+2

A

c+2_m−1

+ · · · + π

c+m+1

A

c+m+1₀

= π

m+1

π

c+2

A

c+2m

+ π

c+3

A

c+3m−1

+ · · · + π

c+m+2

A

c+m+20

= π

m+2

· · · ·

π

K−m

A

K−mm

+ π

K−m+1

A

K−m+1m−1

+ · · · + π

K

A

K0

= π

K−c

π

K−m+1

A

K−m+1m

+ π

K−m+2

A

K−m+2m−1

+ · · · + π

K

A

K1

= π

K−c+1

· · · ·

π

K−m+c

A

K−m+cm

+ π

K−m+c+1

A

K−m+c+1m−1

+ · · · + π

K

A

Kc

= π

K

Hence, we can determine

π

n

=























c−1

X

j=0

π

j

A

jn

+

n

X

j=0

π

c+j

A

c+jn−j

, for n = 0, 1, ..., m,

δn

X

j=0

π

n−m+c+j

A

n−m+c+jm−j

,

for n = m + 1, ..., K,

(5)

where δ

n

= min{m, K − n + m − c}. Solving the above balance equations and

P

K

n=0

(π

n

(1) + π

n

(2)) = 1, we

can obtain the stationary probability π

n

= (π

n

(1), π

n

(2)), for all n = 0, 1, . . . , K.

IV. G

ENERATING

F

UNCTION

M

ETHOD

Next, we define the probability generating functions

G(z) = (G

1

(z), G

2

(z)) =

K

X

n=0

π

n

z

n

,

(6)

where G

1

(z) =

P

_K n=0

π

n

(1)z

n

, and G

2

(z) =

P

_K n=0

π

n

(2)z

n

. We also define

P

n

(z) =

m

X

`=0

P

n

(`)z

`

,

(7)

for n = 0, 1, 2, · · · , K. For simplicity, let

[α] =

·

α 1 − α

0

0

¸

and [β] =

·

0

0

1 − β

β

¸

.

Then it is clear that

A

n

0

= P

n

(0)[α] + [β]

(8)

and

A

n`

= P

n

(`)[α]

(9)

for 0 ≤ n ≤ K and 1 ≤ ` ≤ m. Moreover, we have

m

X

`=0

A

n

By using the probability gerenating functins (6) and (7), we can rewrite the balance equations (5) as fellows:

c

X

n=0

π

n

(P

n

(z)[α] + [β]) +

K

X

n=c+1

π

n

z

n−c

(P

n

(z)[α] + [β]) = G(z).

(10)

Then we have

G

1

(z) = αX(z) + (1 − β)Y(z)

(11)

G

2

(z) = (1 − α)X(z) + βY(z),

(12)

where

X(z) =

c

X

n=0

P

n

(z)π

n

(1) +

K

X

n=c+1

P

n

(z)π

n

(1)z

n−c

(13)

and

Y(z) =

c

X

n=0

π

n

(2) +

K

X

n=c+1

π

n

(2)z

n−c

.

(14)

Theorem 4.1: If z → 1 and α + β 6= 2, then we have

G

1

(1) = X(1) =

1 − β

2 − α − β

,

(15)

and

G

2

(1) = Y(1) =

1 − α

2 − α − β

.

(16)

Proof: When z = 1,

X(1) =

c

X

n=0

P

n

(1)π

n

(1) +

K

X

n=c+1

P

n

(1)π

n

(1) =

K

X

n=0

π

n

(1) = G

1

(1)

(17)

and

Y(1) =

c

X

n=0

π

n

(2) +

K

X

n=c+1

π

n

(2) = G

2

(1),

(18)

where P

n

(1) =

P

m

`=0

P

n

(`) = 1. Next, from equations (11) and (12), we obtain

(1 − α)G

1

(1) = (1 − β)G

2

(1).

(19)

Because G1

(1) + G

2

(1) = 1, we can determine G

1

(1) = X(1) = (1 − β)/(2 − α − β) and G

2

(1) = Y(1) =

(1 − α)/(2 − α − β).

Theorem 4.2: If z → 0, then we have

G

1

(0) = π

0

(1) =?????,

(20)

and

G

2

(0) = π

0

(2) =?????.

(21)

Proof: When z = 0,

X(0) =

c

X

n=0

P

n

(0)π

n

(1)

(22)

and

Y(0) =

c

X

n=0

π

n

(2),

(23)

where P

n

(0) = P

n

(0). Next, from equations (11) and (12), we obtain

V. P

ERFORMANCE

M

EASURE

The average load of one source is defined as the fraction to time (slots) this source spends in the active state,

ad is thus given by

p =

E[active period]

E[active period] + E[passive period]

=

1 1−α 1

1−α

+

1−β1

= G

1

(1),

where E[...] denotes the expected value of the expression between square brackets. This implies that, in general,

the mean lengths of active and passive periods can be expressed as

E[active period] =

1

1 − α

=

a

1 − p

and

E[passive period] =

1

1 − β

=

a

p

,

For some value of the real quantity a. It is clear that the statistical properties of a source can be fully characterized

by the parameters p and a (instead of α and β): the load p is a measure for the ratio of the active and passive

periods, where as the constant a (in the sequel referred to as the “burstiness factor”) is representative for the absolute

lengths of these periods. High values of a are indicative of a high degree of correlation in the packet arrival process.

β = 1 −

p

a

α = 1 −

1 − p

a

Consider the average number of packets that enter the system effectively, c,

c =

K

X

i=0

E

i

π

i

(1)

which is

c = p − π

0

(i) +

1

1 − β

+

α

(1 − β)

2

Thus, we have

π

0

(1) = p − c +

1

1 − β

+

α

(1 − β)

2

Effective mean arrival rate is computed by

λ =

K

X

k=0 m

X

`=1

(π

n

(1) + π

n

(2))P

n

(`)`.

(25)

The average system size can be computed by

L

s

=

K

X

k=0 m

X

`=1

(π

n

(1) + π

n

(2))k.

(26)

The average queue size can be computed by

L

q

=

K

X

k=c+1 m

X

`=1

(π

n

(1) + π

n

(2))(k − c).

(27)

The average system waiting time can be computed by

W

s

=

L

s

λ

.

(28)

The average queue waiting time can be computed by

W

q

=

L

q

λ

.

(29)

The average dropping probability can be computed by

P

RED

=

m − λ

TABLE I

NUMBER OF SERVERScVERSUS AVERAGE QUEUE SIZELqWITHm = 8, maxth= 8, minth= 0, maxp= 1, α = 0.8AND

β = 0.4.

Number of servers c ProModel Matlab

2 5.72 5.78 3 4.64 4.68 4 3.34 3.55 5 1.18 1.25 6 0.07 0.08 TABLE II

NUMBER OF SERVERScVERSUS AVERAGE QUEUE WAITING TIMEWqWITHm = 8, maxth= 8, minth= 0, maxp= 1,

α = 0.8ANDβ = 0.4.

Number of servers c ProModel Matlab

2 2.91 2.94

3 1.64 1.65

4 0.94 0.94

5 0.30 0.32

6 0.02 0.02

VI. S

IMULATION AND

E

VALUATION

In previous sections, we derived analytic expressions of various measures of interest to evaluate RED. There are

several parameters in this RED gateway. How to decide the value of them is a problem. While the analytic approach is

important to quantify relationships between parameters and performance measures, it must be complemented with

simulation or experiments to validate the hypotheses made during the analysis, and to explore phenomena not

amenable to tractable analysis.

In this section, we focus on simulation results. Some experiments are done to test the different performance

effects. Intuitively, if these values can be decided dynamically with some algorithm, the network can be utilized

better. We use ProModel and Matlab to simulate our proposed model. We have conducted extensive simulation

experiments using ProModel. Numerical results obtained from ProModel and Matlab are summarized in Tables

VI-VI.

Fig. VII shows... Fig. VII... Those figures show that the approximation is very accurate. We verify that the

main conclusions of our analysis are valid.

VII. C

ONCLUSION

We have discussed RED Queue as an congestion control mechanism under End-to-End QoS senario. Using the

mathematical model, we have constructed a forecasting model to regulate network congestion problem. It gives us

a good view of the new and promising idea of congestion avoidance. Still some work need to be done for this

paper.

TABLE III

NUMBER OF SERVERScVERSUS AVERAGE DROPPING PROBABILITYPREDWITHm = 8, maxth= 8, minth= 0, maxp= 1, α = 0.8ANDβ = 0.4.

Number of servers c ProModel Matlab

2 0.755 0.754

3 0.648 0.645

4 0.559 0.557

5 0.512 0.510