Distributed Optimal End-to-End Delay Robustnessand Network Throughput Tradeoff in

Communication-Control NetworksMuhammad Tahir∗, Member, IEEE, and Sudip K. Mazumder†, Senior Member, IEEE

Abstract

A resource optimization framework to achieve an optimal tradeoff between end-to-end delay robustness and

network throughput is proposed. The selection of the objective function for delay robustness, providing the desired

tradeoff, is based on the sensitivity analysis. For maximizing the network throughput an effective link transmission

rate based power control problem is solved. We then extend the resource optimization framework using an

iterative suboptimal cross-layer algorithm to improve the link congestion fairness. Using our distributed resource

optimization algorithm we study the effect of delay threshold imposed by the application layer on the robustness.

Our results show that a small compromise in the network throughput can provide a large delay robustness depending

on the operating point. An improvement in the link congestion fairness performance at the cost of reduced network

throughput is also studied.

Index Terms

Delay, robustness, resource optimization, distributed algorithm.

This work is supported by the National Science Foundation (NSF) CAREER Award (Award No. 0239131) and Office of Naval Research(ONR) Young Investigator Award (Award No. N000140510594) received by Prof. Mazumder in the years 2003 and 2005, respectively.However, any opinions, findings, conclusions, or recommendations expressed herein are those of the authors and do not necessarily reflectthe views of the NSF and ONR.∗National University of Ireland, Maynooth, Ireland. mtahir@eeng.nuim.ie†University of Illinois at Chicago. mazumder@ece.uic.edu

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NOMENCLATURE USED.

ε Delay robustness parameterLout(ni) Set of outgoing links from node ni

Gji Channel gain from transmitter of link i to receiver of link j

Pmax Maximum nodal transmit powerPl Transmitter power for link l

Bmax Maximum packet buffer sizewl Link weightrsi End-to-end rate for transmission session si

Rmin(si) Minimum end-to-end rate requirementθl Threshold for wl to decide presence or absence of a link l in the network

L(si) Shortest route associated with transmission session si

Dmax(si) Maximum end-to-end delay thresholdhi Transmission scheduleRl Average transmission rate at link l

H Transmission cycle comprising set of transmission schedulesγl Signal to interference and noise ratio at link l

ηl Link efficiency functionµsi Packet length used for si

ΩηlOptimal network throughput based on ηl

ΩclOptimal network throughput based on link capacity cl

I. INTRODUCTION

Using wireless networks for information exchange in distributed systems necessitates meeting the end-

to-end delay-threshold requirements imposed by the application layer [1], [2], [3], [4]. Network control

systems [1], hierarchical interactive communication-control networks [2], [3], wireless multimedia sensor

networks [4], to name a few, require delay guarantees from the underlying communication network

to meet the quality of service demands of the application. The task becomes even more challenging

when a wireless communication interface is used for inter-node information exchange. For instance, in

the case of distributed implementation of model predictive control [5], [6], the network performance is

dependent on the coordination among the node controllers using inter-node information exchange [7].

In the case of wireless multimedia sensor networks, the quality of the multimedia stream and the buffer

size requirements on the receiver node [4] are a function of ene-to-end delay performance of the wireless

communication network. Delay dependent performance of a distributed-control system [8], poses new

challenges requiring an optimal utilization of the underlying communication network resources [9]. The

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physical layer design, the MAC protocol and the routing algorithm jointly affect the end-to-end delay

and the packet-drop probability [10]. The sampling rate of the control system from wireless network

standpoint is a parameter of the application layer and determines the minimum rate and the maximum

end-to-end delay-threshold requirements to be met by the wireless communication network.

For a given sampling rate and the corresponding minimum-rate requirements by the application layer,

the objective of a wireless-network design is to maximize the network-resource-utilization while meet-

ing the end-to-end delay thresholds. But, an optimal resource utilization problem with proportional

fairness as discussed in [11], [12] can provide a solution with end-to-end delays approaching delay

thresholds. This results in an optimal wireless network throughput at an expense of vulnerable distributed

control-system, which is prone to performance degradation and/or control-system instability due to delay-

threshold violations. To prevent such an event from happening, we have proposed a resource optimization

framework, which provides delay margin (a measure of the gap between optimal end-to-end delay and

the corresponding delay-threshold) by introducing the end-to-end delay-robustness parameter ε (will be

simply called robustness parameter). The introduction of ε leads to delay margin at the expense of slightly

degraded throughput performance of the wireless communication network. The parameter ε achieves an

optimal tradeoff between network-throughput and delay-robustness for a given delay-threshold set by the

application layer.

To realize the above mentioned tradeoff, we have formulated a resource-optimization problem, which

captures the delay-robustness in the end-to-end delay constraints through parameter ε and the price for that

robustness is penalized in the objective function for each transmission session between a source-destination

pair. For a given throughput objective, we have used sensitivity analysis to define the robustness objective

function leading to an optimal tradeoff between contending robustness and network-throughput param-

eters. In a relevant work the authors in [13] have used an energy-robustness tradeoff while performing

the distributed network power control. Our distributed resource optimization algorithm (DROA) achieves

an optimal tradeoff between network-throughput and the delay-robustness for any source-destination pair

independently in contrast to the framework proposed in [13], which achieves the tradeoff at the network

level by considering the total power. We also provide an efficient distributed power control based on the

link-efficiency function [14]. When compared to link capacity based power control [15], the proposed

power control provides better end-to-end delay guarantees at the cost of degradation in the network-

throughput performance.

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The DROA is extended to an integrated cross-layer framework, which incorporates the effect of queuing

delays while evaluating the link weights to achieve link congestion fairness. This results in further

robustness exploiting the cross coupling between the network and the physical layers of the protocol

stack [16], [17]. For instance, a power-and-congestion-aware routing protocol relies on the current power

assignments and the resulting link delays at the transmitting nodes and at the same time optimal power

assignment depends on the current network topology, which is dependent on routing [18]. This leads

to a strong cross coupling between power control and routing due to the fact that both of them are

dependent on the interference distribution in the network. Recently, the problem of joint power control at

the physical layer and power-aware routing at the network layer is discussed in [16], [19]. The iterative

solution proposed in [16] adapts the routes after computing optimal powers without considering the effect

of route switching on scheduling. On the other hand, the framework proposed in [19] is based on the

idea of flow splitting by assigning the rates to the links based on their transmission power levels. The

proposed iterative cross-layer algorithm (ICLA) is different from [16] as it is based on iterative updates

of both the routes and transmission schedules after achieving convergence of DROA. In contrast to [19],

we do not allow flow splitting to reduce the implementation complexity. In the proposed solution, once

DROA is converged, we use the optimal power allocation as well as the current congestion price (in the

form of queuing delay) to update the routes leading to an improved congestion fairness. Rest of the paper

is organized as follow.

We provide a network model and construct the resource optimization problem incorporating the robust-

ness tradeoff in detail in Section II. In Section III, first the objective function for robustness is obtained

using sensitivity analysis followed by the development of DROA for solving the resource optimization

problem distributively, for a given set of routes and transmission schedules. Using the optimal parameters

obtained from DROA, we next provide ICLA for joint routing, scheduling and resource allocation in

Section IV. Numerical results for optimal tradeoff and convergence performance are provided in Section

V. Finally, we conclude our contributions and discuss some of the possible future research directions in

Section VI.

II. SYSTEM MODEL AND PROBLEM FORMULATION

In this section we outline the network model and develop the constraint set and the multi-objective

function leading to the network resource-optimization problem formulation.

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A. Network Model

We model the wireless network as a weighted directed graph with N and L representing the sets of

nodes and links, respectively. Define Lout(ni) ∈ L as the set of outgoing links from node ni ∈ N . Each

link lij , from node nj to node ni abbreviated as l, has two associated attributes: a) the receiver output

power from the intended transmitter of link l, given by GllPl, where Gll represents the channel-gain and

Pl is the transmitter power for link l s.t. Pl ∈ P, 0 ≤ Pl ≤ Pmax∀l; and b) the average queue length Ql.

Using the received-power and average-queue-length attributes associated with each link we define the

link weight wl as

wl =

Pmax

GllPl+ Ql

Bmax

(1

GllPl/Pmax+ Ql/Bmax

)≤ θl

∞ otherwise∀l. (1)

The parameter θl is user defined threshold and depends on the maximum transmitter power Pmax and the

node buffer size Bmax. A possible initialization for wl in (1) is Pl = Pmax/2 and Ql = 0.

In a multi-hop wireless network, the transmission sessions are denoted by set S, where each trans-

mission session si ∈ S represents an ongoing transmission between a source-destination pair through the

intermediate nodes. Each transmission session si is characterized by the following attributes:

• A shortest directed route consisting of a subset of links L(si) ⊆ L ∀si, which can be obtained using

the initialized wl ∀l and employing Dijkistra’s shortest route algorithm [20];

• An associated end-to-end session rate rsi∈ r, where r is the set of rates for active transmission

sessions;

• The minimum rate Rmin(si) requirement ∀si;

• An end-to-end delay-threshold Dmax(si) ∀si imposed by the application layer (in this case given by

the delay dependent performance or stability criteria of the distributed system).

Different minimum data rates Rmin(si) and maximum end-to-end delays Dmax(si) required by different

si constitute heterogeneous wireless network traffic. At the medium access control (MAC) layer we

define transmission cycle H , to be the set of transmission schedules hi ∈ H . We allow more than one

transmission in each hi leading to an interference-limited wireless-network. As a result each transmission

schedule hi has an associated subset of simultaneously transmitting links L(hi) ⊆ L. Simultaneous

transmission between node pairs are allowed when the distance Γkj between the transmitter of link j and

the receiver of link k satisfies Γkj ≥ νΓkk, where the choice of ν is based on the acceptable interference

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level.

We define average transmission rate Rl at link l in one transmission cycle as Rl =∑

hiIl(hi)Rl/|H|.

In the expression for Rl, |H| is the number of transmission schedules hi in one transmission cycle, Rl

is the instantaneous transmission rate and Il(hi) is an indicator function defined by

Il(hi) =

1 l ∈ L(hi)

0 otherwise. (2)

The packet success rate (PSR) at link l is a function of the received signal-to-interference-and-noise-ratio

(SINR) given by γl(P) = GllPl

zl+∑

m6=l GlmPm, where zl is the additive noise. The PSR can be obtained from

link-efficiency function ηl(γl(P)) [14], [21], which is an increasing, continuous and S-shaped (sigmoidal

[22]) function with ηl(∞) = 1. The link-efficiency function for narrow-band modulation also satisfies

ηl(0) = 0 and is valid for many practical cases [23]. Now, the effective data rate at link l is obtained by

scaling average data transmission rate Rl with PSR and is Rlηl(γl(P)).

B. Link-Efficiency Function

To obtain an expression for link-efficiency function, which is an effective measure of the PSR, we first

quantify link packet-error rate (PER). The PER, for many modulation and channel-coding schemes, can

be well approximated by the following family of functions [21], [24]:

PER(γl(P)) =1

1 + eb(γ(dB)l (P)−σ)

. (3)

In (3), γ(dB)l (P) = 10 log10(γl(P)) and b and σ are fitting parameters, which mainly depend on the

modulation type, channel-coding scheme and the packet length used, and can be obtained offline. Fig. 1

shows an example packet-error rate curve as a function of link SINR γl for b = 0.8 and σ = 5. It should

be noted that, the PER function is convex in γl for γl(P) > γmin(l) but not in Pl. When PER(γl(P)) is

small we can approximate (3) using

PER(γl(P)) ≈ e−b(γ(dB)l (P)−σ),

= ebσ(γl (P))−cb , (4)

where the constant c is 10(loge 10)−1. Now the link-efficiency function is obtained to be

ηl(γl(P)) = 1− ebσ(γl (P))−cb . (5)

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The link-efficiency function in (5) will be used to derive the expression for the end-to-end delay discussed

in the next subsection.

C. End-to-end Delay Model

For the delay model, we first focus on a single link. The average delay at link l due to queuing and

packet transmission, when using M/D/1 queuing model [20] and a packet length of µsifor session si, is

obtained fromµsi

2

1

Rlηl(γl(P))−∑si: l∈L(si)

rsi

+1

Rlηl(γl(P))

, (6)

where Rlηl(γl(P)) and∑

si: l∈L(si)rsi

represent the effective transmission and average arrival rates,

respectively, for link l. Our link delay formulation is different from the one proposed in [11], [25],

where the authors have modelled the average link-transmission-rate using the link capacity. It is well

known that capacity achieving codes require large word lengths and complex processing and hence cannot

provide delay guarantees. On the other hand our formulation models average link-transmission-rate as the

multiplication of Rl and ηl(γl(P)) to obtain average link delay. Using (6), we obtain end-to-end delay

bounded by Dmax(si) for transmission session si by accumulating the link delays along the shortest route

L(si) as1

2

∑

l∈L(si)

µsi

Rlηl(γl(P))−∑si: l∈L(si)

rsi

+µsi

Rlηl(γl(P))

≤ Dmax(si). (7)

In (7), first term approximately measures the waiting time in the queue and second term corresponds to

the transmission delay. The possibility of an individual link being member of different shortest routes

requires a systematic procedure to adapt Dmax(si) by the application. This is achieved by first perturbing

Dmax(si) for some si and then performing sensitivity analysis leading to the selection of an optimal delay

robustness objective function.

D. Resource Optimization Problem

To formulate the wireless-network resource-optimization problem, we introduce the link transmission

d(t)l ∈ d(t) and queuing d

(q)l ∈ d(q) delay auxiliary variables to decompose the inequality in (7) into the

following two inequalities∑

l∈L(si)

(d

(t)l + d

(q)l

)≤ Dmax(si), (8)

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µsi/2

Rlηl(γl(P))−∑si: l∈L(si)

rsi

≤ d(q)l ,

µsi/2

Rlηl(γl(P))≤ d

(t)l . (9)

We now introduce the robustness parameter εsi∈ Υ, and εsi

∈ [0, 1) for distributed system application

layer by modifying the delay constraint in (8) as∑

l∈L(si)(d

(t)l +d

(q)l ) ≤ (1−εsi

)Dmax(si). The parameter

εsiis upper bounded by 1 due to the fact that the transmission rate can not be increased arbitrarily and it

is not allowed to take on negative values to avoid system instability due to delay threshold violation. In

contrast to the approaches providing delay guarantees [26], the robustness parameter provides delay margin

against wireless communication network performance fluctuations mainly due to time-varying channel

gains, link congestion, route switching and allows the distributed system to respond to these fluctuations.

Next, we formulate constrained resource optimization problem to achieve the tradeoff between robustness

and network-throughput:

maximize J =∑si

αsiU(rsi

) + (1− αsi)φ(εsi

) , (10)

s.t.1

Dmax(si)

∑

l∈L(si)

(d

(t)l + d

(q)l

)≤ (1− εsi

), ∀si (11)

µsi

2d(q)l

≤Rlηl(γl(P))−

∑

si: l∈L(si)

rsi

,

µsi

2d(t)l

≤ Rlηl(γl(P)), ∀l (12)

0 ≤ d(t)l , d

(q)l , γmin(l) ≤ γl(P), 0 ≤ Pl ≤ Pmax, ∀l (13)

0 ≤ εsi< 1, Rmin(si) ≤ rsi

∀si. (14)

In (10), J is the overall objective function, U(.) and φ(.) are concave functions of their respective

parameters, αsiis the user-defined tradeoff parameter to achieve a desired level of robustness. In (13),

γmin(l) is the received SINR threshold at link l to ensure the convexity of PER. For a given U(rsi), the

objective function φ(εsi) will be used to modulate the robustness-throughput tradeoff by adjusting εsi

. We

will derive an expression for φ(εsi) using sensitivity analysis, which will achieve an optimal robustness-

throughput tradeoff. It is worth mentioning that, setting αsi= 1 and replacing the link-efficiency function

with fixed link capacity leads to the simplified version of the problem discussed in [11]. The resource-

optimization problem discussed in [11], and its extended version with power control in [12], lead to an

optimal solution where∑

l∈L(si)(d

(t)l + d

(q)l ) → Dmax(si) for some si. This leaves negligible margin for

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the application layer to respond to any network topological variations or time varying channel gains and

is one of the motivations leading to the robustness-throughput optimal tradeoff problem in (10)–(14).

III. SOLUTION APPROACH AND DISTRIBUTED ALGORITHM

The resource optimization problem in (10)–(14) is nonlinear due to the link-efficiency function in (5),

the choice of the objective functions and the structure of the link delay inequalities in (12). A distributed

solution to the resource optimization problem will be achieved by first decomposing the problem into

sub-problems and then solving the individual sub-problems coupled through the dual variables. The

introduction of auxiliary variables and usage of ‘log’ transformation will translate each of the nonlinear

sub-problems to an equivalent convex form. To obtain this we first associate dual variables λl ∈ Λ, ξl ∈ Ξ

and ψsi∈ Ψ with end-to-end, queuing and transmission delays in (11)–(12), respectively, to form the

Lagrangian given by

L(r, P, d(q), d(t),Υ,Λ,Ξ,Ψ) = maximize

∑si

(αsiU(rsi

) + (1− αsi)φ(εsi

)) +∑si

ψsi((1− εsi

)

− 1

Dmax(si)

∑

l∈L(si)

(d

(t)l + d

(q)l

) +

∑

l

ξl

(Rlηl(γl(P))− µsi

2d(t)l

)

+∑

l

λl

Rlηl(γl(P))−

∑

si: l∈L(si)

rsi− µsi

2d(q)l

| γmin(l) ≤ γl(P),

0 ≤ d(t)l , d

(q)l , 0 ≤ Pl ≤ Pmax, 0 ≤ εsi

< 1, Rmin(si) ≤ rsi. (15)

= maximize

∑si

αsi

U(rsi)−

∑

l∈L(si)

λlrsi

| Rmin(si) ≤ rsi

+

∑si

((1− αsi)φ(εsi

) + ψsi(1− εsi

)) | 0 ≤ εsi< 1

−

∑

l

λlµsi

2d(q)l

+ξlµsi

2d(t)l

+∑

si: l∈L(si)

ψsi

d(t)l + d

(q)l

Dmax(si)

∣∣∣ 0 ≤ d(t)l , d

(q)l

+

∑

l

((λl + ξl)Rlηl(γl(P))

) | γmin(l) ≤ γl(P), 0 ≤ Pl ≤ Pmax

]. (16)

The maximization problem in (16) is decomposable into rate rsi∈ r, delay d

(t)l ∈ d(t) and d

(q)l ∈ d(q), the

robustness control εsi∈ Υ and the link transmitter power control Pl ∈ P sub-problems. The associated

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dual problem is

minimize g(Λ,Ξ,Ψ) s.t λl, ξl, ψsi≥ 0 ∀ l, si. (17)

In (17), g(Λ,Ξ,Ψ) = L(r∗,P∗,d∗(q),d∗(t),Υ∗,Λ,Ξ,Ψ) and r∗, P∗, d∗(q), d∗(t) and Υ∗ are the optimal

primal variables obtained by solving (16). The block diagram representation in Fig. 2 shows a possible

realization of DROA by decomposing the original problem into sub-problems. Next, we will discuss the

solution approaches for these sub-problems and their distributed implementation. The proof of convergence

for the distributed realization of DROA is provided in Appendix A.

A. Robustness Sub-problem

The robustness sub-problem from (16) is described by

maximize∑si

((1− αsi)φ(εsi

) + ψsi(1− εsi

)) s.t. 0 ≤ εsi< 1. (18)

As pointed out earlier, for a given U(rsi), which is log(rsi

) in our case to achieve proportional through-

put fairness [27], the objective function φ(εsi) is responsible for modulating the robustness-throughput

tradeoff. To achieve this tradeoff optimally, we need to choose an appropriate φ(εsi). For that purpose, as

a first step we fix αsi= 1 ∀si in (10)–(14) and use sensitivity analysis to study the effect of perturbing

the end-to-end delay-threshold Dmax(si) on the optimal network-throughput.

1) Step 1 – Sensitivity Analysis: We perturb end-to-end delay constraint for sthi session by usi

∈ u and

observe its effect on optimal network-throughput ρ∗(u) by solving the following perturbed optimization

problem:

ρ∗(u) = maximize∑si

U(rsi)

1

Dmax(si)

∑

l∈L(si)

(d

(t)l + d

(q)l

)≤ usi

, ∀si

and constraints (12)− (14). (19)

The usi= 1 ∀si, represents the unperturbed system, while 0 < usi

≤ 1 and usi> 1, respectively,

correspond to tightening (throughput reduction) and loosening (throughput increment) the sthi constraint.

If ψ∗si, corresponding to ρ∗(1), represent the optimal value of the Lagrange multipliers associated with the

unperturbed end-to-end delay constraints then the fractional change in the optimal network-throughput

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due to sthi constraint perturbation is obtained as

ρ∗(u)− ρ∗(1)

ρ∗(1)=

ρ∗(usiesi

)− ρ∗(1)

ρ∗(1),

= (usi− 1)

∂ρ∗(1)/∂usi

ρ∗(1)+ o(usi

),

= (usi− 1)(ψ∗si

/ρ∗(1)) + o(usi) ≈ (usi

− 1)ψ∗si

ρ∗(1). (20)

In the first equality of (20) esiis a vector with all its entries equal to 0 with the exception of sth

i

entry, which is 1. The second equality follows from the Taylor series expansion and in the third equality

we have used the fact that ∂ρ∗(1)/∂usi= ψ∗si

[22], followed by the first order approximation. Fig. 3

shows the effect of varying usi, for session si, from its nominal value on the optimal network-throughput

performance. The supporting hyper-plane ρ∗(1)+ψ∗si(usi

− 1) shown in Fig. 3 at ρ∗(1) with gradient ψ∗si

illustrates the result in (20).

2) Step 2 – Choice of Delay Tradeoff Objective Function: Differentiating the objective function in

(18) with respect to εsiand evaluating at optimal point we get ∂φ(εsi

)/∂εsi|εsi=ε∗si

= −ψ∗si/(1 − αsi

).

Replacing ψ∗siwith −(1−αsi

)∂φ(εsi)/∂εsi

|εsi=ε∗siand usi

with (1−εsi) (for usi

∈ (0, 1]) in the expression

for fractional change in throughput in (20) we have

(1− αsi)

(εsi

ρ∗(0)

)∂φ(εsi

)

∂εsi

∣∣∣εsi=ε∗si, (21)

where ρ∗(0) is obtained by mapping ρ∗(1) from usito εsi

domain. The reason for mapping usito εsi

in the

interval (0, 1] is to ensure that Dmax(si), defined by the application layer, is not violated. If the maximum

throughput fraction for sthi session, available for tradeoff with the end-to-end delay, is δsi

(1−αsi)/ρ∗(0),

then equating (21) to this maximum available throughput fraction, leads to

∂φ(εsi)

∂εsi

=δsi

εsi

, (22)

and integrating (22) gives

φ(εsi) = δsi

log(εsi). (23)

Using (23), the robustness-throughput tradeoff sub-problem becomes maximize∑

si((1−αsi

)δsilog(εsi

)+

ψsi(1− εsi

)) s.t. 0 ≤ εsi< 1. This problem can be solved distributively in εsi

using efficient algorithms

available for convex optimization [22].

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B. Power Control Sub-problem

The objective of maximizing accumulated link transmission rates for all active links in an interference-

limited wireless-network leads to the following coupled and constrained power control problem:

maximize∑

l

((λl + ξl)Rlηl(γl(P))

)

s.t. γmin(l) ≤ γl(P), 0 ≤ Pl ≤ Pmax ∀l (24)

From (5), the maximization problem in (24) is equivalent to the following minimization problem

minimize∑

l

((λl + ξl)Rle

bσ(γl(P))−cb)

s.t. γmin(l) ≤ γl(P), 0 ≤ Pl ≤ Pmax ∀l (25)

The problem in (25), is not convex optimization problem but, can be transformed into a convex problem

as explained in the sequel. By introducing the auxiliary variables tl, we rewrite the problem in (25) in

epigraph form [22] as

minimize∑

l

(λl + ξl)Rlebσtl s.t. (γl(P))−cb ≤ tl γmin(l) ≤ γl(P), 0 ≤ Pl ≤ Pmax ∀l. (26)

Next, we apply ‘log’ transformation to the inequality constraints (γl(P))−cb ≤ tl ∀l and define tl = log(tl)

and γmin(l) = log(γmin(l)) ∀l. Now the problem in (26) becomes

minimize∑

l

(λl + ξl)Rlebσetl

s.t. − log(γl(P)) ≤ tlcb

γmin(l) ≤ log(γl(P)), 0 ≤ Pl ≤ Pmax ∀l. (27)

The constraint − log(γl(P)) ≤ tlcb

is tight near optimal solution and the feasibility of the problem in (27)

allows the two constraints − log(γl(P)) ≤ tlcb

and γmin(l) ≤ log(γl(P)) to be combined as log(γl(P)) ≥− tl

cb≥ γmin(l) and rewritten as log(γl(P)) ≥ − tl

cband tl ≤ −(cb)γmin(l). Next we assign Lagrange

multipliers πl ∈ Π ∀l to the constraints − tlcb≤ log(γl(P)) to obtain

minimize∑

l

(λl + ξl)Rlebσetl −

∑

l

πl

(log(γl(P)) +

tlcb

)

s.t. tl ≤ −(cb)γmin(l), 0 ≤ Pl ≤ Pmax ∀l. (28)

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The dual problem corresponding to the minimization problem in (28) is given by maximize gP(Π),

s.t. πl ≥ 0 ∀l. The problem in (28) is decomposed into sub-problems in variables tl and Pl ∀l as

minimize∑

l

((λl + ξl)Rle

bσetl − πltlcb

)s.t. tl ≤ −(cb)γmin(l) ∀l, (29)

maximize∑

l

πl log(γl(P)) s.t. 0 ≤ Pl ≤ Pmax ∀l. (30)

The convexity of the sub-problem in (29) can be verified by the condition − 1tl∂f(tl)/∂tl ≤ ∂2f(tl)/∂t2l ,

where f(tl) is the objective function in (29). The sub-problem in (30) is not convex but, can be transformed

into an equivalent convex form by ‘log’ transformation of power vector P [15]. For that, we first define

Pl = log(Pl) ∀Pl ∈ P and then evaluate the Hessian of (30). Once the convexity is ensured, the gradient

of (30) is used to update Pl ∀l. The associated dual problem, maximize gP(Π), s.t. πl ≥ 0 ∀l, is solved

by using the following sub-gradient update

π(k + 1) =

[π(k)− βπ

(log(γl(P)) +

tlcb

)]+

. (31)

In (31), βπ is constant step size and [x]+ is defined as max0, x. A constant step size allows faster

convergence near optimality but can lead to frequent sign reversals of the slope of dual variable updates.

This situation can be avoided by using a diminishing step-size rule at the expense of slower convergence

near optimality [28]. A proof of convergence for the distributed power control can be drawn on the same

lines as given in Appendix A.

C. Rate Allocation Sub-problem

The rate allocation sub-problem from (16) is given by

maximize∑si

αsi

U(rsi)−

∑

l∈L(si)

λlrsi

s.t. Rmin(si) ≤ rsi

. (32)

The objective function U(rsi) is defined to be concave for proportional fairness among different trans-

mission sessions and is U(rsi) = log(rsi

). The rate sub-problem in (32) is separable in rsiand can be

solved for each rsiseparately. It turns out that a closed-form solution for this problem is possible as

discussed in the sequel. Introducing the multipliers ϑsifor the rate constraint Rmin(si) ≤ rsi

, we obtain

13

the KKT conditions for the sub-problem in (32) as follows:

Rmin(si) ≤ r∗si, 0 ≤ ϑ∗si

∀si,

(r∗si−Rmin(si)

)ϑ∗si

= 0 ∀si,

αsi

r∗si

−∑

l∈L(si)

λl + ϑ∗si= 0 ∀si. (33)

In (33), x∗ shows the optimal value of the variable x. It is observed that ϑ∗si∀si are slack variables and

can be eliminated, reducing (33) to

Rmin(si) ≤ r∗si,

1

r∗si

≥∑

l∈L(si)

λl ∀si, (34)

∑

l∈L(si)

λl − αsi

r∗si

(

r∗si−Rmin(si)

)= 0 ∀si. (35)

Now if Rmin(si) ≥(αsi

/(∑

l∈L(si)λl)

)then the condition in (35), will hold for r∗si

= Rmin(si); and for

the opposite case it will hold for r∗si= αsi

(∑l∈L(si)

λl

)−1

. So we have a closed-form solution, for r∗si,

given by

r∗si=

Rmin(si) Rmin(si) ≥ αsi

(∑l∈L(si)

λl

)−1

αsi

(∑l∈L(si)

λl

)−1

otherwise. (36)

D. Link Delay Sub-problem

The link delay minimization sub-problem from (16) is given by

minimize f(d(t),d(q)) =∑

l

λlµsi

2d(q)l

+ξlµsi

2d(t)l

+∑

si: l∈L(si)

ψsi

d(t)l + d

(q)l

Dmax(si)

s.t. 0 ≤ d(t)l , d

(q)l ∀l. (37)

The optimization sub-problem in (37), is convex if the Hessian of the objective function is positive

semi-definite. Since the objective function is not coupled in the link delay variables, the convexity for

f(d(t),d(q)) can also be verified by using the inequality

− 1

d(i)l

∂f(d(t),d(q))

∂d(i)l

≤ ∂2f(d(t),d(q))(∂d

(i)l

)2 ,

where i ∈ q, t, ∀i. The delay subproblem is then solved by gradient projection method [29].

14

E. Dual Problem

The dual problem in (17) associated with the original problem, can be solved using a sub-gradient

method [28]. The iterative updates for λl and ξl link dual variables, are given by

λl(k + 1) =

λl(k)− βλ

Rlηl(γl(P))−

∑

si:l∈L(si)

rsi− µsi

2d(q)l

+

, (38)

ξl(k + 1) =

[ξl(k)− βξ

(Rlηl(γl(P))− µsi

2d(t)l

)]+

, (39)

and the sub-gradient updates for the ψsidual variables, associated with robustness-parameter, are obtained

as

ψsi(k + 1) =

ψsi

(k)− βψ

(1− εsi

)− 1

Dmax(si)

∑

l∈L(si)

(d

(t)l + d

(q)l

)

+

. (40)

The variables βλ, βξ and βψ in (38)–(40) are the respective step sizes for the dual updates. The block dia-

gram in Fig. 2 shows the distributed implementation of DROA using dual decomposition. The distributed

DROA is initialized by constructing the network topology using Pl = Pmax/2 along with an associated

feasible transmission schedule hi ∈ H .

IV. ITERATIVE CROSS-LAYER ALGORITHM

In the previous section, we proposed a distributed resource optimization algorithm, leading to an optimal

tradeoff between the network-throughput and corresponding delay-robustness. To achieve convergence,

we initialize DROA with a feasible set of routes L(si) ∀si and transmission schedules hi ∈ H . Once

DROA is converged we update the link weights wl ∀l using (1) based on the optimal link power and

delay components. Employing this updated set of link weights, we reevaluate the shortest routes L(si) ∀si

with the possibility of route switching. The route switching is attributed to higher link congestion, which

is taken into account by using link queuing delays in the expression for wl.

Finding an optimal transmission schedule for the updated routes is NP hard [30]. Using an interference

aware link scheduling approach [31] and updated routes, we provide a suboptimal ICLA outlined in

Algorithm 1. The proposed ICLA, achieves convergence in a small number of iterations (less than 10

iterations for a network of 50 nodes). The algorithm alternates between DROA and route and schedule

updates until further improvement in the performance cannot be achieved. Below we describe the key

steps in ICLA to achieve convergence:

15

1) For a given initial set of routes and transmission schedules, when DROA has converged, we update

wl for all existing links according to (1) using P ∗l , d

∗(t)l and d

∗(q)l . The optimal link delay components

d∗(t)l and d

∗(q)l are used to obtain Q∗

l . An existing link l is removed from the network topology if(1

GllP∗l /Pmax

+ Q∗l /Bmax

)> θl,. This can happen if the transmitting node of that link is highly

congested or the channel is in deep fade;

2) Using updated wl ∀l the shortest routes L(si) ∀si and the corresponding transmission schedules

hi ∈ H ∀i are computed again. To ensure convergence and avoid unnecessary iterations of ICLA,

we require J∗ ≤ J∗new and |H|new ≤ |H|, where |H|new is updated transmission cycle consisting of

recomputed transmission schedules and J∗new is the recomputed optimal objective value for DROA;

3) The ICLA is terminated if one of the following conditions are met:

• Objective condition J∗ ≤ J∗new for DROA is violated;

• The transmission cycle constraint |H|new ≤ |H| is violated;

• For a pre-defined tolerance χ, the convergence of ICLA is achieved.

Algorithm 1 Iterative Cross-Layer Algorithm (ICLA)Require: Choose feasible rsi

, Pl, hi and L(si). Set flag = 1

compute r∗si, d

∗(t)l , d

∗(q)l , P ∗

l , J∗ and set J∗new = J∗

while flag 6= 0 doif J∗ ≤ J∗new or |(J∗new − J∗)/J∗| ≥ χ then

update wl, L(si), hi and set J∗ = J∗new

if |H|new ≤ |H| thenrecompute r∗si

, d∗(t)l , d

∗(q)l , P ∗

l and J∗new

elseflag = 0

end ifelse

flag = 0end if

end while

V. RESULTS

To study the robustness-throughput tradeoff characteristics of DROA for different values of αsiwe use

the example network shown in Fig. 4. Constant packet size of 50 bytes and Pmax of 10 dBm are used.

For simultaneous transmissions distance threshold ν = 2 and the channel-gain Gij = ∆−3ij (∆ being the

distance from node j to node i) are used. Since there can be a large number of possible combinations

16

for αsi, Dmax(si), Rmin(si) and resulting εsi

for different si we use αsi= αs, Dmax(si) = Dmax(s),

Rmin(si) = Rmin(s) and εsi= εs without any loss of generality. We also keep the routes L(si) ∀si

and the corresponding transmission schedules fixed in studying this tradeoff. For performance analysis,

an optimal network-throughput (Ωηl), using the link-efficiency function (ηl) based transmission rate, is

defined as follow

Ωηl=

∑s

r∗s

∣∣∣∣∣µs/2

Rlηl(P)−∑s: l∈L(s) rs

≤ d∗(q)l ,

µs/2

Rlηl(P)≤ d

∗(t)l

. (41)

The optimal network-throughput performance, as a function of end-to-end delay-threshold Dmax(s), is

shown in Fig. 5 for different values of αs. Increasing αs when it is small (for instance αs < 0.3) gives a

considerable throughput performance gain while providing moderate gain for αs > 0.6. The corresponding

delay-robustness performance for different values of parameter αs is shown in Fig. 6. The results in Fig.

5 and Fig. 6 show that, for αs > 0.5 a small compromise in the optimal network-throughput Ωηlcan

provide a significant delay-robustness in the form of delay margin1. For example, by changing αs from

0.9 to 0.7 at Dmax(s) = 20 msec the optimal network-throughput decreases by only 3% but provides an

increase of 30% in the delay margin or equivalently a decrease of 44% in the optimal end-to-end delay

d∗s.

The variation in parameter εs provides an insight into the optimal robustness-throughput tradeoff.

Fig. 7 plots εs as function of Dmax(s) and parameter αs. Reducing αs is equivalent to compromising

the throughput for an improvement in the delay margin and is achieved by an increase in εs for a given

delay-threshold Dmax(s) as observed in Fig. 7. In other words for fixed Dmax(s) an increase in εs forces

d∗(q)l and/or d

∗(t)l to decrease to satisfy (11) and hence improving the delay margin. On the other hand

an improvement in delay margin with an increase in Dmax(s) will depend on the choice of αs. A lower

value of parameter αs will result in higher delay margin with an increase in Dmax(s) and vice versa.

We next compare the throughput Ωηl(computed for effective link-transmission-rate) with the link

capacity based throughput measure Ωclemploying the transmission model of [11], [12]. For that purpose,

Ωclis computed as follow

Ωcl=

∑s

r∗s

∣∣∣∣∣µs/2

cl(P)−∑s: l∈L(s) rs

≤ d∗(q)l ,

µs/2

cl(P)≤ d

∗(t)l

. (42)

1We define the delay margin as Dmax(s)− d∗s , where d∗s is the optimal end-to-end delay given by d∗s =∑

l∈L(si)

(d∗(q)l + d

∗(t)l

).

17

Using the throughput definitions in (41) and (42), the normalized throughput performance gap, obtained

as Ωcl− Ωηl

/Ωcl, is shown in Fig. 8. The superior throughput performance in case of link capacity is

attributed to the flexibility in the choice of any capacity-achieving channel-coding techniques and may

require higher Dmax(s) to be viable. On the other hand, the effective-transmission-rate based model, not

limited by capacity-achieving channel codes, will perform better in situations with stringent requirements

on Dmax(s).

Next we study the convergence performance of ICLA by updating the routes and the transmission

schedule once the convergence of DROA is achieved. The network-throughput Ωηlas a function of the

number of ICLA iterations is shown in Fig. 9 for different number of nodes. It can be seen from Fig. 9

that the iterative algorithm converges in less than ten iterations for an arbitrary network of 50 nodes. The

reduction in the network-throughput during convergence is mainly due to the fact that the link queuing

delay at the start of the algorithm is negligible due to Ql = 0. The reduction in throughput performance

in ICLA results in an improved link queuing delay fairness defined as

Link Queuing Delay Fairness =mind(q)

l | d(q)l ∈ d(q)

maxd(q)l | d

(q)l ∈ d(q)

, (43)

The percentage improvement in the link queuing delay fairness as a result of convergence of ICLA

is shown in Fig. 10, along with the percentage reduction in the network-throughput Ωηl. The result in

Fig. 10 shows the tradeoff between throughput optimality and the resulting link congestion fairness and

can be used in tuning the network parameters for the desired performance.

VI. SUMMARY AND CONCLUSIONS

A distributed resource optimization framework for delay-robustness and network-throughput tradeoff

using sensitivity analysis is proposed. Network-throughput maximization is achieved by solving an effec-

tive link-transmission-rate based power control problem. The proposed resource optimization algorithm is

extended to an iterative cross-layer algorithm by solving the resource allocation, routing and scheduling

problems iteratively. Our results show that a small compromise in the optimal network-throughput can

provide large delay-robustness. Higher degradation of network-throughput, while improving link queuing

delay fairness, suggests that an arbitrary scaling of the network is not possible when delay fairness is of

interest. A possible future research is to explore, how the clustering can be used as a possible solution

for network scaling.

18

APPENDIX A

CONVERGENCE OF DROA

Proof: The first step in the proof of convergence is to show the convexity of the problem. This is

verified for each of the subproblems either evaluating the Hessian or by validating (in case of single

variable functions) the inequality − 1x

∂f(x)∂x

≤ ∂2f(x)

(∂x)2for a given function f . Once the convexity of the

problem in (10)–(14) is verified, the dual function can be defined as

g(Λ,Ξ,Ψ) = maxr,P,d(q),d(t),Υ

L(r, P, d(q), d(t),Υ,Λ,Ξ,Ψ) (44)

The maximization in (44), as discussed in Section III, is achieved by solving the robustness, power,

rate and delay subproblems. The solution to those subproblems allows to evaluate g(Λ,Ξ,Ψ). By strong

duality [22], the overall resource optimization problem is solved by minimizing the dual as in (17). The

key step in dual minimization is to show that the updates in (38)–(40) indeed solve the dual minimization

problem. We next show that the update in (38) is indeed a subgradient update for dual variables λl ∀l.For given λl ∀l, let r∗, P∗, d∗(q), d∗(t) and Υ∗ are optimal solutions for the respective variables, then

g(Λ,Ξ,Ψ) =

∑si

(αsi

U(r∗si) + (1− αsi

)φ(ε∗si))

+∑

l

ξl

(Rlηl(γl(P

∗))− µsi

2d∗(t)l

)

+∑si

ψsi

(1− ε∗si

)− 1

Dmax(si)

∑

l∈L(si)

(d∗(t)l + d

∗(q)l

)

+∑

l

λl

Rlηl(γl(P

∗))−∑

si: l∈L(si)

r∗si− µsi

2d∗(q)l

, (45)

and for some arbitrary λ′l ∈ Λ′ ∀l we have

g(Λ′,Ξ,Ψ) ≥∑

si

(αsi

U(r∗si) + (1− αsi

)φ(ε∗si))

+∑

l

ξl

(Rlηl(γl(P

∗))− µsi

2d∗(t)l

)

+∑si

ψsi

(1− ε∗si

)− 1

Dmax(si)

∑

l∈L(si)

(d∗(t)l + d

∗(q)l

)

+∑

l

λ′l

Rlηl(γl(P

∗))−∑

si: l∈L(si)

r∗si− µsi

2d∗(q)l

. (46)

19

Subtracting (46) from (45) we have

g(Λ,Ξ,Ψ)− g(Λ′,Ξ,Ψ) ≤∑

l

(λl − λ′l)

Rlηl(γl(P

∗))−∑

si: l∈L(si)

r∗si− µsi

2d∗(q)l

. (47)

Using the definition of subgradient [28] we can verify that(

Rlηl(γl(P∗))−∑

si: l∈L(si)r∗si− µsi

2d∗(q)l

)is

the subgradient of g(Λ,Ξ,Ψ). A similar procedure can be used for verifying the subgradients for ξl

and ψsiin (39) and (40) respectively. By choosing the step sizes βλ, βξ and βψ small enough [28], the

subgradient updates eventually converge to the optimal dual variables. Once the optimal dual variables

are found, the corresponding primal variables can be obtained by solving their respective subproblems.

And due to strong duality, the primal variables must be global optimum providing unique solution.

REFERENCES

[1] W. Zhang, M. Branicky, and S. Phillips, “Stability of networked control systems,” IEEE Control Systems Magazine, vol. 21, no. 1,

pp. 84–99, 2001.

[2] X. Liu and A. Goldsmith, “Wireless communication tradeoffs in distributed control,” in IEEE Conference on Decision and Control,

2003, pp. 688–694.

[3] S. Mazumder, M. Tahir, and K. Acharya, “Master-slave current-sharing control of a parallel DC-DC converter system over an RF

communication interface,” IEEE Transactions on Industrial Electronics, vol. 55, no. 1, pp. 59–66, 2008.

[4] I. Akyildiz, T. Melodia, and K. Chowdhury, “A survey on wireless multimedia sensor networks,” Computer Networks, vol. 51, no. 4,

pp. 921–960, 2007.

[5] S. Mazumder, K. Acharya, and M. Tahir, “Joint optimization of control performance and network resource utilization in homogeneous

power networks,” to appear, IEEE Transactions on Industrial Electronics, 2009.

[6] K. Low and R. Cao, “Model predictive control of parallel-connected inverters for uninterruptible power supplies,” IEEE Transactions

on Industrial Electronics, vol. 55, no. 8, pp. 2884–2893, 2008.

[7] A. Bemporad, S. Di Cairano, E. Henriksson, and K. Johansson, “Hybrid model predictive control based on wireless sensor feedback:

An experimental study,” in IEEE Conference on Decision and Control, 2007, pp. 5062–5067.

[8] X. Liu and A. Goldsmith, “Wireless network design for distributed control,” in IEEE American Control Conference, 2004, pp. 2823–

2829.

[9] K. Acharya, M. Tahir, and S. Mazumder, “Communication fault-tolerant wireless network control of a load-sharing multiphase

interactive power network,” in IEEE Power Electronics Specialists Conference, 2006, pp. 1–8.

[10] R. Cruz and A. Santhanam, “Optimal routing, link scheduling and power control in multihop wireless networks,” in IEEE INFOCOM,

2003, pp. 702–711.

[11] M. Saad and A. Yu, “Optimal network rate allocation under end-to-end quality-of-service requirements,” IEEE Transactions on Network

and Service Management, vol. 4, no. 3, pp. 40–49, 2007.

[12] M. Tahir and S. Mazumder, “Delay constrained optimal resource utilization of wireless networks for distributed control systems,”

IEEE Communications Letters, vol. 12, no. 4, pp. 289–291, 2008.

20

[13] C. Tan, D. Palomar, and M. Chiang, “Energy-robustness tradeoff in cellular network power control,” To appear IEEE/ACM Transactions

on Networking, 2009.

[14] F. Meshkati, H. Poor, S. Schwartz, and N. Mandayam, “An energy-efficient approach to power control and receiver design in wireless

data networks,” IEEE Transactions on Communications, vol. 53, no. 11, pp. 1885–1894, 2005.

[15] M. Chiang, “Balancing transport and physical layers in wireless multihop networks: jointly optimal congestion control and power

control,” IEEE Journal on Selected Areas in Communications, vol. 23, no. 1, pp. 104–116, 2005.

[16] C. Comaniciu and H. Poor, “On energy efficient hierarchical cross-layer design: joint power control and routing for ad hoc networks,”

EURASIP Journal on Wireless Communications and Networking, vol. 2007, no. Article ID 60707, p. 9, 2007.

[17] R. Madan, S. Cui, S. Lall, and A. Goldsmith, “Modeling and optimization of transmission schemes in energy-constrained wireless

sensor networks,” IEEE/ACM Transactions on Networking, vol. 15, no. 6, pp. 1359–1372, 2007.

[18] A. Yener and S. Kishore, “Distributed power control and routing for clustered CDMA wireless ad hoc networks,” in IEEE Vehicular

Technology Conference, 2004, pp. 2951–2955.

[19] L. Xiao, M. Johansson, and S. Boyd, “Simultaneous routing and resource allocation via dual decomposition,” IEEE Transactions on

Communications, vol. 52, no. 7, pp. 1136–1144, 2004.

[20] D. Bertsekas and R. Gallager, Data Networks. Prentice-Hall, NJ, USA, 1991.

[21] T. Ren and R. La, “Power control algorithm for providing packet error rate guarantees in ad hoc networks,” in IEEE Conference on

Decision and Control and European Control Conference, 2005, pp. 6040–6045.

[22] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004.

[23] F. Meshkati, M. Chiang, H. Poor, and S. Schwartz, “A game-theoretic approach to energy-efficient power control in multi-carrier

CDMA systems,” IEEE Journal on Selected Areas in Communications, vol. 24, no. 6, pp. 1115–1129, 2006.

[24] K. Leung, P. Driessen, and K. Chawla, “Link adaptation and power control for streaming services in EGPRS wireless networks,” IEEE

Journal on Selected Areas in Communications, vol. 19, no. 10, pp. 2029–2039, 2001.

[25] J. Yuan, Z. Li, W. Yu, and B. Li, “A cross-layer optimization framework for multihop multicast in wireless mesh networks,” IEEE

Journal on Selected Areas in Communications, vol. 24, no. 11, pp. 2092–2103, 2006.

[26] A. Orda, “Routing with end-to-end QoS guarantees in broadband networks,” IEEE/ACM Transactions on Networking, vol. 7, no. 3,

pp. 365–374, 1999.

[27] J. Lee, M. Chiang, and R. Calderbank, “Jointly optimal congestion and contention control in wireless ad hoc networks,” IEEE

Communication Letters, vol. 10, no. 3, pp. 216–218, 2006.

[28] N. Shor, K. Kiwiel, and A. Ruszcaynski, Minimization Methods for Non-differentiable Functions. Springer-Verlag, New York, 1985.

[29] R. Haftka and Z. Gurdal, Elements of Structural Optimization. Kluwer Academic Pub, 1992.

[30] J. Yeo, H. Lee, and S. Kim, “An efficient broadcast scheduling algorithm for TDMA ad hoc networks,” Computers and Operations

Research, vol. 29, no. 13, pp. 1793–1806, 2002.

[31] W. Wang, Y. Wang, X. Li, W. Song, and O. Frieder, “Efficient interference-aware TDMA link scheduling for static wireless networks,”

in Proc. of International Conference on Mobile Computing and Networking, 2006, pp. 262–273.

FIGURES 21

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

γl(dB)

PER

(γl)

γl > γ

min(l)

Fig. 1. PER as a function of link SINR. The dashed line marks the region for which link SINR is larger than a certain threshold and asa consequence, ensures the convexity of the PER in (3) for the region γl > γmin(l).

FIGURES 22

Rate Sub-problem

)(

)(maximize

i

iii

sLl

slss rrU

isi srsRi

)(s.t min

Stop if

convergence

or maximum

iterations

reached

isr

lP

Delay Sub-problem

l sLls i

t

l

q

lst

l

sl

q

l

sl

ii

i

ii

sD

dd

dd )(: max

)()(

)()( )(

)(

22minimize

Dual Problem

),(minimize ,g

isll sli

,0,,s.t

Initialize

0

0,

is

ll

Robustness Sub-problem

i

iiii

s

ssss )()1()1(maximize

is si

10s.t

is

Power Sub-problem

isll ,,

l

lllll R ))(()(maximize P

)()(min Pll lPPl0 max)()( , t

lq

l dd

Fig. 2. Block diagram representation for distributed implementation of the network resource-optimization problem using dual decomposition.

FIGURES 23

Delay robustness

increase

Network

throughput increase

)1()( **

ii ss u1

)(*u

1isu

isu

Fig. 3. Robustness-throughput tradeoff using sensitivity analysis. The optimal value ρ∗(1) correspond to usi = 1 ∀si.

FIGURES 24

t6

s6

t7

t10

s5, s

9

s2

s3

s7

t9

s1

t5

t2 t

3t8

s10

t4, s

8s

4

t1

Fig. 4. An example network used in performance evaluation with solid lines representing the selected links participating in scheduledtransmission. The nodes marked by si and ti represent the starting and ending nodes for transmission session i.

FIGURES 25

0.005 0.01 0.015 0.02 0.025 0.03

1.5

2

2.5

3

3.5

Dmax

(s) (sec)

Opt

imal

Net

wor

k T

hrou

ghpu

t (M

bps)

αs = 0.9

αs = 0.7

αs = 0.5 α

s = 0.3 α

s = 0.1

Fig. 5. Optimal network-throughput Ωηl for different αs values as a function of maximum end-to-end delay-threshold (Dmax(s)). Rmin(s)of 100 Kbps is used.

FIGURES 26

0.005 0.01 0.015 0.02 0.025 0.030

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Dmax

(s) (sec)

Opt

imal

End

−to

−en

d D

elay

(se

c)

αs = 0.5

αs = 0.7

αs = 0.9

αs = 0.3 α

s = 0.1

Fig. 6. Optimal end-to-end delay performance corresponding to an arbitrarily chosen transmission session for different values of αs as afunction of maximum end-to-end delay-threshold (Dmax(s)). Rmin(s) of 100 Kbps is used.

FIGURES 27

00.2

0.40.6

0.81 0

0.01

0.02

0.03

0

0.2

0.4

0.6

0.8

1

Dmax

(s)

Parameter αs

Para

met

er ε

s

Fig. 7. The robustness parameter variation as a function of end-to-end delay-threshold and optimization objective weighting parameter αs.

FIGURES 28

5 10 15 20 25 30 35 40 45 500.47

0.48

0.49

0.5

0.51

0.52

0.53

0.54

0.55

0.56

Dmax

(s) (msec)

Nor

mal

ized

Thr

ough

put P

erfo

rman

ce G

ap

Fig. 8. Optimal network-throughput performance comparison of the link-efficiency function (5) based effective transmission rate modelwith the link capacity based effective transmission rate model of [11], [12] using the example network of Fig. 4.

FIGURES 29

1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12

13

14

Number of Route Update Iterations

Opt

imal

Net

wor

k T

hrou

ghpu

t, Ω

η l

|N| = 15 |N| = 30 |N| = 50

Fig. 9. Optimal network throughput as a function of route update iterations. The parameter αs = 0.8, Dmax(s) = 20 msec and Rmin(s)of 100 Kbps are used for this result.

FIGURES 30

0

10

20

30

40

50

60

Number of Nodes |N|

Perf

orm

ance

in %

Network Throughput Reduction

Link Delay Fairness Improvement

15 30 50

Fig. 10. Average percentage performance improvement in link delay fairness leading to congestion avoidance. The price to avoid thecongestion in terms of reduction in network-throughput Ωηl is also shown.