1920 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY 2009
An Effective Power Conservation Scheme for
IEEE 802.11 Wireless Networks
Chai-Hien Gan, Member, IEEE, and Yi-Bing Lin, Fellow, IEEE
Abstract—In IEEE 802.11 wireless networks, a mobile station (MS) can power down the transceiver to save energy. To achieve power conservation, an access point (AP) periodically broadcasts beacons to notify the associated MSs about their traffic indications, and the MSs periodically wake up to listen to the beacons. The MS awake time scheduling is an important issue, which affects the number of simultaneously awake MSs to compete for access. We propose a power conservation scheme to optimally schedule the awake times among the MSs, such that the number of MSs awaking at the same time is minimal. Our study indicates that the proposed scheme demonstrates good performance in terms of frame loss and delay time.
Index Terms—Access point (AP), power conservation, power save mode (PSM), wireless local area network (WLAN).
I. INTRODUCTION
T
HE IEEE 802.11 wireless local area network (WLAN) [1]–[3] has been widely deployed to provide low-cost broadband wireless Internet access. Fig. 1 shows the WLAN architecture, which consists of access points [APs; Fig. 1(1)] and mobile stations [MSs; Fig. 1(2)]. Each MS must associate with an AP to obtain the wireless access service [Fig. 1(3)]. An MS can be a personal digital assistant, a WiFi phone, or a notebook computer. Due to size restriction, the battery capacity of an MS is limited. Therefore, power conservation becomes an important issue in MS design [4], [5].An MS can power down the transceiver to save energy [1], [4]. When the transceiver is on, it is said to be awake or active. When the transceiver is off, it is said to be sleeping, dozing, or in the power-saving mode. To achieve power conservation, the AP periodically broadcasts beacons to notify the associated MSs about their traffic indications (i.e., whether there are frames buffered for these MSs). In the IEEE 802.11 WLAN, a time line is partitioned into beacon intervals. Every beacon interval starts Manuscript received January 29, 2008; revised May 21, 2008. First pub-lished August 12, 2008; current version pubpub-lished April 22, 2009. Y.-B. Lin’s work was supported in part by Grant NSC-96N079, Grant NSC-96N576, Grant NSC-96N230, Chung-Hwa Telecom, the Industrial Technology Research Institute and National Chiao Tung University joint research center, and the Ministry of Education, Aiming for Top University plan. The review of this paper was coordinated by Dr. J. Misic.
C.-H. Gan is with the Information and Communications Research Labs, In-dustrial Technology Research Institute, Hsinchu 310, Taiwan (e-mail: chgan@ itri.org.tw).
Y.-B. Lin is with the Department of Computer Science and Information Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, and also with the Institute of Information Science, Academia Sinica, Taipei 115, Taiwan (e-mail: liny@cs.nctu.edu.tw).
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TVT.2008.2003962
Fig. 1. WLAN architecture.
with a beacon slot to broadcast the beacon. To connect to an AP, an MS issues to the AP the association request containing a listen interval parameter. This parameter is used to specify when the MS should wake up to listen to beacons. Upon receipt of the association request, the AP may grant or deny access based on the content of the request (e.g., the listen interval). If the AP agrees with the listen interval that is specified by the MS, the MS is granted access to the AP. A connected MS issues the disassociation request to the AP when it wishes to exit the wireless access service. When there are frames that are buffered for the MS, the AP notifies the MS through the traffic indication
map (TIM) contained in the beacon. Since the MS may sleep
when the AP delivers the TIM, the AP must wait for at least one listen interval of the MS before discarding the frames. When the MS wakes up, and the TIM indicates that there are incoming frames, the MS will issue a power-saving (PS) poll (PS-poll) message to the AP to retrieve the buffered frames. To obtain the permission for issuing the PS-poll message, the MS may compete with other MSs in a time period called the contention
window. During a contention window, only one MS is permitted
to issue the PS-poll message (and then retrieve the frame). After frame transmission, other MSs may compete again in the next contention window. If the MS cannot issue the PS-poll message within the listen interval due to contention failure, the AP may discard the buffered frames without notification, which significantly affects the quality-of-service (QoS).
Power conservation significantly reduces the MS’s power consumption at the cost of QoS degradation (e.g., frame loss, delivery delay, etc.). Previous studies [6]–[10] have addressed this problem. In [6], the adaptive power save mode (PSM) al-gorithm was proposed to estimate the current frame interarrival time to adapt the interval for issuing PS-poll messages. In [7], 0018-9545/$25.00 © 2008 IEEE
Fig. 2. Buffered frame retrieval process.
the authors analyzed the integration of power conservation and IEEE 802.11 QoS enhancement (i.e., IEEE 802.11e [3]) and studied the impact of the PSM on the enhanced distributed
channel access QoS mechanism. In [8], the authors proposed
a self-tuning power management scheme that dynamically adjusts the power management mechanism for IEEE 802.11 devices based on access patterns of applications. In [9], the authors proposed a smart PSM scheme to adjust the active and sleeping modes according to the desired delay performance and energy consumption. In [10], the authors proposed a power saving backoff prediction to predict the time that the MSs will wait before accessing the IEEE 802.11 WLAN. The above approaches focus on the listen interval adjustment to reduce the power consumption of MSs, which do not address the PS-poll contention issues. In this paper, we propose a power conservation scheme to minimize the PS-poll contention among MSs. This scheme can complement the previous approaches to reduce the power consumption of MSs without significantly degrading the QoS.
In our previous study [11], [12], the shared channel assign-ment and scheduling (SCAS) algorithm was proposed to sched-ule connections for packet transmission in a shared channel. The SCAS algorithm periodically schedules the time slots in a shared channel for each connection, where period Iiis derived
from its requested transmission rate. That is, if the first time slot scheduled to a connection is the kth time slot, then the (k +
nIi)th time slots (for n > 0) are also scheduled to the
connec-tion. In SCAS, Iiis assumed to be a two’s-power number of a
basic time unit. The SCAS algorithm schedules the connections with smaller periods first and schedules in any order if tie break-ing is needed. By usbreak-ing the SCAS algorithm, we have shown that the connections can be scheduled into the time slots without conflict (i.e., any two connections will not be scheduled at the same time slot) if the total requested transmission rate does not exceed the supported transmission rate of the shared channel.
In this paper, we extend the SCAS algorithm to resolve PS-poll contention. We note that, in the IEEE 802.11 WLAN, more than one MS are allowed to listen to a beacon from the AP. On the other hand, SCAS only allows one MS to be scheduled at a time slot to listen to the beacon. Some MSs are blocked if they cannot be scheduled in the free slots (i.e., the slots that have not been scheduled for other MSs). Directly applying the SCAS algorithm in the IEEE 802.11 WLAN may restrict the number of MSs that are served by the AP. By enhancing the SCAS algorithm, we propose a new scheme to schedule the awake times of each associated MS. This scheme minimizes the pos-sibility of PS-poll contention.
II. POWERCONSERVATION
An MS requesting longer listen interval consumes less bat-tery energy at the cost of requiring more buffer space at the AP and longer frame delivery delays [4], [5]. The MS awake time scheduling affects the number of simultaneously awake MSs that compete for access. Fig. 2 illustrates how the simulta-neously awake MSs affect the QoS in the IEEE 802.11 WLAN. In this figure, three MSs Q1, Q2, and Q3are associated with the AP. The listen intervals that are specified by Q1, Q2, and
Q3are 2, 4, and 4, respectively. The AP broadcasts the beacon at each beacon slot [see Fig. 2(1)] in every beacon interval [see Fig. 2(2)]. Depending on the design, multiple PS-poll messages can be issued without conflict within a beacon interval. In this example, we assume that at most one PS-poll message can be is-sued within a beacon interval. If an MS fails to issue the PS-poll message in this beacon interval, it will try to compete for the next beacon interval. If the MS cannot issue the PS-poll mes-sage within its listen interval, the buffered frame is discarded. At the first beacon slot, the AP broadcasts a beacon containing the TIM indicating that there is a frame that is buffered for Q1 [see Fig. 2(3)]. Since Q1is sleeping [see Fig. 2(4)], it does not
1922 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY 2009
Fig. 3. Example for element allocation in the scheduling lists (C = 16), where I1= I2= I8= I9= I10= 4, I3= I4= I5= I11= I12= I13= 8, and
I6= I7= 16.
receive this indication. At the second beacon slot, the TIM indi-cates that there are frames for Q1and Q3[see Fig. 2(5)]. Since only Q1is awake to listen to the TIM [see Fig. 2(6)], it success-fully issues the PS-poll message to retrieve the buffered frame. At the third beacon slot, the TIM indicates that there is a frame for Q3again [see Fig. 2(7)]. At this moment, Q3is sleeping and, thus, cannot retrieve the buffered frame. At the fourth beacon slot, the TIM indicates that there are frames that are buffered for
Q1, Q2, and Q3[see Fig. 2(8)]. These three MSs are awake and compete to issue the PS-poll messages during the contention window [see Fig. 2(9)]. In our example, Q1is granted for access during the contention window and issues a PS-poll message to retrieve the frame. At this period, Q2 and Q3 are deferred to issue the PS-poll message [see the busy period; Fig. 2(10)].
Q2and Q3 will keep awake until their frames are received or dropped. At the fifth beacon slot, the TIM indicates that there are frames that are buffered for Q2and Q3[see Fig. 2(11)]. Q2 and Q3compete to issue the PS-poll messages again. Assume that Q2 is granted to issue the PS-poll message and retrieve the frame. Q3still cannot retrieve its frame during this beacon interval. At the sixth beacon slot, the buffer time for Q3’s frame exceeds the listen interval (of Q3). Therefore, the AP discards the frame, and the TIM indicates that no frame is buffered [see Fig. 2(12)]. In this case, the frame for Q3is lost, and Q3returns to sleep at this beacon interval [Fig. 2(13)].
It is obvious that if more MSs compete for the permis-sion to issue PS-poll messages [e.g., the fourth beacon slot in Fig. 2(8)], the frame loss possibility will be increased. Therefore, the smaller the number of simultaneously awake MSs at the same beacon slot, the better the QoS (i.e., PS-poll contention can be reduced). Based on the above discussion, we make the following definition.
Definition 1: Let M be the maximal number of MSs that are
allocated in the same beacon slot in an awake time scheduling scheme X. X is optimal in terms of potential PS-poll con-tention if the scheme satisfies the following two conditions.
1) M is minimal among all awake time scheduling schemes. 2) The number of the beacon slots that are allocated for M
MSs is minimal.
The IEEE 802.11 standard does not specify how to schedule the awake times of each associated MS, and a typical schedul-ing scheme (called the basic scheme) for power conservation is defined as follows.
Definition 2—[The Basic Scheme]: Suppose that an MS Qi
associates with the AP at the time between (k− 1)th and kth beacon slots. If Qihas the listen interval Ii, then it is scheduled
to awake at (k + nIi)th beacon slots for n > 0.
Intuitively, the basic scheme is not “optimal.” Now, we de-scribe the conditions when the awake time scheduling satisfies Definition 1. We first note that the IEEE 802.11 standard only specifies the listen interval parameter to be a 16-bit value, which ranges from 0 to 216− 1 [1]. For discussion purposes, we define
k classes of the listen intervals: 21, 22, . . . , 2k, where k < 16. Let S be the set of the MSs that are associated with the AP. Each MS Qi∈ S requests a listen interval
Ii = 2ri, where 0≤ ri≤ k. (1)
It is clear that if Ii= 2riand Ij = 2rj, then
Ii= 2rIj, where r = ri− rj. (2)
We note that the constraint for the specified listen intervals (to be 2ri form) can be released to any positive integer by
applying the scheme in our previous study [12]. Such details will complicate the discussion and are omitted.
For an MS Qiwith the listen interval Ii, if Qifirst wakes up
to listen to the beacon from the AP at the jth beacon slot, then it must wake up to listen to the beacons at the (j + nIi)th beacon
slots for n > 0. Let C be the minimal common multiple of the requested listen interval ranges. Without loss of generality, define a beacon cycle as C consecutive beacon slots. Every MS will, at least, wake up once per beacon cycle. For description purposes, we introduce the concept of scheduling list to map the beacon slots that are assigned to the MSs in a beacon cycle. Assume that C = 2k. The positions of elements in a scheduling list are sequentially labeled as 0, 1, 2, . . . , 2k− 1. Element j records the identity of an MS that must wake up at the (j + n2k)th beacon slots (for n≥ 0). If a beacon cycle
cannot accommodate all MSs without conflicts, then a beacon slot in that cycle must be assigned to multiple MSs (and these MSs will compete for access at this slot). We use multiple scheduling lists (denoted by List[1], List[2], . . .) to represent multiple MS assignment. When we say “element l,” it means element l of all scheduling lists in a scheduling scheme. The number of scheduling lists is the maximal number of MSs that must wake up at the same beacon slot. Fig. 3 shows an example for element allocation in the scheduling lists. In this example, we assume that k = 4, and the positions of the elements in a scheduling list are labeled from 0 to 15.
Definition 3: An MS Qi is said to have the listen interval
Ii if element j in a scheduling list is first allocated to Qi,
and then, elements j + nIi(for n > 0 and j + nIi< C) in the
Thus, Qi will wake up at the (j + nIi)th beacon slots for
n≥ 0. Since C is a multiple of Ii, Qi will also wake up at
the (j + nIi+ C)th beacon slots, and the allocated positions
(i.e., j + nIi) for Q2 during the current and the next beacon cycles are identical. Therefore, when the AP broadcasts at the
lth beacon slot, all MSs that are assigned at element (l mod C) must wake up. For the scheduling lists shown in Fig. 3, if
the AP broadcasts at the 16th beacon slot, the MSs Q1, Q8, and Q13[assigned at element (16 mod 24) = 0] must wake up (see Fig. 3).
By utilizing the scheduling list concept, we show how to satisfy the optimal scheduling conditions in Definition 1.
Lemma 1: In a scheduling scheme X, the maximal number
of MSs that are allocated in the same beacon slot (i.e., M ) is minimal if the number of scheduling lists isQi∈S1/Ii.
Proof: Since M is also the number of scheduling lists that
are used in X, it suffices to prove that the minimal number of scheduling lists that can accommodate all MSs in S is
Qi∈S1/Ii. Since the listen interval for Qi is Ii, Qi will
consume a 1/Ii portion of a scheduling cycle. For all MSs
in S, they will consume aQi∈S1/Ii portion of the cycle.
Therefore,Qi∈S1/Ii is the minimal number of scheduling
lists that can accommodate all MSs in S (i.e., M = L =
Qi∈S1/Ii).
Lemma 2: Consider a scheduling scheme X that uses L
scheduling lists in scheduling a set of MSs. If each element
l (0≤ l ≤ 2k− 1) has been allocated to at least L − 1 MSs,
and at least one element is allocated to L MSs, then the number of beacon slots that are allocated to L MSs is minimal.
Proof: Let Uj be the set of elements that are allocated to
exact j MSs. Denote the number of elements in Uj as |Uj|.
Suppose that|UL| = m > 0 and |UL−1| = C − m (where C is
the length of the scheduling list) in X. The total number of MSs that are assigned at elements in UL−1is (C− m)(L − 1).
Suppose that|UL| can be reduced (i.e., X is not optimal). Then,
at least one MS that is assigned at an element in UL must be
reassigned to an element at UL−1 (to reduce |UL|). Without
loss of generality, consider that one MS that is assigned at an element in UL is reassigned to an element in UL−1 (i.e., to
reduce|UL| by one). Therefore, (C − m)(L − 1) + 1 MSs are
assigned at elements in UL−1. By pigeonhole principle [13], at least one element l in UL−1will be allocated to L MSs. After the reallocation of the MS, element l will be moved to UL,
and|UL| is increased by one. In other words, |UL| cannot be
reduced.
Theorem 1: Let L be the number of scheduling lists that are
used in a scheduling scheme X. X is optimal if we have the following.
1) L =Qi∈S1/Ii.
2) each element l (0≤ l ≤ 2k− 1) has been assigned to at least L− 1 MSs, and at least one element is assigned to
L MSs.
Proof: The proof can be directly shown from Lemma 1,
Lemma 2, and Definition 1.
III. PROPOSEDPOWERCONSERVATIONSCHEME This section proposes an optimal power conservation scheme for the MS awake time scheduling. Let S be the set of the MSs
that are associated with an AP, and let L be the number of scheduling lists that are required in the scheme. Initially, L = 0 and S =∅.
When an MS Qijoins set S through the association request, it
is assigned the awake beacon slots through the Join procedure, which is described as follows.
Procedure Join (Qi)
Step J1 S← S ∪ {Qi}.
Step J2 If Qj∈S1/Ij > L, a new scheduling list is
added, and L← L + 1.
Step J3 Let Smbe the set of all MSs that are scheduled in a
scheduling list List[m], whereQj∈Sm1/Ij< 1.
Let SR=∅. For every Qj∈ Sm, if Ij> Ii, then
SR← SR∪ {Qj} and S ← S − {Qj}. At this
point, the elements in List[m] that are occupied by MSs in SRbecome vacant.
Step J4 Suppose that element l is the first vacant element in List[m]. Note that for all Ij≥ Ii, Qj have
been removed, and all elements l + nIi (for n≥ 0
and l + nIi ≤ 2k) are vacant (to be proved in
Lemma 3). These elements are allocated to Qi.
Step J5 For all Qj∈ SR, execute Join (Qj).
After performing the Join procedure, the AP notifies Qi
about its awake beacon slots through the association response message. To efficiently execute the Join procedure, all Qjin SR
with smaller listen intervals are selected earlier in Step J5 for execution. In this MS selection order, we ensure that SR=∅
in Step J3 when the Join procedure is invoked recursively. In Step J3, a list with least vacant elements (i.e., List[m]) is selected from the scheduling lists. This selection ensures that newly incoming MSs or MSs in SR can always be scheduled
in the same scheduling list until the list is allocated fully. The following fact and lemma show that the elements to be allocated to Qiare all vacant after Step J3 is executed.
Fact 1: Let Sm be the set of MSs that are scheduled in
List[m], and let element l be the first vacant element in List[m]. Consider Ii for an MS Qi∈ Sm. If Ii≥ Ij for all
Qj ∈ Sm, then l < Ii.
Proof: See the Appendix.
Lemma 3: Suppose that a set Sm of MSs is scheduled in
List[m] using the Join procedure, where Qj∈Sm1/Ij< 1,
and element l is the first vacant element in List[m]. Consider an MS Qi∈ Smsuch that Ii≥ Ijfor all Qj∈ Sm. Let element
x be the first allocated element for any Qj ∈ Sm. Then, for all
n and n values
l + nIi= x + n Ij. (3)
In other words, elements l + nIi in List[m] are vacant, and,
therefore, Step J4 of the Join procedure is justified.
Proof: See the Appendix.
Next, we prove that, by exercising the Join procedure, there is at most one scheduling list containing vacant elements.
Lemma 4: Consider a set S of MSs and any integer r≥ −k
(where k≥ 0 is an integer). For all Qj ∈ S, if 2−k≤ 1/Ij≤ 2r
andQj∈S1/Ij≥ 2r, then there exists a subset Sm⊂ S such
1924 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY 2009
Proof: See the Appendix.
Corollary 1: Consider a set S of MSs. For all Qj∈ S,
if 1/Ij≤ 1 and
Qj∈S1/Ij≥ 1, then there exists a subset Sm⊆ S such that
Qj∈Sm1/Ij= 1.
Proof: The proof can be directly shown from Lemma 4
with r = 0.
Lemma 5: Consider a set Snof n MSs. Label these MSs as
Q1, Q2, . . . , Qnsuch that
1/I1≥ 1/I2≥ · · · ≥ 1/In (4)
1≤j≤n
1/Ij≥ 1. (5)
Then, there exists a subset Sm=
1≤i≤m{Qi} ⊆ Snsuch that
1≤j≤m1/Ij= 1.
Proof: See the Appendix.
Lemma 6: Consider a set Sn of n MSs labeled as
Q1, Q2, . . . , Qn, where
1/I1≥ 1/I2≥ · · · ≥ 1/In. (6)
If
1≤j≤n
1/Ij= 1 (7)
then all MSs in Sncan be scheduled in one scheduling list by
using the Join procedure.
Proof: See the Appendix.
Theorem 2: By exercising the Join procedure, when there are L scheduling lists that are used in the scheme, there is at most
one scheduling list containing vacant elements.
Proof: We prove by induction on L.
Basis: When L = 1, it is obvious that the hypothesis holds. Induction step: Assume that the hypothesis holds for L≥ 1. We prove that the hypothesis also holds for L + 1. When there are L scheduling lists that are used in the scheme, from the induction hypothesis, there is at most one scheduling list containing vacant elements. Two cases are considered.
Case 1) None of the L scheduling lists contains vacant elements. By Step J2 of the Join procedure,
List[L + 1] is added when a new MS Qi arrives,
and Qiis scheduled in List[L + 1]. Therefore, only
List[L + 1] contains vacant elements. The
hypo-thesis holds.
Case 2) One scheduling list List[m] contains vacant ele-ments. Let Smbe the set of MSs that are scheduled
in List[m]. There are two possibilities when a new MS Qiarrives.
Case 2.1) Qj∈S1/Ij ≤ L. The MS Qi will be
scheduled in List[m], which does not affect the fully allocated scheduling lists. Therefore, only one scheduling list
List[m] may contain vacant elements.
Case 2.2) Qj∈S1/Ij > L. This implies
1/Ii+
Qj∈Sm
1/Ij> 1. (8)
At Step J2 of the Join procedure, a schedul-ing list List[L + 1] will be added. At Step J3 of the Join procedure, MSs Qj ∈
Sm (where Ij > Ii) will be rescheduled
such that all MSs Qj∈ Sm∪ {Qi} can be
scheduled into List[m] in a nondecreasing
Ij order. Since (8) holds, Lemma 5
en-sures that a set Sn exists such that Sn⊆
Sm∪ {Qi} and
Qj∈Sn1/Ij = 1, and
Lemma 6 ensures that all MSs in Sn
can be scheduled in List[m]. Therefore,
List[m] will be fully allocated, and only List[L + 1] contains vacant elements.
From Cases 2.1 and 2.2, the hypothesis holds. When an MS Qi leaves set S through the disassociation
request, Qi is removed through the Leave procedure, which is
described as follows. Procedure Leave (Qi)
Step L1 S← S − {Qi}, and Qi is removed from its
scheduling list List[m] (i.e., the elements that are occupied by Qibecome vacant).
Step L2 Let Sm be the set of MSs that are allocated in
List[m]. If Sm=∅, then List[m] is removed, L ←
L− 1, and the procedure exits. If Sm= ∅, then
Step L3 is executed.
Step L3 Let SR=∅. For all Ij > Ii, Qj ∈ Sm are
re-moved from List[m] and are added into SR (i.e.,
SR← SR∪ {Qj} and S ← S − {Qj}). Let
ele-ment xi (xj) be the first allocated element in
List[m] for Qi (Qj). For all Ij= Ii, if xi< xj,
then SR← SR∪ {Qj} and S ← S − {Qj}.
Step L4 If there exists u= m such that the scheduling list
List[u] contains vacant elements, then all MSs of
the set Suthat are scheduled in List[u] are removed
from the scheduling list and collected into SR, i.e.,
SR← SR∪ Suand S← S − Su. At this point, the
MSs that are occupied by MSs in SRbecome vacant
and will be rescheduled in Step L6 to ensure at most one scheduling list containing vacant elements after rescheduling (to be proved in Lemma 7).
Step L5 If Qj∈S1/Ij < L, List[u] is removed, and
L← L − 1.
Step L6 For all Qj∈ SR, execute Join (Qj).
Similar to Step J5 of the Join procedure, Step L6 executes the Join procedure for Qj∈ SRin the nondecreasing Ijorder.
Note that when an MS Qi joins or leaves the set, the awake
beacon slots of already scheduled MSs (i.e., the MSs in SR)
may be rescheduled. The AP will notify these MSs about their new awake beacon slots. Now, we justify the Leave procedure with the following lemma.
Lemma 7: After exercising the Leave procedure, there is at
most one scheduling list containing vacant elements.
Proof: See the Appendix.
We show that the power conservation scheme satisfies Definition 1.
Theorem 3: The proposed power conservation scheme is
optimal.
Proof: In the power conservation scheme, the number of
scheduling lists that are used in the scheme isQj∈S1/Ij
(see Step J2 of the Join procedure and Step L3 of the Leave pro-cedure). From Theorem 2 and Lemma 7, at most one scheduling list contains vacant elements. From Theorem 1, Definition 1 is satisfied (i.e., this scheme is optimal). We note that the proposed power conservation scheme can be easily integrated with the IEEE 802.11 WLAN. To implement our scheme, we only need to add an indication of the first awake beacon interval at the standard association response message [1].
IV. PERFORMANCEEVALUATION
We develop a simulation model to investigate the perfor-mance of the proposed power conservation scheme. The simu-lation model follows the event-driven approach that is widely adopted in mobile network studies [14], [15]. In the simu-lation, the MSs are classified into k classes, where class i (1≤ i ≤ k) has the listen interval 2i≤ C. We assume that only one PS-poll message can be successfully issued within one beacon interval, and a frame can be buffered in the AP within one listen interval (i.e., the frame is dropped after one listen interval).
Based on simulation experiments, we compare the proposed power conservation scheme (scheme p) with the basic scheme (scheme b; see Definition 2) in terms of two output measures. The following input parameters are considered:
T period of a beacon interval, which is assumed to be fixed;
k number of MS classes;
ti inter-MS arrival time for class i MSs. We assume that
ti follows the gamma distribution with mean 1/λi and
variance Λi;
τi interframe arrival time for a class i MS. We assume that
τi follows the gamma distribution with mean 1/γi and
variance Γi;
β probability that a frame arrival continues the MS session. In other words, the MS ends the session with probability 1− β.
We note that the MSs are periodically awake to listen to the beacons from the AP; however, the traffic of the MSs can arrive in general form. We consider the gamma distribution in our traf-fic model, which has been widely used in telecom performance studies [5], [16]. It has been proven that any experimental data can be fit by a mixture of gamma distributions [17]. Therefore, one may measure the traffic data, fit them into a mixture of gamma distributions, and then use our performance model for analysis.
The following output measures are studied: • Pl: the frame loss probability;
• Tw: the average waiting time between when the frame
arrives and when the frame is delivered.
For discussion purposes, we introduce a “secondary” input parameter U =Qi∈S1/Ii, which is the average number of
MSs that are scheduled to be awake at a beacon slot. The
relation between U and the “primary” input parameters can be expressed as follows: U = k i=1 1 Ii ⎡ ⎣ β 1−β τi+ τi ti ⎤ ⎦ (9) where β 1−β τi+ τi ti
is the average number of class i MSs stayed in the network. Note that the term (β/(1− β))τi is the time period for
gen-erating β/(1− β) frame arrivals, and the term τi is the time
period between when the last frame occurs and when the MS disconnects from the AP. The impacts of the input parameters on the output measures are elaborated upon as follows.
Effects ofU and γi: Fig. 4 plots Pland Twas functions of U
and γi. We assume that the interframe arrival time τifollows the
gamma distribution with mean 1/γi and variance Γi= 1/γi2,
and the inter-MS arrival time tifollows the gamma distribution
with mean 1/λiand variance Λi= 1/λ2i.
As U increases, more MSs will simultaneously wake up at a beacon slot, which increases the possibility of PS-poll contention. It also delays the time for retrieving the buffered frames, and, thus, more buffered frames are dropped due to buffer timeout (i.e., each frame is dropped after one listen interval). Consequently, Pland Twincrease as U increases [see
Fig. 4(a) and (b)].
As γi decreases, it is less likely that there is a frame to be
delivered to an MS when the MS wakes up. Therefore, the possibility of PS-poll contention decreases. Pland Twdecrease
as γidecreases.
Effects of Γi: Fig. 5 shows the effects of variance Γi for
interframe arrival time τi. Pland Twdecrease as Γiincreases.
When Γi is large, more small and large τi intervals are
ob-served, and either more frames for an MS arrive within a short period or no frame arrives within a long period [18]. These frames (that arrive within one listen interval) can be retrieved at the same time when the MS is granted to issue the PS-poll message. Therefore, more frames can be delivered when one PS-poll message is issued, and the waiting times for the delivered frames are reduced.
Comparison Between Scheme p and Scheme b: Fig. 4 shows
that when U or γi are small (i.e., the traffic is not heavy),
scheme p significantly outperforms scheme b. When U or γi
are large (i.e., the traffic is heavy), scheme p performs slightly better than scheme b. In Fig. 5, when Γi is small, scheme p
significantly outperforms scheme b. When Γi increases, the
advantage of scheme p becomes less significant. These results indicate that scheme p demonstrates excellent performance when the traffic load is not heavy.
V. CONCLUSION
In this paper, we have proposed a power conservation scheme for IEEE 802.11 wireless networks to schedule the awake
1926 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY 2009
Fig. 4. Effects of U and τi. (a) Plperformance. (b) Twperformance.
Fig. 5. Effects of the variance of the τidistribution. (a) Plperformance. (b) Twperformance.
times among the MSs that are served by an AP. We have formally proved that both the maximal number of MSs that are allocated in the same beacon slot and the number of the beacon slots that are allocated for the maximal number of MSs are minimal when the proposed scheme is executed. Simulation experiments have been developed to investigate the frame loss probability and frame waiting time measures. The results indi-cate that the proposed power conservation scheme outperforms an existing basic scheme and demonstrates excellent perfor-mance when the traffic load is not heavy.
APPENDIX
Fact 1: Let Sm be the set of MSs that are scheduled in
List[m], and let element l be the first vacant element in List[m]. Consider Ii for an MS Qi∈ Sm. If Ii≥ Ij for all
Qj∈ Sm, then l < Ii.
Proof: We prove by contradiction. Suppose that l≥ Ii. It
implies that all elements x (where 0≤ x < Ii) are allocated to
MSs Qj ∈ Sm. From (2), Ii= 2rIjfor r≥ 0. We have
Since Qj must wake up for every Ij beacon slot, all elements
x + nIj (for n≥ 0) are allocated to Qj. In other words,
el-ements x + n2rI
j are allocated to Qj. From (10), elements
x + nIi must be allocated to Qj. This implies that for all
elements y≥ Ii, y have been allocated to MSs Qj∈ S, which
contradicts the hypothesis that l≥ Ii.
Lemma 3: Suppose that a set Sm of MSs is scheduled in
List[m] using the Join procedure, whereQj∈Sm1/Ij< 1,
and element l is the first vacant element in List[m]. Consider an MS Qi∈ Smsuch that Ii≥ Ijfor all Qj ∈ Sm. Let element
x be the first allocated element for any Qj∈ Sm. Then, for all
n and n values
l + nIi= x + n Ij. (11)
In other words, elements l + nIi in List[m] are vacant, and,
therefore, Step J4 of the Join procedure is justified.
Proof: To prove that hypothesis (11) holds, it suffices to
prove that (l + nIi) mod Ij= (x + n Ij) mod Ij. From (2),
Ii = 2rIjfor r≥ 0, and
(l + nIi) mod Ij = (l + 2rnIj) mod Ij= l mod Ij. (12)
For all n ≥ 0, since elements x + n Ij are allocated to Qj,
from Fact 1, we have x≤ Ij, and
(x + n Ij) mod Ij = x. (13)
Equation (13) implies that if y mod Ij= x, then elements y are
allocated to Qj. Since element l is vacant in the scheduling list,
we must have
l mod Ij= x. (14)
From (12)–(14), we have (l + nIi) mod Ij= (x + n Ij) mod
Ij, and hypothesis (11) holds.
Lemma 4: Consider a set S of MSs and any integer r≥ −k
(where k≥ 0 is an integer). For all Qj∈ S, if 2−k ≤ 1/Ij ≤ 2r
andQj∈S1/Ij≥ 2r, then there exists a subset Sm⊂ S such
thatQj∈Sm1/Ij= 2r.
Proof: We prove by induction on r.
Basis: We pick r =−k. From the hypothesis, for any Qj ∈
S, we have 1/Ij = 2−k. (15) Let Sm={Qj}. From (15), Qj∈Sm1/Ij= 2 −k. The basis step holds.
Induction step: Assume that the hypothesis holds for r >−k. We prove that the hypothesis also holds for r + 1. If S contains an MS Qi, where 1/Ii = 2r+1, then let Sm={Qi}, and the
hypothesis holds. Otherwise (i.e., S does not contain any MS
Qi, where 1/Ii= 2r+1), for all Qj ∈ S
1/Ij ≤ 2rand
Qj∈S
1/Ij≥ 2r+1> 2r. (16)
From (16) and because the hypothesis holds for r, we can find a subset Sm1⊂ S such that
Qj∈Sm1
1/Ij= 2r. (17)
From (16) and (17), for all Ql∈ S − Sm1
1/Il≤ 2rand
Ql∈S−Sm1
1/Il≥ 2r+1− 2r= 2r. (18)
From (18) and the hypothesis, we can find another subset
Sm2⊂ S − Sm1such that
Ql∈Sm2
1/Il= 2r. (19)
Let Sm= Sm1∪ Sm2. From (17) and (19),
Qj∈Sm1/Ij=
2r+ 2r= 2r+1. Therefore, the hypothesis holds for all
r≥ −k.
Lemma 5: Consider a set Sn of n MSs. Label these MSs as
Q1, Q2, . . . , Qnsuch that
1/I1≥ 1/I2≥ · · · ≥ 1/In (20)
1≤j≤n
1/Ij≥ 1. (21)
Then, there exists a subset Sm=
1≤i≤m{Qi} ⊆ Snsuch that
1≤j≤m1/Ij = 1.
Proof: We prove by contradiction. Suppose that for any m, where 1≤ m ≤ n
1≤j≤m
1/Ij= 1. (22)
From (21) and (22), we have 1≤j≤n
1/Ij > 1. (23)
From (22) and (23), there exists m≤ n such that 1≤j≤m 1/Ij > 1 (24) 1≤j≤m−1 1/Ij < 1. (25)
From Corollary 1, there exists a subset S ⊂ Smsuch that
Qj∈S
1/Ij= 1. (26)
Equations (24) and (26) imply that there exists a subset S ⊂
Smsuch that ⎛ ⎝ 1≤j≤m 1/Ij ⎞ ⎠ − ⎛ ⎝ Ql∈Sm−S 1/Il ⎞ ⎠ = 1. (27)
1928 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 4, MAY 2009
From (25) and (27), we have 1≤j≤m−1 1/Ij = ⎛ ⎝ 1≤j≤m 1/Ij ⎞ ⎠ − 1/Im< ⎛ ⎝ 1≤j≤m 1/Ij ⎞ ⎠ − ⎛ ⎝ Ql∈Sm−S 1/Il ⎞ ⎠ ⇒ 1/Im> Ql∈Sm−S 1/Il
which contradicts (20) because 1/Imis minimal among those
for MSs in Sm.
Lemma 6: Consider a set Sn of n MSs labeled as
Q1, Q2, . . . , Qn, where
1/I1≥ 1/I2≥ · · · ≥ 1/In. (28)
If
1≤j≤n
1/Ij= 1 (29)
then all MSs in Sncan be scheduled in one scheduling list by
using the Join procedure.
Proof: For all MSs in Sn, the Join procedure schedules
these MSs into the scheduling list(s) in the nondecreasing listen interval order (i.e., Q1, Q2, . . . , Qn). We prove that Qm∈ Sn
can be scheduled in one scheduling list by induction on m, where 1≤ m ≤ n.
Basis: For m = 1, since there is no other MS that is sched-uled in the scheduling list, Qmcan be accommodated. The basis
step holds.
Induction step: Assume that the hypothesis holds for 1≤
m < n. We prove that the hypothesis also holds for m + 1.
From the induction hypothesis, Q1, Q2, . . ., and Qmhave been
scheduled in the scheduling list. From (29) and because m < n, we have 1≤i≤m 1/Ii< 1≤j≤n 1/Ij= 1. (30)
Since 1≤i≤m1/Ii< 1 [from (30)] and Im+1≥ Ij for
1≤ j ≤ m [from (28)], by Lemma 3, Qm+1can be scheduled
in the scheduling list. Therefore, the hypothesis holds for
1≤ m ≤ n.
Lemma 7: After exercising the Leave procedure, there is at
most one scheduling list containing vacant elements.
Proof: From Theorem 2, there is at most one scheduling
list containing vacant elements before the Leave procedure is executed. Consider the following two cases when the Leave procedure is executed.
Case 1) The leaving MS Qi is previously scheduled in the
scheduling list List[m] that contains vacant ele-ments. In this case, Step L4 of the Leave procedure is not executed, which will not affect other schedul-ing lists. Therefore, after the Leave procedure is executed, only List[m] contains vacant elements. Note that if Sm=∅ (see Step L2 of the Leave
procedure), then List[m] is removed, and thus, no scheduling list contains vacant elements.
Case 2) The leaving MS Qi is previously scheduled in
a fully allocated scheduling list List[m] (i.e.,
Qj∈Sm1/Ij = 1). If no scheduling list contains
vacant elements, Step L4 of the Leave procedure will not be executed. Similar to case 1, only List[m] contains vacant elements. Otherwise (i.e., there is a scheduling list List[u] containing vacant ele-ments), all the MSs that are scheduled in List[u] must be rescheduled (see Step L4 of the Leave procedure). At Step L5 of the Leave procedure, if Qj∈Sm∪Su1/Ij ≤ 1 (i.e.,
Qj∈S1/Ij < L), List[u] is removed, and thus, only List[m]
contains vacant elements. On the other hand, if
Qj∈Sm∪Su1/Ii> 1, then Theorem 2 guarantees
that, after Step L6 is executed, List[m] will be allocated fully, and only List[u] contains vacant elements.
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Chai-Hien Gan (M’05) received the B.S. degree in
computer science from Tamkang University, Taipei, Taiwan, in 1994 and the M.S. and Ph.D. degrees in computer science and information engineering from the National Taiwan University, Taipei, in 1996 and 2005, respectively.
From March 2005 to July 2007, he was a Re-search Assistant Professor with the Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan. Since July 2007, he has been a Researcher with the Information and Communica-tions Research Labs, Industrial Technology Research Institute, Hsinchu. His current research interests include wireless and mobile computing, personal communications services, IP multimedia subsystems, and wireless Internet.
Yi-Bing Lin (M’96–SM’96–F’04) received the
B.S. degree in electrical engineering from National Cheng Kung University, Tainan, Taiwan, in 1983 and the Ph.D. degree in computer science from the University of Washington, Seattle, in 1990.
He is currently the Dean and the Chair Professor with the College of Computer Science, National Chiao Tung University, Hsinchu, Taiwan. He is the author of the books Wireless and Mobile Network
Architecture (Wiley, 2001), Wireless and Mobile All-IP Networks (Wiley, 2005), and Charging for Mobile All-IP Telecommunications (Wiley, 2008).
Dr. Lin is a Senior Technical Editor of IEEE Network. He serves on the editorial boards of the IEEE TRANSACTIONS ON WIRELESS
COMMUNICATIONS and the IEEE TRANSACTIONS ON VEHICULAR
TECHNOLOGY. He has been the General or Program Chair for many prestigious conferences, including the 2002 Association for Computing Machinery (ACM) Mobile Computing and Networking Conference. He has been a Guest Editor for several first-class journals, including the IEEE TRANSACTIONS ON
COMPUTERS. He is listed in ISIHighlyCited.Com among the top 1% most cited computer science researchers. He has also been the recipient of numerous research awards, including the 2005 National Science Council Distinguished Researcher and the 2006 Academic Award of the Ministry of Education. He is a Fellow of the ACM, the American Association for the Advancement of Science, and the Institution of Engineering and Technology.
