Optimality of Frame Aggregation-Based
Power-Saving Scheduling Algorithm for
Broadband Wireless Networks
Wen-Jiunn Liu, Student Member, IEEE, Kai-Ten Feng, Senior Member, IEEE,
and Po-Hsuan Tseng, Member, IEEE
Abstract—The limitation on battery lifetime has been a critical
issue for the advancement of mobile computing. Different types of power-saving techniques have been proposed in various fields. In order to provide feasible energy-conserving mechanisms for the mobile subscriber stations (MSSs), three power-saving types have been proposed for the IEEE 802.16e broadband wireless net-works. However, these power-saving types are primarily targeting for the cases with a single connection between the base station (BS) and the MSS. With the existence of multiple connections, the power efficiency obtained by adopting the conventional scheduling algorithm can be severely degraded. In this paper, with the consideration of multiple connections and their quality-of-service (QoS) constraints, a frame aggregation-based power-saving scheduling (FAPS) algorithm is proposed to enhance the power efficiency by aggregating multiple under-utilized frames into fully-utilized ones. The optimality on the minimum number of listen frames in the proposed FAPS algorithm is also provided, and is further validated via the correctness proofs. Performance evaluation of proposed FAPS scheme is conducted and compared via simulations. Simulation results show that the power efficiency of FAPS algorithm outperforms the other existing protocols with tolerable frame delay.
Index Terms—Power-saving, scheduling, frame aggregation,
optimality, broadband wireless networks. I. INTRODUCTION
T
HE IEEE 802.16 standard for the broadband wirelessnetworks (BWNs) is designed to fulfill various demands for higher capacity, higher data rate, and advanced multimedia services [1]–[5]. In order to prolong the battery lifetime of the mobile subscriber stations (MSSs), the design of a feasible power-saving mechanism is considered a major issue in the IEEE 802.16e standard [6]–[9]. Three power-saving types are specified, i.e., types I, II, and III, in the IEEE 802.16e Manuscript received October 5, 2012; revised March 22 and September 21, 2013; accepted November 19, 2013. The associate editor coordinating the review of this paper and approving it for publication was G. Bianchi.
W.-J. Liu and K.-T. Feng (corresponding author) are with the Department of Electrical and Computer Engineering, National Chiao Tung University, Hsinchu, Taiwan (e-mail: jiunn.cm94g@nctu.edu.tw; ktfeng@mail.nctu.edu.tw).
P.-H. Tseng is with the Department of Electronic Engineering, National Taipei University of Technology, Taipei, Taiwan (e-mail: phtseng@ntut.edu.tw).
This work was in part funded by the Aiming for the Top University and Elite Research Center Development Plan, NSC 102-2221-E-009-018-MY3, the MediaTek research center at National Chiao Tung University, and the Telecommunication Laboratories at Chunghwa Telecom Co. Ltd, Taiwan. This material was also based on works by P.-H. Tseng supported in part by the NSC 102-2221-E-027-004.
Digital Object Identifier 10.1109/TW.2013.123013.121540
point-to-multipoint (PMP) mode such as to meet different demands of traffic between the base station (BS) and the MSSs. Two specific time intervals are defined as the sleep interval for saving energy and the listen interval for listening to the BS and conducting packet transmission. The power-saving class of type I defines the exponential-growing sleep intervals associated with fixed listen intervals. On the other hand, periodic occurrences of both the sleep and listen intervals are considered in type II. The power-saving class of type III consists of pre-determined longer sleep intervals without the existence of the listen period.
There are significant amount of research work [10]–[17] focusing on the energy-saving issues for battery-powered mobile devices. Different types of energy-efficient algorithms have been studied in [10] for generic central-controlled wire-less data networks. Based on the IEEE 802.11 power-saving mechanism [11], several energy conservation schemes have been proposed in both centralized [12]–[16] and decentralized [17] manners. However, these existing techniques are not intended to satisfy the requirements as defined in the IEEE 802.16e standard. In recent research studies, performance analysis of the different IEEE 802.16e power-saving types are investigated. Most of the work concentrates on constructing the analytical models for the power-saving class of type I [18]– [22]; while the enhanced model as proposed in [23] switches the power-saving class between types I and II according to the network traffic. A longest virtual burst first (LVBF) scheduling algorithm has been proposed in [24], which considers both the energy conservation and resource allocation between the BS and multiple MSSs. In [25], the optimal traffic indica-tion interval is proposed to replace the original negotiaindica-tion messages. A power-saving mechanism with periodic traffic indications is proposed to reduce the delay for state transition in order to enhance the power-saving efficiency. Nevertheless, these analytical results and scheduling schemes only assume a single connection between the BS and each MSS, i.e., a single connection is assigned to each MSS.
In view of the multi-connection scenarios, connection-oriented methods have been investigated in different stud-ies [26]–[31]. The maximum unavailability interval (MUI) scheme [26][27] is designed for the connections of power-saving class of type II in the IEEE 802.16e standard. Based on the Chinese remainder theorem, the proper start frame will be identified for each type II connection in order to reduce total 1536-1276/14$31.00 c 2014 IEEE
Sleep Interval Listen Interval
(c) Power Saving Class Type III Operation (b) Power Saving Class Type II Operation (a) Power Saving Class Type I Operation
Normal Mode Sleep Mode Normal Mode
Fig. 1. Three power-saving classes defined in the IEEE802.16e standard.
energy consumption. The extended maximizing unavailability interval (eMUI) [28] is further proposed to extend the MUI method for the mixture of power-saving classes of types I and II. By jointly considering traffic from multiple MSS, a Load-Based Power Saving (LBPS)[29] is proposed to design an adjustable sleep window size for each MSS based on measured traffic parameters. The periodic on-off scheme (PS) and the aperiodic on-off scheme (AS) are another two connection-oriented methods proposed in [30]. The objective of the PS scheme is to provide a scheduling algorithm with periodic sleep and listen intervals, which can elongate the total sleep intervals. However, since connections may have aperiodic traffic pattern, the PS scheme with periodic pattern is not ideal to accommodate those traffic in terms of power efficiency. For further enhancement, the AS scheme with aperiodic sleep and listen intervals is therefore suggested. According to the delay constraint of each connection, the connection-oriented AS scheme schedules each connection from the one with the tightest quality-of-service (QoS) requirement, i.e., the one with the smallest required delay time. With the QoS constraints, the AS scheme delays the packet as much as possible in order to acquire the opportunity of aggregating with other packets, which can improve the power efficiency.
In this paper, the packet-level frame aggregation-based power-saving scheduling (FAPS) algorithm is proposed with the considerations of multiple connections and their corre-sponding QoS constraints. Instead of adopting the connection-oriented methods, the proposed FAPS scheme considers power-saving scheduling at finer granularity from the packet-level perspective. The FAPS algorithm consists of two stages of processes, including the frame aggregation (FA) and the backward adjustment (BA) procedures. The proposed FA procedure is served as the default routine of maximizing the number of sleep frames by aggregating multiple under-utilized frames into fully-under-utilized ones; while the BA method is adopted when the FA process encounters procedure excep-tions. The optimality on the minimum number of listen frames produced by the proposed FAPS algorithm can be obtained and verified via the proof of correctness under the consideration of stepwise grant space set. Performance evaluation of the FAPS scheme is subsequently conducted and compared via simula-tions under different situasimula-tions. Simulation results show that better power efficiency with tolerable delay can be obtained by adopting the proposed FAPS algorithm compared to the baseline protocols.
The rest of the paper is organized as follows. The targeted problem and the corresponding system model are formulated in Section II. Section III explains the proposed FAPS algo-rithm. The optimality of the proposed FAPS protocol for the stepwise grant space set is further described and verified in Section IV; while performance evaluation of the FAPS scheme is conducted in Section V. Section VI draws the conclusions.
II. PROBLEMFORMULATION
A. IEEE 802.16e Sleep Mode Operation
According to the IEEE 802.16e specification [2], the sleep mode is defined to reduce the power consumption of an MSS. As a connection is established between the BS and the MSS, the MSS can be switched into the sleep mode if there is no packet to be transmitted or received during a certain time period. The time duration within the sleep mode is divided into cycles, where each cycle can contain both the sleep and the listen intervals. In the listen interval, the MSS can either transmit/receive data or listen to the medium access control (MAC) messages acquired from the BS. During the sleep interval, on the other hand, the MSS may turn into its sleep power state or associate with other neighbor BSs for handover scanning purpose. It is noticed that the sleep mode can be initiated by either the MSS or the BS. For the MSS-initiated process, the MSS sends a MOB_SLP-REQ massage to the BS for requesting the permission of entering the sleep mode. The BS will reply with a MOB_SLP-RES massage, which also includes the parameters of the connection type, the size of the sleep and listen intervals, and the starting time for the sleep mode.
As mentioned in Section I, three power-saving types are specified for the connections between the BS and the MSS in order to facilitate different characteristics of services. As shown in Fig. 1, the sleep mode of the MSS with the power-saving class of type I consists of exponential-growing sleep intervals and fixed-length listen intervals. Within the listen intervals, the MSS will listen for the MOB_TRF-IND massage obtained from the BS in order to determine if it should return back to the normal mode. In the case that the MSS is determined not to switch back to the normal mode, the length of the next sleep interval will be doubled until the pre-defined maximum sleep window size is reached. Based on the QoS requirements as defined in [1], this type is suitable for non-realtime traffic variable-rate (NRT-VR) connections and the best-effort (BE) services between the BS and the MSSs. The power-saving class of type II defines the repetitive occurrences of the sleep and listen intervals, where the sizes of both intervals are pre-determined fixed parameters. The MSS is allowed to transmit/receive data periodically within the listen intervals. It is noticed that this power-saving type is especially suitable for QoS-guaranteed services, e.g., the unsolicited grant service (UGS) and the realtime traffic variable-rate (RT-VR) connections. Furthermore, without the assignment of listen interval, the power-saving class of type III pre-specifies a long sleep interval for the MSS before it returns back to the normal mode. This type is suggested to be utilized for multicast connections and management operations.
T f
CID 1
CID 2
CID 3
MSS
Listen Frame Sleep Frame
Listen Interval Sleep Interval Listen Interval Sleep Interval Listen Interval Data Burst D1 D2 D3 Data Burst Data Burst
Fig. 2. Three connections with sleep mode operation between the BS and the MSS by adopting the conventional IEEE802.16e power-saving algorithm.
B. Inefficiency of IEEE 802.16e Sleep Mode Operation Since the power-saving types are defined based on a single connection between the BS and the MSS, the inefficiency for energy conservation from the allowable multiple connections has not been considered in the IEEE 802.16e specification. For example, three realtime UGS connections are considered in Fig. 2. The parameter Diis denoted as the delay constraint for the ith connection, which indicates that the data burst in this connection should be transmitted in the defined Di interval. The time interval Tf is defined as the duration of a frame as shown in Fig. 2. Noted that the power-saving class of type II is considered for all the three connections with connection ID (CID) 1, 2, and 3, which are characterized as follows: (a) CIDs 1 and 2 are with packet arrival period = 3· Tf, sleep interval = 2· Tf, listen interval = Tf, and D1= 3 · Tf; and CID 3 is with packet arrival period = 4· Tf, sleep interval = 3 · Tf, listen interval = Tf, and D3= 4 · Tf.
It can be observed that only one sleep frame per four frames will be obtained by directly combining the sleep intervals from these three connections, i.e., with the adoption of conventional IEEE 802.16e scheme as shown in Fig. 2. It can easily be extended that the sleep interval may become zero frame in certain multi-connection scenarios, which can severely degrade the efficiency for power conservation. Therefore, it is necessary to provide a feasible scheduling algorithm in order to reschedule the sleep intervals based on the combined effects from multiple connections.
C. Problem Formulation of Packet-based Power-saving Scheduling (PPS) Algorithm
Considering the aforementioned inefficiency problem of the conventional IEEE 802.16e sleep mode operation, a feasible scheduling algorithm should be proposed to enhance the efficiency of power scheduling under the scenario of multiple connections. Before diving into the design of scheduling al-gorithm, both the system model and the targeted problem will be described first. In order to model the combined effects of both the multiple connections and the QoS delay constraints, a packet-based modeling technique is suggested since all the connections can be individually partitioned into a series of data packets. The proposed grant space (GS) is utilized as the QoS data packet model to represent each QoS data burst in the multiple connections, which is defined as follows.
T f
Frame with Data Burst Frame without Data Burst
Data Burst
Di
Start Termination
Fig. 3. The grant space Gi(si, gi, ti) with the delay constraint Diand its start and termination frames.
Definition 1 (grant space). Given a frame si with a
pre-scheduled grant for a data burst, a grant space Gi(si, gi, ti)
is defined as the adjacent frames ranging from si to ti =
si+ Di− 1, where Di is the maximum QoS delay constraint
for this data burst. The frames siand tiare respectively called
the start and the termination for this grant space, and the grant frame gi is the frame that contains the data burst.
For practical applicability in the WiMAX system, the real-time connection with unsolicited grant service (UGS) can be directly modeled by the grant spaces since the data bursts can be exactly predicted and the delay constraint is also given in the parameter of maximum latency. Moreover, for the connections with realtime Polling Service (rtPS) and extended realtime Polling Service (ertPS), the mandatory parameters of maximum latency (Dps) and minimum reserved traffic rate (T Rmin) [1][2] can also be used to remodel them into pseudo UGS connections as follows: Separate each polling service into time frames with each frame of time length equal to Dps/2. The expected data bursts in each time frame will be Dps/2 multiplied by T Rmin. These bursts in a single time frame will at least have the delay constraint of Dps/2. Two pseudo UGS connections can therefore be formed by the odd-numbered time frame group and the even-numbered time frame group. Each pseudo UGS connection has the delay constraint of Dps/2 with predictable payload of (Dps/2)×T Rmin. With the help of pseudo UGS connections, the grant space model of both rtPS and ertPS connections can therefore be obtained, which validates model applicability for the WiMAX systems.
Fig. 3 illustrates the grant space Gi(si, gi, ti) with the delay constraint Di. The start and the termination of Gi(si, gi, ti) are also indicated at the two terminal frames respectively. The data burst should be scheduled within the grant frame gi, where
si ≤ gi ≤ ti, i.e., between the start and the termination frames. If the data burst is successfully scheduled within Gi(si, gi, ti), it is considered that the QoS requirement for this data burst can be satisfied. With the adoption of grant spaces, the entire system can therefore be modeled by the composition of grant spaces that are acquired from multiple connections. As shown in Fig. 4, there are nine connections with CIDs from 1 to 9 and each connection consists of multiple data bursts with their QoS delay constraints. Based on Definition 1, each data burst with its delay constraint can be modeled as a grant space. For example, the first data burst u of CID 1 with delay constraint 5· Tf is modeled as a grant space with
Du= 5 · Tf and its start and termination are at frame su= 3 and frame tu= 7 respectively. The second data burst d of CID
MSS Frame Frame Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 CID 1 CID 2 CID 3 CID 4 CID 5 CID 6
Frame with Data Burst
T f
Frame without Data Burst Data Burst
p s t CID 7 e CID 8 b CID 9 a m v d f r k u i c g q j n h w
Fig. 4. The entire system modeled by multiple grant spaces that are acquired from multiple connections.
8 with delay constraint 6 · Tf is represented as a grant space with Dd = 6 · Tf and its start and termination are at frame
sd = 10 and frame td= 15 respectively. After completing the representation of data bursts via the grant spaces, the listen and sleep frames of the MSS can therefore be determined based on the arrangement of each data burst.
For example, frame 1 of the MSS should be a listen frame since the first data packet a of CID 9 is scheduled within this frame; while frame 6 is a sleep frame since no packet is scheduled in this frame. As shown in Fig. 4, all the data packets from different connections of the MSS are categorized and result in the listen frames in grey color; while the sleep frames are identified with white color. Moreover, it is considered that the maximum allowable number of data bursts in each MSS frame will be limited, which is specified by the parameter of frame capacity Fmax. After the description of system model, the targeted packet-based power-saving scheduling (PPS) problem can be formulated as follows.
Problem 1 (packet-based power-saving scheduling).
Given a set G = Gi(si, gi, ti) ∀i of multiple grant spaces
and the frame capacity Fmax, how to arrange the data bursts
within the grant spaces in order to maximize the power efficiency of the MSS, i.e., to maximize the total number of sleep frames?
III. PROPOSEDFRAMEAGGREGATION-BASED POWER-SAVINGSCHEDULING(FAPS) ALGORITHM With the consideration of multiple connections and purely packet-level scheduling method, a frame aggregation-based power-saving scheduling (FAPS) algorithm is proposed to resolve the PPS problem as was formulated in Problem 1. The proposed FAPS algorithm can enhance the power efficiency of the MSS with proper arrangement of data bursts under the constraint of pre-specified QoS delay requirement. Two procedures are utilized in the FAPS scheme, including the frame aggregation (FA) and the backward adjustment (BA)
mechanisms. The FA procedure is served as the default routine to maximize the number of sleep frames in the proposed FAPS scheme; while the BA process is utilized if the FA procedure encounters exceptions. These two procedures will be described in the following subsections. The practicability of FAPS scheme will be addressed in the last subsection. A. Frame Aggregation (FA) Procedure
The major design concept of proposed FA procedure is to aggregate those under-utilized listen frames into fully-utilized ones, which can consequently provide more sleep frames. For example, it is assumed that the frame capacity, i.e., the maximum allowable number of data bursts in an MSS frame Fmax, is set to be 3. As shown in Fig. 4, the two under-utilized listen frames, i.e., frame 7 and frame 8, can be aggregated into frame 8 together since the three corresponding data bursts, i.e., bursts j and q in frame 7 and burst n in frame 8, can be scheduled at frame 8 without breaking the QoS delay constraints specified in their grant spaces. Additional sleep frame at frame 7 can therefore be acquired, which improves the power efficiency of the system. The proposed FA procedure is described as follows.
1) Forward Collection Mechanism: In order to acquire more sleep frames, a data burst λ should be delayed as much as possible to obtain more chance to aggregate with other data bursts. However, according to the QoS delay constraint, the data burst λ must be scheduled before the termination tλ specified within its grant space Gλ(sλ, gλ, tλ). Therefore, the first step of the proposed FA procedure is to delay all data bursts from left to right until some of the data bursts reach their delay constraints. For example, as shown in Fig. 4, all the data bursts within the setD = {a, c, e, g, i, m, p, s, u} defined by their grant spaces with yellow background color can be delayed to the maximum at frame 5. Note that data bursts m and u can further be delayed to frames 6 and 7 respectively based on the requirements from their grant spaces. This procedure can be analogues to the action of a windshield wiper as represented by the solid bar at frame 0 in Fig. 4. The windshield wiper will be stuck at the so-called stuck frame xG = 5 since both the data bursts p and s in this group can not be further proceeded based on their QoS constraints. Therefore, the unscheduled grant spaces in the set D will be stuck at frame xG, which can be formally defined as the stuck group as follows.
Definition 2 (stuck group). Given a set G = Gi(si, gi, ti)
∀i of unscheduled grant spaces, the stuck group of G is defined as the set
SG= {Gζ ∈ G | sζ ≤ xG}, (1)
where sζ is the start of grant space Gζ and xG= min(ti) ∀i
is called the stuck frame ofG with tidenoting the termination
of grant space Gi.
As was mentioned in the previous paragraph, even in the stuck groupSG, there may still exist data bursts that can be delayed beyond the stuck frame xG based on their grant spaces, e.g., the data bursts m and u as shown in Fig. 4 can be delayed to frames 6 and 7 respectively. In order to identify this type
Frame with Data Burst T f
Frame without Data Burst Data Burst p s e a m u i c g
Strictly Stuck Subgroup
Non-Strictly Stuck Subgroup
Stuck Group Stuck Frame S S G S G S S G x G
Fig. 5. The strictly stuck subgroupSSGand non-strictly stuck subgroupSSG.
of data bursts and their corresponding grant spaces, the stuck groupSGcan further be divided into two sorted subgroups as follows.
Definition 3 (strictly and non-strictly stuck subgroups).
Given the grant spaces Gζ(sζ, gζ, tζ) in a stuck group
SG that are sorted in ascending order according to the termination of each grant space. If two grant spaces possess the same termination, the start of the grant spaces is utilized as the parameter to perform sorting. The strictly and non-strictly stuck subgroups of SGare respectively defined as the set
SS
G= {Gζ ∈ SG| tζ = xG}, (2)
and the set
SS
G= SG− SSG, (3)
where xGis the stuck frame defined for this stuck groupSG. 2) Backward Push Mechanism: As shown in Fig. 5, the stuck group SG obtained from the data burst set D = {a, c, e, g, i, m, p, s, u} in Fig. 4 is sorted and separated into the strictly stuck subgroupSS
Gand the non-strictly stuck subgroup
SS
G. It can be observed that all data bursts inD will be stuck at the stuck frame xG by adopting the forward collection mechanism. However, it is impossible to schedule all the data bursts into frame xG if the maximum allowable number of data bursts Fmax of an MSS frame is less than the size of
SG. Therefore, the constraint originated from the parameter Fmaxwill be considered in the second step of the proposed FA procedure. Moreover, the scheduling of data bursts specified in the grant spaces ofSS
Gshould be performed prior to that of
SS
G since all the data bursts inSSG must be scheduled before the stuck frame xG.
Frame with Data Burst
Tf
Frame without Data Burst Data Burst
p s e a i c g
Strictly Stuck Subgroup
Stuck Frame SS G xG Fmax = 3 Ns = 7 Pn = 3 Partition 1 Partition 2 Partition 3 p s e a i c g
Strictly Stuck SubgroupSS
G
Partition 1
Partition 2
Partition 3
Fig. 6. The arrangement of data bursts in the strictly stuck subgroupSSG. The arrangement of data bursts in the strictly stuck subgroup
SS
G is explained as follows. Let Ns be the number of grant spaces in SS
G and Pn = Ns/Fmax be the number of
partitions utilized to separateSS
Gcounting from the sorted first grant space. Note that each partition inSS
Gwill have at most Fmax data bursts in order to meet the requirement of frame capacity. Finally, referenced from the stuck frame xG, the data bursts in each partition i will be moved back Pn− i frames gradually if it is allowed based on their corresponding grant space constraints. If certain data burst can not be moved all the way to the designated frame, it will stay at the frame unable to proceed back. If no exception occurs, the arrangement of data bursts specified inSS
Gis completed.
For example, as shown in Fig. 6, the frame capacity is Fmax = 3, the number of grant spaces in SSG is Ns = 7, and the number of partitions is Pn = Ns/Fmax = 3. The data bursts in partition 1 should be moved back Pn− 1 = 2 frames. However, the data burst c will only be moved back one frame since it is limited by the start frame sc of its grant space Gc(sc, gc, tc). In partition 2, same situation can be observed at the data burst p which is constrained by its grant space Gp(sp, gp, tp). Finally, the data bursts a and e will be scheduled in the same frame. The bursts c, g, and i can also be aggregated; while the stuck frame can accommodate the remaining data bursts p and s. It can be observed that all data bursts inSS
Gis properly scheduled without breaking their QoS delay constraints and the maximum allowable number of data bursts of an MSS frame.
3) Packet-Padding for Backward Push Mechanism: On the other hand, the scheduling of non-strictly stuck subgroup
SS
G will be considered according to the packet-padding for backward push mechanism. In order to improve the power efficiency, the main concept is to pad the data bursts ofSSGinto those under-utilized listen frame after the arrangement ofSS
G. Therefore, the data bursts specified in SS
G can be scheduled within some of the listen frames that are under-utilized in
SS
G, which is explained as follows. Acquiring a under-utilized listen frame δ starting from the left-most frame, the design procedure is to find a proper number of grant spaces in SS
G whose data bursts can be scheduled in the frame δ. Note that the acquisition order of the grant spaces is the same as the order defined in Definition 3.
Frame with Data Burst T f
Frame without Data Burst Data Burst p s e a m u i c g
Strictly Stuck Subgroup
Non-Strictly Stuck Subgroup
Stuck Group Stuck Frame S S G S G S S G x G
Fig. 7. The complete arrangement of the strictly stuck subgroupSSGand the non-strictly stuck subgroupSSG.
As shown in Fig. 7, the first under-utilized listen frame is the frame with data bursts a and e, and the second one is the frame with data bursts p and s. Based on the aforementioned technique, the data burst m will be arranged together with the data bursts a and e; while the burst u will be accommodated with bursts p and s, leading to the fully-utilized listen frames. Finally, if there still remain data bursts specified in the grant spaces ofSS
G, these data bursts should be scheduled with the other unscheduled grant spaces as the new input grant spaces for the next round of the FA procedure. It can be observed that the set of data burstD = {a, c, e, g, i, m, p, s, u} in Fig. 4 can be successfully scheduled within 3 listen frames, which is smaller than the original 5 listen frames. Furthermore, Fig. 8 shows the scheduling result of the complete power-saving system exemplified in Fig. 4 after executing five rounds of the FA procedure. It can be seen that the number of required listen frames is decreased from the original 12 to 7 listen frames by adopting the FA algorithm, which effectively enhances the power efficiency of the system.
4) Complexity Discussion for FA Procedure: Let ngs be the number of all unscheduled GSs. In the forward collection mechanism, the stuck group can be obtained by (a) finding the smallest termination, (b) grouping the suitable GSs, and then (c) sorting these GSs. The time complexity required for steps (a), (b), and (c) are respectively O(ngs), O(ngs), and O(ngs· log(ngs)). Let nSgsbe the GS number of obtained strictly stuck subgroup and Dmax the maximum delay for all connections. In the backward push mechanism, the time complexity will be O(nSgs· Dmax) since all GSs in SSGshould do schedulability test backwardly in its delay constraint. The same reason can be applied to the packet padding, i.e., the last step of FA procedure. The time complexity will be O(nS
gs·Dmax), where
nS
gsis the GS number of obtained non-strictly stuck subgroup.
MSS Frame Frame Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 CID 1 CID 2 CID 3 CID 4 CID 5 CID 6
Frame with Data Burst
T f
Frame without Data Burst Data Burst
p s t CID 7 CID 8 b CID 9 v d f k u i c g h w r q j n e a m
Fig. 8. The scheduling result of the proposed FA procedure for the example system in Fig. 4 with multiple connections.
B. Backward Adjustment (BA) Procedure
As shown in Fig. 9, there are eight data bursts from a to h with their own grant spaces defined from their individual connections. The maximum allowable number of data bursts Fmaxof an MSS frame is assumed to be 2. In the first round of the FA procedure, data bursts a and b can be intuitively and completely scheduled. Note that each round of the FA procedure is conducted according to the stuck group as defined in Definition 2. Data bursts c and d are also successfully aggregated in the second round without exceptions. In the third round of execution, there are four unscheduled grant spaces for data bursts e, f , g, and h, which are separated into the strictly and non-strictly stuck subgroupsSS
G and SSG. Based on the FA procedure, the grant spaces ofSS
Gwill be divided into two partitions, including partition 1 consisting of data bursts e and f and partition 2 containing g as shown in Fig. 9.
In partition 1, each data burst should be moved backward Pn− 1 = Ns/Fmax − 1 = 1 frame. However, it is unable to move these bursts backward since frame 5 is already full of data bursts, i.e., data bursts c and d. Data bursts e and f will still be placed at frame 6. On the other hand, data burst g specified in partition 2 will also be scheduled at frame 6. Therefore, the exception of proposed FA procedure
occurs since frame 6 with frame capacity Fmax = 2 is
utilized to allocate data bursts e and f and frame 6 is the only choice for partition 2. It will violate the frame capacity constraint if no proper scheduling adjustment is performed. The proposed BA procedure is therefore utilized to solve the scheduling exception problems that are encountered by the aforementioned FA process. The BA procedure is described in the following two steps.
1) Recursive Backward Movement: The recursive backward movement of the proposed BA procedure can be found in Algorithm 1 and is described as follows. Given a data burst λ and the targeted frame θ, a function makespace(λ, θ) is utilized to allocate a space for λ at frame θ by recursively moving the data burst λ backward. If λ can be scheduled in
MSS Frame Frame Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 CID 1 CID 2 CID 3 CID 4 CID 5 CID 6 CID 7 CID 8 h b d c a g
Frame Aggregation Procedure First Round (Completed)
Frame Aggregation Procedure Second Round (Completed)
Frame Aggregation Procedure Third Round (On-Going)
Frame with Data Burst Tf
Frame without Data Burst Data Burst
Scheduling Exception Partition 1 Partition 2 Fmax = 2 Ns = 3 Pn = 2 makespace(g, 6) e f e f makespace(h, 6) Packet Padding S S G S S G Stuck Frame x G
Fig. 9. The proposed backward adjustment procedure for the scheduling exceptions of the FA procedure.
θ, the function makespace will return a true value which consequently solves the exception problem resulting from the FA procedure. If λ can not be scheduled into frame θ, another data burst ξ in θ whose grant space Gξ(sξ, gξ, tξ) has the smallest start frame will be selected to perform the recursive backward movement. Subsequently, based on the same function makespace, the process will be continued by backward searching the frame π ranging from frame (θ− 1) to the start frame sξ of data burst ξ. If a frame π is found to be non-saturated, data burst ξ will be scheduled into this corresponding frame π. Finally, the unscheduled data burst λ can be allocated into frame θ, which completes the recursive backward movement of the proposed BA procedure for scheduling the data burst λ.
As shown in Fig. 9, data burst g in partition 2 will encounter the exception problem while adopting the third round of the FA procedure. Therefore, based on the recursive back-ward movement of the proposed BA procedure, the function makespace(λ, θ) will be invoked, where λ = g and θ = 6. For those data bursts e and f scheduled in frame 6, the grant space Ge(se, ge, te) of data burst e has the smallest start frame, i.e., se = 2. Consequently, data burst e will be selected as the burst that should be moved backward. Subsequently, the function makespace(λ, π) will be re-conducted for the new inputs λ = e and π = θ − 1, . . . , se, i.e., π = 5, 4, 3, and 2 for each time. As the first function makespace(e, 5) for scheduling data burst e is invoked, the next selected data burst which may be moved back is burst c in frame 5 since it has the smallest start frame, i.e., frame 4. Recursively, when makespace(c, 4) for arranging data burst c is initiated, burst ais also chosen as the candidate to move backward. However, burst a can not be moved back since it is at the start frame 4 of the grant space Ga(sa, ga, ta). Since burst a can not be moved, data burst c can only be scheduled at its current frame 5. Moreover, since it is not possible to move burst c backward, the data burst e should not be scheduled at the frame 5 which indicates that makespace(e, 5) can not allocate a space for e at the frame 5.
The second function makespace(e, 4) for scheduling data burst e will continually be invoked to verify if burst e can be scheduled at frame 4. For the same reason as explained in the previous paragraph, burst e also can not be arranged at frame 4. Finally, data burst e will be scheduled at frame 3 since the third function makespace(e, 3) for scheduling burst ewill return a true value, representing the available vacancy of frame 3. As data burst e is placed into frame 3, there exists a vacancy in frame 6 for data burst g to be scheduled which completes the recursive backward movement.
Algorithm 1:Recursive Backward Movement: makespace(λ, θ) Input: λ: data burst, θ: targeted frame
Output: true or false: can λ be scheduled in θ ? 1 begin
2 if λ can be placed into frame θ then 3 return true;
4 else
5 let ξ be the data burst in frame θ whose grant space
Gξ(sξ, gξ, tξ) has the smallest start frame;
6 for π= θ − 1 tosξ do 7 if makespace(ξ, π) then
8 put data burst ξ into frame π;
9 return true; 10 end 11 end 12 return false; 13 end 14 end
2) Packet-Padding for Recursive Backward Movement: After the execution of recursive backward movement, there may still exist under-utilized frames that are capable to allocate additional data bursts. For example, in the scheduling of data burst g as shown in Fig. 9, data burst e will be pushed backward to frame 3 based on the makespace algorithm, which makes the frame 3 as an under-utilized frame. In order to enhance the power efficiency, some data bursts should be aggregated into the frame 3. It can also be observed that the conduct of makespace(h, 6) can achieve this goal. The data burst f can be moved to the frame 3 due to the makespace algorithm. Based on this observation, the packet padding mechanism for the recursive backward movement can be designed to adopt the makespace with function inputs as the data burst in the non-strictly stuck subgroupSS
G and the stuck frame xG. This packet padding process is continuously conducted until either a f alse return value is obtained or before an additional listen frame is introduced. In other words, this packet padding mechanism will not lead to the creation of new listen frames. As shown in Fig. 9, based on the packet padding mechanism, data burst h in SS
G can be scheduled
at the stuck frame xG = 6 by conducting the function
makespace(h, 6) which effectively makes frame 6 to be a fully-utilized frame. The original under-utilized frame 3 also reaches its maximum capacity. With the recursive backward movement and the packet padding scheme of the proposed BA procedure, the third round of proposed FA procedure can therefore be completed with enhanced power efficiency.
3) Complexity Discussion for BA Procedure: The re-cursive backward movement is based on the function makespace(λ, θ) in Algorithm 1. It can be observed that
θ will lie in the range of grant space Gξ(sξ, gξ, tξ), i.e.,
sξ ≤ θ ≤ tξ. The iteration number of the for loop with parameter π will be accordingly less than Dmax times. In the worst case, the makespace(λ, θ) function will be exe-cuted O((Dmax)n
S
gs) times since there are at most nS gs grant spaces in the strictly stuck subgroup which need to run the BA procedure. Finally, in the packet padding for recursive backward movement, there are at most nSgs grant spaces in the non-strictly stuck subgroup which have the chance to be padded into under-utilized frames. Similar time complexity O((Dmax)n
S
gs) can also be acquired.
C. Practicability of FAPS Scheme
The benefits and usage scenarios of proposed FAPS scheme is summarized as follows. In this paper, a complicated schedul-ing problem has been decomposed into several parts in ac-cordance with the procedures in IEEE 802.16 networks. The original scheduling mechanism includes three different stages as follows:
(S1) Admission control to map each application to its connec-tion ID;
(S2) Request bandwidth and design sleep pattern from packet scheduling for each connection; and
(S3) System scheduling at BS side.
With proper admission control to decide the grant space for each connection, the proposed FAPS scheme mainly focuses on the second item (S2) for packet scheduling. With this prob-lem decomposition, the proposed FAPS scheme is proven to be optimal in the sub-problem (S2). The MSS can apply the FAPS algorithm for concurrent CIDs to achieve power conservation. A practical situation for higher number of concurrent CIDs will be the inter-networking scenario for WiFi and WiMAX networks. The WiFi-AP conserves energy by aggregating its traffic flows from WiFi users to request an optimal sleep pattern to WiMAX network.
Considering the DL scenario, the proposed FAPS scheme can also be implemented at BS side providing that the entire scheduling method is decomposed as the same procedure above. The FAPS scheme can aggregate multiple connections from multiple MSSs for achieving an optimal power-saving pattern to conserve the BS’s power, which fulfills the require-ments for green communications. Furthermore, without addi-tional modifications, the FAPS algorithm can also be utilized for (S1) to aggregate multiple applications into one connection ID. It is intuitive that we can implement this stage as follows: (a) virtually map one service flow from the application to one CID for parameter calculation only, i.e., to form a grant space for each service flow without assigning real CID; and (b) aggregate different grant spaces into one CID using the FAPS scheme. This is considered a useful scenario since there are more and more applications with social networking (e.g., Facebook) or real-time information (e.g., Weather) which would periodically update its latest information. In summary, a feasible power-saving scheme (e.g., the proposed FAPS algorithm) is required to deal with the three cases as stated above, where the total number of multiple connections even up to 20 might still be reasonable. Consequently, the FAPS
Frame with Data Burst
T f
Frame without Data Burst Data Burst m d c p q g f Strictly Stuck Subgroup Non-Strictly Stuck Subgroup Stuck Group Stuck Frame S S G S G S S G x G b a e h First Round
Other Grant Spaces r u Fmax = 3 Ns = 10 Pn = 4 n
Stepwise Grant Space Set G
No-Follower Frame WA WB Pn B A = G- (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)
Fig. 10. The exemplified packet arrangement in the first round of the proposed backward push mechanism.
scheme can provide the optimal sleep pattern to these usage cases, which can be practically implemented.
IV. OPTIMALITY OFPROPOSEDFAPS ALGORITHM
In this section, the optimality for the proposed FAPS algorithm to result in the minimum number of listen frames will be proven under the input scenario with stepwise grant space set. The stepwise grant space set is described as follows.
Definition 4 (stepwise grant space set). A grant space set G = Gi(si, gi, ti) is called stepwise if the following two
properties can be satisfied:
1) The order for each grant space inG is identified in the ascending manner according to the value of termination, and is further sorted by the start if same termination is encountered.
2) For each pair of consecutive grant spaces Gi(si, gi, ti)
and Gj(sj, gj, tj) in G, the start and the termination of
the former are respectively less than or equal to those of the latter, i.e., si≤ sj and ti≤ tj.
As shown in Fig. 10, all grant spaces in the stepwise grant space set G are listed based on the criterions in Definition 4, and the sequential numbers are also specified at the left. If there exists the stepwise grant space set G, minimum number of listen frames can be obtained by adopting the proposed FAPS algorithm that provides the optimal packet arrangement under the frame capacity constraint Fmax. The flow of correctness proof for the FAPS algorithm is depicted in Fig. 11, and is explained as follows.
In the first round, based on the given stepwise grant space setG, the proposed FAPS algorithm will perform the forward collection mechanism to construct the stuck group and further
First Round for
Backward Push Mechanism Complete
No Feasible Algorithm
Packet Padding for
First Round Backward Push Mechanism Grant Space Removal Due to
Observed No-Follower Frames
Second Round for Backward Push Mechanism
Packet Padding for
Second Round Backward Push Mechanism Start of Backward
Adjustment Mechanism
Grant Space Removal for All Scheduled Data Bursts
Recursive Backward Movement
Packet Padding for Recursive Backward Movement
SS G No Grant Space in No Grant Space Scheduling Failure Lemma 1 Lemma 2, Lemma 3 Lemma 4 Lemma 5
Lemma 6, Lemma 7, Lemma 8
Existence of No-Follower Frames
SS G
Still Grant Spaces in
SS G
Still Grant Spaces in
Fig. 11. The flow of the correctness proof for the proposed FAPS algorithm.
determine both the strictly and non-strictly stuck subgroups. Subsequently, the backward push mechanism will be executed to properly rearrange the data bursts according to the frame capacity Fmax = 3 data bursts. As illustrated in Fig. 10, the stepwise grant space set G consists of the grant spaces of data bursts {a, b, . . . r, u, . . .}. Based on the backward push mechanism, the first partition {a, b, c} should be scheduled at the frame xG− 3; while the second partition {d, e, f} will be placed at frame xG− 2. However, due to the grant space constraint on the start frame sf, data burst f will be scheduled at frame xG− 1. The remaining data bursts in the strictly stuck subgroupSS
Gare arranged as shown in Fig. 10. In order to facilitate the correctness proof, it is required to define the no-follower frame as follows.
Definition 5 (no-follower frame). With the adoption of the
proposed FAPS algorithm on a stepwise grant space set G, some grant spaces can be scheduled together in a frame θn
and the last grant space to be scheduled in θn is specified as
Gξ(sξ, gξ, tξ). The frame θnis defined as a no-follower frame
if there does not exist any other data burst after the grant space Gξ(sξ, gξ, tξ) that has the chance to be scheduled in
this frame.
As shown in Fig. 10, due to the properties of stepwise grant space set in Definition 4, the frame (xG− 2) is a no-follower frame since there is no other data burst after the grant space Ge(se, ge, te) which can be scheduled in this frame.
Fact 1. Given the frame capacity Fmaxand a setG with MG
grant spaces, all the start and the termination frames of the grant spaces in G are bounded in the frame range [θs, θt]
with width W = θt− θs+ 1. If the number MGcan fulfill the
equality ofMG/Fmax = W , all the frames ranging from θs
to θtmust be necessary listen frames. On the other hand, if the
number MG results in the inequality as MG/Fmax > W ,
there does not exist any scheduling algorithm that can arrange
these data bursts based on the QoS constraints in the grant space setG.
Fact 2. Assuming that at least UL necessary listen frames
exist before the end of an arbitrary frame θn in [θs, θt], the
entire grant space setG is partitioned into two disjoint sets, including the removable set G− and the remaining set Gr.
The set G− can be scheduled within the minimum number of listen frames and can be removed without interfering the scheduling ofGrif the following two conditions are satisfied:
(a) the set G− can be feasibly scheduled within UL frames
before the end of frame θn; and (b) it is not possible to place
the data bursts in Gr into those UL necessary listen frames.
Lemma 1. In the first round of the backward push
mecha-nism, given the first observed no-follower frame θn, the set
G−consisting of all grant spaces inG whose data bursts are
scheduled before the end of frame θn can be removed without
interfering the scheduling of the remaining grant spacesGr.
The set G− is considered completely scheduled within the minimum number of listen frames.
Proof: The strictly stuck subgroup is denoted as SS G with Ns grant spaces after the first round of the backward push mechanism. Based on the backward push mechanism, the number of partitions is obtained as
Pn= Ns/Fmax. (4)
For example, as shown in Fig. 10, the values can be obtained as Ns = 10, Fmax = 3, and Pn = 4. The listen frames for accommodating all the data bursts specified in the grant spaces ofSS
Gwill be within the range [xG−Pn+1, xG] since the maximum backward movement in the proposed backward push mechanism is Pn − 1 frames. Moreover, since the first no-follower frame θncan be observed during the first round of the backward push mechanism, the frame θn must be within the same listen frame range, i.e., θn∈ [xG− Pn+ 1, xG].
Based on the last grant space Gξ(sξ, gξ, tξ) scheduled in the no-follower frame θn, SSG can further be divided into the following two parts: (a) the set A consisting of all the grant spaces before and including Gξ(sξ, gξ, tξ); and (b) the other set B containing the remaining grant spaces after Gξ(sξ, gξ, tξ). As shown in Fig. 10, for example, the last grant space in the no-follower frame θn is Ge(se, ge, te). The data bursts of the set A will be {a, b, c, d, e}; while those of the set B will be {f, g, h, m, n}. Given the size MA of the set
A and based on the backward push mechanism, set A must
be accommodated in the range [xG− Pn+ 1, θn] with width
WA, which can be determined as
WA= MA/Fmax. (5)
The grant spaces in the set B are constrained in the range [θn+ 1, xG] due to the stuck frame and the no-follower frame definitions in Definitions 2 and 5, respectively. The width of the range [θn+ 1, xG] can be denoted and determined as
WB= Pn− WA. (6)
Furthermore, the size MBof the setB is obtained as
MB= Ns− MA. (7)
Therefore, based on (4) to (7), the relationship between MB and WB can be derived as
MB/Fmax = (Ns− MA)/Fmax
≥ Ns/Fmax − MA/Fmax = Pn− WA= WB. (8) Based on (8) and Fact 1, since setB is bounded in the range [θn+ 1, xG] with width WB, the frames in the range [θn+ 1, xG] must be necessary listen frames if a feasible scheduling method exists, i.e., WB = MB/Fmax. Furthermore, there are at least Pn = Ns/Fmax listen frames required by all
Nsdata bursts in SSG before the end of the stuck frame xG. Therefore, there must be at least UL= Pn−WB= WAlisten frames before the end of the no-follower frame θn since those
WB frames in the range [θn+ 1, xG] have been proven to be necessary listen frames.
The setA can fulfill with the requirement of set G− since all data bursts of A are scheduled before the end of θn and no more grant space can be arranged within that range due to Definition 5. Furthermore, all data bursts in A = G−
can be scheduled before the end of θn with WA = UL
frames. Therefore, based on Fact 2, the set G− can be removed without interfering the scheduling of the remaining grant spaces. The scheduling of setG− is completed with the minimum number of listen frames. It completes the proof.
Based on each observed no-follower frame, the correspond-ing grant space set G− can be repeatedly removed from the stepwise grant space set G according to Lemma 1. All data bursts in the removed grant spaces are properly scheduled within the minimum number of listen frames.
Lemma 2. After completing the potential grant space
re-moval in SS
G based on Lemma 1, the remaining frames with data bursts are all fully-utilized except for the stuck frame which can be an under-utilized frame.
Proof: Based on the backward push mechanism, all grant spaces will be partitioned by the frame capacity Fmax. All partitions are full of Fmax grant spaces except for the last partition that can have the number of grant spaces less than Fmax. Furthermore, all data bursts in the same partition will have the same pre-determined target frame, i.e., the ith partition will be moved back Pn− i frames from the stuck frame xG. If all data bursts in the same partition can reach the target frame, this target frame must be fully-utilized. On the other hand, if the data burst ξ can not be scheduled in its target frame θξt, two possible situations can occur as follows. The first situation is that there does not exist additional vacancy in frame θt
ξ which denotes full utilization of frame θ t ξ. The other situation happens when the start frame index sξ of data burst ξ in the grant space Gξ(sξ, gξ, tξ) is larger than frame
θt
ξ, i.e., sξ > θξt. For example, as shown in Fig. 10, the data bursts in the first partition{a, b, c} will be scheduled in frame (xG− 3) and occupy the entire frame. On the other hand, the second partition{d, e, f} should all be scheduled in the frame xG− 2 as planned. However, due to the start frame constraint of the grant space, data burst f can not be scheduled to reach frame xG− 2.
As the reason stated for the second situation, the target frame θξt can become an under-utilized frame. Nevertheless, this frame must be a no-follower frame according to Defi-nition 5 since there is no other data burst after grant space Gξ(sξ, gξ, tξ) that has the chance to be scheduled in frame
θt
ξ. For example, as illustrated in Fig. 10, target frame θ t
ξ of
data burst ξ = f corresponds to frame xG− 2, and there is no data burst after burst f that can be placed at this frame xG−2. Based on Lemma 1, all the grant spaces with the grant frame gi≤ θξtcan be removed. Therefore, after completing all possible grant space removals, the remaining frames with data bursts are fully-utilized except for the potential under-utilized stuck frame. It completes the proof.
After the execution of the backward push mechanism, the packet padding in the proposed FA mechanism will be conducted. The data bursts specified in the non-strictly stuck subgroupSS
Gwill be placed in the vacant locations within the under-utilized frame. As shown in Fig. 10, data burst p will be scheduled at the same frame along with the data bursts m and n.
Lemma 3. If all data bursts specified in the non-strictly stuck
subgroupSSGare scheduled within the under-utilized frames in the first round of backward push mechanism, the stuck group
SGcan be removed without interfering the scheduling of the remaining grant spaces. All the removed grant spaces are considered completely scheduled within the minimum number of listen frames.
Proof: Based on Lemma 1, all the grant spaces in
G whose data bursts are scheduled before the end of the
no-follower frame can be removed without interfering the scheduling of the remaining grant spaces. This leads to the size reduction for the strictly stuck subgroup SS
G. Based on Lemma 2, the remaining frames with data bursts will all be fully-utilized except for the stuck frame xG. In other words, the stuck frame xGis the only frame which can accommodate the data bursts specified in the non-strictly stuck subgroup
SS
G. If all data bursts specified in SSG are scheduled within the stuck frame xG, the size Mn of the grant spaces in the stuck groupSGand the total number of listen frames Ln will possess the relationship as Ln = Mn/Fmax. Note that Ln is also denoted as the minimum number of required frames before the end of the stuck frame xG. Moreover, due to the stuck group definition in Definition 2, there is no other grant space in (G − SG) which can be scheduled at any frame θξ ≤ xG. The listen frames for accommodating the stuck group SG will not be further updated or influenced by the scheduling of (G − SG). Therefore, based on Fact 2, the stuck group SG = G− can be removed without interfering the scheduling of the remaining grant spaces. All the removed grant spaces are considered completely scheduled with the minimum number of listen frames. It completes the proof.
If the stuck group SG can be removed based on Lemma 3, the proposed FA mechanism will be re-conducted as the first round with the new stepwise grant space set (G − SG). On the other hand, after the execution of the packet padding mechanism, if there still exist some grant spaces in SSG, the second round of backward push mechanism will be performed. The new stuck frame xG will therefore be updated.
Lemma 4. In the second round of the backward push
mech-anism, given the first observed no-follower frame θn, the set
G− consisting of all the grant spaces inG whose data bursts are scheduled before the end of frame θn can be removed
without interfering the scheduling of the remaining grant spaces. The setG−is considered completely scheduled within the minimum number of listen frames.
Proof: In the second round of the backward push mech-anism, based on the last grant space Gξ(sξ, gξ, tξ) that is scheduled in θn, the current strictly stuck subgroup can be divided into the grant space setsA and B. The set A contains the grant space Gξ(sξ, gξ, tξ) and those before Gξ(sξ, gξ, tξ); while the remaining grant spaces are in set B. Moreover, the grant space setC is also introduced to include the previously scheduled and not-yet removed grant spaces. Note that set C is perfectly scheduled within LCfully-utilized frames, i.e., the following relationship will hold as MC= LC· Fmax where
MCis the number of grant spaces in setC. The listen frames that contain set C, i.e., the previous scheduling results, may or may not influence the second round of the backward push mechanism. Therefore, the integer variable δ is introduced to represent the number of frames blocked by set C. The relationship in (6) will therefore be updated as
Pn = δ + WA+ WB. (9)
With the new Pn in (9), the inequality specified in (8) will still be satisfied since WB+ δ ≥ WB. Based on (8) and Fact 1, since setB is bounded in the range [θn+1, xG] with width
WB, the frames in the range [θn+ 1, xG] must be necessary listen frames if there exists a feasible scheduling method, i.e., WB = MB/Fmax. In other words, the variable δ must be zero in order to have a feasible solution.
Note that there are at least V =(MC+MA+MB)/Fmax listen frames before the end of the stuck frame xGin order to contain sets A, B and C. Moreover, since the WB frames in the range [θn + 1, xG] have been proven as necessary
listen frames, there must be at least UL = V − WB =
LC+ Pn − WB = LC+ WA listen frames before the end of the no-follower frame θn. It can be observed that all data bursts in sets A and C can be scheduled before the end of the no-follower frame θnwith UL= LC+ WAlisten frames. Furthermore, based on Definition 5, there will be no other grant space after the last grant space Gξ(sξ, gξ, tξ) scheduled in θn which can affect the scheduling of sets A and C. Therefore, based on Fact 2, the set G− = A + C can be removed without interfering the scheduling of the remaining grant spaces. The scheduling of G− is completed with the minimum number of listen frames. It completes the proof.
If there exists a no-follower frame in the second round of the backward push mechanism, the corresponding grant space set G− can be removed from the stepwise grant space set
G based on Lemma 4. All data bursts in the removed grant
spaces are properly scheduled in the minimum number of listen frames. The remaining grant spaces will result in a new stepwise grant space set which initiates a first round execution of the backward push mechanism. On the other hand, if no-follower frames are not encountered, either the packet padding for the proposed FA mechanism (Lemma 5) or the proposed BA mechanism (Lemma 6) will be executed. The first case that conducts packet padding is considered as follows.
Lemma 5. If all data bursts specified in the non-strictly stuck
subgroup SSG are scheduled within the under-utilized frames of the second round backward push mechanism, the stuck group SG and all previously scheduled grant spaces can be removed without interfering the scheduling of the remaining grant spaces. All the removed grant spaces are considered completely scheduled within the minimum number of listen frames.
Proof: It can be observed that all listen frames occupied by the data bursts of SG and all previously scheduled grant spaces are fully-utilized except for the possible under-utilized stuck frame. Since all data bursts ofSS
Gcan be placed within the stuck frame, the stuck groupSGand all previously sched-uled grant spaces are properly schedsched-uled within the minimum number of listen frames. Moreover, based on Definition 2, the remaining grant spaces do not have the chance to be placed in either the stuck frame or the frames before. Finally, based on Fact 2, the stuck groupSGand all previously scheduled grant spaces can be removed without interfering the scheduling of the remaining grant spaces. All those removed grant spaces are completely scheduled with the minimum number of listen frames. It completes the proof.
If the grant space removal can be done based on Lemma 5, the remaining unscheduled grant spaces can form a new stepwise grant space set as the input for the first round of the backward push mechanism. The second case that executes the proposed BA mechanism is addressed as follows.
Lemma 6. If the recursive backward movement of the
pro-posed BA mechanism can not properly schedule the data bursts, there does not exist a feasible scheduling algorithm.
Proof: When the recursive backward movement of the proposed BA mechanism is employed, the stuck frame xG must be full of data bursts and there is still at least one data
