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Departmental Papers (ESE) Department of Electrical & Systems Engineering
July 2008
Throughput-Optimal Scheduling in Multichannel Access Point Throughput-Optimal Scheduling in Multichannel Access Point
Networks under Infrequent Channel Measurements Networks under Infrequent Channel Measurements
Koushik Kar
Rensselaer Polytechnic Institute
Xiang Luo
Rensselaer Polytechnic Institute
Saswati Sarkar
University of Pennsylvania
, swati@seas.upenn.edu
Follow this and additional works at: https://repository.upenn.edu/ese_papers
Recommended Citation Recommended Citation
Koushik Kar, Xiang Luo, and Saswati Sarkar, "Throughput-Optimal Scheduling in Multichannel Access
Point Networks under Infrequent Channel Measurements", . July 2008.
Copyright 2008 IEEE. Reprinted from
IEEE Transactions on Wireless Communications
, Volume 7, Issue 7, July 2008,
pages 2619-2629.
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Throughput-Optimal Scheduling in Multichannel Access Point Networks under Throughput-Optimal Scheduling in Multichannel Access Point Networks under
Infrequent Channel Measurements Infrequent Channel Measurements
Abstract Abstract
We consider the problem of uplink/downlink scheduling in a multichannel wireless access point network
where channel states differ across channels as well as users, vary with time, and can be measured only
infrequently. We demonstrate that, unlike infrequent measurement of queue lengths, infrequent
measurement of channel states reduce the maximum attainable throughput.We then prove that in
frequency division multiplexed systems, a dynamic scheduling policy that depends on both the channel
rates (averaged over the measurement interval) and the queue lengths, is throughput optimal. We also
generalize the scheduling policy to solve the joint power allocation and scheduling problem. In addition,
we provide simulation studies that demonstrate the impact of the frequency of channel and queue state
measurements on the average delay and attained throughput.
Keywords Keywords
infrequent channel measurements, multichannel access point networks, throughput-optimal scheduling
Comments Comments
Copyright 2008 IEEE. Reprinted from
IEEE Transactions on Wireless Communications
, Volume 7, Issue 7,
July 2008, pages 2619-2629.
This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way
imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or
personal use of this material is permitted. However, permission to reprint/republish this material for
advertising or promotional purposes or for creating new collective works for resale or redistribution must
be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document,
you agree to all provisions of the copyright laws protecting it.
This journal article is available at ScholarlyCommons: https://repository.upenn.edu/ese_papers/449
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY 2008 2619
Throughput-Optimal Scheduling in
Multichannel Access Point Networks under
Infrequent Channel Measurements
Koushik Kar, Xiang Luo, and Saswati Sarkar
Abstract—We consider the problem of uplink/downlink
scheduling in a multichannel wireless access point network where
channel states differ across channels as well as users, vary with
time, and can be measured only infrequently. We demonstrate
that, unlike infrequent measurement of queue lengths, infrequent
measurement of channel states reduce the maximum attainable
throughput. We then prove that in frequency division multiplexed
systems, a dynamic scheduling policy that depends on both the
channel rates (averaged over the measurement interval) and
the queue lengths, is throughput optimal. We also generalize
the scheduling policy to solve the joint power allocation and
scheduling problem. In addition, we provide simulation studies
that demonstrate the impact of the frequency of channel and
queue state measurements on the average delay and attained
throughput.
Index Terms—Infrequent channel measurements, multi-
channel access point networks, throughput-optimal scheduling.
I. INTRODUCTION
F
UTURE wireless networks are likely to provide each
user access to multiple channels. The dynamic scheduling
problem at any given time in such networks is to determine
(i) the set of users that can transmit/receive, and (ii) the set
of channels that a user can use. Our goal is to optimally
determine the above so as to maximize the system throughput
using on-line adaptive policies. The availability of multiple
channels gives rise to several unique challenges in attaining
the above goal. Channel characteristics at any given time
will typically be different for different channels, and these
characteristics will also vary with time. In a system with a
large number of users and channels, an individual user could
use only a small number of channels at any time. Therefore,
measuring the channel quality perceived by each user for
each channel would require additional probe packets, which
introduces a significant measurement overhead. Thus unlike
single-channel networks, scheduling in multichannel networks
Manuscript received December 10, 2006; revised December 19, 2007;
accepted March 19, 2008. The associate editor coordinating the review of
this paper and approving it for publication was M. Buehrer. This work was
supported by the National Science Foundation under grants NCR-0238340,
CNS-0435306, NCR-0448316 and CNS-0435141. This is an extended version
of a work presented at IEEE INFOCOM 2007, Anchorage, AK, USA, May
2007.
K. Kar and X. Luo are with the Department of Electrical, Computer and
Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA
(e-mail: {kark, luox3}@rpi.edu).
S. Sarkar is with the Department of Electrical Engineering, University of
Pennsylvania, Philadelphia, PA 19104, USA (e-mail: swati@seas.upenn.edu).
Digital Object Identifier 10.1109/TWC.2008.061044.
must be done under inaccurate channel state information,
resulting from infrequent channel measurements. Moreover,
in a multichannel wireless system, the scheduling questions
depend strongly on the transmission mechanisms. Specifically,
the scheduling constraints differ significantly based on whether
the transmission by each user is single-channel or multi-
channel, and the manner in which power is allocated across
channels. Our contribution in this paper is to develop optimal
scheduling policies that address the above challenges.
Our first contribution is to demonstrate that infrequent
channel state measurements affect the system throughput in
a fundamentally different way than infrequent measurements
of other state variables. Specifically, it is well-known that
infrequent measurements of queue lengths of users do not
alter the maximum attainable throughput region, as long as
the measurement intervals are upper bounded by a constant.
We however show that infrequent measurement of channel
states does reduce the maximum attainable throughput region.
We further prove that a weighted queue-length based schedul-
ing policy attains the maximum attainable throughput region
under partial information about channel states. The weights
must be chosen based on the average channel rates till the
next measurement instant. We also investigate the structure
of the optimal scheduling policy under specific scheduling
constraints. We show that for single-channel transmission by
users, the throughput-optimal scheduling policy is a maximum
weighted matching between the users and the channels, and
for multi-channel transmission by users, on the other hand, the
scheduling policy corresponds to a maximum weighted poly-
matching. We then show how our results can be extended to
jointly optimize the scheduling and power allocation under
multi-channel transmission. From a practical perspective, the
algorithms that we present in this paper can be used for
uplink/downlink scheduling and power assignment for mul-
tichannel wireless systems like 802.16 access point networks.
II. R
ELATED WORK
There is a rich body of literature on the subject of
throughput-optimal scheduling in a wide variety of queueing
networks [1], [2], [6], [7], [15], [19], [20], [21], [22], [23],
[24], [25], [26], [29], [31], [30]. These papers either assume
that the service rates of the queues do not vary with time,
or if the service rates vary, the schedulers know the service
rates of the queues before each scheduling decision. The
equivalent assumption in our context is that the schedulers
1536-1276/08$25.00
c
2008 IEEE
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2620 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY 2008
know the instantaneous channel states. Recently, Neely at al.
have addressed the problem of jointly selecting the queues
to serve and determining the service rates of the selected
queues by appropriately regulating the transmission power
levels [18]. They also assume that the scheduler always knows
the instantaneous states of the channels. Our main contribution
is to develop throughput optimal scheduling policies when
the scheduler knows the channel states only infrequently. We
also demonstrate that the impact of infrequent knowledge of
channel states is substantially different from that of infrequent
knowledge of queue lengths. While infrequent knowledge
of queue lengths does not alter the maximum achievable
throughput region (as shown by several previous results in
different settings [1], [2], [20], [21], [22], [23], [24], [25],
[26], [27]), we show in this paper that infrequent knowledge
of service rates substantially reduces the maximum achievable
throughput region.
Several interesting medium access control protocols, e.g.,
[9], [14], [8], [16], [28], [32], have been proposed for se-
lecting channels in context of specific wireless technologies,
e.g., IEEE 802.11, which do not however guarantee through-
put optimality. Our contribution lies in the development of
scheduling algorithms that provably maximize throughput in
presence of time variations, asymmetry in the rates of different
channels, and infrequent measurements.
For the case where each user can transmit over multiple
channels simultaneously, there have been several recent papers
that address a problem that is closely related to ours [5],
[33], [10], [12]. The authors in [5], [33] have addressed the
question of how resources (like bandwidth and power) should
be allocated to users in an multi-channel transmission system
to maximize system throughput. However, in these works, the
resource allocation problem is not considered in a stochastic
setting, and therefore the problem addressed in [5], [33]
is quite different from the stochastic dynamic optimization
problem that we consider here. In [10], [12], the authors
address the multi-channel transmission case of our problem
for two-state (on-off) channel models. In contrast, we consider
channel models that are much more general (can have any
number of states) and address both the cases of single-channel
and multi-channel transmission by users. More importantly,
unlike our work, the results in [10], [12] assume that the
instantaneous channel states are always known, and do not
jointly optimize the channel and power allocations.
III. F
ORMULATION
A. System Model and Assumptions
Our system consists of a set of users sharing a set of
channels to communicate with an access point (AP). Let
M denote the set of channels, and N denote the set of
users. The access point network that we consider is a cen-
tralized network, where the scheduling decisions (both uplink
and downlink) are taken by the AP. In the following, we
focus most of our discussion on uplink scheduling, where
the users are transmitting data to the AP; the formulation
and approach presented here can easily be extended to the
downlink case. We assume that the AP is equipped with a
separate transceiver for each channel, and is thus capable of
receiving data simultaneously from multiple users provided
they receive on different channels. However, the AP cannot
successfully receive data from multiple users over a single
channel. In this scenario, whether a user can simultaneously
transmit on multiple channels or not, depends on the specific
system considered, and is discussed in Section III-B.
We allow channel conditions to vary across channels as
well as users. Channel conditions depend on various factors
like fading and interference (from neighboring access point
networks), which typically depend on the channel frequency,
as well as the user location. Therefore, the attainable rate on a
channel may be different for different channels; moreover, the
attainable rate may also depend on the user using the channel.
Let α
ij
(0 ≤ α
ij
≤ 1) denote the packet success probability
when user i transmits a packet on channel j. In the rest of
the paper, we will therefore refer to α
ij
as the channel rate of
user i on channel j. Note that the channel rates are typically
functions of time, since fading and interference levels at any
location can vary with time. These variations will be more
pronounced when the users are mobile.
We assume that time is slotted, and the slots are denoted
by t =1, 2, .... All packets have the same length, and the
transmission time of a packet equals a slot length. We assume
that packet arrivals occur at the beginning of any time slot,
and packet departures occur at the end of the time slot. At
any given time slot, the number of packet arrivals for different
users can be arbitrarily correlated. For user i, the number of
arrivals in any slot follows an i.i.d. process, with mean λ
i
.
Let
λ =(λ
i
,i ∈ N ) denote the vector of average arrival
rates. Note that while our results assume i.i.d. traffic arrival
patterns, they can be extended to more general arrival patterns
using fluid flow techniques [4]. We assume that each channel
rate, α
ij
, evolves in time according to a finite-state Markov
chain. At any given time, the different α
ij
s can be arbitrarily
correlated. Finally, we state our assumptions on the sampling
of channel and queue states. Let the time slots be grouped
into intervals of time T . Thus the (k +1)th interval consists
of slots kT, ..., (k +1)T − 1. Although the channel conditions
and queue lengths can change in each slot, these are measured
only at the beginning of each interval, i.e., at the beginning of
slot kT,fork =0, 1, .... Thus the interval length T denotes the
duration between successive sampling instances of the channel
conditions and queue lengths.
B. Scheduling Constraints
Next, we describe the constraints on our scheduling policy.
At the beginning of each interval, for each channel, a user is
selected to transmit on that channel during the interval. Note
that a channel cannot be assigned to multiple users in the
same interval. Under single-channel transmission, a user can
transmit on only one channel at any given time. Therefore, in
this case, the scheduling policy across channels corresponds
to a matching [3] in a bipartite graph, where the users and
the channels represent the two sets of vertices that need to be
matched. Under multi-channel transmission, however, a user
can transmit on multiple channels at the same time. Thus in
this case, a user can be matched to multiple channels, but
not vice versa. In this paper, we refer to such a one-to-many
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KAR et al.: THROUGHPUT-OPTIMAL SCHEDULING IN MULTICHANNEL ACCESS POINT NETWORKS UNDER INFREQUENT CHANNEL MEASUREMENTS 2621
(b) Poly−matching
users
channels
2
2
1
3
1
users
channels
2
2
1
3
1
(a) Matching
Fig. 1. Matching vs. poly-matching: The figure shows one possible matching
and one possible poly-matching for 3 users and 2 channels. (Note that the
matching/poly-matching is represented by the bold edges.)
matching between the users and channels as a poly-matching.
Figure 1 explains the difference between matchings and poly-
matchings.
Note that there can be multiple matchings or poly-matchings
in the bipartite graph of users and channels (the total number
of matchings or poly-matchings is in fact, exponential in the
size of the user-channel graph), and different matchings and
poly-matchings will provide significantly different through-
puts. A good choice of matching or poly-matching is critical to
attaining high system throughput. Therefore, the key challenge
in the dynamic scheduling question considered here is to select
the right matching or poly-matching at any time slot, so as to
maximize the long-term system throughput.
C. Stability Region and Throughput-optimal Scheduling
The notion of throughput-optimal scheduling is based on
the notion of a “stability region”; so we define the latter first.
Asystemissaidtobestable for an arrival rate vector
λ
under a scheduling policy Ψ, if the expected lengths of all
queues in the system remain bounded over all time, when
the packet arrival rate vector is
λ and Ψ is used as the
scheduling policy. In such a case, scheduling policy Ψ is said
to stabilize the system for arrival rate vector
λ.Thestability
region of the system is the set of all arrival rate vectors
for which the system can be stabilized by some scheduling
policy. Intuitively, the arrival rate vector belonging to the
stability region is “attainable”, since there exists a scheduling
policy under which the system is stable for that arrival rate
vector. Moreover, a rate vector outside the stability region
is not attainable, since all scheduling policies would lead to
unbounded queues in the system for that arrival rate vector.
As we argue later in the paper, the stability region in our
system depends on the measurement interval T .LetΛ
T
denote
the stability region of the system for interval length T .An
analytical characterization of the stability region of the system
that we consider can be found in the appendix (refer to (8)-
(10)).
A scheduling policy is said to be throughput-optimal if it
stabilizes the system for all arrival rate vectors that are strictly
within the stability region. In other words, a throughput-
optimal scheduling policy can “attain” all arrival rate vectors
that belong to the interior of the stability region Λ
T
.Inthe
next few sections, we present throughput-optimal scheduling
policies for the multichannel wireless system described above.
IV. T
HROUGHPUT-OPTIMAL SCHEDULING
Before we present our scheduling policy and argue about
its throughput-optimality, we discuss some properties of the
stability region Λ
T
.
A. Characterization of the Stability Region
In the following lemma, we prove that the stability region
reduces with increase in T. Let Int(Λ
T
) represent the interior
of the stability region, Λ
T
.
Lemma 1: For any T ≥ 1, Λ
lT
⊆ Λ
T
∀ positive integers
l. If l>1, there exists systems where Int(Λ
lT
) ⊂ Int(Λ
T
).
Lemma 1 is proved in the appendix. Intuitively, Lemma 1
states that the stability region “shrinks” as the measurement
interval increases.
Note that in practice, some inference on the channel states
can be drawn from the success or failure of packets transmitted
during an interval. However, in our definition of Λ
T
,we
assume that such information is not used by the scheduling
policy.
Let us now consider a scenario where the queue states are
measured only at the beginning of each interval (of T time
slots), but the channel states are measured at the beginning
of every time slot. Let
ˆ
Λ
T
denote the stability region in this
case. The following result can be easily shown, and has been
observed in the existing literature in different contexts [21],
[23], [24], [25], [26], [27]:
Observation 2: For any T ≥ 1, Int(
ˆ
Λ
T
) = Int(Λ).
The above observation (proof outline in appendix) states
that the stability region remains the same if the queue mea-
surement interval is increased, as long as the channel states
are measured every time slot.
From the lemma and observation stated above, we can
conclude that the shrinking of the stability region Λ
T
with
increasing T , is a result of the reduction in the channel
rate measurement frequency, and not due to the reduction
in the frequency of queue-length measurements. Increasing
the queue measurement interval (while keeping the channel
measurement interval fixed) does not affect the maximum
achievable throughput; it usually results only in an increase in
the average packet delay. Increasing the channel measurement
interval, however, not only increases the average delay, but
also leads to a reduction in the maximum achievable through-
put. Thus the reduction in the frequency of measurement
in the channel rates affects the system in a fundamentally
different way than that of the queue-lengths. The optimal
scheduling policy which we state in the next section provides
more intuition behind these results. We also substantiate these
observations through simulation results in Section V.
B. Scheduling Policy
We now describe our scheduling policy Ψ
T
,whichis
parameterized by the length T of the measurement interval.
The scheduling policy consists of two components: (i) packet
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