Page 1

LIFO-Backpressure Achieves Near Optimal

Utility-Delay Tradeoff

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

Abstract—There has been considerable recent work developing

a new stochastic network utility maximization framework using

Backpressure algorithms, also known as MaxWeight. A key open

problem has been the development of utility-optimal algorithms

that are also delay efficient. In this paper, we show that the

Backpressure algorithm, when combined with the LIFO queueing

discipline (called LIFO-Backpressure), is able to achieve a utility

that is within O(1/V ) of the optimal value for any scalar

V ≥ 1, while maintaining an average delay of O([log(V )]2)

for all but a tiny fraction of the network traffic. This result

holds for general stochastic network optimization problems and

general Markovian dynamics. Remarkably, the performance of

LIFO-Backpressure can be achieved by simply changing the

queueing discipline; it requires no other modifications of the

original Backpressure algorithm. We validate the results through

empirical measurements from a sensor network testbed, which

show good match between theory and practice.

Index Terms—Queueing, Dynamic Control, LIFO scheduling,

Lyapunov analysis, Stochastic Optimization

I. INTRODUCTION

Recent developments in stochastic network optimization

theory have yielded a very general framework that solves a

large class of networking problems of the following form: We

are given a discrete time stochastic network. The network state,

which describes current realization of the underlying network

randomness, such as the network channel condition, is time

varying according to some probability law. A network con-

troller performs some action based on the observed network

state at every time slot. The chosen action incurs a cost,1

but also serves some amount of traffic and possibly generates

new traffic for the network. This traffic causes congestion, and

thus leads to backlogs at nodes in the network. The goal of the

controller is to minimize its time average cost subject to the

constraint that the time average total backlog in the network

be kept finite.

This general setting models a large class of networking

problems 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 to

this 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 with

the 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 following

grants: DARPA IT-MANET W911NF-07-0028, NSF CAREER CCF-0747525,

and continuing through participation in the Network Science Collaborative

Technology 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 provable

performance guarantees, robustness to stochastic network con-

ditions and, most importantly, their ability to achieve the de-

sired performance without requiring any statistical knowledge

of the underlying randomness in the network.

Most prior performance results for Backpressure are given

in the following [O(1/V ),O(V )] utility-delay tradeoff form

[6]: Backpressure is able to achieve a utility that is within

O(1/V ) of the optimal utility for any scalar V ≥ 1, while

guaranteeing a average network delay that is O(V ). Although

these results provide strong theoretical guarantees for the

algorithms, the network delay can actually be unsatisfying

when 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. Previous

works [7] and [8] show improved tradeoffs are possible

for single-hop networks with certain structure, and develops

optimal [O(1/V ),O(log(V ))]and [O(1/V ),O(√V )] utility-

delay tradeoffs. However, the algorithms are different from

basic Backpressure and require knowledge of an “epsilon”

parameter that measures distance to a performance region

boundary. Work [9] uses a completely different analytical

technique 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 learning

phase, 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 a

remarkable feature that distinguishes LIFO-Backpressure from

previous algorithms in [7] [8] [9], and provides a deeper

understanding of backpressure itself, and the role of queue

backlogs as Lagrange multipliers (see also [2] [9]). However,

this performance improvement is not for free: We must drop

a small fraction of packets in order to dramatically improve

delay for the remaining ones. We prove that as the V parameter

is increased, the fraction of dropped packets quickly converges

to zero, while maintaining O(1/V ) close-to-optimal utilitiy

and O([log(V )]2) average backlog. This provides an analytical

justification for experimental observations in [10] that shows a

related LIFO-Backpressure rule serves up to 98% of the traffic

with delay that is improved by 2 orders of magnitude.

arXiv:1008.4895v2 [math.OC] 3 Apr 2011

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LIFO-Backpressure was proposed in recent empirical work

[10]. The authors developed a practical implementation of

backpressure routing and showed experimentally that applying

LIFO queuing discipline drastically improves average packet

delay, but did not provide theoretical guarantees. Another

notable recent work providing an alternative delay solution is

[11], which describes a novel backpressure-based per-packet

randomized routing framework that runs atop the shadow

queue structure of [12] while minimizing hop count as ex-

plored in [13]. Their techniques reduce delay drastically and

eliminates the per-destination queue complexity, but does not

provide O([log(V )]2) average delay guarantees.

Our analysisof the delay

Backpressure is based on the recent “exponential attraction”

result developed in [9]. The proof idea can be intuitively

explained by Fig. 1, which depicts a simulated backlog process

of a single queue system with unit packet size under Backpres-

sure. The left figure demonstrates the “exponential attraction”

performanceof LIFO-

0 10002000 300040005000

t

60007000 80009000 10000

0

!*

V

O([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 Backpressure

deviate from some fixed point with probability that decreases

exponentially in the deviation distance. Hence the queue size

will mostly fluctuate within the interval [QLow,QHigh] which

can be shown to be of O([log(V )]2) size. This result holds

under both FIFO and LIFO, as they result in the same queue

process. Now suppose LIFO is used in this queue. Then from

the right figure, we see that most of the packets will arrive at

the queue when the queue size is between QLowand QHigh,

and these new packets will always be placed on the top of

the queue due to the LIFO discipline. Most packets thus enter

and leave the queue when the queue size is between QLowand

QHigh. Therefore, these packets “see” a queue with average

size no more than QHigh−QLow= O([log(V )]2). Now let λ

be the packet arrival rate into the queue, and let˜λ be the

arrival rate of packets entering when the queue size is in

[QLow,QHigh] and that eventually depart. Because packets

always occupy the same buffer slot under LIFO, we see that

these packets can occupy at most QHigh− QLow+ δmax

buffer slots, ranging from QLow to QHigh+ δmax, where

δmax= 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 arrive

when the queue size is in [QLow,QHigh] satisfies:

D ≤QHigh− QLow+ δmax

˜λ

=O([log(V )]2)

˜λ

.

(1)

Finally, the exponential attraction result implies that λ ≈˜λ.

Hence for almost all packets entering the queue, the average

delay is D = O([log(V )]2/λ).

This paper is organized as follows. In Section II, we set up

our notations. We then present our system model in Section

III. We provide an example of our network in Section IV. We

review the Backpressure algorithm in Section V. The delay

performance of LIFO-Backpressure is presented in Section VI.

Simulation results are presented in Section VII. We then also

present experimental testbed results in Section VIII. Finally,

we comment on optimizing a function of time averages in

Section IX.

II. NOTATIONS

Here we first set up the notations used in this paper: R

represents the set of real numbers. R+ (or R−) denotes the

set of nonnegative (or non-positive) real numbers. Rn(or Rn

is the set of n dimensional column vectors, with each element

being in R (or R+). bold symbols a and aTrepresent column

vector and its transpose. a ? b means vector a is entrywise

no less than vector b. ||a−b|| is the Euclidean distance of a

and b. 0 and 1 denote column vector with all elements being

0 and 1. [a]+= max[a,0] and log(·) is the natural log.

+)

III. SYSTEM MODEL

In this section, we specify the general network model we

use. We consider a network controller that operates a network

with the goal of minimizing the time average cost, subject

to the queue stability constraint. The network is assumed to

operate in slotted time, i.e., t ∈ {0,1,2,...}. We assume there

are 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, or

a collection of other relevant information about the current

network channels and arrivals. We assume that S(t) evolves

according a finite state irreducible and aperiodic Markov chain,

with a total of M different random network states denoted

as S = {s1,s2,...,sM}. Let πsidenote the steady state

probability of being in state si. It is easy to see in this case that

πsi> 0 for all si. The network controller can observe S(t)

at the beginning of every slot t, but the πsiand transition

probabilities are not necessarily known.

B. The Cost, Traffic, and Service

At each time t, after observing S(t) = si, the controller

chooses an action x(t) from a set X(si), i.e., x(t) = x(si)for

some x(si)∈ X(si). The set X(si)is called the feasible action

set for network state siand is assumed to be time-invariant and

compact for all si∈ S. The cost, traffic, and service generated

by the chosen action x(t) = x(si)are as follows:

(a) The chosen action has an associated cost given by the

cost function f(t) = f(si,x(si)) : X(si)?→ R+ (or

X(si)?→ R−in reward maximization problems);

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3

(b) The amount of traffic generated by the action to

queue j is determined by the traffic function Aj(t) =

Aj(si,x(si)) : X(si)?→ R+, in units of packets;

(c) The amount of service allocated to queue j is given by

the rate function µj(t) = µj(si,x(si)) : X(si)?→ R+, in

units of packets;

Note that Aj(t) includes both the exogenous arrivals from

outside the network to queue j, and the endogenous arrivals

from other queues, i.e., the transmitted packets from other

queues, to queue j. We assume the functions f(si,·), µj(si,·)

and Aj(si,·) are continuous, time-invariant, their magnitudes

are uniformly upper bounded by some constant δmax∈ (0,∞)

for all si, j, and they are known to the network operator. We

also assume that there exists a set of actions {x(si)k}k=1,2,...,∞

with x(si)k∈ X(si)and some variables ϑ(si)

and k with?

πsi

k

for some η > 0 for all j. That is, the stability constraints

are feasible with η-slackness. Thus, there exists a stationary

randomized policy that stabilizes all queues (where ϑ(si)

represents the probability of choosing action x(si)kwhen

S(t) = si) [6].

i=1,...,M

k

≥ 0 for all si

kϑ(si)

k

ϑ(si)

k

= 1 for all si, such that

[Aj(si,x(si)k) − µj(si,x(si)k)]?≤ −η, (2)

?

si

??

k

C. Queueing, Average Cost, and the Stochastic Problem

Let q(t) = (q1(t),...,qr(t))T∈ Rr

the queue backlog vector process of the network, in units of

packets. We assume the following queueing dynamics:

qj(t + 1) = max?qj(t) − µj(t),0?+ Aj(t)

and q(0) = 0. By using (3), we assume that when a queue does

not have enough packets to send, null packets are transmitted.

In this paper, we adopt the following notion of queue stability:

+, t = 0,1,2,... be

∀j,

(3)

E?

r

?

j=1

qj

?? limsup

avto denote the time average cost induced by

an action-choosing policy Π, defined as:

t→∞

1

t

t−1

?

τ=0

r

?

j=1

E?qj(τ)?< ∞.

(4)

We also use fΠ

fΠ

av? limsup

t→∞

1

t

t−1

?

τ=0

E?fΠ(τ)?,

(5)

where fΠ

We call an action-choosing policy feasible if at every time

slot t it only chooses actions from the feasible action set

X(S(t)). We then call a feasible action-choosing policy under

which (4) holds a stable policy, and use f∗

optimal time average cost over all stable policies. In every

slot, the network controller observes the current network state

and chooses a control action, with the goal of minimizing the

time average cost subject to network stability. This goal can

be mathematically stated as: (P1)

the following, we will refer to (P1) as the stochastic problem.

Note that in some network optimization problems, e.g.,

[15], the objective of the network controller is to optimize

a function of a time average metric. In this case, we see

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

avto denote the

min : fΠ

av, s.t.(4). In

max-weight with a control parameter V , there exists some fixed point γ∗= (γ∗

pressure algorithm for utility optimization problems [6].

Backpressure: At every time slot t, observe the current

network state S(t) and the backlog q(t). If S(t) = si, choose

x(si)∈ X(si)that solves the following:

that the Backpressure algorithm and the deterministic problem

presented in the next section can similarly be constructed, but

will be slightly different. We will discuss these problems in

Section IX.

IV. 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 network

operator decides whether or not to allocate one unit of power

to serve packets at each queue, so as to support all arriving

traffic, i.e., maintain queue stability, with minimum energy

expenditure. We assume the network state S(t), which is

the quadruple (R1(t),R2(t),CH1(t),CH2(t)), evolves ac-

cording to the finite state Markov chain with three states

s1 = (1,1,G,B),s2 = (1,1,G,G), and s3 = (0,0,B,G).

Here Ri(t) denotes the number of exogenous packet arrivals

to 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 arriving

at queue i at time t. CHi(t) = G/B means that channel i

has a “Good” or “Bad” state. When a link’s channel state is

“Good”, one unit of power can serve 2 packets over the link,

otherwise it can only serve one. We assume power can be

allocated to both channels without affecting each other.

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.

In this case, we see that there are three possible network

states. 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 is

f(si,x(si)) = x1+ x2, for all si. The network states, the

traffic functions, and the service rate functions are summarized

in 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 depends

on x(si). Also note that A2(t) equals µ1(t) + R2(t) instead

of min[µ1(t),q1(t)]+R2(t) due to our idle fill assumption in

Section III-C.

NETWORK STATE, TRAFFIC AND RATE

On Using LIFO in Max-Weight Scheduling

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

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

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

TABLE I

S(t)

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

s1

11GB1

2x1+ 12x1

x2

s2

11GG1

2x1+ 12x1

2x2

s3

00BG0

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 util

V. BACKPRESSURE AND THE DETERMINISTIC PROBLEM

In this section, we first review the Backpressure algorithm

[6] for solving the stochastic problem. Then we define the

deterministic problem and its dual. We first recall the Back-

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

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

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

1,...,γ∗

r)T= Θ(V ) su

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

Pr{?q(t) − γ∗? > D + m} ≤ e−cm,

(

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

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

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

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

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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 be

decomposed into separate parts that are easier to solve, e.g.,

[3], [4]. Also, when the network state process S(t) is i.i.d., it

has been shown in [6] that,

fBP

av = f∗

av+ O(1/V ),qBP= O(V ),

(7)

where fBP

expected average network backlog size under Backpressure,

respectively. When S(t) is Markovian, [3] and [4] show that

Backpressure achieves an [O(log(V )/V ),O(V )] utility-delay

tradeoff if the queue sizes are deterministically upper bounded

by Θ(V ) for all time. Without this deterministic backlog

bound, it has recently been shown that Backpressure achieves

an [O(? +T?

and T?representing the proximity to the optimal utility value

and the “convergence time” of the Backpressure algorithm

to that proximity [16]. However, there has not been any

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]:

?

s.t.

Aj(x) ?

si

≤ Bj(x) ?

av

and qBPare the expected average cost and the

V),O(V )] tradeoff under Markovian S(t), with ?

min :

F(x) ? V

si

πsif(si,x(si))

(8)

?

πsiAj(si,x(si))

?

si

πsiµj(si,x(si)), ∀j,

x(si)∈ X(si)

∀i = 1,2,...,M,

where πsicorresponds to the steady state probability of S(t) =

siand x = (x(s1),...,x(sM))T. The dual problem of (8) can

be obtained as follows:

max : g(γ),s.t. γ ? 0,

(9)

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

?

+γj

g(γ) =inf

x(si)∈X(si)

si

πsi

?

V f(si,x(si))

(10)

?

j

?Aj(si,x(si)) − µj(si,x(si))??

.

Here γ = (γ1,...,γr)Tis the Lagrange multiplier of

(8). It is well known that g(γ) in (10) is concave in the

vector γ, and hence the problem (9) can usually be solved

efficiently, particularly when cost functions and rate functions

are separable over different network components. Below, we

use γ∗

of the problem (9) with the corresponding V .

V= (γ∗

V 1,γ∗

V 2,...,γ∗

V r)Tto denote an optimal solution

VI. 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 Backpressure

is first proposed in [10], although they did not provide any

theoretical performance guarantee. We will show, under some

mild conditions (to be stated in Theorem 3), that under LIFO-

Backpressure, the time average delay for almost all packets

entering the network is O([log(V )]2) when the utility is

pushed to within O(1/V ) of the optimal value. Note that the

implementation complexity of LIFO-Backpressure is the same

as the original Backpressure, and LIFO-Backpressure only

requires the knowledge of the instantaneous network condi-

tion. This is a remarkable feature that distinguishes it from

the previous algorithms achieving similar poly-logarithmic

tradeoffs in the i.i.d. case, e.g., [7] [8] [9], which all require

knowledge of some implicit network parameters other than the

instant network state. Below, we first provide a simple example

to demonstrate the need for careful treatment of the usage of

LIFO in Backpressure algorithms, and then present a modified

Little’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 time

0, and one packet periodically arrives every slot thereafter (at

times 1,2,3,...). The system is initially empty and can serve

exactly one packet per slot. The arrival rate λ is clearly 1

packet/slot (so that λ = 1). Further, under either FIFO or

LIFO service, there are always 2 packets in the system, so

Q = 2.

Under FIFO service, the first packet has a delay of 1 and

all packets thereafter have a delay of 2:

WFIFO

1

= 1 , WFIFO

i

= 2 ∀i ∈ {2,3,4,...},

where WFIFO

(WLIFO

i

i

is similarly defined for LIFO). We thus have:

is the delay of the ithpacket under FIFO

W

FIFO?= lim

K→∞

1

K

K

?

i=1

WFIFO

i

= 2.

Thus, λW

indeed holds.

Now consider the same system under LIFO service. We still

have λ = 1, Q = 2. However, in this case the first packet never

departs, while all other packets have a delay equal to 1 slot:

FIFO= 1 × 2 = 2, Q = 2, and so λW

FIFO= Q

WLIFO

1

= ∞ , WLIFO

i

= 1 ∀i ∈ {2,3,4,...}.

Thus, for all integers K > 0:

1

K

K

?

i=1

WLIFO

i

= ∞.

and so W

hand, if we ignore the one packet with infinite delay, we note

that all other packets get a delay of 1 (exactly half the delay

in the FIFO system). Thus, in this example, LIFO service

significantly improves delay for all but the first packet.

For the above LIFO example, it is interesting to note that

if we define˜Q and ˜ W as the average backlog and delay

associated only with those packets that eventually depart, then

we have˜Q = 1,˜ W = 1, and the equation λ˜ W =˜Q indeed

holds. This motivates the theorem in the next subsection,

which considers a time average only over those packets that

eventually depart.

LIFO= ∞. Clearly λW

LIFO?= Q. On the other

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5

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 queueing

system. Let N(t) be the number of arrivals that use a buffer

location within set B up to time t. Let D(t) be the number of

departures from a buffer location within the set B up to time

t. Let Wibe the delay of the ith job to depart from the set

B. Define W as the limsup average delay considering only

those jobs that depart:

W?=limsup

t→∞

1

D(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:

liminf

t→∞

N(t)

t

≥ λmin,

Further suppose that limt→∞D(t) = ∞ with probability 1 (so

the number of departures is infinite). Then the average delay

W satisfies:

W?=limsup

t→∞

1

D(t)

D(t)

?

i=1

Wi≤ |B|/λmin,

where |B| is the size of the finite set B.

Proof: See Appendix A.

C. LIFO-Backpressure Proof

We now provide the analysis of LIFO-Backpressure. To

prove our result, we first have the following theorem, which

is the first to show that Backpressure (with either FIFO or

LIFO) achieves the exact [O(1/V ),O(V )] utility-delay trade-

off under a Markovian network state process. It generalizes

the [O(1/V ),O(V )] performance result of Backpressure in

the i.i.d. case in [6].

Theorem 2: Suppose S(t) is a finite state irreducible and

aperiodic Markov chain2and condition (2) holds, Backpres-

sure (with either FIFO or LIFO) achieves the following:

fBP

av = f∗

av+ O(1/V ), qBP= O(V ),

(11)

where fBP

backlog under Backpressure.

Proof: See [17].

Theorem 2 thus shows that LIFO-Backpressure guarantees

an average backlog of O(V ) when pushing the utility to within

O(1/V ) of the optimal value. We now consider the delay

performance of LIFO-Backpressure. For our analysis, we need

the following theorem (which is Theorem 1 in [9]).

Theorem 3: Suppose that γ∗

condition (2) holds, and that the dual function g(γ) satisfies:

av

and qBPare the expected time average cost and

Vis unique, that the slackness

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 constants

2In [17], the theorem is proven under more general Markovian S(t)

processes that include the S(t) process assumed here.

D,K,c∗= Θ(1), i.e., all independent of V , such that for any

m ∈ R+,

P(r)(D,Km)

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

P(r)(D,Km)

? limsup

t→∞

τ=0

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 indeed

the steady state probability that there exists a queue j whose

queue value deviates from γ∗

In this case, Theorem 3 states that qj(t) deviates from γ∗

Θ(log(V )) distance with probability O(1/V ). Hence when

V is large, qj(t) will mostly be within O(log(V )) distance

from γ∗

very restrictive. The condition (12) can usually be satisfied in

practice when the action space is finite, in which case the dual

function g(γ) is polyhedral (see [9] for more discussions). The

uniqueness of γ∗

utility optimization problems, e.g., [2].

We now present the main result of this paper with respect

to the delay performance of LIFO-Backpressure. Below, the

notion “average arrival rate” is defined as follows: Let Aj(t)

be the number of packets entering queue j at time t. Then the

time average arrival rate of these packets is defined (assuming

it exists): λj = limt→∞

t

we assume that time averages under Backpressure exist with

probability 1. This is a reasonable assumption, and holds

whenever the resulting discrete time Markov chain for the

queue vector q(t) under backpressure is countably infinite

and irreducible. Note that the state space is indeed countably

infinite if we assume packets take integer units. If the system

is also irreducible then the finite average backlog result of

Theorem 2 implies that all states are positive recurrent.

Let D,K,c∗be constants as defined in Theorem 3, and

recall that these are Θ(1) (independent of V ). Assume V ≥ 1,

and define Qj,Highand Qj,Lowas:

≤

c∗e−m,

(13)

(14)

1

t

t−1

?

Pr{∃j,|qj(τ) − γ∗

V j| > D + Km}.

V jby more than D+Km distance.

V jby

V j. Also note that the conditions of Theorem 3 are not

Vcan usually be satisfied in many network

1

?t−1

τ=0Aj(τ). For the theorem,

Qj,High

?=

?=

γ∗

max[γ∗

V j+ D + K[log(V )]2,

V j− D − K[log(V )]2,0].Qj,Low

Define the interval Bj?=[Qj,High,Qj,Low]. The following the-

orem considers the rate and delay of packets that enter when

qj(t) ∈ Bjand that eventually depart.

Theorem 4: Suppose that V ≥ 1, that γ∗

the slackness assumption (2) holds, and that the dual function

g(γ) satisfies:

Vis unique, that

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 Bjas above. Then for any queue

j with a time average input rate λj> 0, we have under LIFO-

Backpressure that: