Stability Analysis of the PRDR Algorithm: A new Congestion Control Scheme for Wireless Networks

Abir Ben Ali , Ilhem Lengliz, , Farouk Kamoun CRISTAL Laboratory , National School of Computer Science

2010, La Manouba Abir.BenAli@isetr.rnu.tn , Ilhem.Lengliz@cristal.rnu.tn, Farouk.Kamoun@ensi.rnu.tn

Abstract — This paper investigates the congestion control issue in the Internet, especially in wireless environments, and proposes a new congestion control scheme, called Proportional and Derivative (PRDR). Based on the Active Queue Management (AQM) principle, the designed PRDR controller embedded to the router monitors the level of network congestion through the occupancy of the buffer which is maintained within a prefixed control target. Being based on the difference between them, the PRDR controller associated with each link computes periodically at time a fair rate and forwards the result to the next gateways till the destination. Using a feedback scheme, a destination supplies the source with the minimal received. In their turn, sources have to adapt their transmission rate according to received fair rate. Borrowing its modelling from automatic control theory, the design of stability and stabilization conditions of the PRDR algorithm are studied in depth. All theoretical results are tested through simulations under the network simulator ns-2 in a wireless environment.

Keywords — congestion control, wireless networks, PRDR, performance evaluation, simulations.

I. INTRODUCTION Traffic engineering encloses the modelling, characterization, measurement and control of network traffic in order to achieve specific performance purposes, including the effective and reliable data transfer as well as the scheduling of network resources under QoS guarantee. Network management and control can be considered a very complex task and may include issues regarding resource management, congestion control, connection admission control and active queue management. Congestion control schemes are introduced in order to regulate the traffic and to avoid the network collapse point. Such control has to fulfil a list of key requirements: stability, fairness and complexity of implementation. First, a congestion control scheme must converge to a fixed operational point and admit controlled oscillations, i.e. limited cycles with bounded variations. In both situations, the goal of the congestion control is to achieve a stable and effective state in terms of utilization, throughputs and delays. Second, the fair bandwidth allocation between competing flows has become a major problem for two reasons: the first one is the exponential growth of the internet, where users constantly seek for high rate and low delays communications. The other reason is the increase of best-effort traffic which does not apply any control and is harmful towards TCP connections. This kind of unfairness is called TCP-unfriendliness. Hence, the aforesaid problem has stimulated new proposals from both protocol and network designers,

which adopt two main approaches, not necessarily exclusive from each other. The first approach is to implement end-to-end congestion control, specifically conceived for real-time applications. Generally, the presented mechanisms, such as TCP [1], TFRC [2], don’t invoke any network assistance and observe the loss events as a network congestion indication. The second approach suggests ton involve the network into the congestion and shared resources management and consists in implementing an intelligent queue management, also called Active Queue Management, at Internet routers considered as network bottlenecks. Besides, the preservation of QoS guarantees in wireless networks has been one of the hot topics of networking research communities in the few recent years. Such preservation includes mechanisms and algorithms at different layers of the OSI reference model, in particular Physical layer, Medium Access Control (MAC) layer, IP layer and Transport layer. Because of high topology and link capacity fluctuation of wireless ad hoc networks, besides the contenting links, traffic and congestion control has to be investigated in different approaches. The rest of the paper is organized as follows: sections 2 surveys the congestion control issues and related proposals in wireless ad hoc environments. We present in section 3 the design of a new router-based congestion control scheme, called Proportional and Derivative (PRDR), which is based on the router occupancy level to monitor the network congestion. In concert with a rate explicit approach and feedback scheme, the PRDR algorithm uses the RTP protocol as an underlying transport protocol as well as RTCP protocol to convey control and feedback messages. The stability and stabilization conditions analysis of the PRDR algorithm are extensively investigated and studied in section 4. Section 5 provides simulation results aiming to evaluate the efficiency of the PRDR algorithm in wireless scenarios. II. ON CONGESTION CONTROL IN WIRELESS NETWORKS Due to their convenience and mobility, wireless networks represent an important trend for Internet access. Although, and despite extensive research and development, transport protocol still suffer in multihop wireless problems of stability and unpredictability, especially in presence of high traffic load [3, 4]. Reported deficiencies such as to high packet drop rates, unfair channel allocation and starvation and the particularly fluctuating topologies and bandwidths [5, 7, 8, 9] are the major cause of performance degradation of wireless multihop networks.

978-1-4244-8435-5/10/$26.00 ©2010 IEEE

Congestion control concerns in particular real-time flows and mobile multimedia applications (video, audio,…) which are based mostly on the UDP protocol. This transport protocol is known to provide an unreliable service and without any congestion control, making generally these applications non-reactive to a congestion state in the network, and thus leading to the network collapse and the throttling of TCP connections established yet.

Most of congestion control mechanisms proposed for wired Internet are no longer applicable in wireless environments, since they don’t take into consideration the new imminent features for such kind of networks. Thus, in end-to-end congestion control techniques as TFRC, the transmission rate adaptation is accomplished according the network state observed at the end hosts (loss events, RTTs). Such techniques assume a best-effort network and don’t deal with the intermediate routers behaviour, nor with the way do they allocate the shared bandwidth among the competing flows.

Hence, on one hand, the end hosts reaction could be biased in presence of wireless random losses, which could be mistaken for congestion losses, which results in inadequate transmission rate reduction and channel under-utilization.

On the other hand, large values of RTT triggered by the random backoffs in CSMA/CA based networks meaning a rate reduction of correspondent flows, which aggravates the unfairness situation in the network. It is shown in [10] that the TFRC algorithm presents a performance degradation when deployed in wireless environments, and this because of the RTS/CTS mechanism. Many researches intend to allocate bandwidths both fairly and dynamically for adaptive UDP-based applications in a wireless context [11, 12, 13], but present several weaknesses and disadvantages that we try to explicit briefly in the following: A first family of congestion control mechanisms that operates in a TFRC-like and end-to-end approach include at the end hosts loss discrimination algorithms (LDAs) in order to distinguish between random and congestion-related losses [11, 13, 14]. Nevertheless, in addition to difficulties experienced by theses algorithms as described for TFRC above, the LDAs are in most of the cases inaccurate and don’t work for all types of wireless topologies and constraints (wireless last hop, heterogeneous, ad-hoc,…) [6]. The second family assembles router-assisted algorithms, where bandwidth allocation is performed at the intermediate nodes (mobile nodes or gateways). These algorithms prove their efficiency and robustness since every node has the best knowledge of local information about crossing flows and the available bandwidth [15, 16]. But none of these proposals deals with the stability criterion and the convergence to a steady state. Motivated by this, we present in this paper the PRDR algorithm that we propose for UDP-based congestion control in a wireless and mobile context. Already

implemented for the wired Internet, the performance study of the PRDR algorithm by simulation showed its ability to guarantee a fair bandwidth allocation between the competing flows, as well as its convergence to a steady state [17]. PRDR relies on RTP as a transport protocol and its underlying control protocol RTCP to convey end-to-end control and feedback information. In order to accurately calculate the feedback, the PRDR controller must know the exact bandwidth capacity in advance. This parameter plays an important role in our algorithm but is hard to set in advance for a wireless link. Thus, we enhanced PRDR with an adaptive algorithm that estimates and computes a proper value of the quested capacity. III. DESIGN OF THE PRDR ALGORITHM A. THE PRDR ARCHITECTURE The PRDR scheme adopts an explicit rate feedback control, equivalent to ATM’s ABR and is enhanced with additional mechanisms to fit into a wireless multihop context. In the underlying data streaming transport layer, the Real-Time Protocol (RTP) [18] is used since it can be implemented on top of UDP/IP stack and is used to carry various multimedia traffics. Control and feedback information is sent using the RTCP protocol in usual sender and receiver RTCP reports packets (SR and RR). This prevents us from introducing a new header type in IP packets. . The behaviour of the PRDR algorithm is illustrated in figure 1 and is as follows: an incoming traffic source s connected to a destination d expresses its initial desired rate Rd in a field of an RTCP “Application-specific” control packet which is forwarded to next router until it reaches the destination d. Each intermediate node m on the path from s to d captures the value of Rd carried in the RTCP control packet and substitutes it to the locally computed fair share rate Rm if smaller and forwards this information to its neighbour node m+1. Finally, the control packet reaches the destination d with the smallest value of Rm on the connection path. Then, the destination sends the received fair share back to source s in an RTCP RR packet, which replaces its actual transmission rate Rd by the received Rm.

Figure 1. PRDR framework

Different sources periodically send RTCP control packets every TControl. The choice of the TControl value affects sensibly both the transient response (settling time and initial connection parameters) and the control overhead due to the computation and the transmission of the feedback information. Faster updates periods lead to shorter settling time, more rapid steady state and smaller buffer overshoot, whereas a smaller TControl value increases the control overhead. The

ds

Feedback

Rd

work in [19] provides further discussion on the major trade-offs involved in selecting the control period TControl. B. THE CONTROLLER EQUATIONS The PRDR algorithm was originally designed for ABR traffic and is based on control rules used in the automatic systems [20]. Every router uses a proportional and derivative controller in order to compute the supported rate taking into account the current and the past queue occupancy. We have adapted in a previous work the PRDR algorithm to the congestion control of UDP traffic in the Internet using the RTP/RTCP protocol suite as an underlying transport layer. Each source sends periodically a RTCP report containing a special field indicating the application desired rate Rd at which it desires to transmit. Intermediate routers, belonging to the path from the source to the destination adjust the Rd value according a locally computed fairshare, denoted RF, and forward the control message to next router until reaching the destination. This latter sends back to the source the actual rate value RF at which it is authorized to transmit. At its turn, the source adjusts its transmission rate to the value mentioned in the RTCP feedback message.

Figure 2. Functioning of the PRDR controller As depicted in figure 2, each node has a congestion controller associated to its outgoing link i, this controller calculates at each control period n a supported fair rate qi(n) packets/s based on local information: the difference between the buffer occupancy xi(n) and a fixed threshold x0, as well as the control decision at present and in the finite past : qi(n-1), qi(n-2), …., qi(n-k). Then, the dynamics of buffer i is described by the following equation:

0

)0)(()()1( 0

J

jjnijniqni xxqSatq

N,0

)( iK

kknik q (1)

and x(n+1) = x(n) + q(n+1) - (n) (2)

Where J and K are nonnegative integers. The saturation function is such that:

otherwise if

0 if 0)(

zaza

zzaSat

The discrete-time system is modelled by the couple of recurrent equations described in (1) and (2). The recurrence function determines the system state at sampling times n, n-1, n-2,… 1 , having in mind its state in the past The saturation function prevents qi(n) from being negative or growing infinitely in the case of non-congested paths. An appropriate choice of q0 value could be:

NCq0

where C is the link capacity in packets/s and N is the number of active connections crossing the link i. The term x0 is a threshold on the buffer occupancy and is chosen heuristically. Indeed, for a buffer capacity of 50 packets, a threshold of 40 packets could be selected. Conditions on control gains are established in Ben Mohamed’s work in [19], in order to ensure the steady state properties of this system: if xs and qs denote the steady state values corresponding to (1) and (2) under the assumption that the input traffic is constant, then:

NrC

qs0

qsJ

jj

K

k kxxs

0

00 (3)

where r0 is the rate (in packets per slot) of the offered traffic of the link i. In order to ensure that xs = x0, we choose j and

k so that:

00 0=

,0J

j

K

kkj (4)

C. WIRELESS ENHANCEMENTS The major challenge faced by Active Queue Management (AQM) schemes, where flow and congestion control are performed by the routers is the unknown wireless link bandwidth. In presence of highly dynamic topologies, interferences and contentions, it is roughly hard to speculate about the accurate value of the wireless link capacities. Recently, a number of capacity estimation proposals have begun to address this issue: packet-pair probing techniques [21, 22] aim to estimate a maximal achievable bandwidth on a source-destination path. This approach has the disadvantage to be biased and does not take into account the fairness issue between competing flows. Other approaches [23, 24] attempt to construct contention graphs and to fairly allocate the bandwidth between links belonging to the same vertex (called contention clique). Nevertheless, the implementation of graph theories and coloring still costy and complex.

xi(n) x0

Outgoing linkqi(n) Outgoing queue

In the PRDR equation, the saturation rate q0 ensures the

fair allocation between the flows, since its presents the maximal quota that could be given to a flow:

In wired networks, the target rate is simple to compute, since it represents the outgoing link capacity, but it is, as explained below non evident to guess in wireless multihop networks. We present here a straightforward distributed capacity estimation based on outgoing throughput observation at the queue level: assume N active flows at a router k, the measured throughput at the outgoing queue represents the maximal amount of data that can be sent by this router. Thus, it represents the maximal bandwidth of this intermediate router, which will be fairly shared between the N competitive flows. This estimation is accurate only for a lossless queue, that is, for a non overflowed buffer. Thus, we need to resize the Target rate’s value according to the fixed threshold x0, which represents an acceptable value of buffer occupation. Then, the estimation equation is as follows:

where x is the queue length in packets observed in the last control interval and IR is an incremental rate used to increase the link bandwidth estimation, and consequently the buffer occupancy x, till reaching the buffer threshold x0.

IV. DESIGN AND STABILITY ANALYSIS OF THE SECOND ORDER CONTROLLER We present in this section a design example of the PRDR algorithm by setting J = 1 and K =1 and we consider the Lyapunov linear and discrete system’s stability and the stabilizing conditions of the PRDR controller. The second-order proportional and derivative closed-loop system, for J = 1 and K =1 could be written as the following:

x(n+1) = x(n) + q(n+1) - (n) (5)

q(n+1) = Sat q0{ q(n) – 0 (x(n)- x0) - 1 (x(n-1)- x0 )

- ß0 q(n) – ß1 q(n-1) } (6)

The system state is represented by the state vector Y, a 4-dimension column-vector supporting the system’s state variables:

)1()(

0)1(0)(

)(

nqnq

nxnx

nY xx

It is shown that expressions (5) and (6) can be rewritten in the following canonical form:

Y(n+1) = A Y(n) + B (7)

The system’s transfer matrix A is such as:

010010110

0001101101

A

and

0001

B

The closed loop system described by equation (7) is steady if and only if the poles of the characteristic polynomial P( ), defined as the determinant of I - A have negative real parts [25].

The characteristic polynomial is defined as:

10010110

001101101

)(P

yields: P( )= [ 3 + ( 0-2+ 0) 2+( 1- 0+ 0+ 1+ 1)

+ 1- 1] (8)

Resolving the polynomial P, we draw conditions on values of control gains 0, 1, 0 and 1 according to feedback systems stability criterion stipulated in [25]. The simplified conditions are as follows:

-1< 0< 1 (9) 0 ( 0 +4) - 1 > 0 (10) 0 + 1> 0 (11) 0 - 1-4< 0 (12) 1 = - 0 (13)

The choice of the gain coefficients could then be driven among the definition domain defined by the five above conditions. V. PERFORMANCE OF THE PRDR ALGORITHM IN A WIRELESS ENVIRONMENT A. Network topology and test configurations In this section, we investigate the performances of the PRDR algorithm in a wireless network of an heterogeneous topology with base station (BS) on the ns2.31 network simulator that implements the CMU Monarch Project’s Wireless and Mobility Extensions [26].

As depicted in figure 3, a number k of wired nodes, connected to the BS via a gateway G with 15 Mbps and 10

Nq0 =

Target_Rate = Outgoing_Throughput, if x x0

Target_Rate + IR, else

Target Rate

ms-delayed links, initiate k CBR/RTP traffic to k wireless destinations that share a wireless channel of 11 Mbps. The payload packet size is 1500 bytes, whereas the DSDV protocol is used.

All results are given from five times simulations with 300 seconds duration each.

The BS implements the first-order PRDR algorithm, depicted in equation (6). The BS queue length is set to 50 packets (default parameter in the ns2 simulator). The threshold value on queue occupancy x0 sensibly affects both the queuing delays and the throughputs: smaller x0

results in smaller steady state queuing delays. Nevertheless, smaller value of x0 might lead to underutilization of the link capacity which results in a decrease in the network throughput. Therefore, x0 has to be large enough to ensure full utilization of the network capacity. Hence, we believe that the value of 40 packets is a suitable choice. The control period Tcontrol is set heuristically to 100 ms and corresponds to the maximal value of the round trip times over all the connections.

In our simulations, the following several important performance metrics are evaluated:

good end-to-end throughput ri: the amount of data delivered to the destination for flow i (1 i N)

BS buffer occupancy x

Throughput stability measured as the standard deviation for ri series, denoted i

Buffer occupation stability measured as the standard deviation for the occupation x, denoted

x

fairness index: we use the Jain’s fairness index used in [25] and defined by:

Figure 3. Network topology

B. Simulation results

1) Channel allocation We first consider the network described in section

3.1 where the BS uses a classical drop-tail queuing algorithm. Figure 4 plots the instantaneous throughputs

of six RTP connections as well as the instantaneous BS buffer occupation and shows the unfair channel allocation between the competing flows. The analysis of the ns2 traces revealed that the major cause of packet drops is buffer overflow (IFQ). The drop-tail queuing discipline is makes the traffic synchronised, which allows the first and the fourth flow to monopolize the queue space and consequently get the maximal channel bandwidth allocation. The mean value of the fairness index samples is 0.7. Moreover, the BS buffer is saturated all the time (x = 50 packets).

2) Channel allocation with PRDR algorithm

We present in this section simulation results with six competing RTP flows using the same network configuration described in section 5, A. A set of simulations were conducted for different values of the control gains 0, 1, 0

and 1, summarized in table 1.

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

0 50 100 150 200 250 300

time (s)

Rat

e (M

bp

s)

Flow1

Flow2

Flow3

Flow4

Flow5

Flow6

0

10

20

30

40

50

60

0 50 100 150 200 250 300time (s)

X

Figure 4: Instantaneous RTP rates and buffer occupation

TableI. Simulation Configurations for different values of the control gains

Configurations 0 1 0 1 C1 3 1.5 0.5 -0.5 C2 3 2 0.5 -0.5 C3 3.5 1.5 0.25 -0.25 C4 3.5 2 0.5 -0.5 C5 1 2 0.5 -0.5 C6 0.5 1.75 0.5 -0.5 C7 0.5 1.5 0.5 -0.5

The saturation rate q0 measured by the wireless enhanced PRDR mechanism is 780 kbps, which represents the maximal achievable fairshare. The table 2 below provides the performance criteria described in section 5, A for the seven configurations. The stability of each flow in term is expressed by standard deviations i, where smaller values of i

(tending to zero) characterize a more stable system.

N

iiN

N

ii

f

r

r

1

2

1

2

11Mbps

PRDR 15 Mbps 10 ms

Base station Gateway N RTP destinations

N RTP sources

As illustrations, we report the instantaneous RTP throughputs as well as the related buffer occupancy respectively for configurations 1, 2 and 4. Obviously, the six competing flows succeed in all cases to share the channel bandwidth fairly, where each flow gets its fairshare of 780 kbps (figures 5, 6 and 7) and the BS buffer occupancy is very close to the fixed threshold x0

of 40 packets. However, according to the plots in figure 7, the fourth configuration exhibits an instable behaviour, where several burst losses can be observed (greatest values of the throughput standard deviations). Moreover, the buffer shows frequent deviations from the threshold x0

(standard variation of 11.35). Besides, the best system behaviours are shown for the first and second

configurations, for which the throughputs are more stable (figures 5 and 6) and the occupations are closer to 40 packets.

3) Queuing delays with PRDR algorithm The PRDR controller regulates the congestion state by monitoring the buffer occupation level x keeping it nearby the threshold x0. By maintaining x at the level of 40 packets, the queuing delays are smaller (0.1 ms) than the case without any control (0.23 s). Figure 8 plots the averaged instantaneous delay (one-way trip delay) of the six RTP competing connections with and with PRDR control. Obviously, the PRDR scheme succeeds to reduce at half the queuing delay of the connections.

TableII. Performance parameters for the simulated configurations i (kbps) Configurations x

(packets) f

i=1 i=2 i=3 i=4 i=5 i=6 C1 9.01 0.99 52.60 41.64 36.74 57.07 32.56 41.00 C2 8.24 0.98 39.34 35.13 44.99 17.75 25.14 14.95 C3 10.71 0.98 53.43 62.65 47.91 35.34 58.80 41.42 C4 11.35 0.96 54.53 44.67 67.26 58.35 65.50 50.35 C5 9.11 0.98 31.62 45.80 52.12 75.66 44.46 52.76 C6 10.97 0.94 69.16 70.91 74.28 40.81 50.44 53.45 C7 8.82 0.97 37.30 13.86 30.66 36.03 35.97 37.39

0

0,5

1

1,5

2

2,5

0 50 100 150 200 250 300

Time (s)

Rat

e (M

bp

s)

Flow1

Flow2

Flow3

Flow4

Flow5

Flow6

0

10

20

30

40

50

60

0 50 100 150 200 250 300time(s)

Occ

up

ancy

(p

acke

ts)

Figure 5. Instantaneous RTP rates and buffer occupation for configuration 1

0

0,5

1

1,5

2

2,5

3

0 50 100 150 200 250 300

time(s)

Rat

(M

bp

s)

Flow1

Flow2

Flow3

Flow4

Flow5

Flow6

0

10

20

30

40

50

60

0 50 100 150 200 250 300

time (s)

Occ

up

ancy

(p

acke

ts)

Figure 6. Instantaneous RTP rates and buffer occupation for configuration 2

0

0,5

1

1,5

2

2,5

3

0 50 100 150 200 250 300

time (s)

Rat

e (M

bp

s)

Flow1

Flow2

Flow3

Flow4

Flow5

Flow6

0

10

20

30

40

50

60

0 50 100 150 200 250 300

time(s)

Occ

up

ancy

(p

acke

ts)

Figure 7. Instantaneous RTP rates and buffer occupation for configuration 4

0

0,05

0,1

0,15

0,2

0,25

0,3

0 50 100 150 200 250 300

time (s)

del

ay (

s)

UDP delays

PRDR delays

Figure 8. Instantaneous delays with and without PRDR control for configuration 1

VII. CONCLUSION AND FUTURE WORK This paper presents the design and an analysis of a congestion control mechanism (PRDR) for multimedia traffic over wireless networks. We have implemented the new algorithm and presented some simulation results in ns2. Results show that our mechanism can fairly allocate wireless bandwidth resource in heterogeneous networks and converges to a steady state. An in-depth study is conducted for the choice of the control gains, in order to insure the system stability in a fair way. Moreover, the buffer occupancy is better controlled and kept close to the fixed threshold x0. There are several further study topics. First, the behaviour of PRDR should be investigated in more realistic network topologies including mobile devices and random input traffic. Second, it is important to carry out a comparative performance study of PRDR with similar congestion control, such a cited in [15, 16].

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