Access Point Buffer Management for Power Saving

8/18/2019 Access Point Buffer Management for Power Saving

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IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 9, NO. 4, DECEMBER 2012 473

Access Point Buffer Management forPower Saving in IEEE 802.11 WLANs

Yi-hua Zhu, Senior Member, IEEE, Han-cheng Lu, and Victor C. M. Leung, Fellow, IEEE

Abstract—It is crucial to save power and prolong the run-time of mobile stations (STAs) in wireless local area networks(WLANs). In an infrastructure WLAN, a STA cannot be con-nected until it is associated with an access point (AP), whichis responsible for buffering frames for all the associated STAsoperating in the power saving mode. Hence, efficient memoryutilization is critical for an AP to accommodate as many power-saving STAs as possible. The basic power management (BPM)scheme introduced in the IEEE 802.11 standard achieves powersaving by allowing STAs not engaging in data delivery to operatein doze mode, but it does not consider the efficient use of the memory in the AP. To tradeoff power consumption formemory usage, we present an AP-priority timer-based powermanagement (APP-TPM) scheme and develop a novel model forstochastic analysis of the proposed scheme. Based on this model,the probability distributions of the numbers of frames bufferedat the AP and the average numbers of frames buffered at the

AP are obtained. Moreover, a power-aware buffer managementscheme (PBMS), which is based on the derived statistics, isproposed to accommodate as many STAs as possible given afixed amount of memory in the AP while maintaining low powerconsumption. Simulation results show that the proposed schemeperforms better than BPM in terms of memory usage in the AP.

Index Terms—Power management, WLAN, IEEE 802.11,power saving.

I. INTRODUCTION

W IRELESS local area networks (WLANs) based on theIEEE 802.11 standard [1] are becoming increasinglypopular since devices in these networks can communicate over

shared radio channels in license-free frequency bands usingthe Distributed Coordination Function (DCF) for medium

access control (MAC). In infrastructure WLANs, access points(APs) are used to relay data packets between stations (STAs)

and the global Internet. Saving power is critical for battery-

operated portable STAs to have a long runtime. The basic

power management (BPM) scheme in the IEEE 802.11 stan-

dard allows a STA in an infrastructure WLAN to operate in

power saving mode, whereby the STA can go into doze mode

Manuscript received on August 27, 2011; revised on March 20 and June3, 2012. The Associate Editor coordinating the review of this paper andapproving it for publication was P. Bellavista.

This work was supported in part by the National Natural Science Foun-dation of China under Grant 61070190; in part by the Zhejiang ProvincialNatural Science Foundation of China under Grant Z1100455; and in part bythe Zhejiang Provincial Key Science & Technology Project of China undergrant No. 2009C14033.

Y.-H. Zhu and H.-C. Lu are with the School of Computer Science andTechnology, Zhejiang University of Technology, Zhejiang 310023, P. R. China(e-mail: [email protected], [email protected]).

V. C. M. Leung is with the Department of Electrical and ComputerEngineering, The University of British Columbia, Vancouver, BC, CanadaV6T 1Z4 (e-mail: [email protected]).

Digital Object Identifier 10.1109/TNSM.2012.062512.110188

and power down its radio transceiver when it is not engaged

in data delivery. A STA operating in power saving mode is

referred as a power-saving STA in this paper. Thus, a power-

saving STA can operate in one of four modes: transmission,

reception, idle, and doze, in which the doze mode has the least

power consumption. Usually, power management algorithms

aim to achieve power savings by maximizing the doze periods

of STAs.

Before connecting to an infrastructure WLAN, a STA is

required to associate with an AP of the WLAN by sending

an Association Request (AR). In the case of a power-saving

STA, the AR contains a Listen Interval (LI) parameter usedto indicate how often the STA wakes up to listen to beacon

frames from the AP [1]. The AP assigns an AssociationID (AID) to the STA if the AR is accepted by the AP.

Under BPM, when a power-saving STA has no frame totransmit/receive, it switches to doze mode for the duration

equal to its LI. Meanwhile, the AP buffers incoming data

frames for the dozing STAs and periodically broadcasts a

beacon that contains a Traffic Indication Map (TIM) including

AIDs of the dozing STAs to announce which STAs have

pending data frames in the AP. When a power-saving STAwakes up, it listens to the beacon to see if the bit corresponding

to its AID is set in the TIM, in which case the STA sends the

AP a Power-Saving-poll (PS-poll) frame to retrieve the dataframe; otherwise, it dozes again for another LI period.

As far as power saving is concerned, a longer LI is preferred

to allow the power-saving STA to take a longer doze period,but this causes usage of more memory in the AP to buffer

frames destined to the STA, in addition to longer packet de-lays. It is required that an AP should hold the buffered frames

for at least one LI before discarding them, or if the AP is short

of memory, the pending frames are dropped according to an

aging function [1]. In fact, any power management scheme

is not expected to gain a high throughput if it does not take

available buffer size into account, because the buffered framesin the AP will be discarded if the AP is short of memory,

which produces lots of retransmissions. Hence, the AP needs abuffer management scheme to allocate an appropriate amount

of memory to buffer the frames for a dozing STA, when an

AR including an LI parameter is initially received from the

power-saving STA.

Although memory is cheap nowadays, the memory size of

an AP is fixed and limited when the AP is put in use. It maynot be practical to put the WLAN temporary out of service

in order to upgrade the AP with more memory. Currently, the

memory size in an AP is typically tens of megabytes. For

instance, the Netgear WG103 AP, which is currently available

1932-4537/12/$31.00 c 2012 IEEE


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474 IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 9, NO. 4, DECEMBER 2012

for sale, has 32MB SDRAM. Usually, an AP inside a WLAN

is associated with multiple STAs. Consequently, if there are

multiple power-saving STAs in the network, the AP is only

able to allocate a small amount of memory to buffer frames

for each power-saving STA. For example, with the device

mentioned above only about 3MB memory on average canbe allocated for a STA if there are 10 power-saving STAs in

the WLAN. Such a tight limitation of the memory size makesit almost impossible for a STA to enter sleep mode via BPM

without dropping some packets, which is undesirable when

it is involved in multimedia communications with quality of

service (QoS) constraints. The reason is that, when BPM is

applied with an inappropriate LI, the incoming frames may be

discarded at the AP due to buffer overflow, which degradesQoS. To avoid this condition, the only choice is to disable

BPM and keep the STA awake, which may be unsatisfactoryfrom an energy-saving point of view. Our AP-Priority Timer-

based Power Management (APP-TPM) proposed in this paper

can remedy the shortcoming of BPM, by enabling a STA’s

sleep time and idle time to be controlled so that frames

destined to it are not discarded by the AP due to bufferoverflow.

APP-TPM is based on our previous timer-based power

management (TPM) [2]. TPM can considerably reduce the

number of buffered frames by letting STAs stay in the idle

state for a period longer than the idle period of STAs under

BPM. Hence, with TPM, it is possible for STAs to enter

sleep mode to save power even when the AP has only a

small amount of available memory. Two timers called the idletimer and the doze timer are incorporated in TPM. Like BPM,

the doze timer T D is set to equal to LI each time the STAenters the doze mode, and the STA wakes up when the doze

timer times out. Unlike BPM, which allows an idle STA to

switch to doze mode at the beginning of the next beacon,TPM extends the idle period of BPM by multiple Beacon

Intervals (BIs) such that the idle STA is not allowed to enter

doze mode until a preset amount of time T I , specified by theidle timer, has elapsed. In TPM, the idle timer is started as

soon as the STA becomes idle. The idle timer is reset if the

STA transmits/receives a frame before the timer times out. TheSTA switches to doze mode when the timer times out while

the STA continues to stay idle. The initial values of the twotimers, i.e., T I and T D, are negotiated between the STA andthe AP when the STA is associated with the AP.

Both TPM and BPM set its doze period to LI as specified

by the power-saving STA. The main difference in the activities

of them is shown in Fig. 1, in which the vertical bars representthe beacons, and we assume that the STA becomes active at

time A and goes idle at time B. Under BPM, the STA enters

sleep mode at time C, i.e., the beginning of the next beacon.Under TPM, however, its sleep is postponed to time E when

the idle timer with value T I times out, where T I is set to twomore BIs than the idle period of BPM.

At the cost of slightly higher power consumption than BPM,

the number of frames buffered at the AP and the STA under

TPM could be considerably reduced by increasing T I and/ordecreasing T D [2]. These adjustments supported by TPMmake it possible for an AP with limited available memory to

fully buffer the frames destined to power-saving STAs during

Fig. 1. TPM vs. BPM.

their sleeping periods, so that it is possible to enable STAs

to operate in the power-saving mode without sacrificing QoS

support.

In fact, the IEEE 802.11 standard does not specify how

the AP determines whether an AR should be granted, and

this decision process is implementation-specific. One common

consideration for granting AR is the amount of memory

required for frame buffering, a rough estimate of which basedon the LI in the AR frame is possible [3]. Hence, deriving the

Number of Buffered Frames at the AP (NBF-AP) is significantfor an AP to make an appropriate decision on granting ARs.

Although the statistics of the sum of NBF-AP and the number

of frames buffered at the STA can be obtained from the

model of TPM presented in [2], it has the drawback that the

statistics of NBF-AP cannot be separated from those of thesum. Therefore the results in [2] cannot be applied towards

the design of a buffer management scheme that is capable

of predicting the number of frames buffered at the AP for

each STA and optimizing each STAs power consumption.

This drawback is overcome by the model presented in this

paper. In addition, the average NBF-AP (ANBF-AP), i.e., theexpected NBF-AP, is derived and used in the proposed buffer

management scheme.

Since the available memory in the AP determines how

many power-saving STAs are allowed to be associated with

the AP, the overall power saving of the system depends not

only on the power management scheme, but also on the buffermanagement scheme. To the best of our knowledge, none

of the power management schemes presented for the 802.11

infrastructure WLAN in the literature have considered buffer

management in the AP. This paper fills the gap.

The main contributions of the paper are as follows:

1)We propose a power-aware buffer management scheme(PBMS) for the AP, which allocates buffers based on thestatistics of numbers of frames queued in the AP for sleeping

STAs, in order to accommodate as many STAs as possible

given a fixed amount of memory in the AP, while minimizing

the total of all the STAs.

2)We propose an AP-priority TPM (APP-TPM) scheme

and present a novel model for analyzing the statistics of the

proposed scheme, which takes into account of separate queues

for incoming frames and outgoing frames.

3)We derive the statistics including the averages and the

probability distributions of NBF-AP.

The remainder of this paper is organized as follows. The


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ZHU et al.: ACCESS POINT BUFFER MANAGEMENT FOR POWER SAVING IN IEEE 802.11 WLANS 475

TABLE IACRONYMS AND NOTATIONS

Acronyms and notations Definition

AP Access Point

AR Association Request

AID Association ID

APP-TPM AP-Priority Timer-based Power Management

LI Listen Interval

BI Beacon Interval

BPM Basic Power Management

NBF-AP The Number of Buffered Frames at the AP

ANBF-AP The average NBF-AP

PS-poll Power-Saving-poll

PFD Percentage of Frame Discarded

TPM Timer-based Power Management

TIM Traffic Indication Map

T I The initial value of the idle timer

T D The initial value of the doze timer

MIT Mandatory Idle Time

PIT Prolonged Idle Time

Γ Memory size constraint of the AP

PBMS is proposed in Section II and the APP-TPM scheme

applicable to IEEE 802.11 infrastructure WLANs is presentedin Section III. Based on stochastic analysis, we model APP-

TPM and derive statistics in Section IV. Simulation results for

validating the derived statistics and the performance analysis

of PBMS are presented in Section V. Related work is surveyed

in Section VI. Section VII concludes the paper. In addition,

some acronyms and notations are listed in Table I.

I I . POWER-AWARE B UFFER M ANAGEMENT S CHEME

As mentioned in the previous section, if the AP grants apower-saving STA in response to its AR, it has to buffer

all the frames destined to the STA for a duration of at leastone LI when the STA goes to sleep. One of the key factors

that impact the AP’s decision on whether to accept an AR

is its available memory. The proposed PBMS helps the AP

make decisions and aims to minimize STAs average power

consumption by adjusting T I so that the available memory inthe AP is sufficient to hold all the frames destined to the STAsoperating in doze mode for at least one LI.

Assume there are N active STAs and the memory size con-straint of the AP is Γ. Suppose the k -th STA, k = 1, 2,...,n,has a pending AR in the AP, and the remaining N − n STAshave previously granted ARs. For k = 1, 2,...,n, let E (k),

L(k)

AP , T (k)I and T

(k)D represent the average power consumption,

ANBF-AP, T I and T D, respectively, of the k-th STA, andx(k) is an indicator used to represent whether the AR of thek-th STA is granted (x(k) = 1) or not (x(k) = 0). Notingthat maximizing 1/E (k) is equivalent to minimizing E (k), wepropose the following optimization problem for our PBMS,

which will be referred to as PBMS-OPT in the sequel.

Max Φ(x(1), x(2),...,x(n); T (1)I , T

(2)I ,...,T

(n)I ) ≡

nk=1

x(k) 1

E (k)

w.r.t. x(1), x(2),...,x(n); T (1)I , T

(2)I ,...,T

(n)I

s.t.

n

k=1

x(k)L(k)

AP ≤ Γ −N

i=n+1

L(i)

AP ;

x(k) ∈ {0, 1}, k = 1, 2,...,n(1)

Especially, when n = 1, i.e., there is only one pending ARat the AP, we set x(n) = 1 in (1).

Clearly, to increase the objective function Φ(·) =nk=1 x

(k)[E (k)]−1, we prefer more x(k)s (k = 1, 2,...,n) tobe set to 1, i.e., more ARs are granted. But this also increases

the ANBF-AP. As a result, only some x(k)s are allowed to beset to 1 if the available memory at the AP is not enough (see

the first line of the constraints in (1)). It should be pointed

out that, for a given k, E (k), L(k)

AP depend on both T (k)I and

T (k)D ( see (21)-(24) and (26) in Section IV). Consequently, it

is feasible to choose and set as many x(k)s to 1 as possible

by adjusting T (k)I (k = 1, 2,...,n). This is the main objective

of the proposed PBMS-OPT given in (1).

PBMS-OPT is evoked each time the AP makes the decisionto accept some AR(s). When the optimal solution of PBMS-

OPT is found, for each k in {1, 2,...,n}, if x(k) = 1, then theAR from the k-th STA is granted; otherwise x(k) = 0 and it

is rejected. In addition, the optimal T (k)I s for the granted ARs

are carried in the last field, called “Vendor Specific”, in theassociation response frame [1] notifying the k -th STA of the

acceptance (T (k)I is used to set the value of the idle timer in

the k -th STA).

The PBMS is based on the improved TPM, i.e., APP-TPM,which is presented in the next section. Additionally, in Section

IV we derive some important statistics of APP-TPM, including

those crucial for PBMS-OPT, such as E (k) and L(k)

AP .

III. AP-PRIORITY T IMER-BASED P OWER M ANAGEMENT

Rather than only considering the total number of frames

in both the incoming and outgoing queues as in [2], we

consider a more realistic model that represents the two queues

individually, referred as queue-AP and queue-STA, to hold the

incoming frames at the AP (to the STA) and the outgoing

frames at the STA (to the AP), respectively. That is, when

the STA operates in doze mode, the frames destined to it areplaced in queue-AP while those generated by the STA are

placed in queue-STA. To alleviate the possibility of shortage

of buffers in the AP, transmissions of the frames in queue-AP

are given priority over those in queue-STA, i.e., the frames at

the AP are transmitted prior to those at the STA. This can be

realized by having the STA delay transmitting the frames in

queue-STA until it receives the last frame buffered in the AP,

which is indicated by the More Data field in the frame header

being set to 0 [1]. Accordingly, our scheme is referred to as

AP-Priority TPM, which consists of two components: APP-TPM-AP, which runs in the AP, and APP-TPM-STA, which

runs in the STA and includes an idle timer and a doze timer.


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Fig. 2. Flowchart of APP-TPM-AP.

The main logic of APP-TPM-AP is as follows. When the AP

begins to handle the pending ARs, for each k in {1, 2,...,n}, it

sets T (k)D to the LI received from the AR of the k-th STA (the

two-octet field “Listen interval” in the frame header containsthe LI parameter [1]), and then, it finds the optimal T

(k)I (k =

1, 2,...,n) via PBMS-OPT shown in (1).After the AP receives from a STA a frame in which the

“Power management bit” field is set to 1, which indicates that

the STA will enter the doze mode after the completion of the

current frame exchange [1], the AP starts buffering the frames

destined to the STA according to the logic shown in Fig. 2.

The main logic of APP-TPM-STA is as follows. After aSTA sends an AR to the AP, the STA waits for the AP to

reply with the decision to accept the AR or not. If accepted,

the STA gets T I from the AP and sets T D to the LI parametercontained in the AR frame previously sent to the AP. The STA

sets the value of the idle timer to T I as soon as it becomes idle.It switches to the active state (i.e., receives/transmits frames)

immediately if a frame arrives before the idle timer timeouts.

If the STA remains idle till the idle timer expires, it switches

to the doze mode after sending the AP a frame with “Power

management bit” field set to 1. The value of the doze timeris set to T D as soon as the STA enters doze mode and itwakes up when the doze timer expires to listen for the next

TIM. If no frame is buffered at the AP and the STA, the

STA sleeps for another period of T D; otherwise, it switchesto active mode to receive the incoming frames first and then

transmit the outgoing frames. Fig. 3 shows the flowchart of

APP-TPM-STA.

IV. KEY STATISTICS OF PBMS

As in Section II, we let N be the number of STAs accessingthe AP in the WLAN. In this section, by extending the model

used in [2], we derive some statistics of APP-TPM for the

k-th STA, including E (k) and L(k)

AP (k = 1, 2,...,N ), whichare critical for PBMS-OPT given in (1).

Fig. 3. Flowchart of APP-TPM-STA.

At any time, each STA can operate in transmission, recep-

tion, idle, or doze mode. As in [2], we combine transmissionand reception into one state, called active state.

In APP-TPM, the doze period of the k-th STA is deter-

ministic, i.e., it is a constant T (k)D set to LI that is equal to a

multiple of BIs [1]. In order to reduce the difficulty of deriving

the statistics by stochastic analysis, we first consider the doze

period as a random variable (assumption (i) below) and then

let the variable take a deterministic value T (k)D to derive the

statistics for the APP-TPM.

A. Assumptions

In general, the doze period of the STA, the time of an

incoming frame arriving at the STA, the time of an outgoing

frame generated at the STA, and the time for the STA to

transmit/receive a frame are all random variables. Naturally,the best way of modeling APP-TPM is to let all the related

random variables be generally distributed. Unfortunately, it

does not seem feasible to derive the system statistics based

on general distributions. Considering that the Poisson process

has been used for modeling the incoming and outgoing traffic

[2][4], and the service time distribution of MAC queues in

802.11 ad-hoc networks has been modeled by an exponen-

tial distribution [5], for analytical tractability, we make thefollowing assumptions.

(i) The doze period of the k-th STA is a random vari-

able χ(k)D with expectation E [χ

(k)D ] = 1/η

(k), 2nd moment

E [(χ(k)D )

2] ≡ α(k), probability density function (pdf) g(k)(x),and cumulative distribution function (CDF) G(k)(x) [2]. The

hazard rate function [6] of χ(k)D is τ

(k)(x) ≡ g(k)(x)

G(k)

(x), where

G(k)

(x) ≡ 1 − G(k)(x).(ii) Incoming frames arriving at the AP and destined to the

k-th STA form a Poisson process with rate λ(k)1 and outgoing

frames generated at the k-th STA form a Poisson process with

rate λ(k)2 [2][4].


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(iii) The time for the k-th STA to transmit/receive aframe, including the time to compete for channel access, is

an exponentially distributed random variable χ(k) [5] withparameter µ(k), i.e., the pdf, CDF, and hazard rate function

of χ(k) are f (k)(x) = µ(k)e−µ(k)x, F (k)(x) = 1 − e−µ

(k)x

and σ(k)(x) = µ(k), respectively.We assume all the above random variables are mutually

independent. In addition, we introduce the following notations:

λ(k)1,2 ≡ λ

(k)1 + λ

(k)2 , ρ

(k) ≡ λ(k)1,2/µ

(k) (2)

B. State Transitions of APP-TPM

The conditional probability that the STA has dozed for atime period of x and then terminates doze mode within a verysmall time interval ∆t is given by the following equation.

Pr{χ(k)D < x + ∆t|χ

(k)D ≥ x} =

Pr{x ≤ χ(k)D < x + ∆t}

Pr{χ(k)D ≥ x}

= g(k)(ux)∆t/G(k)

(x), ux ∈ [x, x + ∆t].

We have lim∆t→0 g(k)(ux)/G(

k)(x) = τ (k)(x) aslim∆t→0 ux = x. Hence, τ (k)(x) is the probability densityof the event that “the STA terminates doze mode after it has

dozed for a time period of x”, which yields the result: “theprobability that the STA terminates dozing within ∆t after ithas dozed for a time period of x is τ (k)(x)∆t”.

We use notation Ai,j to represent the joint event that theSTA is in active state, and there are i outgoing frames inqueue-STA waiting for transmission to the AP and j incomingframes in queue-AP waiting for transmission to the STA. In

addition, Di,j is used to represent the event that the STA is indoze state and there are i outgoing frames and j incoming

frames buffered in queue-STA and queue-AP, respectively,where i, j = 0, 1,.... Moreover, we use letter I for the idlestate. All the possible transitions among the states Ai,j , Di,j(i, j = 0, 1, 2,...) and state I are shown in Fig. 4, wherecircles, squares, and triangles represent the active states, dozestates, and idle states, respectively. In addition, state transitions

are indicated by arrows connecting the respective states, and

each arrow is labeled with the corresponding hazard ratefunction. For the sake of conciseness, in the figure, we omit

superscript (k). That is, λ1, λ2, τ (x), and σ(x) stand for λ(k)1 ,

λ(k)2 , τ

(k)(x), and σ(k)(x), respectively.

C. Model of APP-TPM

Clearly, µ(k), defined in assumption (iii), is the rate of transmitting/receiving frames between the AP and the STA,

i.e., the average number of frames transmitted/received per

unit time. From assumption (ii), the combined rate of incoming

and outgoing frames arriving at the AP and the STA is λ(k)1,2 .

Hence, we assume λ(k)1,2 < µ

(k), i.e., ρ(k) < 1 to preventthe number of frames to be transmitted from growing without

bound.

Let ξ (t) be the state of the STA at time t, and Y (t) be theaccumulated doze time (ADT) at time t, which is the timeperiod from the instant when the STA starts dozing to time

t. Use pDi,j (t, x) to represent the probability density of the

event that the STA is in state Di,j with ADT x at time t,which satisfies [2]:

pDi,j(t, x)dx = Pr{ξ (t) = Di,j , x < Y (t) ≤ x + dx} (3)

where i, j = 0, 1,.... Define P Di,j (t) ≡ Pr{ξ (t) = Di,j},q Ai,j (t) ≡ Pr{ξ (t) = Ai,j} and P I (t) ≡ Pr{ξ (t) = I }.From (3), we have P Di,j (t) =

∞

0 pDi,j(t, x)dx. Let P Di,j ≡

limt→∞P Di,j (t), q Ai,j ≡ limt→∞ q Ai,j (t), and P I ≡ limt→∞P I (t).Besides, we use P

(k)I , P

(k)A , and P

(k)D to represent the prob-

ability that the k-th STA is in the idle state, active state,and doze state, respectively. Thus, P

(k)I = P I , P

(k)A =

∞

i=0

∞

j=0 q Ai,j and P (k)D =

∞

i=0

∞

j=0 P Di,j . Further, weintroduce z-transforms:

Qi(z) ≡∞j=0

q Ai,jzj(i = 0, 1,...; 0 < z


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Fig. 4. State transition diagram.

q Ai,j(t) = −q Ai,j(t)(µ(k) + λ

(k)1,2) + λ

(k)2 q Ai−1,j (t)

+ µ(k)

q Ai,j+1 (t) + ∞0 p

Di,j+1(t, x)τ (k)

(x)dx,

i, j = 1, 2,...

(9)

pD0,0(t, 0) = P I (t)λ(k)1,2e

−λ(k)1,2T

(k)I

1 − e−λ(k)1,2T

(k)I

+

∞

0

pD0,0(t, x)τ (k)(x)dx

(10)

pD0,j (t, 0) = 0, j = 1, 2,... (11)

∂

∂t pD0,0(t, x) +

∂

∂x pD0,0(t, x)

= −[λ

(k)

1,2 + τ (k)

(x)] pD0,0(t, x), t > 0, x > 0

(12)

∂

∂t pD0,j (t, x) +

∂

∂x pD0,j (t, x)

= −[λ(k)1,2 + τ

(k)(x)] pD0,j (t, x) + λ(k)1 pD0,j−1 (t, x),

t > 0, x > 0, j = 1, 2,...

(13)

∂

∂t pDi,0(t, x) +

∂

∂x pDi,0(t, x)

= −[λ(k)1,2 + τ

(k)(x)] pDi,0(t, x) + λ(k)2 pDi−1,0(t, x),

t > 0, x > 0, i = 1, 2,...

(14)


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∂

∂t pDi,j(t, x) +

∂

∂x pDi,j (t, x)

= −[λ(k)1,2 + τ

(k)(x)] pDi,j (t, x) + λ(k)1 pDi,j−1(t, x)

+ λ(k)2 pDi−1,j (t, x),

t > 0, x > 0, i = 1, 2,...,j = 1, 2,...

(15)

We assume that the STA is in idle at time t = 0, i.e.,

P I (0) = 1; pDi,j (0, x) = 0, i , j = 0, 1,... (16)

D. Statistics of APP-TPM

From (4)-(16), we obtain the following equations (Refer to

Appendix II for the detailed derivations).

K (y, z) = P (k)I

ρ(k)yzyy − zy

[1 − 1

z0(1 − e−λ

(k)1,2T

(k)I )

]

+ λ

(k)2 y(zy)

2(z − y)

µ(k)(y − zy)(z − zy)Q0(zy) +

zy[(1 + ρ(k))z − 1]

z − zyQ0(z)

+ P I ρ(k)e−λ

(k)

1,2T I

[1 − e−λ(k)1,2T

(k)I ][1 − e−λ

(k)1,2T

(k)D ]

{zyye−(λ

(k)

1,2−λ

(k)

1 z0)T (k)

D

z0(y − zy)

+ zy(y − z)e

−(λ(k)1,2−λ

(k)1 zy)T

(k)D

(z − zy)(y − zy) −

zye−(λ

(k)1,2−λ

(k)1 z)T

(k)D

z − zy

− zyye

−λ(k)1,2T

(k)D

z0(y − zy) +

zy(z − y)e−(λ

(k)1,2−λ

(k)1 zy−λ

(k)2 y)T

(k)D

(y − zy)(z − zy)

+ zye

−(λ(k)1,2−λ

(k)1 z−λ

(k)2 y)T

(k)D

z − zy}

(17)

where

z0 ≡ 12λ

(k)1

(µ(k)+λ(k)1,2−

(µ(k) + λ(

k)1,2)

2 − 4λ(k)1 µ

(k)) (18)

zy ≡ µ(k)

µ(k) + λ(k)1,2 − λ

(k)2 y

(19)

Q0(z) = P I λ(k)1,2{(z0 − z)[1 − e

−λ(k)1,2T

(k)D + e−λ

(k)1,2(T

(k)I +T

(k)D )]

+ e−λ(k)1,2T

(k)I [ze−(λ

(k)1,2−λ

(k)1 z0)T

(k)D − z0e

−(λ(k)1,2−λ

(k)1 z)T

(k)D ]}

/{z0(1− e−λ

(k)1,2T

(k)I )(1− e−λ

(k)1,2T

(k)D )

· [λ(k)1 z

2 − (µ(k) + λ(k)1,2)z + µ

(k)]}(20)

and

P (k)I = {1 +λ(k)1,2(1 − z0)

λ(k)2 z0+

λ(k)1,2e

−λ(k)1,2T

(k)I

(1− e−λ(k)1,2T

(k)I )(1− e−λ

(k)1,2T

(k)D )

· 1 − z0 + λ

(k)2 z0T

(k)D + z0e

−λ(k)2 T D − e−(λ

(k)1,2−λ

(k)1 z0)T

(k)D

λ(k)2 z0

}−1

(21)

Moreover,

P (k)D =

P (k)I λ(k)1,2T

(k)D e

−λ(k)1,2T

(k)I

[1 − e−λ(k)1,2T

(k)I ][1 − e−λ

(k)1,2T

(k)D ]

(22)

P (k)A = K (1, 1) = P

(k)I

λ(k)1,2

λ(k)2 z0

{(1− z0) + e−λ

(k)1,2T

(k)I

· [1 − z0 + z0e−λ

(k)2 T

(k)D − e−(λ

(k)1,2−λ

(k)1 z0)T

(k)D ]

(1 − e−λ(k)1,2T

(k)I )(1− e−λ

(k)1,2T

(k)D )

}.

(23)

Thus, the average power consumption of the k

-th STA is as

follows:

E (k) = P (k)I E I + P

(k)A E A + P

(k)D E D (24)

where E A, E I , and E D are the power consumption of the STAwhen it stays in the active, idle, and doze state, respectively.Further, we have the following results.

Under APP-TPM, when the WLAN is in steady state, the

probability that there are j incoming frames destined to thek-th STA and buffered at the AP during the doze period of the STA is

P (k)

AP ( j) =

∞

i=0

P Di,j

=P (k)I λ

(k)1,2e

−λ(k)1,2T

(k)I [1 − e−λ

(k)1 T

(k)D

js=0

(λ(k)1 T

(k)D )

s

s! ]

λ(k)1 (1− e−λ

(k)1,2T

(k)I )(1− e−λ

(k)1,2T

(k)D )

(25)

ANBF-AP during the doze period of the k-th STA is

L(k)AP =

∞j=1

jP (k)AP ( j)

= P (k)I

λ(k)1,2e

−λ(k)1,2T

(k)I λ

(k)1 (T

(k)D )

2

2(1 − e−λ(k)1,2T

(k)I )(1− e−λ

(k)1,2T

(k)D )

(26)

V. PBMS A ND P ERFORMANCE E VALUATIONS

With BPM, when a STA finishes transmitting/receiving its

frames, the STA is required to stay in the idle state till the

next beacon comes [1]. We refer to the time period from the

instant when the STA becomes idle to the instant when the

next beacon arrives as the mandatory idle time (MIT). As

mentioned in Section I, the main difference between BPM and

APP-TPM is that, when a STA becomes idle, BPM allows the

STA to operate in doze mode after MIT expires whereas APP-

TPM requires the STA to stay in the idle state for a period

of T I , which is equal to MIT plus a period of time called the prolonged idle time (PIT) that is equal to a multiple of BIs.

Obviously, APP-TPM is reduced to BPM if PIT is set to 0.As MIT is a random variable uniformly distributed in [0, BI],

the average MIT is 0.5 BI. Thus, T I = (0.5 + m)BI wherem is a positive integer [2]. We fix BI=0.1 s [1]. Additionally,for the k-th STA, we introduce the notation R

(k)λ ≡ λ

(k)1 /λ

(k)2 .

Thus, download (or upload) traffic of the STA is heavier than

upload (or download) traffic when R(k)λ ≥ 1 (or R

(k)λ ≤ 1).

In the IEEE 802.11 standard, a two-octet field in the ARframe is used for LI [1], which indicates that LI ranges from 0

to 216−1 BI. Thus, T (k)D takes a value over [0, 6553.5] s when

BI =0.1 s. As in [4], we set the power consumptions of theSTA in active, idle, and doze states to E A=1W, E I =0.83W,E D=0.13W, respectively. Obviously, power consumption of


8/14


TABLE IISIMULATION PARAMETERS

Parameter Value

PLCP Preamble Length 144 b

PLCP Header Length 48 b

BI (Beacon interval) 0.1 s

Data rate 11 Mbps

Size of data frame 1300 B

Size of ACK frame 34 B

SlotTime (Length of a time slot) 20 us

SIFS (Short interframe space) 10 us

DIFS (Distributed interframe space) 50 us

CWmin (Minimum contention window) 31

CWmin (Maximum contention window) 1023

N (Number of STAs) 20

Duration of simulation time 100 s

different wireless interfaces is different, but the way of finding

the optimal T (k)I for PBMS-OPT is similar.

Noting that the foundation of the proposed PBMS-OPTis the fact that ANBF-AP can be considerably reduced by

adjusting T I and/or T D, we first validate this fact by simula-tions. The simulation program is written in MATLAB and thesimulation parameters are listed in Table II.

Setting PIT=0, 1, 2, 3, 4, 5 BI, T D =30, 60, 90 s, andrunning the simulation for 100 s, which takes about 11 hours

on a Lenovo T400 laptop, we obtain Fig. 5, where the valuescorresponding to PIT=0 are those of BPM. It can be clearly

seen from the figure that the total ANBF-AP for the 20 STAscan be reduced if we choose a suitable pair of T D and PIT.For instance, we can choose T D=30 s and PIT=2, 3, 4, 5BI, or T D=60 s and PIT=3, 4, 5 BI, or T D=90 s and PIT=

3, 4, 5 BI if we need to control the total ANBF-AP to beless than 0.5 × 104. Consequently, in PBMS-OPT where T Dis fixed in the AR, we can find a suitable PIT to reducethe total ANBF-AP, which underlies PBMS-OPT. We have

repeated the simulations with different parameters, and theabove observations remain valid.

From Fig. 5, we obtain another important observation

when T D=90. The total ANBF-AP in BPM (correspondingto PIT=0) is about 3.2 × 104, which occupies 32000 × 1300B ≈ 40 MB as the size of data frame is set to 1300 B.This indicates that, if all the ARs are granted, buffer overflow

occurs in a Netgear WG103 AP with 32 MB SDRAM, as

mentioned in Section I, when BPM is applied. However,

this problem disappears by the proposed APP-TPM in whichPBMS-OPT is used.

Now, we move on to investigate the performance of PBMS-

OPT. Currently, 802.11a/g-based WLANs support data rates

of 6, 9, 12, 18, 24, 36, 48, and 54Mbps. In [7], it is shown

that at these data rates and without considering backoff pro-

cedure and RTS/CTS control frames, the transmission times

of a frame with a 1300-byte payload under 802.11a stan-

dard are 0.001936, 0.001331, 0.001011, 0.000707, 0.000554,

0.000402, 0.000326, and 0.000302 s, respectively, which are

equivalent to frame transmission rates of approximately 516,751, 989, 1414, 1805, 2488, 3068, and 3311 frames per

second (fps). Recent 802.11n WLAN standard increases the

Fig. 5. Simulation results for total ANBF-AP with various T D and PIT.

maximum data rate to more than 500 Mbps and increases theframe transmission rate accordingly. Therefore, in the numericevaluations, we select µ=2000 fps [2] as the nominal value.

Intuitively, increases in transmitting/receiving rate µ shouldreduce the ANBF-AP. However, this turns out to be incorrect.The experiments, in which µ is set to 1000, 1500, 2000, ...,

and 5000, respectively, indicate that, for given λ(k)1 and λ

(k)2 ,

increases in µ only exert a small influence on the ANBF-AP (we omit the figure of this experiment due to space

limitation). The main reason is that, increasing µ only makesthe STA finish transmitting/receiving frames and enter doze

state sooner, but the ANBF-AP depends on the arrival rate of

the incoming frames. Hence, increasing µ does not help much

in reducing the ANBF-AP except that it increases the powerconsumption. Hence, the observations from the experiments

with µ=2000 can be applied to other cases in which a differentµ is chosen.

Next, we investigate the impact of parameters R(k)λ and µ

on the ANBF-AP of the k-th STA, given in (26). Fixing m=1,i.e., T

(k)I =1.5 BI=0.15 s, setting T

(k)D =30 s, λ

(k)2 =10, 20, ...,

50 fps and R(k)λ =1, 2, ..., 16, we obtain Fig. 6. In addition,

setting R(k)λ to 1, 1/2, ..., and 1/16 leads to Fig. 7.

From Fig. 6, we observe that in the case where the download

traffic is heavier than the upload traffic, for a given λ(k)2 (e.g.,

λ(k)2 =10 fps), ANBF-AP first goes up and then down as R

(k)λ

increases (or λ(k)1 increases due to λ

(k)2 fixed). The reason

is that, for a given doze period, more frames arrive at the

AP when λ(k)1 is increased, which causes ANBF-AP to grow.

But, when λ(k)1 continues to increase, the idle time of the

STA decreases, which causes the STA to have less probability

of entering doze state, i.e., the doze period of the STA isshortened, making ANBF-AP decrease. The same observation

can be obtained from Fig. 7 when upload traffic is heavierthan download traffic. The above observations imply that, the

AP does not need to enlarge its buffer size if the number of

ARs of STAs increases (i.e., if more STAs intend to connect to

the WLAN), which contributes to make λ(k)1 as well as R

(k)λ

increase.


9/14


Fig. 6. Impacts of R(k)λ

on the ANBF-AP when R(k)λ ≥ 1.

Fig. 7. Impacts of R(k)λ

on the ANBF-AP when R(k)λ ≤ 1.

Then, we compare APP-TPM with BPM in terms of ANBF-

AP. Setting T (k)D =60 s, λ

(k)2 =10 fps, R

(k)λ =2, 4, 6, and letting

T (k)I =0.15, 0.25, ..., 1.05 s (i.e., PIT is set to 0.1, 0.2, ..., 1.0

s, respectively) yield Fig. 8, which depicts the reduction ratio

of ANBF-AP, defined as (N BPM − N TPM )/N BPM where

N BPM and N TPM represent ANBF-APs under BPM andAPP-TPM, respectively. Especially, ANBF-AP for R

(k)λ =2 in

Fig. 8 is depicted in Fig. 9, in which each group has two bars

that represent ANBF-AP under BPM on the left and APP-

TPM on the right. It can be observed that ANBF-AP can be

reduced significantly by increasing PIT slightly. For instance,

when R(k)λ =2, if we select T

(k)I =0.25 or 0.45 s (i.e., set PIT

to 0.2 or 0.4 s), ANBF-AP is reduced from 600 (for BPM)to 300 or close to 0, respectively (see Fig. 9). Equivalently,

the reduction ratio of ANBF-AP can reach 50% or nearly

100%, respectively (see Fig. 8). It implies that, if the buffer

in the AP is enough to hold 300 frames, AR with LI set to

60 s (i.e., T (k)D =60 s) may not be granted by the AP under

Fig. 8. Reduction ratio of ANBF-AP in APP-TPM to that in BPM.

Fig. 9. Comparison of ANBF-AP in APP-TPM with that in BPM.

BPM without a high probability of packet drops, but it can be

granted under the APP-TPM with T (k)I =0.25 or more while

guaranteeing a low probability of packet drops. In other words,

APP-TPM is more flexible and can accommodate more STAs

than BPM. It should be noted that APP-TPM aims to trade

STAs power consumption for their ANBF-AP, i.e., APP-TPM

expends more energy than BPM as the STAs under APP-TPM stay idle longer than those under BPM, as mentioned

in Section I.

Finally, we consider PBMS given in (1). It is easy to show

that PBMS is equivalent to the well-known Knapsack prob-

lem, which can be solved by heuristic or genetic algorithms

[18] with acceptable complexity due to the small number of

STAs in a typical WLAN. In fact, it also can be solved by

enumeration when the number of pending ARs, i.e. n, is small.Assume there are 3 pending ARs in the AP, i.e., n=3, and

the LIs contained in the ARs are T (1)D =30 s, T

(2)D =60 s, and

T (3)D =90 s, respectively. Fix λ

(1)2 = λ

(2)2 = λ

(3)2 =10 fps and

R(1)λ = R

(2)λ = R

(3)λ =1, 2, or 3; set the upper bound of T

(1)I ,


10/14


Fig. 10. The optimal power consumption with buffer size constraint.

T (2)I , T

(3)I to 2000; and let the upper bound of the available

memory size Γ be 200, 400, ..., and 1200, respectively. Thus,PBMS-OPT in (1) can be recast as (27).

Max x(1) 1

E (1) + x(2)

1

E (2) + x(3)

1

E (3)

w.r.t. x(1), x(2), x(3); T (1)I , T

(2)I , T

(3)I

s.t. x(1)L(1)AP + x

(2)L(2)AP + x

(3)L(3)AP ≤ Γ;

x(1), x(2), x(3) ∈ {0, 1};

T (1)I , T

(2)I , T

(3)I ≤ 2000

(27)

Via genetic algorithm, we obtain the optimal power con-

sumption of the above PBMS-OPT, which is shown in the

upper part in Fig. 10, where Rλ

= i represents the case

when R(1)λ = R(2)λ = R

(3)λ = i(i = 1, 2, 3). Moreover, the

corresponding best T (1)I , T

(2)I and T

(3)I for Rλ=3 are depicted

in the lower part in the figure.

From Fig. 10, we observe that, as the buffer size constraint

Γ is gradually relaxed, power consumption can be gradu-ally decreased by finding the optimal T

(1)I , T

(2)I and T

(3)I .

For example, when Γ=200, for the case Rλ=3, by settingT (1)I = T

(2)I =0.25 and T

(3)I =0.45 (see the lower part of the

figure), the power consumption is about 2400 mW (see the

upper part of the figure), which can be reduced to about 1700

mW by choosing T (1)I =0.05 and T

(2)I =T

(3)I =0.25 when Γ=600.

This supports the key idea of our APP-TPM that trades power

consumption for memory usage.As mentioned above, APP-TPM is able to trade off more

power consumption for less frame discarding by adjusting

the pair of parameters T I and T D, whereas BPM does nothave this flexibility. To illustrate this tradeoff, we evaluate

the Percentage of Frame Discarded (PFD) by the AP under

BPM due to insufficient memory. Let D1 represent ANBF-AP under BPM. Since the AP buffer size is Γ, D1 − Γ isthe average number of the frames discarded by the AP due to

buffer overflow. Hence, PFD = max{(D1−Γ)/D1, 0}×100%since PFD must be non-negative. Additionally, we evaluate theratio of the power consumption with APP-TPM to that with

BPM.

Fig. 11. The PDF in BPM and the ratio of power consumption betweenAPP-TPM and BPM.

Fig. 11 illustrates the PFD in BPM (see the upper part of thefigure) and the ratio of power consumption between APP-TPM

and BPM (see the lower part of the figure) when Γ=200, 400,..., 1200. From Fig. 11, we observe that: 1) PFD decreases as Γgrows and no frame discarding occurs when Γ is sufficient forbuffering frames (e.g., Γ=1000 in the case when Rλ=1), whichagrees with our intuitions; and 2) when memory shortage

exists in the AP (e.g., Γ


11/14


in the following equation to the constraint of (1)

∞j=Γ+1

x(k)P (k)

AP ( j) < P 0, k = 1, 2,...,n (28)

where the sum on the left side represents the probability of

memory overflow in the AP for the k-th STA. Obviously, thenew optimization problem can be solved by searching in the

optimal solution of PBMS-OPT given in (1) for the STAssatisfying (28).

It should be pointed out that the above observations arefrom the proposed model based on Poisson traffic. In practice

traffic may be bursty, in which case the model may not matchvery well. In future work we shall validate the model using

real traffic traces to quantify the performance of the proposed

algorithm under more realistic conditions.

VI . RELATED WOR K

Hitherto, many power saving schemes for infrastructure

WLANs have been proposed and most of them remedy the

shortcomings existing in the BPM introduced in the 802.11standard [1]. Noting that, in BPM, all the STAs having pending

frames are informed by the AP at the beginning of each beaconinterval, which may cause multiple STAs to compete for the

channel and hence reduce the system throughput, Aste et al.[8] proposed a power saving scheme that let the AP select

several of the STAs with pending frames to be woken up at

a time while deferring transmissions of the pending frames

of the other STAs to successive beacon intervals, based on a

cost function that takes power consumption and packet latency

into account. Gan et al. [9] presented a scheme that schedulesawake times among STAs optimally such that the number

of STAs competing to access the channel at the same time

as well as frame loss and delay time are reduced, and theyhave proved that both the maximal number of STAs that are

allocated in the same beacon slot and the number of the beaconslots that are allocated for the maximal number of STAs are

minimal when the proposed scheme is executed. Additionally,

He et al. [10] presented a scheme to deliver pending data

packets, which divides a beacon period into multiple slices and

uses scheduled transmission for delivery of buffered frames.

An AP-centric power saving scheme was proposed by Xie et

al. [11], which lets the AP choose the optimal BI and LIs

based on the traffic patterns of STAs and schedule the STAsto wake up at different times to increase energy efficiency

by reducing STAs simultaneous wake-ups. Moreover, frame

buffering delay, together with the metrics of power saving,throughput, and energy efficiency, are used to evaluate the

performance of the proposed scheme. Sarkar [12] proposed an

adaptive algorithm that dynamically adjusts the sleep durations

according to average packet arrival rates and packet delay

constraints.

Evidently, the prerequisite of implementing the above sur-

veyed schemes is that the AP has a sufficiently large buffer

size. These schemes may cause too many buffered frames to

be discarded by the AP due to the lack of a sufficient amount

of memory to buffer them, since prolonging the sleep period ordeferring transmissions of the pending frames both contribute

to an increase in the number of buffered frames in the AP.

Under the BPM introduced in the IEEE 802.11 standard,

the STA operating in power saving mode wakes up at every

LI to listen to the beacon to see whether the TIM contained in

the beacon frame indicates its frames are buffered at the AP or

not. If yes, the STA contends for the channel to transmit a PS-

poll frame to the AP to retrieve the buffered frames. Hence,BPM causes the downlink (AP to STA) packet delay to depend

on the BI. That is, a larger BI may yield a larger downlink packet delay. Perez-Costa et al. [13] presented a scheme called

adaptive power saving mode algorithm (APSM) that reduces

the downlink packet delay according to the downlink frame

interarrival time observed at the AP MAC layer. In addition,

Lo et al. [14] presented a multipolling mechanism, called con-

tention period multipoll (CP-Multipoll), which incorporatesthe DCF access scheme into the polling scheme. The proposed

CP-Multipoll scheme is able to guarantee the bounded delayrequirements of real-time flows. In addition, Hsieh et al. [15]

presented an energy-efficient multipoll (EE-Multipoll) MAC

scheme, which combines power management with a low MAC

protocol overhead; they also determined a suitable wake-up

time schedule to achieve a desirable guarantee of bandwidthutilization. Again, the buffer size in the AP impacts on whether

the above proposed schemes can be realized.As mentioned in the introduction section, buffer size of the

AP should be considered in a power management scheme so

that the frames buffered in the AP for a dozing STA do not

get discarded, which reduces the needs of the upper layers

in the protocol stack to retransmit the packets and maintains

a high system throughput. It is shown in [16] that, the use

of fixed-size buffers in 802.11 networks inevitably leads toeither undesirable channel underutilization or unnecessarily

high delays; high throughput and low delay can be achieved by

dynamic buffer-sizing algorithms. Unfortunately, none of the

power management schemes in the literature have consideredbuffer size or buffer utilization. We finally stress again thatour proposed PBMS, which is based on dynamic buffer size,

fills this gap. There exist a couple of models for analyzing

the performance of power saving schemes. Zheng et al.

[4] proposed a time-out driven power management scheme

and presented a multiple vacation M/G/1/K queuing model

to analyze the performance of this scheme. He et al. [17]provided a Markov chain model to analyze the performance

of the power saving protocols for multicast services in WLANsin addition to a theoretical framework for several power saving

protocols including IEEE 802.11 a/b/g/n. In our previous work

[2] and this paper, we model TPM based on a vector Markov

process and provide the stochastic analysis.Since we could not find any paper in the literature that

deals with both buffer management and power managementin IEEE 802.11-based infrastructure WLAN, we only compare

our proposed APP-TPM with BPM as specified in the IEEE

802.11 standard, i.e., a comparison between our APP-TPM

and other schemes is not presented in this paper.

VII. CONCLUSIONS

Conserving energy is an important topic for WLAN. We

have proposed the AP-priority TPM scheme and investigatedit extensively by developing a realistic model with two queues

to separately hold incoming frames and outgoing frames. The


12/14


model enables derivations of the key statistics of APP-TPM,

based on which we have proposed PBMS. Compared with

BPM given in the IEEE 802.11 standard, under a fixed amount

of memory in the AP, the proposed PBMS can accommodate

as many ARs as possible by trading power consumption for a

reduced frame dropping probability. This is accomplished bychanging the values of the timers T I and T D.

APPENDIX I

For the sake of conciseness, we omit the superscripts, i.e.,

(k)s, in λ(k)1 , λ

(k)2 , λ

(k)1,2 , τ

(k)(x), and σ(k)(x).Firstly, the state transitions relevant to state I in Fig. 4 reveal

that the probability of the STA in state I at time t + ∆t, i.e.,P I (t + ∆t), equals to the sum of the following:

1) the probability that the STA is in state I at time t, andduring ∆t, the accumulated idle time (AIT) of the STA doesnot exceed T I and no new frames comes;

2) the probability that the STA is in state A0,0 at time t,and during ∆t, the AIT of the STA does not exceed T I and

no new frames comes.The above statement yields the following according to (4).

[P I (t + ∆t) − P I (t)]/∆t = −λ1,2P I (t)/(1 − e−λ1,2T I )

+ q A0,0(t)µ(1 − λ1∆t)(1− λ2∆t) + o(∆t)/∆t

Letting ∆t approach 0 leads to (5).Secondly, from state A0,0 in Fig. 4, the probability that the

STA in state A0,0 at time t + ∆t equals to the sum of thefollowing six items:

1) the probability that the STA is in state A0,0 at time t,and during ∆t, the STA continues TX/RX and no new framescomes;

2) the probability that the STA is in state A0,1 at time t,and during ∆t, the STA finishes TX/RX and no new framescomes;

3) the probability that the STA is in state A1,0 at time t,and during ∆t, the STA finishes TX/RX and no new framescomes;

4) the probability that the STA is in state D0,1 at time t, andduring ∆t, the STA stops dozing and no new frames comes;

5) the probability that the STA is in state D1,0 at time t, andduring ∆t, the STA stops dozing and no new frames comes;

6) the probability that the STA is in state I at time t, andduring ∆t, the AIT does not exceed T I but an incoming oroutgoing frame comes.

Summing up, we have[q A0,0(t + ∆t) − q A0,0(t)]/∆t = q A0,0(t)[−(µ + λ1,2)

+ o(∆t)/∆t] + q A0,1(t)µ(1 − λ1∆t)(1 − λ2∆t)

+ q A1,0(t)µ(1 − λ1∆t)(1 − λ2∆t)

+ (1 − λ1∆t)(1− λ2∆t)

∞

0

pD0,1(t, x)τ (x)dx

+ (1 − λ1∆t)(1− λ2∆t)

∞

0

pD1,0(t, x)τ (x)dx

+ P I (t)λ1,2[1 − λ1,2e−λ1,2T I ∆t/(1 − e−λ1,2T I ) + o(∆t)]

Then, letting ∆t approach 0 leads to (6). Moreover, (7)-(9)are similarly derived.

Thirdly, the probability that the STA is in state D0,0 withADT s ≤ ∆t at time t, expressed as pD0,0(t + ∆t, s)∆taccording to (3), equals to the sum of the following:

1) the probability that the STA is in state I at time t, andduring ∆t, the AIT exceeds T I but no new frame comes, i.e.,(1 − λ1∆t)(1− λ2∆t)P I (t) · Pr{χI + ∆t ≥ T I |χI < T I };

2) the probability that the STA is in state D0,0 withADT x ∈ (0,∞) at time t, and during ∆t, the STAstops dozing and no new frame comes, which equals to ∞

0 pD0,0(t, x)τ (x)∆t(1 − λ1∆t)(1 − λ2∆t)dx;Summing up, using (4), dividing the derived equation by

∆t, and letting ∆t → 0 (s → 0 as well), we have (10).Clearly, the STA cannot change its state to D0,j when there

is a frame buffered at the AP. In other words, it is impossible

that the STA is in state D0,j with ADT s ≤ ∆t when j =1, 2,.... Consequently,

pD0,j (t + ∆t, s)∆t = P {ξ (t + ∆t) = D0,j ,

s < Y (t + ∆t) ≤ s + ∆t} = 0, j = 1, 2,...

Dividing by ∆t and letting ∆t → 0, we obtain (11).

Fourthly, the probability that the STA is in state D0,0 withADT x + ∆t(x > 0) at time t + ∆t, equals to the probabilitythat the STA is in state D0,0 with ADT x at time t, and during∆t, the STA continues dozing and no new frame comes. Thus,we have (12).

Lastly, the probability that the STA is in state D0,j( j =1, 2,...) with ADT x + ∆t(x > 0) at time t + ∆t, equals tothe sum of two items: 1) the probability that the STA is in stateD0,j with ADT x at time t, and during ∆t, the STA continuesdozing and no new frame comes; and 2) the probability that

the STA is in state D0,j−1 with ADT x at time t, and during∆t, the STA continues dozing and there is an incoming framearrives at the AP but no outgoing frame generated at the STA.

Thus we obtain (13). Similarly, we obtain (14) and (15).

APPENDIX I I

As did in Appendix I, we omit all the superscripts of (k) in the following description. Given a pdf h(x), its Laplace-Stieltjes Transform (LST) is denoted by h∗(s) = ∞

0 e−sxh(x)dx(s > 0).In the APP-TPM, doze period χD takes a deterministic

value T D, i.e., Pr{χD = T D}=1 and Pr{χD = T D}=0,which implies that the pdf of χD is a Dirac function; i.e.,g(x) = δ (x − T D) [2] satisfying

∞

0

h(x)g(x)dx = ∞

0

h(x)δ (x − T D)dt = h(T D) (a.1)

Additionally, we have

G(x) = Pr{χD ≤ x} =

0, x < T D;

1, x ≥ T D(a.2)

Performing LST on t in (12)-(15) and using (16) leads to(a.3). The correctness of (a.3) can be validated by directly sub-

stituting it into the equations derived from (12)-(15) performed

by LST.

p∗Di,j (s, x) = λi2λ

j1

i! j! xi+j p∗D0,0(s, 0)e

−(s+λ1,2)xG(x),

(i, j = 0, 1, 2...)

(a.3)


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Let θ ≡ lims→0

sp∗D0,0(s, 0). Then (a.1) and (a.3) leads to

limt→∞

∞

0

pDi,j(t, x)τ (x)dx

= lims→0

s

∞

0

p∗Di,j(s, x)τ (x)dx

= λi2λ

j1

i! j! (T D)

i+jθe−λ1,2T D(i, j = 0, 1,...)

(a.4)

Letting t →∞ in (6)-(9), using (a.4), we have

− q A0,0(µ + λ1,2) + µq A0,1 + µq A1,0 + λ1,2P I

+ λ1T Dθe−λ1,2T D + λ2T Dθe

−λ1,2T D = 0(a.5)

− q A0,j (µ + λ1,2) + µq A0,j+1 + λ1q A0,j−1

+ θe−λ1,2T D(λ1T D)j+1/( j + 1)! = 0, j = 1, 2,...

(a.6)

− q Ai,0(µ + λ1,2) + λ2q Ai−1,0 + µq Ai+1,0

+ µq Ai,1 + θe−λ1,2T D(λ2T D)

i+1/(i + 1)!

+ λ1(λ2)i(T D)

i+1θe−λ1,2T D/i! = 0, i = 1, 2,...

(a.7)

− q Ai,j(µ + λ1,2) + λ2q Ai−1,j + µq Ai,j+1 + θ(T D)i+j+1

· e−λ1,2T D(λ2)i(λ1)

j+1/[i!( j + 1)!] = 0, i , j = 1, 2,...(a.8)

Letting t →∞ in (5) and using (2) produce

q A0,0 = P I ρ/(1− e−λ1,2T I ) (a.9)

Using (10) and (a.4), we have

θ = P I λ1,2e−λ1,2T I

(1 − e−λ1,2T I )(1 − e−λ1,2T D) (a.10)

From (a.3) and (a.2), we have

pDi,j = θλi2λj1i! j!

T D

0

xi+je−λ1,2xdx,i,j = 0, 1, 2,... (a.11)

Multiplying (a.6) by z j , summing the derived equations for j = 1, 2,..., and using ex =

∞

n=0 xn/n!, we obtain

Q0(z) = {[µ− (µ + λ1,2)z]q A0,0 + µzq A0,1

− θe−λ1,2T D(eλ1T Dz − λ1T Dz − 1)}

/[λ1z2 − (µ + λ1,2)z + µ]

(a.12)

Obviously, the denominator of (a.12) has a unique root z0 inthe interval (0, 1), which is shown in (18). Clearly, z-transformQ0(z) is an analytic function over (0, 1). Thus, the numerator

of (a.12) has to be 0 when z = z0, which yields

q A0,1 = 1

µz0{[(µ + λ1,2)z0 − µ]q A0,0

+ θe−λ1,2T D(eλ1T Dz0 − λ1T Dz0 − 1)}(a.13)

Substituting (a.13) into (a.12) and applying (a.9) and (a.10)

produce Q0(z) shown in (20). Multiplying (a.8) by zj , sum-

ming the derived equations for j = 1, 2,..., we obtain

θe−λ1,2T D

z

(λ2T D)i

i! (eλ1T Dz − λ1T Dz − 1)

− (µ + λ1,2 − µ

z)(Qi(z) − q Ai,0)

+ λ2(Qi−1(z)− q Ai−1,0) − µq Ai,1 = 0, i = 1, 2,...

(a.14)

Multiplying (a.14) by yi, summing the derived equationsfor i = 1, 2,... yields

K (y, z) =

∞i=0

yiq Ai,0 + [(µ + λ1,2 − λ2y)z − µ]−1

· {[(µ + λ1,2)z − µ]Q0(z)− [(µ + λ1,2)z − µ]q A0,0

− µz∞

i=1

yiq Ai,1

+ θe−λ1,2T D(eλ2T Dy − 1)

· (eλ1T Dz − λ1T Dz − 1)}

(a.15)

Evidently, zy shown in (19) is the root of the denominator of the second term in (a.15). Hence, the numerator of the second

term (the expression within the pair of curly braces) equals to0 when z = zy, which leads to

∞i=1

yiq Ai,1 = λ2yQ0(zy)/µ− λ2yq A0,0/µ

+ θe−λ1,2T D(eλ2T Dy − 1)(eλ1T Dzy − λ1T Dzy − 1)/(µzy)(a.16)

Multiplying (a.7) by yi, summing the derived equations fori = 1, 2,... leads to

∞i=0

yiq Ai,0 = q A0,0 + zy

µ(y − zy){−µyq A1,0 + λ2y

2Q0(zy)}

+ θe−λ1,2T D [(eλ2T Dy − 1)(eλ1T Dzy − λ1T Dzy − 1)y/zy

+ λ1T Dy(eλ2T Dy − 1) + (eλ2T Dy − λ2T Dy − 1)]}

(a.17)

From (a.5) and (a.13), we have

q A1,0 = 1

z0q A0,0 − ρP I

− θe−λ1,2T D [ρT D + e

λ1T Dz0

− λ1T Dz0 − 1µz0]

(a.18)

From (a.15)-(a.18), we obtain (17). In addition, letting z =1 in (20), we obtain Q0(1). Then, setting y = 1 and z = 1 in(17) yields (23). Moreover, using (a.11) and (a.10), we have

(22). Further, using (22), (23), and P (k)D + P

(k)A + P

(k)I = 1

leads to (21).

Using (a.11), we obtain

P AP ( j) =∞i=0

P Di,j = θ

T D0

(λ1x)j

j! e−λ1xdx

which yields (25) through using (a.10) and integration by

parts. Moreover, the preceding equation also yields (26) byusing Talor series of ex as follows:

LAP =

∞i=0

jP AP ( j) = θλ1(T D)2

2

which is equivalent to (26) after substituting (a.10).

REFERENCES

[1] LAN MAN Standards Committee of the IEEE Computer Society, “IEEE802.11 standard—wireless LAN medium access control and physicallayer specifications,” June 2007.

[2] Y.-H. Zhu and V. C. M. Leung, “Efficient power management forinfrastructure IEEE 802.11 WLANs,” IEEE Trans. Wireless Commun.,vol. 9, no. 7, pp. 2196–2205, July 2010.


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[3] M. S. Gast, 802.11 Wireless Networks: The Definitive Guide, 2nd edition.O’Reilly, 2005.

[4] R. Zheng, J. C. Hou, and L. Sha, “Performance analysis of power man-agement policies in wireless networks,” IEEE Trans. Wireless Commun.,vol. 5, pp. 1351–1361, June 2006.

[5] S. G. Sitharaman, “Modeling queues using Poisson approximation inIEEE 802.11 ad-hoc networks,” in Proc. 2005 IEEE Workshop on Localand Metropolitan Area Networks, pp. 1–6.

[6] S. M. Ross, Introduction to Probability Models, 9th edition. Elsevier(Singapore) Pte LTD, 2007.

[7] Y. Rong, A. Y. Teymorian, L. Ma, X. Cheng, and H.-A. Choi, “A noveladaptation scheme for 802.11 Networks,” IEEE Trans. Wireless Commun.,vol. 8, no. 2, pp. 862–870, Feb. 2009.

[8] N. Aste, L. Atzori, and L.S. Randaccio, “WLC24-2: selective powermanagement in IEEE 802.11 infrastructure WLANs,” in Proc. 2006 GLOBECOM , pp. 1–6.

[9] C.-H. Gan and Y.-B. Lin, “An effective power conservation scheme forIEEE 802.11 wireless networks,” IEEE Trans. Veh. Technol., vol. 58, no.4, pp. 1920–1929, May 2009.

[10] Y. He, R. Yuan, X. Ma, J. Li, and C. Wang, “Scheduled PSM forminimizing energy in wireless LANs,” in Proc. 2007 IEEE InternationalConference on Network-Protocols, pp. 154–163.

[11] Y. Xie, X. Luo, and R. K. C. Chang, “Centralized PSM: an AP-centricpower saving mode for 802.11 infrastructure networks,” in Proc. 2009

IEEE Sarnoff Symposium, pp. 1–5.[12] M. Sarkar and R. L. Cruz, “A MAC layer dynamic power management

scheme for multiple users in a WLAN,” in Proc. 2007 IEEE Wireless

Communications and Networking Conference, pp. 2126–2130.[13] X. Perez-Costa and D. Camps-Mur, “APSM: bounding the downlink

delay for 802.11 power save mode,” in Proc. 2005 IEEE InternationalConference on Communications, vol. 5, pp. 3616–3622.

[14] S.-C. Lo, G. Lee, and W.-T. Chen, “An efficient multipolling mechanismfor IEEE 802.11 wireless LANs,” IEEE Trans. Computers, vol. 52, no.6, pp. 764–778, 2003

[15] J.-R. Hsieh, T.-H. Lee, and Y.-W. Kuo, “Energy-efficient multi-pollingscheme for wireless LANs,” IEEE Trans. Wireless Commun., vol. 8, no.3, pp. 1532–1541, Mar. 2009.

[16] T. Li, D. Leith, and D. Malone, “Buffer sizing for 802.11-basednetworks,” IEEE/ACM Trans. Netw., vol. 19, no. 1, pp. 156–169, 2011.

[17] Y. He, R. Yuan, and W. Gong, “Modeling power saving protocolsfor multicast services in 802.11 wireless LANs,” IEEE Trans. MobileComputing, vol. 9, no. 5, pp. 657–671, 2010.

[18] J. H. Holland, Adaptation in Natural and Artificial Systems: An In-troductory Analysis with Applications to Biology, Control, and Artificial

Intelligence. The MIT Press/Bradford Books edition, 1992.

Yi-Hua Zhu (M01-SM07) received his B.S. degreein mathematics from Zhejiang Normal University,Zhejiang, China, in July 1982; his M.S. degree inoperation research and cybernetics from ShanghaiUniversity, Shanghai, China in April 1993; and hisPh.D. degree in computer science and technologyfrom Zhejiang University, Zhejiang, China, in March2003.

Dr. Zhu is a professor at Zhejiang University of Technology, Hangzhou, Zhejiang, China. He is a

member of China Computer Federation TechnicalCommittee on Sensor Network. His current research interests include infor-mation dissemination, stochastic modeling and analysis, power management,mobility management for wireless networks, and network coding. He hasserved as technical program committee members in the international confer-ences ICC, WCNC, GlobeCom, etc. He is the recipient of the Best PaperAward of Chinacom 2008. He has published more than 120 research papersin proceedings and journals including IEEE TRANSACTIONS ON W IRELESSCOMMUNICATIONS, IEEE TRANSACTIONS ON V EHICULAR TECHNOLOGY,and more.

Han-Cheng Lu received his B.S. degree in information science and technol-ogy from Nangjing University of Posts and Telecommunications in 2008. Heis currently a candidate student for his master degree in computer science.

Victor C. M. Leung (S75-M89-SM97-F03) re-ceived the B.A.Sc. (Hons.) degree in electrical en-gineering from the University of British Columbia(U.B.C.) in 1977, and was awarded the APEBCGold Medal as the head of the graduating classin the Faculty of Applied Science. He attendedgraduate school at U.B.C. on a Natural Sciences andEngineering Research Council Postgraduate Schol-arship and completed the Ph.D. degree in electricalengineering in 1981.

From 1981 to 1987, Dr. Leung was a SeniorMember of Technical Staff at MPR Teltech Ltd. In 1988, he was a Lecturerin the Department of Electronics at the Chinese University of Hong Kong.He returned to U.B.C. as a faculty member in 1989, and currently holdsthe positions of Professor and TELUS Mobility Research Chair in AdvancedTelecommunications Engineering in the Department of Electrical and Com-puter Engineering. Dr. Leung has co-authored more than 500 technical papersin international journals and conference proceedings, and several of thesepapers had been selected for best paper awards. His research interests are inthe areas of wireless networks and mobile systems.

Dr. Leung is a registered professional engineer in the Province of BritishColumbia, Canada. He is a Fellow of IEEE, a Fellow of the EngineeringInstitute of Canada, and a Fellow of the Canadian Academy of Engineering.He is a Distinguished Lecturer of the IEEE Communications Society. Hehas served on the editorial boards of the IEEE JOURNAL ON SELECTEDAREAS IN C OMMUNICATIONS – Wireless Communications Series, the IEEETRANSACTIONS ON W IRELESS C OMMUNICATIONS and the IEEE TRANS-ACTIONS ON VEHICULAR TECHNOLOGY, and is serving on the editorialboards of the IEEE T RANSACTIONS ON COMPUTERS, IEEE WIRELESSCOMMUNICATIONS LETTERS, Computer Communications, the Journal of Communications and Networks, as well as several other journals. He hasguest-edited many journal special issues, and served on the technical programcommittee of numerous international conferences, and contributed to theorganization of many international conferences. He is a winner of the IEEEVancouver Section Centennial Award and a 2011 U.B.C. Killam ResearchPrize.

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Access Point Buffer Management for Power Saving