EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONSEuro. Trans. Telecomms. 2003; 14:155–160 (DOI: 10.1002/ett.907)

Letter

Mobile Networks

An analytical model for an RIO queue fed with On/Off

sources over a wireless link

Edward Chan*, Xinwei Hong and Pei Zhang

Department of Computer Science, City University of Hong Kong, Kowloon, Hong Kong

SUMMARY

The diverse nature of Internet applications gives rise to a need for a network architecture that is robust andcan support a wide range of quality of service. Differentiated service is one such architecture. Although ithas been the subject of much recent research, existing work focuses primarily on wired networks. Aswireless networks become more important, it is necessary to examine the performance of differentiatedservice over these types of networks. In this paper, we analyze the performance of an RIO queue fed withhomogeneous On/Off sources over a two-state Markov wireless channel. The system is used to model theprovision of assured forwarding differentiated service over wireless links. A novel stepped-RIO scheme isused to provide an analytically tractable representation, and simulation results confirm the accuracy of theanalytical model. Copyright # 2003 AEI.

1. INTRODUCTION

The current structure of the Internet does not provide the

capability to support a wide range of quality of service

(QoS). This problem will become even more acute when

wireless services are incorporated into the Internet. A new

architecture known as differentiated service (DiffServ)

has been proposed by IETF to extend the Internet to be a

QoS-capable, scalable and efficient network [1]. A consi-

derable amount of research on the implementation and

performance of DiffServ has already appeared [2, 3].

However, rapid development in wireless network appli-

cations highlights the need to determine how well the

DiffServ model will work in a wireless environment.

Unfortunately, there are as yet very few studies that exam-

ine how DiffServ can be extended to provide QoS assur-

ance in a wireless environment. A study of the issues

and some proposals have beem made [4], but no analytical

model is available. The objective and main contribution of

this paper is to formulate an analytical model of a category

of DiffServ known as assured forwarding, and to evaluate

its performance. The rest of the paper is organized as fol-

lows. Section 2 will discuss our model of DiffServ over a

wireless link. In Section 3 we apply a fluid-flow-based

technique to analyze the performance of the model. The

results of the model are compared with those obtained in

simulation experiments in Section 4, and the paper con-

cludes with Section 5.

2. SYSTEM MODEL

In this section we describe the details of our system model.

We focus on two major aspects: characterization of the

wireless link and modeling of differentiated service.

The channel characteristics of wireless systems, such as

high error rate and low bandwidth capacity, present severe

Received 1 December 2000

Revised 20 May 2001

Copyright # 2003 AEI Accepted 17 December 2001

* Correspondence to: Edward Chan, Department of Computer Science, City University of Hong Kong, Kowloon, Hong Kong.E-mail: csedchan@cityu.edu.hk

Contract/grant sponsor: City University of Hong Kong; contract/grant number: Strategic Grant 7000795.

challenges to network designers. To guarantee the perfor-

mance requirements of Internet services, more sophisti-

cated and complex radio layer protocols (RLP), such as

automatic repeat request (ARQ) and forward error correc-

tion (FEC) [5], have been used over wireless links in an

attempt to improve higher-layer performance. We will

model the wireless channel (with its characteristic of high,

and fluctuating bit error rate) with a fluid version of the

Gilbert–Elliott model [6, 7]. This model is Markovian with

two states: Good and Bad, and the durations of the Good

and Bad states are exponentially distributed with means

1=� and 1=g.

In general, because the error rate in the Bad state is

higher than that in the Good state, the real transmission

rate in the Bad state will be lower than that in the Good

state. A useful model of these techniques [8] will be

adopted in this paper, and we simply assume that the real

transmission rates in the Good and Bad states are wg and

wb respectively.

The DiffServ architecture is composed of a number of

functional elements, namely packet classifiers, traffic con-

ditioners and per-hop forwarding behaviours (PHB) [1]. In

a DiffServ model, the type of service (TOS) field of IP

packet header, which has been redefined as the differen-

tiated service code point (DSCP), is used for classification.

Each IP packet is then assigned a PHB, which represents a

particular forwarding treatment, at the boundary node. The

PHB determines the priority, the maximum delay in

the transmission queues, the link-sharing bandwidth and

the probability of a packet being dropped.

The DiffServ model ensures high scalability by separat-

ing the operations performed in the borders of the network

from those accomplished in the core network. Packets are

classified and conditioned (marked, policed and shaped) at

the edge of the network in order to receive a particular

PHB forwarding on nodes along their path. The core rou-

ters need not do the aforementioned operations, but just

forward the packets accordingly. Since the number of

PHB is relatively small, the design of the core router can

be greatly simplified.

To support different QoS, DiffServ uses the TOS byte to

tag packets for preferential treatment. Besides the provi-

sion of a best-effort service using the default PHB, two

other kinds of PHBs have been proposed by IETF, namely

expedited forwarding (EF) PHB and assured forwarding

(AF) PHB. EF PHB is a high-priority behavior typically

used for network control traffic such as routing updates.

EF can offer the user a performance level that is similar

to that of a leased line, as long as the user’s traffic is limited

to a nominal bandwidth. AF, on the other hand, forwards

packets based on some assigned level of queuing resources

and three drop precedences. We will focus on the AF

model in this paper.

A simple model for assured forwarding differentiated

service has been presented [9]. In this model, a profile is

pre-defined for each source. A regulator is then applied to

mark packets as belonging to the IN profile (higher priority)

or the OUT profile (lower priority). We assume that the

packets from a source are tagged as an IN packet or an

OUT packet with probability p and �p ¼ 1 � p respectively.

For the wireless link, we use the random early detection

with IN and OUT packets (RIO) scheme [9] which is based

on the widely used random early detection (RED) queue

management scheme first proposed by Floyd and Jacobson

[10]. RIO uses the same mechanism as in RED, but con-

ceptually uses two RED algorithms: one for IN packets

and one for OUT packets. By choosing the parameters

for the IN and OUT algorithms differently, RIO can prefer-

entially drop OUT packets when the link is congested to

guarantee the performance of the IN packets.

For analytical purposes, we use a stepped-RIO scheme

as an approximation of the actual RIO scheme in this

paper. This scheme retains the key characteristics of the

original scheme and is simpler to analyze than the original

RIO scheme. [9]. Assuming the queue size of the wireless

link is B, we divide the queue occupancy into B=d possible

levels where d is length of a level. B is assumed to be mul-

tiples of d. Let Lk be the kth level, i.e.,

Lk ¼ fn j kd4n < ðk þ 1Þdg for k ¼ 0; 1; . . . ;B=d � 1

ð1Þ

In the stepped-RIO scheme, the drop probability of

packets will remain unchanged in any given level, and var-

ies between levels. We define the pass ratio to be �INðnÞ for

IN, �OUTðnÞ for OUT and �ðnÞ for all (ALL) packets when

queue occupancy is n. Then

�INðnÞ ¼#bndc

IN

�OUTðnÞ ¼#bndc

OUT

�ðnÞ ¼ p�INðnÞ þ �p�OUTðnÞ

ð2Þ

where #IN and #OUT are design parameters for IN and OUT

packets and 04#IN; #OUT41 and #IN4#OUT, bxc is the

maximum integer less than or equal to x. Obviously, the

drop probabilities for INs, OUTs and ALLs are ��INðnÞ ¼1 � �INðnÞ; ��OUTðnÞ ¼ 1 � �OUTðnÞ, and ��ðnÞ ¼ 1 � �ðnÞ,respectively.

Using the stepped-RIO scheme as an approximation, we

can now study the performance of our system.

156 E. CHAN ET AL.

Copyright # 2003 AEI Euro. Trans. Telecomms. 2003; 14:155–160

3. PERFORMANCE EVALUATION

The fluid-flow technique is applied to analyze the system

described in the previous section. N homogeneous On/Off

sources are assumed to be buffered and multiplexed for

transmission on the wireless link. Each On/Off source

has an On state and an Off state. The arrival rate in the

On state is constant R. No packet arrives in the Off state.

The transmission rate from the On state to the Off state is

a, and b vice versa. Then, the whole system can be char-

acterized by a Markov process �. The states of the process

are the mixtures of the states of channel (Good and Bad)

and the states of arrival sources (the number of sources in

On state). � will have 2ðN þ 1Þ states. We can enumerate

the states in our analysis as follows. States s ¼ 0; 1; . . . ;Nrepresent the state when channel is in the Good state

and the number of On sources is s. States

s ¼ N þ 1;N þ 2; . . . ; 2N þ 1 represent the state when

channel is in the Bad state and the number of On sources

is s� ðN þ 1Þ. Let S be the state space and M the state tran-

sition matrix. Then, we have

Mðs1; s2Þ ¼

ðN � s1Þb s1 2 A; s2 ¼ s1 þ 1

ð2N þ 1 � s1Þb s1 2 ANþ1 s2 ¼ s1 þ 1

s1a s1 2 A1; s2 ¼ s1 � 1

ðs1 � N � 1Þa s1 2 ANþ2 s2 ¼ s1 � 1

� s1 2 B; s2 ¼ s1 þ N þ 1

g s1 2 BNþ1 s2 ¼ s1 � N � 1

0 otherwise

8>>>>>>>>><>>>>>>>>>:

ð3Þwhere A ¼ f0; 1; . . . ;N � 1g, B ¼ f0; 1; . . . ;Ng, and Ax

is defined as a set whose elements are mapped from set

A by increasing x.

Let ~p ¼ ½pð0Þ; pð1Þ; . . . ; pð2N þ 1Þ� be the stationary

state distribution. We can obtain ~p by

~pM ¼ 0X

~p ¼ 1 ð4Þ

The fluid-flow method is now applied to analyze the sys-

tem. Let �t denote the state of � at time t, and qt denote

the queue length at time t. The state distribution of the sys-

tem in equilibrium is given by

�sðxÞ ¼ limt!1

Pr½�t ¼ s; qt4x� ðs 2 S; 0 � x � BÞ

ð5Þ

The distribution is continuous, except possibly at the

level boundaries and the threshold. In what follows, we

drop the subscript t when specifying stationary distribu-

tions. Let HðxÞ ¼ ½�0ðxÞ;�1ðxÞ; . . . ;�2Nþ1ðxÞ�. Follow-

ing the fluid-flow method, the governing differential

equations are readily obtained, for k ¼ 0; 1; . . . ;B=d � 1

d

dxHðxÞDðkÞ ¼ HðxÞM for x 2 Lk; ð6Þ

where DðkÞ ¼ diagf�wg; �ðkdÞR� wg; . . . ; �ðkdÞNR�wg;�wb; �ðkdÞR� wb; . . . ; �ðkdÞNR� wbg. The diagonal

element of the matrix DðkÞ is the drift rate of change of the

buffer content (away from the boundaries) when � is in

the corresponding state. Equation (6) yields a system of

2ðN þ 1ÞB=d equations.

The system of differential equations in Equation (6) for

each individual level of buffer occupancy has been treated

elsewhere [11]. Following this treatment we obtain, for

k ¼ 0; 1; . . . ;B=d � 1

HðxÞ ¼ HðkÞðxÞ ¼

¼P

s2S aðkÞs wðkÞ

s expðzðkÞs xÞ; x 2 LkPs2S a

ðB=d�1Þs wðB=d�1Þ

s expðzðB=d�1Þs xÞ x ¼ B

ð7Þ(

Here, fzðkÞs ;wðkÞs g are solutions to B=d sets of eigenvalue

problems

zðkÞs wðkÞs DðkÞ ¼ wðkÞ

s M ðs 2 SÞ ð8Þ

The coefficients faðkÞs g can be obtained from the bound-

ary conditions. Let SðkÞD denote the set of states of � which

give a downward drift to the buffer content when the level

of buffer occupancy is k; similarly, let SðkÞU be the set of

states of � which giving an upward drift. That is, for

k ¼ 0; 1; . . . ;B=d � 1

SðkÞD ¼ fs 2 S jDðkÞðs; sÞ < 0g

SðkÞU ¼ fs 2 S jDðkÞðs; sÞ > 0g

ð9Þ

We make the simplifying assumption that there exists

no s such that DðkÞðs; sÞ ¼ 0; Mitra [11] has shown how

exceptions may be handled. It can be proved that

Sð0ÞD � S

ð1ÞD � � � � � S

ðB=d�1ÞD

Sð0ÞU � S

ð1ÞU � � � � � S

ðB=d�1ÞU

ð10Þ

On the basis of the behavior of d Xt=dt at the level

boundaries, we obtain the following complete system of

boundary conditions, for k ¼ 0; 1; . . . ;B=d � 2

Hð0Þs ð0Þ ¼ 0 s 2 S

ð0ÞU

HðkÞs ððk þ 1ÞdÞ ¼ Hðkþ1Þ

s ððk þ 1ÞdÞ s 2 SðkÞD [ S

ðkþ1ÞU

HðB=d�1Þs ðBÞ ¼ pðsÞ s 2 S

ðB=d�1ÞD

ð11Þ

These boundary conditions form a system of

2ðN þ 1ÞB=d equations. On substituting the expressions

RIO QUEUE FED WITH ON/OFF SOURCES OVER A WIRELESS LINK 157

Copyright # 2003 AEI Euro. Trans. Telecomms. 2003; 14:155–160

for HðxÞ in Equation (7), we obtain a system of linear

equations in the 2ðN þ 1ÞB=d coefficients faðkÞs g which

could be solved numerically. As shown elsewhere [12],

the equilibrium state distributions HðxÞ exhibit jumps at

the boundaries of the buffer occupancy levels, and the fol-

lowing probabilistic interpretation applies to the jumps:

(k ¼ 1; 2; . . . ;B=d � 1)

Prð� ¼ s; q ¼ kdÞ ¼ �ðkÞs ðkdÞ ��ðk�1Þ

s ðkdÞ s 2 S

ð12ÞFrom Equations (7) and (12), we can get the queue

length distribution qsðxÞ ¼ Prð� ¼ s; q ¼ xÞ, s 2 S; 04x4B. Note also that

Prð� ¼ s; buffer emptyÞ ¼ �sð0ÞPrð� ¼ s; buffer fullÞ ¼ ps ��sðBÞ

ð13Þ

We now consider the packet loss probability. The loss

rate for IN packets due to RIO scheme (when the buffer

is not full) isPN

s¼0

PBn¼0 psR

��INðnÞðqsðnÞ þ qsþNþ1ðnÞÞ.The loss rate for IN packets due to a full buffer is r�,

where � ¼PN

s¼0ðsR�ðBÞ � wgÞfps ��sðBÞg þP2Nþ1

s¼Nþ1

fðs� N � 1ÞR�ðBÞ � wbÞðps ��sðBÞg is the loss rate

for all packets, and r ¼ fp�INðBÞ=p�INðBÞ þ �p�OUTðBÞgis the ratio of IN packets which enter the queue after being

dropped by the RIO scheme, to all packets. Then the loss

probability for IN packets is

LIN ¼ r�þPN

s¼0

PBn¼0 psR

��INðnÞfqsðnÞ þ qsþNþ1ðnÞgPNs¼0 psRfpðsÞ þ pðsþ N þ 1Þg

ð14ÞSimilarly, we have the packet loss probabilities for OUT

and All packets as follows:

LOUT ¼ ð1 � rÞ�þPN

s¼0

PBn¼0 �psR

��OUTðnÞfqsðnÞ þ qsþNþ1ðnÞgPNs¼0 �psRfpðsÞ þ pðsþ N þ 1Þg

LALL ¼ �þPN

s¼0

PBn¼0 sR

��ðnÞfqsðnÞ þ qsþNþ1ðnÞgPNs¼0 sRfpðsÞ þ pðsþ N þ 1Þg

ð15ÞBased on the definition of effective throughput, i.e., the

rate of packet generation minus the rate of packets loss, we

have the effective throughput of IN, OUT and all packets

as follows

T IN ¼�

1 � LIN�XN

s¼0

psRfpðsÞ þ pðsþ N þ 1Þg

TOUT ¼�

1 � LOUT�XN

s¼0

�psRfpðsÞ þ pðsþ N þ 1Þg

TALL ¼�

1 � LALL�XN

s¼0

sRfpðsÞ þ pðsþ N þ 1Þg

ð16Þ

Applying Little’s law, we have the mean delay of the

model

EðdÞ ¼XNs¼0

sðpðsÞ þ pðsþ N þ 1ÞÞ=TALL ð17Þ

4. NUMERICAL AND SIMULATION RESULTS

In this section, we will compare the results obtained from

our model with those obtained from simulating an equiva-

lent system [13] to demonstrate the accuracy of our analy-

tical model. The effect of some system parameters on

performance will also be studied.

All simulations were done using the ns-2 simulation

package [13]. We model a wireless access point which

implements the RIO scheduling method. All simulation

runs have at least 106 packets which go through the access

point, thus ensuring that the confidence interval for all

results is 95%.

We based the simulation experiments on the Gilbert–

Elliott model with the following parameters [6, 7]:

wg ¼ 10 Mbps, wb ¼ 3:333 Mbps, 1=� ¼ 0:1s, 1=g ¼0:0333s. Buffer size and level are set to 50 and 5 respec-

tively. The RIO parameters #IN and #OUT are set to 0.99

and 0.8. The packet size is 100 bytes. A hybrid FEC/

ARQ scheme is used, where the FEC mechanism used

can recover up to 8 error bits in a packet. The maximum

number of retransmissions for the ARQ scheme is 3. In

all our numerical results, we treat the CRC code of FEC

scheme as part of the payload. In all simulations, the sim-

plified stepped-RIO has been used.

The On and Off periods of the On/Off sources are both

exponentially distributed with the mean sojourn time of

the On state set to 0.731 s, and that of the Off state

0.385 s. The arrival rate R in the On period is 491:8 kbps.

The packets are tagged as IN packet with probability 0.5.

Packet loss probabilities for IN, OUT and ALL packets

are shown in Figure 1. The most important observation is

that the analytical model is very accurate. It can also be

seen that, in all cases, the OUT packets have higher packet

loss probability than the IN packets. This confirms that the

RIO scheme does give preference to IN packets and their

QoS. When the number of sources is low, i.e., utilization is

low, the RIO scheme is very efficient (a small increase in

the loss probability for the OUT packets can substantially

reduce the loss probability for the IN packets). However,

as the number of sources increases, the efficiency of the

RIO scheme decreases.

158 E. CHAN ET AL.

Copyright # 2003 AEI Euro. Trans. Telecomms. 2003; 14:155–160

In Figure 2, we plot the effective throughput against the

number of sources. An interesting observation is that, as

the number of sources increases, the throughput of the

OUT packets decreases.

In Figure 3, the average packet delay is presented. As

expected, packet delay increases as the utilization

increases. The figure also shows that our results are quite

accurate.

We also performed some experiments to test the impact

of various parameters. In Figure 4, we show the impact of

one important design parameter, i.e., #OUT for the OUT

packets, on the packet loss probability. #IN is fixed at

0.99 for this experiment. The number of sources is also

fixed at ten. All other parameters are the same as these

in the former simulation experiments. From the figure,

we can see that as the #OUT decreases, packet loss prob-

ability decreases for the IN packets and increases for the

OUT packets. Thus, we conclude that to protect the QoS

of the IN packets, a small #OUT which can meet the

QOS requirement of the OUT packets can be chosen.

Figure 1. Packet loss probabilities vs number of sources.

Figure 2. Effective throughput vs number of sources.

Figure 3. Packet delay vs number of sources.

Figure 4. Packet loss probabilities vs # for OUT packets.

RIO QUEUE FED WITH ON/OFF SOURCES OVER A WIRELESS LINK 159

Copyright # 2003 AEI Euro. Trans. Telecomms. 2003; 14:155–160

5. CONCLUSION

We have presented a new method to analyze the perfor-

mance of a wireless assured forwarding differentiated ser-

vice, which is modeled by a RIO queue fed with On/Off

sources over a wireless link. Applying a fluid-flow

approach on a novel stepped-RIO approximation, we

obtain the packet loss probability and effective throughput

for IN, OUT and ALL packets, which are in excellent

agreement with simulation results. This analytical model

should be a useful tool for network designers who need

to study the effect of various network parameters on the

performance of a wireless differentiated service.

ACKNOWLEDGEMENTS

This research work is supported in part by City University ofHong Kong Strategic Grant 7000795.

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AUTHORS’ BIOGRAPHIES

Xinwei Hong received the BS and MS degrees in electrical engineering from Chongqing University of Post and Telecommunication,Chongqing, China, and the PhD degree in electrical engineering from Huazhong University of Science and Technology, Wuhan,China. Since 1998 he has been with the Department of Electronic and Information at Huazhong University. His research interestsare in ATM/IP networking, wireless ad hoc networks, queueing and stochastic systems.

Edward Chan received the BS and MS degrees in electrical engineering from Stanford University, and his PhD in computer sciencefrom the University of Sunderland. He is currently an Associate Professor in the Computer Science Department of the City Universityof Hong Kong. His research interests are in high-speed computer networks and mobile computing systems.

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