LIFOBackpressure achieves near optimal utilitydelay tradeoff
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 utilityoptimal algorithms that are also delay efficient. In this paper, we show that the Backpressure algorithm, when combined with the LIFO queueing discipline (called LIFOBackpressure), 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 LIFOBackpressure 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.

Conference Paper: When heavytailed and lighttailed flows compete: The response time tail under generalized maxweight scheduling
INFOCOM, 2013 Proceedings IEEE; 01/2013  SourceAvailable from: cttc.cat
Conference Paper: Studying Practical AnytoAny Backpressure Routing in WiFi Mesh Networks from a Lyapunov Optimization Perspective.
IEEE 8th International Conference on Mobile Adhoc and Sensor Systems, MASS 2011, Valencia, Spain, October 1722, 2011; 01/2011  01/2012;
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LIFOBackpressure Achieves Near Optimal
UtilityDelay 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 utilityoptimal algorithms
that are also delay efficient. In this paper, we show that the
Backpressure algorithm, when combined with the LIFO queueing
discipline (called LIFOBackpressure), 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
LIFOBackpressure 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 ITMANET W911NF070028, NSF CAREER CCF0747525,
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 )] utilitydelay 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 utilitydelay tradeoffs. Previous
works [7] and [8] show improved tradeoffs are possible
for singlehop 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 polylogarithmic 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 predetermined learning
phase, which adds additional complexity to the implemen
tation. The current work, following the line of analysis in
[9], instead shows that similar polylogarithmic 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 FirstinFirstOut (FIFO) to LastInFirst
Out (LIFO) (called LIFOBackpressure below). This is a
remarkable feature that distinguishes LIFOBackpressure 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 ) closetooptimal utilitiy
and O([log(V )]2) average backlog. This provides an analytical
justification for experimental observations in [10] that shows a
related LIFOBackpressure 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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2
LIFOBackpressure 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 backpressurebased perpacket
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 perdestination queue complexity, but does not
provide O([log(V )]2) average delay guarantees.
Ouranalysisofthedelay
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”
performanceofLIFO
01000 20003000 40005000
t
6000 70008000900010000
0
!*
V
O([log(V)]2)
" (V)
!"#$%&'()*+(
,./%0
,1$2
!&'(
344/56#
,78789:/;8
Fig. 1. The LIFOBackpressure 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 LIFOBackpressure 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 nonpositive) 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 timeinvariant 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, timeinvariant, 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 ηslackness. 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 actionchoosing 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 actionchoosing 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 actionchoosing 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
maxweight 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 2queue 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 IIIC.
NETWORK STATE, TRAFFIC AND RATE
On Using LIFO in MaxWeight Scheduling
This note provides a brief summary of the development of using LIFO in maxweight scheduli
(called LIFO scheduling in short) in stochastic network optimization. The idea of using LIFO with t
maxweight 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+ 1 2x1
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 maxweight 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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4
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 )] utilitydelay
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 LIFOBackpressure is the same
as the original Backpressure, and LIFOBackpressure only
requires the knowledge of the instantaneous network condi
tion. This is a remarkable feature that distinguishes it from
the previous algorithms achieving similar polylogarithmic
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. LIFOBackpressure Proof
We now provide the analysis of LIFOBackpressure. 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 )] utilitydelay 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 LIFOBackpressure 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 LIFOBackpressure. 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) + Lγ∗
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 LIFOBackpressure. 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) + Lγ∗
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: