PROC. WIOPT 2011. 1

LIFO-Backpressure Achieves Near OptimalUtility-Delay Tradeoff

Longbo Huang, Scott Moeller, Michael J. Neely, Bhaskar Krishnamachari

Abstract—There has been considerable recent work developinga new stochastic network utility maximization framework usingBackpressure algorithms, also known as MaxWeight. A key openproblem has been the development of utility-optimal algorithmsthat are also delay efficient. In this paper, we show that theBackpressure algorithm, when combined with the LIFO queueingdiscipline (called LIFO-Backpressure), is able to achieve a utilitythat is within O(1/V ) of the optimal value for any scalarV ≥ 1, while maintaining an average delay of O([log(V )]2)for all but a tiny fraction of the network traffic. This resultholds for general stochastic network optimization problems andgeneral Markovian dynamics. Remarkably, the performance ofLIFO-Backpressure can be achieved by simply changing thequeueing discipline; it requires no other modifications of theoriginal Backpressure algorithm. We validate the results throughempirical measurements from a sensor network testbed, whichshow good match between theory and practice.

Index Terms—Queueing, Dynamic Control, LIFO scheduling,Lyapunov analysis, Stochastic Optimization

I. INTRODUCTION

Recent developments in stochastic network optimizationtheory have yielded a very general framework that solves alarge class of networking problems of the following form:We are given a discrete time stochastic network. The networkstate, which describes the current realization of the underlyingnetwork randomness, such as the network channel condition,is time varying according to some probability law. A networkcontroller performs some action based on the observed net-work state at every time slot. The chosen action incurs a cost,1 but also serves some amount of traffic and possibly generatesnew traffic for the network. This traffic causes congestion, andthus leads to backlogs at nodes in the network. The goal of thecontroller is to minimize its time average cost subject to theconstraint that the time average total backlog in the networkbe kept finite.

This general setting models a large class of networkingproblems ranging from traffic routing [1], flow utility max-imization [2], network pricing [3] to cognitive radio applica-tions [4]. Also, many techniques have also been applied tothis problem (see [5] for a survey). Among the approaches

Longbo Huang, Scott Moeller, Michael J. Neely, and Bhaskar Krishna-machari (emails: {longbohu, smoeller, mjneely, bkrishna}@usc.edu) are withthe Department of Electrical Engineering, University of Southern California,Los Angeles, CA 90089, USA.

This material is supported in part under one or more of the followinggrants: DARPA IT-MANET W911NF-07-0028, NSF CAREER CCF-0747525,and continuing through participation in the Network Science CollaborativeTechnology Alliance sponsored by the U.S. Army Research Laboratory.

1Since cost minimization is mathematically equivalent to utility maximiza-tion, below we will use cost and utility interchangeably

that have been adopted, the family of Backpressure algorithms[6] are recently receiving much attention due to their provableperformance guarantees, robustness to stochastic network con-ditions and, most importantly, their ability to achieve the de-sired performance without requiring any statistical knowledgeof the underlying randomness in the network.

Most prior performance results for Backpressure are givenin the following [O(1/V ), O(V )] utility-delay tradeoff form[6]: Backpressure is able to achieve a utility that is withinO(1/V ) of the optimal utility for any scalar V ≥ 1, whileguaranteeing an average network delay that is O(V ). Althoughthese results provide strong theoretical guarantees for thealgorithms, the network delay can actually be unsatisfyingwhen we achieve a utility that is very close to the optimal,i.e., when V is large.

There have been previous works trying to develop algo-rithms that can achieve better utility-delay tradeoffs. Previousworks [7] and [8] show improved tradeoffs are possiblefor single-hop networks with certain structure, and developsoptimal [O(1/V ), O(log(V ))]and [O(1/V ), O(

√V )] utility-

delay tradeoffs. However, the algorithms are different frombasic Backpressure and require knowledge of an “epsilon”parameter that measures distance to a performance regionboundary. Work [9] uses a completely different analyticaltechnique to show that similar poly-logarithmic tradeoffs, i.e.,[O(1/V ), O([log(V )]2)], are possible by carefully modify-ing the actions taken by the basic Backpressure algorithms.However, the algorithm requires a pre-determined learningphase, which adds additional complexity to the implemen-tation. The current work, following the line of analysis in[9], instead shows that similar poly-logarithmic tradeoffs,i.e., [O(1/V ), O([log(V )]2)], can be achieved by the orig-inal Backpressure algorithm by simply modifying the ser-vice discipline from First-in-First-Out (FIFO) to Last-In-First-Out (LIFO) (called LIFO-Backpressure below). This is aremarkable feature that distinguishes LIFO-Backpressure fromprevious algorithms in [7] [8] [9], and provides a deeperunderstanding of backpressure itself, and the role of queuebacklogs as Lagrange multipliers (see also [2] [9]). However,this performance improvement is not for free: We must dropa small fraction of packets in order to dramatically improvedelay for the remaining ones. We prove that as the V parameteris increased, the fraction of dropped packets quickly convergesto zero, while maintaining O(1/V ) close-to-optimal utilitiyand O([log(V )]2) average backlog. This provides an analyticaljustification for experimental observations in [10] that shows arelated LIFO-Backpressure rule serves up to 98% of the trafficwith delay that is improved by 2 orders of magnitude.

PROC. WIOPT 2011. 2

LIFO-Backpressure was proposed in recent empirical work[10]. The authors developed a practical implementation ofbackpressure routing and showed experimentally that apply-ing the LIFO queuing discipline drastically improves aver-age packet delay, but did not provide theoretical guarantees.Another notable recent work providing an alternative delaysolution is [11], which describes a novel backpressure-basedper-packet randomized routing framework that runs atop theshadow queue structure of [12] while minimizing hop countas explored in [13]. Their techniques reduce delay drasticallyand eliminates the per-destination queue complexity, but doesnot provide O([log(V )]2) average delay guarantees.

Our analysis of the delay performance of LIFO-Backpressure is based on the recent “exponential attraction”result developed in [9]. The proof idea can be intuitivelyexplained by Fig. 1, which depicts a simulated backlog processof a single queue system with unit packet size under Backpres-sure. The left figure demonstrates the “exponential attraction”

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

t

!*

VO([log(V)]

2)

" (V)

!"#$%&'()*+(

,-./%0

,-1$2

!&'(

344/56#

,78789:/;8

Fig. 1. The LIFO-Backpressure Idea

result in [9], which states that queue sizes under Backpressuredeviate from some fixed point with probability that decreasesexponentially in the deviation distance. Hence the queue sizewill mostly fluctuate within the interval [QLow, QHigh] whichcan be shown to be of O([log(V )]2) size. This result holdsunder both FIFO and LIFO, as they result in the same queueprocess. Now suppose LIFO is used in this queue. Then fromthe right figure, we see that most of the packets will arrive atthe queue when the queue size is between QLow and QHigh,and these new packets will always be placed on the top ofthe queue due to the LIFO discipline. Most packets thus enterand leave the queue when the queue size is between QLow andQHigh. Therefore, these packets “see” a queue with averagesize no more than QHigh−QLow = O([log(V )]2). Now let λbe the packet arrival rate into the queue, and let λ̃ be thearrival rate of packets entering when the queue size is in[QLow, QHigh] and that eventually depart. Because packetsalways occupy the same buffer slot under LIFO, we see thatthese packets can occupy at most QHigh − QLow + δmaxbuffer slots, ranging from QLow to QHigh + δmax, whereδmax = Θ(1) is the maximum number of packets that can en-ter the queue at any time. We can now apply Little’s Theorem[14] to the buffer slots in the interval [QLow, QHigh + δmax],and we see that average delay for these packets that arrivewhen the queue size is in [QLow, QHigh] satisfies:

D ≤ QHigh −QLow + δmax

λ̃=O([log(V )]2)

λ̃. (1)

Finally, the exponential attraction result implies that λ ≈ λ̃.Hence for almost all packets entering the queue, the averagedelay is D = O([log(V )]2/λ).

This paper is organized as follows. We present our systemmodel in Section II. We then provide an example of ournetwork in Section III. We review the Backpressure algorithmin Section IV. The delay performance of LIFO-Backpressureis presented in Section V. Simulation results are presented inSection VI. We then also present experimental testbed resultsin Section VII.

II. SYSTEM MODEL

In this section, we specify the general network model weuse. We consider a network controller that operates a networkwith the goal of minimizing the time average cost, subjectto the queue stability constraint. The network is assumed tooperate in slotted time, i.e., t ∈ {0, 1, 2, ...}. We assume thereare r ≥ 1 queues in the network.

A. Network State

In every slot t, we use S(t) to denote the current net-work state, which indicates the current network parameters,such as a vector of channel conditions for each link, ora collection of other relevant information about the currentnetwork channels and arrivals. We assume that S(t) evolvesaccording a finite state irreducible and aperiodic Markov chain,with a total of M different random network states denotedas S = {s1, s2, . . . , sM}. Let πsi denote the steady stateprobability of being in state si. It is easy to see in this case thatπsi > 0 for all si. The network controller can observe S(t)at the beginning of every slot t, but the πsi and transitionprobabilities are not necessarily known.

B. The Cost, Traffic, and Service

At each time t, after observing S(t) = si, the controllerchooses an action x(t) from a set X (si), i.e., x(t) = x(si) forsome x(si) ∈ X (si). The set X (si) is called the feasible actionset for network state si and is assumed to be time-invariant andcompact for all si ∈ S. The cost, traffic, and service generatedby the chosen action x(t) = x(si) are as follows:

(a) The chosen action has an associated cost given by thecost function f(t) = f(si, x(si)) : X (si) 7→ R+ (orX (si) 7→ R− in reward maximization problems);

(b) The amount of traffic generated by the action toqueue j is determined by the traffic function Aj(t) =Aj(si, x(si)) : X (si) 7→ R+, in units of packets;

(c) The amount of service allocated to queue j is given bythe rate function µj(t) = µj(si, x(si)) : X (si) 7→ R+, inunits of packets;

Note that Aj(t) includes both the exogenous arrivals fromoutside the network to queue j, and the endogenous arrivalsfrom other queues, i.e., the transmitted packets from otherqueues, to queue j. We assume the functions f(si, ·), µj(si, ·)and Aj(si, ·) are continuous, time-invariant, their magnitudesare uniformly upper bounded by some constant δmax ∈ (0,∞)for all si, j, and they are known to the network operator. Wealso assume that there exists a set of actions {x(si)k}k=1,2,...,∞

i=1,...,M

PROC. WIOPT 2011. 3

with x(si)k ∈ X (si) and some variables ϑ(si)k ≥ 0 for all si

and k with∑k ϑ

(si)k = 1 for all si, such that∑

si

πsi{∑

k

ϑ(si)k [Aj(si, x(si)k)− µj(si, x(si)k)]

} ≤ −η, (2)

for some η > 0 for all j. That is, the stability constraintsare feasible with η-slackness. Thus, there exists a stationaryrandomized policy that stabilizes all queues (where ϑ

(si)k

represents the probability of choosing action x(si)k whenS(t) = si) [6].

C. Queueing, Average Cost, and the Stochastic Problem

Let q(t) = (q1(t), ..., qr(t))T ∈ Rr+, t = 0, 1, 2, ... bethe queue backlog vector process of the network, in units ofpackets. We assume the following queueing dynamics:

qj(t+ 1) = max[qj(t)− µj(t), 0

]+Aj(t) ∀j, (3)

and q(0) = 0. By using (3), we assume that when a queue doesnot have enough packets to send, null packets are transmitted.In this paper, we adopt the following notion of queue stability:

E{ r∑j=1

qj}

, lim supt→∞

1t

t−1∑τ=0

r∑j=1

E{qj(τ)

}<∞. (4)

We also use fΠav to denote the time average cost induced by

an action-choosing policy Π, defined as:

fΠav , lim sup

t→∞

1t

t−1∑τ=0

E{fΠ(τ)

}, (5)

where fΠav(τ) is the cost incurred at time τ by policy Π.

We call an action-choosing policy feasible if at every timeslot t it only chooses actions from the feasible action setX (S(t)). We then call a feasible action-choosing policy underwhich (4) holds a stable policy, and use f∗av to denote theoptimal time average cost over all stable policies. In everyslot, the network controller observes the current network stateand chooses a control action, with the goal of minimizing thetime average cost subject to network stability. This goal canbe mathematically stated as: (P1) min : fΠ

av, s.t. (4). Inthe following, we will refer to (P1) as the stochastic problem.

III. AN EXAMPLE OF OUR MODEL

Here we provide an example to illustrate our model. Con-sider the 2-queue network in Fig. 2. In every slot, the networkoperator decides whether or not to allocate one unit of powerto serve packets at each queue, so as to support all arrivingtraffic, i.e., maintain queue stability, with minimum energyexpenditure. We assume the network state S(t), which isthe quadruple (R1(t), R2(t), CH1(t), CH2(t)), evolves ac-cording to the finite state Markov chain with three statess1 = (1, 1, G,B), s2 = (1, 1, G,G), and s3 = (0, 0, B,G).Here Ri(t) denotes the number of exogenous packet arrivalsto queue i at time t, and CHi(t) is the state of channel i.Ri(t) = x implies that there are x number of packets arrivingat queue i at time t. CHi(t) = G/B means that channel ihas a “Good” or “Bad” state. When a link’s channel state is“Good”, one unit of power can serve 2 packets over the link,

q1(t) q2(t)A1(t)=R1(t)!1(t) !2(t)

CH1(t) CH2(t)

R2(t)

Fig. 2. A two queue tandem example.

otherwise it can only serve one. We assume power can beallocated to both channels without affecting each other.

In this case, we see that there are three possible networkstates. At each state si, the action x(si) is a pair (x1, x2),with xi being the amount of energy spent at queue i,and (x1, x2) ∈ X (si) = {0/1, 0/1}. The cost function isf(si, x(si)) = x1 + x2, for all si. The network states, thetraffic functions, and the service rate functions are summarizedin Fig. 3. Note here A1(t) = R1(t) is part of S(t) and is inde-pendent of x(si); while A2(t) = µ1(t) +R2(t) hence dependson x(si). Also note that A2(t) equals µ1(t) + R2(t) insteadof min[µ1(t), q1(t)] +R2(t) due to our idle fill assumption inSection II-C.

On Using LIFO in Max-Weight Scheduling

This note provides a brief summary of the development of using LIFO in max-weight scheduling

(called LIFO scheduling in short) in stochastic network optimization. The idea of using LIFO with the

max-weight algorithm was first proposed in [1].

TABLE I

NETWORK STATE, TRAFFIC AND RATE

S(t) R1(t) R2(t) CH1(t) CH2(t) A1(t) A2(t) µ1(t) µ2(t)

s1 1 1 G B 1 2x1 + 1 2x1 x2

s2 1 1 G G 1 2x1 + 1 2x1 2x2

s3 0 0 B G 0 x1 x1 2x2

I. THE STATE OF THE ART - LIFO IN UTILITY MAXIMIZATION

So far, the LIFO scheduling results, either theoretical or experimental, have been focusing on utility

maximization in networks. The main reason for this is that when we try to optimize a utility over a network,

one can show, under some mild conditions that can usually be satisfied in practice, that the network backlog

vector under the max-weight algorithm is exponentially attracted to some fixed point. Specifically, under

max-weight with a control parameter V , there exists some fixed point !! = (!!1 , ..., !!r )T = !(V ) such

that the network backlog vector q(t) satisfies the following property in steady state:

Pr{!q(t)" !!! > D + m} # e"cm, (1)

for some c, D = !(1). That is, the network backlog vector size will increase linearly with the V parameter,

and it will mostly be within log(V ) distance to !! when V is large.

In practice, the FIFO queueing discipline is often used in many networking applications. Therefore,

a packet has to wait for all the packets in front of it in the queue before getting served. According to

the above attraction result, we see that under max-weight with FIFO, a packet usually has to wait for a

number of !(V ) packets. Thus the average network delay is !(V ). This analytical result is consistent

with what we see in simulations and experiments.

Now consider using the LIFO discipline. We see then in steady state, most packets will arrive and

leave the queue when the queue size is within log(V ) distance to the fixed point !!. Hence for most of

the packets, they “see” a queue with roughly log(V ) packets. Hence the average delay for most of the

packets is only logarithmic in V .

Fig. 3. The traffic and service functions under different states.

IV. BACKPRESSURE AND THE DETERMINISTIC PROBLEM

In this section, we first review the Backpressure algorithm[6] for solving the stochastic problem. Then we define thedeterministic problem and its dual. We first recall the Back-pressure algorithm for utility optimization problems [6].

Backpressure: At every time slot t, observe the currentnetwork state S(t) and the backlog q(t). If S(t) = si, choosex(si) ∈ X (si) that solves the following:max : −V f(si, x) +

r∑j=1

qj(t)[µj(si, x)−Aj(si, x)

](6)

s.t. x ∈ X (si).

Depending on the problem structure, (6) can usually bedecomposed into separate parts that are easier to solve, e.g.,[3], [4]. Also, when the network state process S(t) is i.i.d., ithas been shown in [6] that,

fBPav = f∗av +O(1/V ), qBP = O(V ), (7)

where fBPav and qBP are the expected average cost and theexpected average network backlog size under Backpressure,respectively. When S(t) is Markovian, [3] and [4] show thatBackpressure achieves an [O(log(V )/V ), O(V )] utility-delaytradeoff if the queue sizes are deterministically upper boundedby Θ(V ) for all time. Without this deterministic backlogbound, it has recently been shown that Backpressure achievesan [O(ε + Tε

V ), O(V )] tradeoff under Markovian S(t), with εand Tε representing the proximity to the optimal utility valueand the “convergence time” of the Backpressure algorithmto that proximity [16]. However, there has not been any

PROC. WIOPT 2011. 4

formal proof that shows the exact [O(1/V ), O(V )] utility-delay tradeoff of Backpressure under a Markovian S(t).

We also recall the deterministic problem defined in [9]:

min : F(x) , V∑si

πsif(si, x(si)) (8)

s.t. Aj(x) ,∑si

πsiAj(si, x(si))

≤ Bj(x) ,∑si

πsiµj(si, x(si)), ∀ j,

x(si) ∈ X (si) ∀ i = 1, 2, ...,M,

where πsi corresponds to the steady state probability of S(t) =si and x = (x(s1), ..., x(sM ))T . The dual problem of (8) canbe obtained as follows:

max : g(γ), s.t. γ � 0, (9)

where g(γ) is called the dual function and is defined as:

g(γ) = infx(si)∈X (si)

∑si

πsi

{V f(si, x(si)) (10)

+∑j

γj[Aj(si, x(si))− µj(si, x(si))

]}.

Here γ = (γ1, ..., γr)T is the Lagrange multiplier of(8). It is well known that g(γ) in (10) is concave in thevector γ, and hence the problem (9) can usually be solvedefficiently, particularly when cost functions and rate functionsare separable over different network components. Below, weuse γ∗V = (γ∗V 1, γ

∗V 2, ..., γ

∗V r)

T to denote an optimal solutionof the problem (9) with the corresponding V .

V. PERFORMANCE OF LIFO BACKPRESSURE

In this section, we analyze the performance of Back-pressure with the LIFO queueing discipline (called LIFO-Backpressure). The idea of using LIFO under Backpressureis first proposed in [10], although they did not provide anytheoretical performance guarantee. We will show, under somemild conditions (to be stated in Theorem 3), that under LIFO-Backpressure, the time average delay for almost all packetsentering the network is O([log(V )]2) when the utility ispushed to within O(1/V ) of the optimal value. Note that theimplementation complexity of LIFO-Backpressure is the sameas the original Backpressure, and LIFO-Backpressure onlyrequires the knowledge of the instantaneous network condi-tion. This is a remarkable feature that distinguishes it fromthe previous algorithms achieving similar poly-logarithmictradeoffs in the i.i.d. case, e.g., [7] [8] [9], which all requireknowledge of some implicit network parameters other than theinstant network state. Below, we first provide a simple exampleto demonstrate the need for careful treatment of the usage ofLIFO in Backpressure algorithms, and then present a modifiedLittle’s theorem that will be used for our proof.

A. A simple example on the LIFO delay

Consider a slotted system where two packets arrive at time0, and one packet periodically arrives every slot thereafter (attimes 1, 2, 3, . . .). The system is initially empty and can serve

exactly one packet per slot. The arrival rate λ is clearly 1packet/slot (so that λ = 1). Further, under either FIFO orLIFO service, there are always 2 packets in the system, soQ = 2.

Under FIFO service, the first packet has a delay of 1 andall packets thereafter have a delay of 2:

WFIFO1 = 1 , WFIFO

i = 2 ∀i ∈ {2, 3, 4, . . .},where WFIFO

i is the delay of the ith packet under FIFO(WLIFO

i is similarly defined for LIFO). We thus have:

WFIFO M= lim

K→∞

1K

K∑i=1

WFIFOi = 2.

Thus, λWFIFO

= 1× 2 = 2, Q = 2, and so λWFIFO

= Qindeed holds.

Now consider the same system under LIFO service. We stillhave λ = 1, Q = 2. However, in this case the first packet neverdeparts, while all other packets have a delay equal to 1 slot:

WLIFO1 =∞ , WLIFO

i = 1 ∀i ∈ {2, 3, 4, . . .}.Thus, for all integers K > 0:

1K

K∑i=1

WLIFOi =∞.

and so WLIFO

= ∞. Clearly λWLIFO 6= Q. On the other

hand, if we ignore the one packet with infinite delay, we notethat all other packets get a delay of 1 (exactly half the delayin the FIFO system). Thus, in this example, LIFO servicesignificantly improves delay for all but the first packet.

For the above LIFO example, it is interesting to note thatif we define Q̃ and W̃ as the average backlog and delayassociated only with those packets that eventually depart, thenwe have Q̃ = 1, W̃ = 1, and the equation λW̃ = Q̃ indeedholds. This motivates the theorem in the next subsection,which considers a time average only over those packets thateventually depart.

B. A Modified Little’s Theorem for LIFO systems

We now present the modified Little’s theorem. Let B rep-resent a finite set of buffer locations for a LIFO queueingsystem. Let N(t) be the number of arrivals that use a bufferlocation within set B up to time t. Let D(t) be the number ofdepartures from a buffer location within the set B up to timet. Let Wi be the delay of the ith job to depart from the setB. Define W as the lim sup average delay considering onlythose jobs that depart:

W M= lim supt→∞

1D(t)

D(t)∑i=1

Wi.

We then have the following theorem:Theorem 1: Suppose there is a constant λmin > 0 such

that with probability 1:

lim inft→∞

N(t)t≥ λmin,

PROC. WIOPT 2011. 5

Further suppose that limt→∞D(t) =∞ with probability 1 (sothe number of departures is infinite). Then the average delayW satisfies:

W M= lim supt→∞

1D(t)

D(t)∑i=1

Wi ≤ |B|/λmin,

where |B| is the size of the finite set B.Proof: See [17].

C. LIFO-Backpressure Proof

We now provide the analysis of LIFO-Backpressure. Toprove our result, we first have the following theorem, whichis the first to show that Backpressure (with either FIFO orLIFO) achieves the exact [O(1/V ), O(V )] utility-delay trade-off under a Markovian network state process. It generalizesthe [O(1/V ), O(V )] performance result of Backpressure inthe i.i.d. case in [6].

Theorem 2: Suppose S(t) is a finite state irreducible andaperiodic Markov chain2 and condition (2) holds, Backpres-sure (with either FIFO or LIFO) achieves the following:

fBPav = f∗av +O(1/V ), qBP = O(V ), (11)

where fBPav and qBP are the expected time average cost andbacklog under Backpressure.

Proof: See [18].Theorem 2 thus shows that LIFO-Backpressure guarantees

an average backlog of O(V ) when pushing the utility to withinO(1/V ) of the optimal value. We now consider the delayperformance of LIFO-Backpressure. For our analysis, we needthe following theorem (which is Theorem 1 in [9]).

Theorem 3: Suppose that γ∗V is unique, that the slacknesscondition (2) holds, and that the dual function g(γ) satisfies:

g(γ∗V ) ≥ g(γ) + L||γ∗V − γ|| ∀ γ � 0, (12)

for some constant L > 0 independent of V . Then un-der Backpressure with FIFO or LIFO, there exist constantsD,K, c∗ = Θ(1), i.e., all independent of V , such that for anym ∈ R+,

P(r)(D,Km) ≤ c∗e−m, (13)

where P(r)(D,Km) is defined:

P(r)(D,Km) (14)

, lim supt→∞

1t

t−1∑τ=0

Pr{∃ j, |qj(τ)− γ∗V j | > D +Km}.

Proof: See [9].Note that if a steady state distribution exists for q(t), e.g.,

when all queue sizes are integers, then P(r)(D,Km) is indeedthe steady state probability that there exists a queue j whosequeue value deviates from γ∗V j by more than D+Km distance.In this case, Theorem 3 states that qj(t) deviates from γ∗V j byΘ(log(V )) distance with probability O(1/V ). Hence whenV is large, qj(t) will mostly be within O(log(V )) distancefrom γ∗V j . Also note that the conditions of Theorem 3 are not

2In [18], the theorem is proven under more general Markovian S(t)processes that include the S(t) process assumed here.

very restrictive. The condition (12) can usually be satisfied inpractice when the action space is finite, in which case the dualfunction g(γ) is polyhedral (see [9] for more discussions). Theuniqueness of γ∗V can usually be satisfied in many networkutility optimization problems, e.g., [2].

We now present the main result of this paper with respectto the delay performance of LIFO-Backpressure. Below, thenotion “average arrival rate” is defined as follows: Let Aj(t)be the number of packets entering queue j at time t. Then thetime average arrival rate of these packets is defined (assumingit exists): λj = limt→∞

1t

∑t−1τ=0Aj(τ). For the theorem,

we assume that time averages under Backpressure exist withprobability 1. This is a reasonable assumption, and holdswhenever the resulting discrete time Markov chain for thequeue vector q(t) under backpressure is countably infiniteand irreducible. Note that the state space is indeed countablyinfinite if we assume packets take integer units. If the systemis also irreducible then the finite average backlog result ofTheorem 2 implies that all states are positive recurrent.

Let D,K, c∗ be constants as defined in Theorem 3, andrecall that these are Θ(1) (independent of V ). Assume V ≥ 1,and define Qj,High and Qj,Low as:

Qj,HighM= γ∗V j +D +K[log(V )]2,

Qj,LowM= max[γ∗V j −D −K[log(V )]2, 0].

Define the interval Bj M=[Qj,High, Qj,Low]. The following the-orem considers the rate and delay of packets that enter whenqj(t) ∈ Bj and that eventually depart.

Theorem 4: Suppose that V ≥ 1, that γ∗V is unique, thatthe slackness assumption (2) holds, and that the dual functiong(γ) satisfies:

g(γ∗V ) ≥ g(γ) + L||γ∗V − γ|| ∀ γ � 0, (15)

for some constant L > 0 independent of V . Define D,K, c∗

as in Theorem 3, and define Bj as above. Then for any queuej with a time average input rate λj > 0, we have under LIFO-Backpressure that:

(a) The rate λ̃j of packets that both arrive to queue j whenqj(t) ∈ Bj and that eventually depart the queue satisfies:

λj ≥ λ̃j ≥[λj − δmaxc

∗

V log(V )

]+

. (16)

(b) The average delay of these packets is at most Wbound,where:

WboundM=[2D + 2K[log(V )]2 + δmax]/λ̃j .

This theorem says that the delay of packets that enter whenqj(t) ∈ Bj and that eventually depart is at most O([log(V )]2).Further, by (16), when V is large, these packets represent theoverwhelming majority, in that the rate of packets not in thisset is at most O(1/V log(V )).

Proof: (Theorem 4) Theorem 2 shows that average queuebacklog is finite. Thus, there can be at most a finite numberof packets that enter the queue and never depart, so the rate ofpackets arriving that never depart must be 0. It follows that λ̃jis equal to the rate at which packets arrive when qj(t) ∈ Bj .

PROC. WIOPT 2011. 6

Define the indicator function 1j(t) to be 1 if qj(t) /∈ Bj , and0 else. Define λ̃cj

M=λj − λ̃j . Then with probability 1 we get: 3

λ̃cj = limt→∞

1t

t−1∑τ=0

Aj(τ)1j(τ)

= limt→∞

1t

t−1∑τ=0

E{Aj(τ)1j(τ)

}.

Then using the fact that Aj(t) ≤ δmax for all j, t:

E{Aj(t)1j(t)

}= E

{Aj(t)|qj(t) /∈ Bj

}Pr{qj(t) /∈ Bj)

≤ δmaxPr(qj(t) /∈ [Qj,Low, Qj,High]).

Therefore:

λ̃cj ≤ δmax limt→∞

1t

t−1∑τ=0

Pr(qj(τ) /∈ [Qj,Low, Qj,High])

≤ δmax limt→∞

1t

t−1∑τ=0

Pr(|qj(τ)− γ∗V,j | > D +Km),

where we define mM=[log(V )]2, and note that m ≥ 0 becauseV ≥ 1. From Theorem 3 we thus have:

0 ≤ λ̃cj ≤ δmaxc∗e−m =δmaxc

∗

V log(V ). (17)

This completes the proof of part (a). Now define B̃j =M=[Qj,High + δmax, Qj,Low]. Since Bj ⊂ B̃j , we see that therate of the packets that enter B̃j is at least λ̃j . Part (b) thenfollows from Theorem 1 and the facts that queue j is stableand that |B̃j | ≤ 2D + 2K[log(V )]2 + δmax.

Note that if λj = Θ(1), we see from Theorem 4 that, underLIFO-Backpressure, the time average delay for almost allpackets going through queue j is only O([log(V )]2). Applyingthis argument to all network queues with Θ(1) input rates, wesee that all but a tiny fraction of the traffic entering the networkonly experiences a delay of O([log(V )]2). This contrasts withthe delay performance result of the usual Backpressure withFIFO, which states that the time average delay will be Θ(V )for all packets [9]. Also note that under LIFO-Backpressure,some packets may stay in the queue for very long time.This problem can be compensated by introducing certaincoding techniques, e.g., fountain codes [19], into the LIFO-Backpressure algorithm.

VI. SIMULATION

In this section, we provide simulation results of the LIFO-Backpressure algorithm. We consider the network shown inFig. 4, where we try to support a flow sourced by Node 1destined for Node 7 with minimum energy consumption.

We assume that A(t) evolves according to the 2-stateMarkov chain in Fig. 5. When the state is HIGH , A(t) = 3,else A(t) = 0. We assume that the condition of each linkcan either be HIGH or LOW at a time. All the links exceptlink (2, 4) and link (6, 7) are assumed to be i.i.d. every time

3The time average expectation is the same as the pure time average bythe Lebesgue Dominated Convergence Theorem, because we assume the puretime average exists with probability 1, and that 0 ≤ Aj(t) ≤ δmax ∀t.

1

2

3

4

6

5

7

A(t)

(0.4, 3)

(0.2, 4)

(0.5, 3)

(0.4, 3)

(0.3, 4)

(0.4, 3)

(0.4, 3)

(0.3, 4)

(0.5, 4)

Fig. 4. A multihop network. (a, b) represents the HIGH probability a andthe rate b obtained with one unit of power when HIGH .

slot, whereas the conditions of link (2, 4) and link (6, 7)are assumed to be evolving according to independent 2-stateMarkov chains in Fig. 5. Each link’s HIGH probability andunit power rate at the HIGH state is shown in Fig. 4. Theunit power rates of the links at the LOW state are all assumedto be 1. We assume that the link states are all independent andthere is no interference. However, each node can only spendone unit of power per slot to transmit over one outgoing link,although it can simultaneously receive from multiple incominglinks. The goal is to minimize the time average power whilemaintaining network stability.

HIGH LOW

0.3

0.7 0.7

0.3

Fig. 5. The two state Markov chain with the transition probabilities.

We simulate Backpressure with both LIFO and FIFO for106 slots with V ∈ {20, 50, 100, 200, 500}. It can be verifiedthat the backlog vector converges to a unique attractor as Vincreases in this case. The left two plots in Fig. 6 show theaverage power consumption and the average backlog underLIFO-Backpressure. It can be observed that the average powerquickly converges to the optimal value and that the averagebacklog grows linearly in V . The right plot of Fig. 6 showsthe percentage of time when there exists a qj whose valuedeviates from γ∗V j by more than 2[log(V )]2. As we can see,this percentage is always very small, i.e., between 0.002 and0.013, showing a good match between the theory and thesimulation results.

0 200 400 6001.48

1.5

1.52

1.54

1.56

1.58

1.6

1.62

1.64

V

0 200 400 6000

200

400

600

800

1000

1200

1400

1600

1800

2000

V

0 200 400 600

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

V

Power Backlog Deviation

Fig. 6. LEFT: average network power consumption. MIDDLE: averagenetwork backlog size. RIGHT: percentage of time when ∃ qj such that|qj − γ∗V j | > 2[log(V )]2.

Fig. 7 compares the delay statistics of LIFO and FIFO for

PROC. WIOPT 2011. 7

more than 99.9% of the packets that leave the system beforethe simulation ends, under the cases V = 100 and V = 500.We see that LIFO not only dramatically reduces the averagepacket delay for these packets, but also greatly reduces thedelay for most of these packets. For instance, when V = 500,under FIFO, almost all packets experience the average delayaround 1220 slots. Whereas under LIFO, the average packetdelay is brought down to 78. Moreover, 52.9% of the packetsonly experience delay less than 20 slots, and 90.4% of thepackets experience delay less than 100 slots. Hence mostpackets’ delay are reduced by a factor of 12 under LIFO ascompared to that under FIFO!

7

TABLE IQLA WITH FIFO VS. QLA WITH LIFO

V=100Case Avg. DL % DL < 20 % DL < 50 % DL < 100LIFO 55.4 55.0 82.1 91.8FIFO 260.6 0 0 0

V=500Case Avg. DL % DL < 20 % DL < 50 % DL < 100LIFO 78.3 52.9 80.4 90.4FIFO 1219.8 0 0 0

VIII. EMPIRICAL VALIDATION

In this section we validate our analysis against empiricalresults obtained from the same testbed and Backpressure Col-lection Protocol (BCP) code developed in [10]. It is importantto note that these experiments are therefore not one-to-onecomparable with the analysis and simulations which we havepreviously presented. We note that BCP runs atop the defaultCSMA MAC for TinyOS which is not known to be throughputoptimal, that the testbed may not precisely be defined bya finite state Markovian evolution, and finally that limitedstorage availability on real wireless sensor nodes mandates theintroduction of virtual queues to maintain backpressure valuesin the presence of data queue overflows.

In order to avoid using very large data buffers, in [10] theforwarding queue of BCP has been implemented as a floatingqueue. The concept of a floating queue is shown in Figure10, which operates with a finite data queue of size Dmax

residing atop a virtual queue which preserves backpressurelevels. Packets that arrive to a full data queue result in adata queue discard and the incrementing of the underlyingvirtual queue counter. Underflow events (in which a virtualbacklog exists but the data queue is empty) results in nullpacket generation, which are filtered and then discarded bythe sink. 3

Despite these real-world differences, we are able to demon-strate clear order-equivalent delay gains due to LIFO usage inBCP in the following experimentation.

Fig. 9. The 40 tMote Sky devices used in experimentation on Tutornet.

A. Testbed and General Setup

To demonstrate the empirical results, we deployed a col-lection scenario across 40 nodes within the Tutornet testbed

3The LIFO floating queue can be shown (through sample path arguments) tohave a discard rate that is still proportional to O( 1

V c0 log(V ) ) with c0 = !(1)derived in [18].

(see Figure 9). This deployment consisted of Tmote Skydevices embedded in the 4th floor of Ronald Tutor Hall atthe University of Southern California.

In these experiments, one sink mote (ID 1 in Figure 9)was designated and the remaining 39 motes sourced trafficsimultaneously, to be collected at the sink. The Tmote Skydevices were programmed to operate on 802.15.4 channel 26,selected for the low external interference in this spectrum onTutornet. Further, the motes were programmed to transmitat -15 dBm to provide reasonable interconnectivity. Theseexperimental settings are identical to those used in [10].

Fig. 10. The floating LIFO queues of [10] drop from the dataqueue during overflow, placing the discards within an underlyingvirtual queue. Services that cause data queue underflows generatenull packets, reducing the virtual queue size.

We vary Dmax over experimentation because the exactvalue of Dmax is not readily apparent in a real system. Thishighlights the difficulty faced by techniques requiring explicitknowledge of this or similar system parameters (e.g., Fast-QLA in [9]). In practice, BCP defaults to a Dmax setting of12 packets, the maximum reasonable resource allocation for apacket forwarding queue in these highly constrained devices.

B. Experiment ParametersExperiments consisted of Poisson traffic at 1.0 packets

per second per source for a duration of 20 minutes. Thissource load is moderately high, since the boundary of thecapacity region for BCP running on this subset of motes haspreviously been documented at 1.6 packets per second persource [10]. A total of 36 experiments were run using thestandard BCP LIFO queue mechanism, for all combinationsof V ! {1, 2, 3, 4, 6, 8, 10, 12} and LIFO storage thresholdDmax ! {2, 4, 8, 12}. In order to present a delay baseline forBackpressure we additionally modified the BCP source codeand ran experiments with 32-packet FIFO queues (no floatingqueues) for V ! {1, 2, 3}. 4

C. ResultsTestbed results in Figure 11 provide the system average

packet delay from source to sink over V and Dmax, and

4These relatively small V values are due to the constraint that the moteshave small data buffers. Using larger V values will cause buffer overflow atthe motes.

Fig. 7. Delay statistics under Backpressure with LIFO and FIFO for packetsthat leave the system before simulation ends (more than 99.9%). %DL < ais the percentage of packets that enter the network and has delay less than a.

Fig. 8 also shows the delay for the first 20000 packets thatenter the network in the case when V = 500. We see that underBackpressure plus LIFO, most of the packets experience verysmall delay; while under Backpressure with FIFO, each packetexperiences roughly the average delay.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

0

2

4

6

8

10

12

14

!"#$%&$

&'()*$!$+,-$./)('$

0%01

+%01

Fig. 8. Packet Delay under Backpressure with LIFO and FIFO

VII. EMPIRICAL VALIDATION

In this section we validate our analysis empirically by carry-ing out new experiments over the same testbed and Backpres-sure Collection Protocol (BCP) code of [10]. This prior workdid not empirically evaluate the relationship between V , finitestorage availability, packet latency and packet discard rate. Wenote that BCP runs atop the default CSMA MAC for TinyOSwhich is not known to be throughput optimal, that the testbedmay not precisely be defined by a finite state Markovianevolution, and finally that limited storage availability on realwireless sensor nodes mandates the introduction of virtualqueues to maintain backpressure values in the presence of dataqueue overflows.

In order to avoid using very large data buffers, in [10] theforwarding queue of BCP has been implemented as a floatingqueue. A floating queue operates with a finite data queueof size Dmax residing atop a virtual queue which preservesbackpressure levels. Packets that arrive to a full data queueresult in a data queue discard and the incrementing of theunderlying virtual queue counter. Underflow events (in whicha virtual backlog exists but the data queue is empty) results innull packet generation, which are filtered and then discardedby the destination.

Fig. 9. The 40 tMote Sky devices used in experimentation on Tutornet.

A. Testbed and General Setup

To demonstrate the empirical results, we deployed a col-lection scenario across 40 nodes within the Tutornet testbed(see Figure 9). This deployment consisted of Tmote Skydevices embedded in the 4th floor of Ronald Tutor Hall at theUniversity of Southern California. In these experiments, onesink mote (ID 1 in Figure 9) was designated and the remaining39 motes sourced traffic simultaneously, to be collected at thesink. The Tmote Sky devices were programmed to operateon 802.15.4 channel 26, selected for the low external inter-ference in this spectrum on Tutornet. Further, the motes wereprogrammed to transmit at -15 dBm to provide reasonableinterconnectivity. These experimental settings are identical tothose used in [10]. We vary Dmax over experimentation.In practice, BCP defaults to a Dmax setting of 12 packets,the maximum reasonable resource allocation for a packetforwarding queue in these highly constrained devices.

B. Experiment Parameters

Experiments consisted of Poisson traffic at 1.0 packetsper second per source for a duration of 20 minutes. Thissource load is moderately high, as the boundary of the ca-pacity region for BCP running on this subset of motes haspreviously been documented at 1.6 packets per second persource [10]. A total of 36 experiments were run using thestandard BCP LIFO queue mechanism, for all combinationsof V ∈ {1, 2, 3, 4, 6, 8, 10, 12} and LIFO storage thresholdDmax ∈ {2, 4, 8, 12}. In order to present a delay baseline forBackpressure we additionally modified the BCP source codeand ran experiments with 32-packet FIFO queues (no floatingqueues) for V ∈ {1, 2, 3}. 4

4These relatively small V values are due to the constraint that the moteshave small data buffers. Using larger V values will cause buffer overflow atthe motes.

PROC. WIOPT 2011. 8

C. Results

Testbed results in Figure 10 provide the system averagepacket delay from source to sink over V and Dmax, andincludes 95% confidence intervals. Delay in our FIFO imple-mentation scales linearly with V, as predicted by the analysisin [9]. This yields an average delay that grows very rapidlywith V , already greater than 9 seconds per packet for V = 3.Meanwhile, the LIFO floating queue of BCP performs muchdifferently. We have plotted a scaled [log(V )]2 target, andnote that as Dmax increases the average packet delay remainsbounded by Θ([log(V )]2).

Fig. 10. System average source to sink packet delay for BCP FIFOversus BCP LIFO implementation over various V parameter settings.

These delay gains are only possible as a result of discardsmade by the LIFO floating queue mechanism that occur whenthe queue size fluctuates beyond the capability of the finitedata queue to smooth. Figure 11 gives the system packetloss rate of BCP’s LIFO floating queue mechanism over V .Note that the poly-logarithmic delay performance of Figure10 is achieved even for data queue size 12, which itselfdrops at most 5% of traffic at V = 12. We cannot stateconclusively from these results that the drop rate scales likeO( 1

V c0 log(V ) ). We hypothesize that a larger V value would berequired in order to observe the predicted drop rate scaling.Bringing these results back to real-world implications, notethat BCP (which minimizes a penalty function of packetretransmissions) performs very poorly with V = 0, and wasfound to have minimal penalty improvement for V greaterthan 2. At this low V value, BCP’s 12-packet forwardingqueue demonstrates zero packet drops in the results presentedhere. These experiments, combined with those of [10] suggeststrongly that the drop rate scaling may be inconsequential inmany real world applications.

REFERENCES

[1] L. Tassiulas and A. Ephremides. Stability properties of constrainedqueueing systems and scheduling policies for maximum throughput inmultihop radio networks. IEEE Trans. on Automatic Control, vol. 37,no. 12, pp. 1936-1949, Dec. 1992.

[2] A. Eryilmaz and R. Srikant. Fair resource allocation in wireless networksusing queue-length-based scheduling and congestion control. IEEE/ACMTrans. Netw., 15(6):1333–1344, 2007.

Fig. 11. System packet loss rate of BCP LIFO implementation overvarious V parameter settings.

[3] L. Huang and M. J. Neely. The optimality of two prices: Maximizingrevenue in a stochastic network. Proc. of 45th Annual Allerton Confer-ence on Communication, Control, and Computing (invited paper), Sept.2007.

[4] R. Urgaonkar and M. J. Neely. Opportunistic scheduling with reliabilityguarantees in cognitive radio networks. IEEE INFOCOM Proceedings,April 2008.

[5] Y. Yi and M. Chiang. Stochastic network utility maximization: A tributeto Kelly’s paper published in this journal a decade ago. EuropeanTransactions on Telecommunications, vol. 19, no. 4, pp. 421-442, June2008.

[6] L. Georgiadis, M. J. Neely, and L. Tassiulas. Resource Allocation andCross-Layer Control in Wireless Networks. Foundations and Trends inNetworking Vol. 1, no. 1, pp. 1-144, 2006.

[7] M. J. Neely. Super-fast delay tradeoffs for utility optimal fair schedulingin wireless networks. IEEE Journal on Selected Areas in Communica-tions (JSAC), Special Issue on Nonlinear Optimization of Communica-tion Systems, 24(8), Aug. 2006.

[8] M. J. Neely. Optimal energy and delay tradeoffs for multi-user wirelessdownlinks. IEEE Transactions on Information Theory vol. 53, no. 9, pp.3095-3113, Sept. 2007.

[9] L. Huang and M. J. Neely. Delay reduction via Lagrange multipliersin stochastic network optimization. IEEE Transactions on AutomaticControl, to appear.

[10] S. Moeller, A. Sridharan, B. Krishnamachari, and O. Gnawali. Routingwithout routes: The backpressure collection protocol. 9th ACM/IEEEInternational Conference on Information Processing in Sensor Networks(IPSN), 2010.

[11] E. Athanasopoulou, L. X. Bui, T. Ji, R. Srikant, and A. Stolyar.Backpressure-based packet-by-packet adaptive routing in communica-tion networks. arXiv:1005.4984v1, May 2010.

[12] L Bui, R Srikant, and A Stolyar. Optimal resource allocation for multi-cast sessions in multi-hop wireless networks. Philosophical Transactionsof The Royal Society, Series A, pages 2059–2074, Jan 2008.

[13] L. Bui, R. Srikant, and A. Stolyar. Novel architectures and algorithms fordelay reduction in back-pressure scheduling and routing. Proceedingsof IEEE INFOCOM 2009 Mini-Conference, April 2009.

[14] D. P. Bertsekas and R. G. Gallager. Data Networks. Prentice Hall, 1992.[15] M. J. Neely, E. Modiano, and C. Li. Fairness and optimal stochastic

control for heterogeneous networks. IEEE INFOCOM Proceedings,March 2005.

[16] M. J. Neely. Stability and capacity regions or discrete time queueingnetworks. arXiv:1003.3396v1, March 2010.

[17] L. Huang, S. Moeller, M. J. Neely, and Bhaskar Krishnamachari.Lifo-backpressure achieves near optimal utility-delay tradeoff. ArXivTechnical Report, arXiv:1008.4895v2., 2010.

[18] L. Huang and M. J. Neely. Max-weight achieves the ex-act [O(1/V ), O(V )] utility-delay tradeoff under Markov dynamics.arXiv:1008.0200v1, 2010.

[19] M. Mitzenmacher. Digital fountains: A survey and look forward. IEEEInformation Theory Workshop (ITW), San Antonio, Texas, Oct, 2004.