Distributed and Energy-Aware MAC for Differentiated Services ...

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Distributed and Energy-Aware MAC for Differentiated Services Wireless Packet Networks: A General Queuing Analytical Framework Afshin Fallahi, Student Member, IEEE, and Ekram Hossain, Senior Member, IEEE Abstract—We present a novel queuing analytical framework for the performance evaluation of a distributed and energy-aware medium access control (MAC) protocol for wireless packet data networks with service differentiation. Specifically, we consider a node (both buffer-limited and energy-limited) in the network with two different types of traffic, namely, high-priority and low-priority traffic, and model the node as a MAP (Markovian Arrival Process)/PH (Phase-Type)/1/K nonpreemptive priority queue. The MAC layer in the node is modeled as a server and a vacation queuing model is used to model the sleep and wakeup mechanism of the server. We study standard exhaustive and number-limited exhaustive vacation models both in multiple vacation case. A setup time for the head-of-line packet in the queue is considered, which abstracts the contention and the back-off mechanism of the MAC protocol in the node. A nonideal wireless channel model is also considered, which enables us to investigate the effects of packet transmission errors on the performance behavior of the system. After obtaining the stationary distribution of the system using the matrix-geometric method, we study the performance indices, such as packet dropping probability, access delay, and queue length distribution, for high-priority packets as well as the energy saving factor at the node. Taking into account the bursty traffic arrival (modeled as MAP) and, therefore, the nonsaturation case for the queuing analysis of the MAC protocol, using phase-type distribution for both the service and the vacation processes, and combining the priority queuing model with the vacation queuing model make the analysis very general and comprehensive. Typical numerical results obtained from the analytical model are presented and validated by extensive simulations. Also, we show how the optimal MAC parameters can be obtained by using numerical optimization. Index Terms—Wireless packet networks, differentiated services, quality of service (QoS), energy efficiency, priority queuing, Markovian arrival process, phase-type distribution, matrix-geometric method.

Ç 1

INTRODUCTION

D

. The authors are with the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, Canada R3T 5V6. E-mail: {afallahi, ekram}@ee.umanitoba.ca.

In this paper, we use the theory of discrete-time Markov chains and develop a novel queuing analytical model to evaluate the performance of an energy-aware distributed MAC protocol considering bursty traffic arrival patterns. The MAC protocol, which in fact works as the server, provides service differentiation between two classes of packets, namely, high-priority and low-priority packets (e.g., a dual queue MAC [3]). It executes back-off procedures, does clear channel assessments, waits for and receives acknowledgements, and handles packet retransmissions in the case of collisions/transmission failure. Each node is distinguished by two operational modes: active and sleep. While in the active state, the node is fully working and is able to transmit/receive data, in the sleep state, the transmission circuit, which is usually the most power consuming part of the node, is disabled. The sleep and wakeup mechanism is modeled by using a vacation queuing model. In our discrete-time queuing system, time is divided into fixed length intervals called slots. During consecutive slots, packets arriving in a node are stored and served on a first-in first-out (FIFO) basis in two different finite-length queues. The arrivals are modeled by a Markovian arrival process (MAP),1 which allows correlated and bursty arrivals.

Manuscript received 30 Aug. 2005; revised 4 May 2006; accepted 4 Aug. 2006; published online 15 Feb. 2007. For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference IEEECS Log Number TMC-0256-0805.

1. MAP was introduced by Neuts [5] as a generalization of the Poisson process, which is unable to represent the correlation structure in a stochastic process.

ISTRIBUTED medium access control (MAC) protocols are necessary in traditional wireless networks, such as wireless local area networks (WLANs) and cellular wireless networks, and also in multihop wireless networks such as wireless mesh, ad hoc, and sensor networks [1]. Energy efficiency is often a major concern in such networks due to the limited battery power of the nodes [2]. Specifically, for multihop wireless networks such as sensor networks, protocol mechanisms to make efficient use of the limited energy of the nodes, and thus, extend the lifetime of the network, are essential. One way to save energy in such a network is to use an efficient sleep and wakeup mechanism to turn off the radio transceiver occasionally so that the desired trade-off between the node energy savings and the network performance (e.g., throughput and data delivery delay) can be achieved. Again, to support traffic with different quality of service (QoS) requirements, a prioritization mechanism is required at the MAC protocol to achieve service differentiation.

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To avoid the intricacies of any specific MAC protocol (e.g., IEEE 802.11 distributed coordination function (DCF) MAC [4]), we use a generic distribution for the time required to initiate service (i.e., data packet transmission) from the system’s idle state and this is referred to as the setup time in this paper. For example, the setup time can model the overhead due to channel sensing, back-off, and handshaking mechanisms in the MAC protocol before the actual packet transmission starts. Again, to model the packet transmission time (which can vary depending on the packet length), as well as the time required for the sender to receive the acknowledgment (ACK) information, a service time distribution is assumed for each packet.2 The setup time and the service time are assumed to have phase-type (PH) distribution.3 While the time required to transmit the first packet (when the system switches to the active mode from sleep mode) includes both the setup time and the service time, the time required for transmission of the subsequent packets consists of only the service time (e.g., a reservation type of MAC protocol). Based on the discrete-time MAP/PH/1 priority queuing model in [6], we analyze a finite queue with vacation to obtain the performance measures of the MAC protocol in terms of queue length distribution, packet dropping probability, and access delay in the queue for high-priority packets, as well as energy saving performance at that node. We assume a nonpreemptive priority queuing model where the MAC protocol completes transmission of the current packet in service regardless of its priority and then decides to serve the next packet. We study two types of vacation systems (or, in other words, two sleeping mechanisms) in this work: standard exhaustive vacation system and number-limited exhaustive vacation system [7]. In the former, the server attends the queues until the system becomes empty (i.e., no waiting packets in the queues) and then goes to vacation (sleep mode). In the latter case, the server serves the queues until it has served M packets or the queues become empty, whichever occurs first. In the presented analytical framework, effects of both ideal and nonideal wireless channel conditions are considered. For the nonideal case, we assume a nonzero probability of packet transmission failure. The major contributions of this paper are as follows: .

.

.

A novel analytical model, which combines priority queuing with vacation queuing model, is developed to analyze the trade-off between the QoS and the energy efficiency of a distributed and energy-aware wireless MAC protocol considering service differentiation. Two different disciplines, namely, standard exhaustive vacation and number-limited exhaustive vacation, are analyzed, which correspond to two different sleeping mechanisms for the service process (i.e., the MAC protocol activity) at a node. The bursty traffic arrival process and nonsaturation MAC protocol state are considered in the queuing analysis.

2. Note that, for MAC protocols such as the IEEE 802.11 DCF, the ACK reception time is, in general, not deterministic. 3. Phase-type distribution can virtually model any practical distribution [8].

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The rest of the paper is organized as follows: In Section 2, we review the related works. The system model, including the arrival and the service processes and the vacation model, is described in Section 3. In Section 4, the queuing analytical model for the standard exhaustive vacation case is developed. Specifically, we establish a Markov chain and present a matrix-geometric analysis of this model for nonpreemptive priority queuing with multiple vacations. Based on the analysis of the steady state distribution of the packets in the system, the different QoS measures are obtained. Similar developments for the number-limited exhaustive case are presented in Section 5. The effects of nonideal channel model and setup time (for the first transmission at the beginning of the active mode) are analyzed in Sections 4 and 5. An application of the proposed analytical model is presented in Section 6. In Section 7, typical numerical results obtained from the analytical models are presented and validated through simulations. Conclusions are stated in Section 8.

2

RELATED WORK

Performance analysis of wireless MAC protocols was done quite extensively in the literature. Performance of Alohatype MAC protocols under a long-range dependent traffic arrival process was investigated in [9]. Performance of carrier sense multiple access with collision avoidance (CSMA/CA) protocols was studied in [10]. Performance evaluation of a class of MAC algorithms in multicell and ad hoc network environments was presented in [11]. Using theoretical upper bounds on network performance, [12] evaluated the effects of various design choices, such as power control, queuing discipline, and choice of routing and media access protocols in a wireless ad hoc network in a multihop environment. Performance of IEEE 802.11 DCF MAC protocol in the saturation state (i.e., when the transmission queue for each station is assumed to be always nonempty) was investigated in [13], [14]. In [15], [16], more practical queuing models for the IEEE 802.11 DCF MAC were proposed, which incorporated practical packet arrival processes. However, the service rate for each node was based on the results obtained in [13], where saturation state was assumed. This limitation was overcome in [17], where performance analysis in the nonsaturation state was considered by introducing probability generating functions, which allow the computation of the probability distribution function (pdf) of the delay. However, computing the pdfs using the proposed method has high computational cost and, therefore, the approach is of limited practical use. Furthermore, service differentiation at the MAC layer was not considered. A great amount of work on wireless MAC protocols in the literature focused on supporting heterogeneous traffic over the shared wireless channel (e.g., in [18], [19]). A number of QoS differentiation mechanisms for heterogeneous wireless networks were discussed in [20]. In [3], dual queue scheme, a software upgrade-based method, was proposed to provide QoS for voice over IP (VoIP) service over legacy 802.11 WLAN. With two separate queues on top of the 802.11 MAC controller for real-time (RT) and non-real-time (NRT) packets, a strict priority queuing discipline was implemented to serve these two queues in order to give higher priority

FALLAHI AND HOSSAIN: DISTRIBUTED AND ENERGY-AWARE MAC FOR DIFFERENTIATED SERVICES WIRELESS PACKET NETWORKS: A...

to the RT packets. In this model, the NRT queue is never served as long as the RT queue is nonempty. Moreover, the energy efficiency issue was not considered. A wide variety of solutions were introduced for power management in wireless multihop networks by putting a node in the sleep mode [21], [22]. The problem of energyefficient protocol design for ad hoc and sensor networks was addressed quite extensively in the literature [2], [23], [24], [25], [26], [27], [28]. Several approaches for QoS support in a multihop relay network, such as a sensor network, were discussed in [29], [30]. In [25], a wakeup scheme was designed to reduce the degradation in end-toend delay due to the sleeping/wakeup mechanism in a sensor network. However, queuing dynamics at a node due to the sleep and wakeup mechanism were not investigated in the above works. A Markov model to analyze the sleep/active dynamics in a sensor node was developed in [31]. In [27], a discretetime Markov chain model was used to represent the sensor dynamics in sleep/active mode while taking into account the channel contention and the routing issues. A vacation model along with an M/G/1 queue for MAC layer service process was used in [32] to analyze the sleeping mechanism in a sensor network. However, no prioritization was assumed between different arrival streams. Allowing for differentiated services would require the consideration of a priority queuing model. In a traditional priority queuing system, the server is ready all the time to serve one of the queues. When the server is serving one of the queues, it is said to be on vacation for the other queues. In the presence of a sleeping mechanism, however, when it is assumed that the server process in a node is on vacation (i.e., in sleep mode), it serves none of the queues. Therefore, a system combining both the priority and the vacation models needs to be considered for modeling the sleep and the wakeup mechanism in the MAC protocol at a node with differentiated services mechanism. Priority queuing and vacation queuing models were analyzed separately in [6] and [7], respectively, using the matrix-geometric method. Also, a few other works used the matrix-geometric method of queuing analysis for wireless networks [33]. A performance analysis model for GPRS (General Packet Radio Service) networks was presented in [34], in which the traffic interarrival time was modeled by a phase-type distribution (to capture the correlation in the arrival process) and the transmission time of IP (Internet Protocol) packets was modeled as an MMAP (Marked Markovian Arrival Process).

3

SYSTEM MODEL

We model the distributed and energy-aware MAC mechanism at a node in a wireless network as a discretetime queuing system. All events which occur between time t and t þ 1 are assumed to become effective at time t þ 1. At least one epoch interval is required to serve one packet. We describe the arrival process, the service process, the channel model, and the vacation model for this queuing system below.

3.1 Packet Arrival Process Modeling the packet arrivals at a node by using a discrete Markovian arrival process (D-MAP), we allow correlation

383

among the interarrival times within each type of packets and between the two types of packets (i.e., high-priority and low-priority packets, which will be referred to as type-1 and type-2 packets, respectively) as well. A MAP is described by four substochastic matrices D0 , D11 , D12 , and D2 , with D1 ¼ D11 þ D12 and D ¼ D0 þ D1 þ D2 , where D is stochastic. The element ðD0 Þij represents the probability of a transition from phase i to phase j with no arrival, ðD11 Þij represents a transition with an arrival of type-1 packet, ðD12 Þij represents a transition with an arrival of type-2 packet, and ðD2 Þij represents a transition with two arrivals (one of each type). The arrival rate i for a type-i packet ði ¼ 1; 2Þ is given as i ¼ ðD1i þ D2 Þe, where is the solution of ¼ D and e ¼ 1 (e is a column vector of 1s with appropriate dimension). D-MAP is an extension of the Markov modulated Bernoulli process (MMBP). In MMBP, a Bernoulli arrival is modulated by a discrete-time Markov chain. A Markovian arrival process works in a similar manner, but arrivals may also occur when the Markov process jumps from one state to another. The Bernoulli process is a special type of D-MAP. We consider one special case of this arrival process where there are two independent discrete arrival streams described by the matrices D0 ðiÞ, D1 ðiÞ, i ¼ 1; 2. In this case, D0 ¼ D0 ð1Þ D0 ð2Þ, D11 ¼ D1 ð1Þ D0 ð2Þ, D12 ¼ D0 ð1Þ D1 ð2Þ, and D2 ¼ D1 ð1Þ D1 ð2Þ, where is the Kronecker product.

3.2

Service Process and MAC Layer Protocol Activity The service process is determined by the MAC protocol activity (i.e., active mode, sleep mode) and the corresponding mechanisms (e.g., channel access mechanism, handshaking protocols etc.). The type-1 and type-2 packets are stored in two separate queues (Fig. 1a). In the active mode, the service process prioritizes type-1 packets over type-2 packets. We study the nonpreemptive priority case here, in which there is no service interruption upon arrival of a type-1 packet when a type-2 packet is being serviced. Packets from each queue are served on a FIFO basis. The server state diagram in Fig. 1b illustrates the service process in more detail. When the server switches to the active mode, a setup time, which includes the overhead due to channel sensing, handshaking, and back-off, etc., is required before the actual data packet transmission occurs. We distinguish between the distribution of transmission latency for the head-of-line (HOL) packet in the queue and the distribution of transmission latency for the packets following the HOL packet. While the transmission latency for the HOL packet consists of both the service time and the setup time, the transmission latency for a non-HOL packet consists only of the service time. We assume that, when the node acquires channel access, the channel is reserved for the node (e.g., reservation MAC protocol) for a period of time and, therefore, the same service time distribution can be assumed for each of the non-HOL packets. The decision regarding staying in active mode or switching back to sleep mode is made based on the queue state and selected strategy. As shown in Fig. 1a, an empty queue causes the server to switch to sleep mode. While the

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Fig. 1. System model: (a) service differentiation and the energy saving mechanism at the MAC layer protocol and (b) server state diagram.

queues are not empty, the decision is made based on M (the number of transmitted packets since last wake up) during the Send Pkt state (in Fig. 1b). The transmission latency for the HOL packet follows a discrete phase-type distribution represented by ðp ðiÞ; Sp ðiÞÞ, for packet type i ði ¼ 1; 2Þ.4 We use ðc ðiÞ; Sc ðiÞÞ, i ¼ 1; 2 to represent the service time distributions for the non-HOL packets. In this representation, p ðiÞ and c ðiÞ are row vectors, Sp ðiÞ and Sc ðiÞ are substochastic matrices, and Sp0 ðiÞ ¼ e Sp ðiÞe and Sc0 ðiÞ ¼ e Sc ðiÞe.

3.3

Channel Model and Packet Error Recovery Process We assume a nonideal wireless channel, where p is the probability of successful transmission of a packet (i.e., 1 p is the probability of transmission failure). For simplicity, we assume that each node knows whether the packet transmission is successful or not at the end of the same slot that transmission occurs. We assume an infinite-persistent automatic repeat request (ARQ) mechanism for retransmission in case of transmission error. 3.4 Vacation Model A single server vacation queue with two arrivals and nonbatch service is considered in discrete time. For the vacation process, the distribution of time spent in vacation (i.e., in sleep mode) is assumed to be phase-type with parameters ð; V Þ. To make the system more energyefficient, we assume a multiple vacation model, in which, upon going from the vacation to the active mode, if the server finds the queues empty, it returns to the vacation mode immediately. We consider two types of vacation systems—the standard exhaustive vacation system and the number-limited (NL) exhaustive vacation system [7]. These two systems correspond to two different sleeping strategies for the node. 4. This assumption on different distributions of the transmission latency for the HOL packets for the two queues makes the model more general.

In the former case, the server goes to the sleep mode after serving all packets from both of the queues. In the latter case, the server process goes to the sleep mode after it has served M packets or the queues have become empty, whichever occurs first.

4

ANALYSIS OF THE STANDARD EXHAUSTIVE VACATION SYSTEM

In this case, all the packets in the queues are served when the server process switches to the active mode from the sleep mode. If the server process (i.e., the MAC protocol) finds the queues empty, it goes to vacation (i.e., the node goes to sleep mode).

4.1 The Markov Chain Consider the following state space: v0 ¼ ð0; 0; j; lÞ; s0 ¼ ð0; i2 ; j; k2 Þ; i2 1; v ¼ ði1 ; i2 ; ; j; lÞ; ði1 þ i2 Þ 1; s1 ¼ ði1 ; i2 ; j; k1 Þ; i1 1; i2 0; s2 ¼ ði1 ; i2 þ 1; j; k2 Þ; i1 1; i2 0: v0 represents the vacation (i.e., sleep or idle mode) while the queues are empty with the arrival in jth phase and the vacation in lth phase. In s0 , there are only type-2 packets in the system with arrival in the jth phase and a type-2 packet is being served while the service is in phase k2 . In states v , s1 , and s2 , there is at least one type-1 packet in the system. In v , the system is on vacation (i.e., sleep mode) while the phases of arrival and vacation are j and l, respectively. s1 and s2 represent the system in service when the packet in the server is of type-1 with k1 being the phase of service and the packet in the server is of type-2 with k2 being the phase of service, respectively.


Considering the setup time, state s0 is partitioned to s;p 0 and s;c 0 , which represent the state for head-of-line packet and other packets, respectively. The same type of partitioning is applicable to s1 and s2 and they will be partitioned s;c s;p s;c into s;p 1 , 1 and 2 , 2 , respectively. Consequently, the state space for the system becomes [ [ s;c [ [ s;p [ s;c [ s;p [ s;c ¼ v0 s;p 0 v 1 1 2 2 : ð1Þ 0 The transition matrix P describing this Markov chain has the form: 3 2 B00 B01 7 6 B10 A1 A0 7 6 7 6 A A A 2 1 0 7 6 ð2Þ P ¼6 7; .. .. .. 7 6 . . . 7 6 5 4 A2 A1 A0 A2 A1 þ A0 where the ith row in the above matrix represents i highpriority packets in the system. Moreover, the jth row in block matrices B00 , B01 :B10 , B11 , A0 , A1 , A2 represents j lowpriority packets in the system. Here, the state space and the matrices are defined in the same way as in [6]. The block matrices are given in Appendix A, which can be found on the Computer Society Digital Library at http://computer. org/tmc/archives.htm.

4.2

Matrix-Geometric Analysis and Steady-State Probability Distribution The matrix P is of the quasi-birth-death (QBD) type and can be analyzed using the matrix-geometric method [35]. For the transition probability matrix P , let x ¼ ½x0 x1 x2 . . . xK (where K is the queue size) represent the steady-state probability vector corresponding to the number of packets in the queue, where x ¼ xP and xe ¼ 1. If matrix R is the minimal nonnegative solution to the following matrix quadratic equation: R ¼ A0 þ RA1 þ R2 A2 ; it can be shown that [35] xiþ1 ¼ xi R;

1 i < K;

where xi ¼ ½xi0 xi1 xi2 . . . xiK , 0 i K, s;c s;p s;c xij ¼ ½xvij xs;p ij ð1Þ xij ð1Þ xi;jþ1 ð2Þ xi;jþ1 ð2Þ; s;c i 1, j 0. Here, xs;p ij ð1Þ and xij ð1Þ represent the probability of a type-1 packet being in service while the HOL or a non-HOL packet is being transmitted, respectively. Also, s;c xs;p i;jþ1 ð2Þ and xi;jþ1 ð2Þ (for a type-2 packet in service) are the s;c same as xs;p ij ð1Þ and xij ð1Þ, respectively.

4.3 Performance Measures 4.3.1 Queue Length Distribution Let qi be the probability that there are i high-priority packets in the system. Then, we have qi ¼ xi e. 4.3.2 Packet Dropping Probability Using the steady state probability vector xi for a maximum queue length of K packets for high-priority packets, the probability of packet dropping ðpd Þ can be found as follows:

pd ¼ 1 1

K X

385

xvKj ½ðD11 þ D2 Þ ðV þ V 0 Þ

j¼0

þ 1 1

K X

0 xs;p Kj ð1Þ½ðD11 þ D2 Þ ð1 pÞSp ð1Þc ð1Þ

j¼0

þ 1 1

K X

xs;p Kj ð1Þ½ðD11 þ D2 Þ Sp ð1Þ

j¼0

þ 1 1

K X

0 xs;c Kj ð1Þ½ðD11 þ D2 Þ ð1 pÞSc ð1Þc ð1Þ

j¼0

þ 1 1

K X

xs;c Kj ð1Þ½ðD11 þ D2 Þ Sc ð1Þ

j¼0

þ 1 1

K X

0 xs;p K;jþ1 ð2Þ½ðD11 þ D2 Þ ð1 pÞSp ð2Þc ð1Þ

j¼0

þ 1 1

K X

xs;p K;jþ1 ð2Þ½ðD11 þ D2 Þ Sp ð2Þ

j¼0

þ 1 1

K X

0 xs;c K;jþ1 ð2Þ½ðD11 þ D2 Þ ð1 pÞSc ð2Þc ð1ÞÞ

j¼0

þ 1 1

K X

xs;c K;jþ1 ð2Þ½ðD11 þ D2 Þ Sc ð2Þ:

j¼0

Note that, based on the dropping probability defined above, the queue throughput can be obtained as 1 ð1 pd Þ.

4.3.3 Probability of Sleep (Energy Saving Factor) This is defined as the probability that the server is in the P v sleep mode and is given by S ¼ K i¼0 xi . Since the longer the time that a node stays in the sleep mode, the more is the amount of energy saved, this is an indicator of energy saving in the node. 4.3.4 Access Delay Distribution Access delay is defined as the time required for a packet to arrive at the head of the queue since its arrival into the queue. Access delay distribution for a high-priority packet is studied here by using the concept of absorbing Markov chains. In this case, an absorbing state is the state in which the target packet arrives at the head of the line. When a high-priority packet arrives, if it finds the system nonempty, then either a high-priority or a low-priority packet is being processed. Since this is a nonpreemptive case, the arriving high-priority packet has to wait either for transmission of all the high-priority packets ahead of it or transmission of the low-priority packet which might be going on at the time of its arrival. We first determine the state probability vectors zsij ðj þ 1Þ, i 1, j ¼ 0; 1, representing the probability of an arriving high-priority packet finding i high-priority (type-1) packets ahead of it and a type-ðj þ 1Þ packet in processing. zvij is the same as above, but it corresponds to the case when an arriving packet finds the server in the sleep mode. Also, note that zs01 refers to the probability of having no type-1 packet in the system and at least one type-2 packet in the system, and zs00 is the probability of having the system empty. Moreover, zv0 is for the arrival of a type-1 packet that

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finds the system empty and the server is on vacation (i.e., sleep mode). We have

þ

K X

xv0j ½ðD11

K X

þ D2 Þ ðV þ V Þ;

þ

K X

þ 1 1

0 xs;c 0j ½ðD11 þ D2 Þ pSc ð2Þ

K X

xs;p 1j ð1Þ½ðD11

1 zs;p 01 ¼ 1

þ D2 Þ

0 xs;c 1j ð1Þ½ðD11 þ D2 Þ pSc ð1Þ;

xs;p 0j ½ðD11 þ D2 Þ Sp ð2Þ

j¼1

þ 1 1

K X

0 xs;p 0j ½ðD11 þ D2 Þ ð1 pÞSp ð2Þc ð2Þ;

j¼1 1 zs;c 01 ¼ 1

K X

xs;c 0j ½ðD11 þ D2 Þ Sc ð2Þ

j¼1

þ 1 1

K X

zvi0 ¼ 1 1

þ D2 Þ ð1 pÞSc0 ð2Þc ð2Þ;

xvij ½ðD11 þ D2 Þ V ;

j¼0

1 zs;p i0 ð1Þ ¼ 1

K X

K X

xs;p ij ð1Þ½ðD11 þ D2 Þ Sp ð1Þ

We study the delay distribution of type-1 packets as the time to absorption in a Markov chain with the following transition matrix: 2 3 e00 B 6B 7 e 6 e10 A1 7 ð3Þ Pe ¼ 6 7; e e A2 A1 4 5 .. .. . .

e00 ¼ B 2 0 V V0 60 I 0 6 6 6 0 I pSp0 ð2Þ I Sp ð2Þ 4 0 I pSc0 ð2Þ

þ 1 1

2

xvij ½ðD11 þ D2 Þ V 0 p ð1Þ;

e10 B

j¼0 1 zs;c i0 ð1Þ ¼ 1

K X

xs;c ij ð1Þ½ðD11 þ D2 Þ Sc ð1Þ

3

0 0 I ð1 pÞSp0 ð2Þc ð2Þ

7 7; 7 7 5

I Sc ð2Þ þ I ð1 pÞSc0 ð2Þc ð2Þ

0

j¼0 K X

0 xs;p ij ð2Þ½ðD11 þ D2 Þ ð1 pÞSp ð2Þc ð2Þ:

where the block matrices are as follows: xs;c 0j ½ðD11

j¼1 K X

0 xs;c ij ð2Þ½ðD11 þ D2 Þ ð1 pÞSc ð2Þc ð2Þ

s;c z0 ¼ ½zv0 zs00 zs;p 01 z01 ; s;c s;p s;c zi ¼ ½zvi0 zs;p i0 ð1Þ zi0 ð1Þ zi1 ð2Þ zi1 ð2Þ; z ¼ ½z0 z1 . . .:

pSp0 ð1Þ

j¼0 K X

K X

Then,

j¼0

þ 1 1

xs;c ij ð2Þ½ðD11 þ D2 Þ Sc ð2Þ

j¼1

j¼1

1 1

xs;p ij ð2Þ½ðD11 þ D2 Þ Sp ð2Þ;

j¼1

0 xs;p 0j ½ðD11 þ D2 Þ pSp ð2Þ

K X

K X

þ 1 1

j¼1

þ 1 1

APRIL 2007

j¼1 0

j¼1

zs00 ¼ 1 1

K X

NO. 4,

j¼1 1 zs;c i1 ð2Þ ¼ 1

0 zv0 ¼ 1 1 x00 ½ðD11 þ D2 Þ ðV þ V Þ

1 1

1 zs;p i1 ð2Þ ¼ 1

VOL. 6,

0 60 6 ¼6 60 40 0

0 I Sp0 ð1Þ I Sc0 ð1Þ 0 0

3 0 0 0 07 7 0 07 7; 0 05 0 0

j¼0

þ 1 1

K X

0 xs;p ij ð1Þ½ðD11 þ D2 Þ ð1 pÞSp ð1Þc ð1Þ

j¼0

þ 1 1

K X

0 xs;c iþ1j ð1Þ½ðD11 þ D2 Þ pSc ð1Þc ð1Þ

j¼0

þ 1 1

K X

0 xs;p ijþ1 ð2Þ½ðD11 þ D2 Þ pSp ð2Þc ð1Þ

j¼0

þ 1 1

K X

0 xs;c ijþ1 ð2Þ½ðD11 þ D2 Þ pSc ð2Þc ð1Þ

j¼0

þ

1 1

K X

xs;p iþ1j ð1Þ½ðD11

þ D2 Þ

pSp0 ð1Þc ð1Þ

j¼0

þ 1 1

K X j¼0

0 xs;c ij ð1Þ½ðD11 þ D2 Þ ð1 pÞSc ð1Þc ð1Þ;

e1 ¼ A 2 IV 6 6 6 6 6 0 6 6 6 6 6 6 6 6 0 6 6 6 6 6 0 6 6 6 6 6 6 6 6 4 0

I V 0 p ð1Þ

0

0

I Sp ð1Þ

I ð1 pÞ

0

Sp0 ð1Þc ð1Þ 0

e1 ðsc sc Þ A 1 1

0

0

I pSp0

I Sp ð2Þ

ð2Þc ð1Þ 0

I pSc0 ð2Þc ð1Þ

0

0

3

7 7 7 7 0 7 7 7 7 7 7 7 7 7 0 7; 7 7 7 I ð1 pÞ 7 7 7 Sp0 ð2Þc ð2Þ 7 7 7 7 7 7 c c e A1 ðs2 s2 Þ 5


e1 ðsc sc Þ ¼ I Sc ð2Þ þ I ð1 pÞS 0 ð2Þc ð2Þ; A 2 2 c c c e A1 ðs s Þ ¼ I Sc ð1Þ þ I ð1 pÞS 0 ð1Þc ð1Þ; 1 1

c

2

0 60 6 e2 ¼ 6 0 A 6 40

0 0 0 0 I pSp0 ð1Þc ð1Þ 0

0 I pSc0 ð1Þc ð1Þ 0 0 0 0 0 0 0 0

3 0 07 7 07 7: 05 0

The following set of equations is used in the computation: e00 þ z1 B e10 ; z0 ¼ z0 B e1 þ ziþ1 A e2 ; zi ¼ zi A zv0 ¼ zv0 V ; s;c 0 0 zs00 ¼ zv0 V 0 þ zs00 þ zs;p 01 ðI pSp ð2ÞÞ þ z01 ðI pSc ð2ÞÞ s;c 0 0 þ zs;p 10 ð1ÞðI pSp ð1ÞÞ þ z10 ð1ÞðI pSc ð1ÞÞ; s;p zs;p 01 ¼ z01 ðI Sp ð2ÞÞ; s;p s;c 0 zs;c 01 ¼ z01 ðI ð1 pÞSp ð2Þc ð2ÞÞ þ z01 ðI Sc ð2Þ

þ I ð1 pÞSc0 ð2Þc ð2ÞÞ; zvi0 ¼ zvi0 ðI V Þ; s;p v 0 zs;p i0 ð1Þ ¼ zi0 ðI V p ð1ÞÞ þ zi0 ð1ÞðI Sp ð1ÞÞ; s;p 0 zs;c i0 ð1Þ ¼ zi0 ð1ÞðI ð1 pÞSp ð1Þc ð1ÞÞ 0 þ I ð1 pÞSc0 ð1Þc ð1ÞÞ þ zs;p i1 ð2ÞðI pSp ð2Þc ð1ÞÞ s;c 0 þ zs;c i0 ð1ÞðI Sc ð1Þ þ zi1 ð2ÞðI pSc ð2Þc ð1ÞÞ 0 þ zs;p iþ1;0 ð1ÞðI pSp ð1Þc ð1ÞÞ 0 þ zs;c iþ1;0 ð1ÞðI pSc ð1Þc ð1ÞÞ; s;p zs;p i1 ð2Þ ¼ zi1 ð2ÞðI Sp ð2ÞÞ; s;p 0 zs;c i1 ð2Þ ¼ zi1 ð2ÞðI ð1 pÞSp ð2Þc ð2ÞÞ 0 þ zs;c i1 ð2ÞðI Sc ð2Þ þ I ð1 pÞSc ð2Þc ð2ÞÞ:

If WT denotes the probability that the delay of a type-1 packet is less than or equal to T , then WT ¼ zT0 e.

5

ANALYSIS OF THE NUMBER-LIMITED EXHAUSTIVE VACATION SYSTEM

In this case, the server serves the queues until it has served M packets or the queues become empty, whichever occurs first. As has been mentioned before, the transmission latency for the first packet in transmission after the idle period (i.e., the HOL packet) includes the setup time and is different from the service time of each of the following packets. Moreover, the channel is nonideal and a packet transmission is assumed to be successful with probability p.

5.1 The Markov Chain The state space in this case is as follows: v0 ¼ ð0; 0; j; lÞ; s0 ¼ ð0; i2 ; u; j; k2 Þ; i2 1; v ¼ ði1 ; i2 ; j; lÞ; ði1 þ i2 Þ 1; s1 s2

¼ ði1 ; i2 ; u; j; k1 Þ; i1 1; i2 0; ¼ ði1 ; i2 þ 1; u; j; k2 Þ; i1 1; i2 0:

387

This is the same as the standard exhaustive case in Section 4 except that the number of the served packets in the state space is different. The vacation states are exactly the same as before. In s0 , s1 , and s2 , u denotes the number of served packets ðu ¼ 0; 1; 2; . . . ; MÞ. The server goes to vacation (sleep mode) after serving M packets even if there are more waiting packets in the system. The transition matrix P describing this Markov chain is the same as (2). However, the matrix blocks are different as shown in Appendix B, which can be found on the Computer Society Digital Library at http://computer. org/tmc/archives.htm. Here, e is a column vector of 1s whose length is M and ej is a column vector of zeros except a 1 as its jth element and its length is M, and ej is the transpose of ej . Moreover, state s0 is partitioned into s;p 0 and s;c 0 to include and exclude the setup time, respectively. The same concept is true for s1 and s2 and they are s;c s;p s;c partitioned into s;p 1 , 1 and 2 , 2 , respectively. The block matrices are given in Appendix B, which can be found on the Computer Society Digital Library at http:// computer.org/tmc/archives.htm.

5.2

Matrix-Geometric Analysis and Steady State Distribution The matrix-geometric analysis to obtain the steady state distribution for the number-limited exhaustive vacation system (using the new block matrices of P) is the same as that for the standard system. 5.3 Performance Measures The performance measures (e.g., queue length distribution, packet dropping probability, and energy saving factor) are obtained in the same way as that for the standard exhaustive case. For access delay distribution we proceed in the same way with the new matrices and obtain the probability vectors in Appendix C, which can be found on the Computer Society Digital Library at http://computer. org/tmc/archives.htm. We evaluate the delay distribution for a type-1 packet as the time to absorption in a Markov chain with the same transition matrix as in (3). The block matrices are as described in Appendix D, which can be found on the Computer Society Digital Library at http://computer. org/ tmc/archives.htm. Then znþ1 ¼ zn Pe;

WT ¼ zT0 e:

The following set of equations are used in the computation: e00 þ z1 B e10 ; z0 ¼ z0 B e1 þ ziþ1 A e2 ; zi ¼ zi A s;c 0 0 zv0 ¼ zv0 V þ zs;p 01 ðeM I pSp ð2ÞÞ þ z01 ðeM I pSc ð2ÞÞ s;c 0 0 þ zs;p 10 ð1ÞðeM I pSp ð1ÞÞ þ z10 ð1ÞðeM I pSc ð1ÞÞ; 0 zs00 ¼ zv0 ðe1 V 0 Þ þ zs00 ðIðMÞ IÞ þ zs;p 01 ðIðM 1Þ I pSp ð2ÞÞ 1Þ I pS 0 ð2ÞÞ þ zs;c ðIðM 01

c

0 þ zs;p 10 ð1ÞðIðM 1Þ I pSp ð1ÞÞ 1Þ I pS 0 ð1ÞÞ; þ zs;c ð1ÞðIðM 10

c

388

IEEE TRANSACTIONS ON MOBILE COMPUTING, s;p zs;p 01 ¼ z01 ðIðMÞ I Sp ð2ÞÞ s;p 0 zs;c 01 ¼ z01 ðIðMÞ I ð1 pÞSp ð2Þc ð2ÞÞ

þ zs;c 01 ðIðMÞ I Sc ð2Þ þ IðMÞ I ð1 pÞSc0 ð2Þc ð2ÞÞ; 0 zvi0 ¼ zvi0 ðI V Þ þ zs;p i1 ð2ÞðeM I pSp ð2ÞÞ 0 þ zs;c i1 ð2ÞðeM I pSc ð2ÞÞ 0 þ zs;p Iþ1;0 ð1ÞðeM I pSp ð1ÞÞ 0 þ zs;c Iþ1;0 ð1ÞðeM I pSc ð1ÞÞ;

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To obtain the access probability, we follow [37] and define the utility function of node i as follows: Y ui ¼ pi pw;j; j6¼i

where pw;j denotes the probability that node j does not transmit a packet (e.g., either the node has no packet to transmit or it has a packet and decides not to transmit). Defining pp;j as the probability of having a nonempty queue in node j at the beginning of a transmission slot, the probability pw;j is given as follows [37]: pw;j ¼ pp;j ð1 pj Þ þ ð1 pp;j Þ ¼ 1 pp;j pj :

s;p v 0 zs;p i0 ð1Þ ¼ zi0 ðe1 I V p ð1ÞÞ þ zi0 ð1ÞðIðMÞ I Sp ð1ÞÞ;

In a network with N nodes, we have zs;c i0 ð1Þ

¼

0 zs;p i0 ð1ÞðIðMÞ I ð1 pÞSp ð1Þc ð1ÞÞ 0 þ zs;c i0 ð1ÞðIðMÞ I Sc ð1Þ þ I ð1 pÞSc ð1Þc ð1ÞÞ s;p 1Þ I pSp0 ð2Þc ð1ÞÞ þ zi1 ð2ÞðIðM s;c 1Þ I pSc0 ð2Þc ð1ÞÞ þ zi1 ð2ÞðIðM s;p 1Þ I pSp0 ð1Þc ð1ÞÞ þ ziþ1;0 ð1ÞðIðM

ui ¼

and the probability that node i accesses the shared medium successfully is

0 þ zs;c iþ1;0 ð1ÞðIðM 1Þ I pSc ð1Þc ð1ÞÞ; s;p zs;p i1 ð2Þ ¼ zi1 ð2ÞðIðMÞ I Sp ð2ÞÞ; s;p 0 zs;c i1 ð2Þ ¼ zi1 ð2ÞðIðMÞ I ð1 pÞSp ð2Þc ð2ÞÞ

þ zs;c i1 ð2ÞðIðMÞ I Sc ð2Þ þ IðMÞ I ð1 pÞSc0 ð2Þc ð2ÞÞ:

6

APPLICATION

OF THE

ANALYTICAL MODEL

As an application of the presented analytical model, we use it for the performance evaluation of the predictive p-persistent CSMA protocol [36]. This protocol is a variant of p-persistent CSMA with the difference that, when the channel is idle, the probability of packet transmission varies according to the traffic condition (instead of being constant, as in p-persistent CSMA). Distribution of service time in this protocol is geometric (a special case of phase-type distribution) with parameter px , which denotes the probability of successful access and transmission to the shared wireless medium. In other words, when a node is in active mode and it has data packets to be transmitted and the channel is idle, it can successfully transmit with probability px . Therefore, Sp ¼ Sc ¼ px and Sp ¼ Sc ¼ 1 px . When either the channel is busy or there is no packet for transmission, the server goes to sleep mode. This is similar to adopting a back-off mechanism to avoid collision, as well as to save energy. We model this back-off mechanism in a node by a geometric distribution in which the back-off ends in each node with probability (i.e., the node returns to active mode with probability ). The value of px is not the same for all nodes in the network. The nodes compute the appropriate values of px and transmit with different probabilities depending on the number of nodes present in the system. At the beginning of each contention period, node i decides to transmit with probability pi .

N Y pi pw;j 1 pp;i pi j¼1

px;i ¼

N pp;i pi Y pw;j : 1 pp;i pi j¼1

The sleep and wakeup dynamics are modeled by a 1-phase PH distribution with V ¼ 1 and V ¼ . We refer to and as the probability of service and the probability of wakeup, respectively. Moreover, we assume that the probability of service is the same for both type-1 and type-2 packets unless it is explicitly stated.

7

NUMERICAL ANALYSIS RESULTS

AND

SIMULATION

7.1 System Parameters For numerical analysis, we consider simpler versions of MAP and PH distributions. Packet arrival is assumed to follow a Bernoulli process, a simple case of MAP with 1 and 2 as the probability of arrival for high-priority and low-priority packets, respectively. It implies that the interarrival times are geometrically distributed. Then, the matrices D0 , D11 , D12 , and D2 become scalars and are given as follows: D0 ¼ ð1 1 Þð1 2 Þ ¼ d0 ; D2 ¼ 1 2 ¼ d2 ; D11 ¼ 1 ð1 2 Þ ¼ d11 ; D12 ¼ ð1 1 Þ2 ¼ d12 : We consider a geometric distribution as a special case of phase-type distribution with one phase: p ¼ c ¼ 1, Sp ¼ p , Sc ¼ c , Sp ¼ 0p ¼ 1 p , and Sc ¼ 0c ¼ 1 c for service time distribution of both types of packets. The vacation period is also modeled as a PH distribution having one phase with V ¼ 0 and V ¼ . For numerical analysis and simulations, we study the standard exhaustive vacation case as a special case of the number-limited exhaustive vacation case with M ¼ 1. For the number-limited case, we assume M ¼ 1. We use the parameter N to denote node density (i.e., the number of neighboring nodes around the “tagged” node is N 1).


389

Fig. 2. Effect of wakeup probability on energy saving factor and packet dropping (for p ¼ 0:5, ¼ 0:2, p ¼ 0:5, c ¼ 0:9).

Fig. 3. Effect of wakeup probability on queue length distribution (for p ¼ 0:5, ¼ 0:2, p ¼ 0:5, c ¼ 0:9).

7.2 Simulation Setup We use MATLAB for an event-driven simulation with the same system parameters for both the standard and the number-limited cases. For each data point, we repeat the simulation 10,000 times and consider 1,000 time slots in each iteration. We assume that, if an arrival event and a service completion event coincide, the latter occurs first, followed by the former. For the predictive p-persistent CSMA protocol, we set up the simulation for different values of node density N and compute the channel access probability for each node at the beginning of each contention period. We assume pi ¼ 0:2 for a high node density scenario ðN 20Þ and pi ¼ 0:8 for a low node density scenario ðN 5Þ. We assume that pi decreases linearly when the number of nodes increases from five to 20. 7.3

Numerical Results

7.3.1 Effect of the Wakeup Probability Figs. 2 and 3 show the effects of the wakeup probability on the energy saving factor, packet dropping probability, and

queue length distribution. Here, the probability of successful transmission is 0.5 and the probability of service completion and the probability of arrival for both types of arrivals are p ¼ 0:5, c ¼ 0:9, and ¼ 0:2, respectively. The simulation results are plotted against the numerical results obtained from the analysis. It is evident that the simulation results follow the numerical results very closely, which validates the accuracy of the analytical model. The maximum difference between simulation and analysis results in Figs. 2a and 3b is 0.04, 0.05, 0.07, and 0.04, respectively. For comparison, the results on energy saving performance and packet dropping probability for the ideal channel case (i.e., no transmission error) and zero setup time are also shown in Fig. 2. The degradation in the energy saving performance due to the nonzero setup time (e.g., due to the contention and back-off) and nonideal channel conditions (when compared to the more ideal system characteristics) is evident from this figure. Figs. 2 and 3 basically reveal the trade-off between the energy savings and the QoS performance. Higher wakeup probability leads to QoS improvement, but at the expense of more energy consumption.

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Fig. 4. Effect of probability of success on energy saving factor and packet dropping probability with fixed wakeup probability (for ¼ 0:25, c ¼ 0:9).

Fig. 5. Effect of probability of success on queue length and delay with fixed wakeup probability (for p ¼ 0:8, c ¼ 0:9, p ¼ 0:8).

7.3.2 Effect of Transmission Error We study the effect of the probability of successful packet transmission (i.e., channel effect) on the performance of the system for different values of wakeup probability. Fig. 4 shows the effect of the probability of successful transmission on the QoS performances and the energy saving factor. Here, the probability of arrival (traffic load) is ¼ 0:25 for both arrival streams and we study the system for two different wakeup probabilities. The channel conditions affect the setup time. A higher probability of channel error tends to destroy the control packets during the setup phase and, therefore, tends to increase the setup time. Hence, increasing the value of p decreases the value of p . To capture this fact, we set p ¼ c a p (with c ¼ 0:9 and a ¼ 0:5) to obtain the numerical results. Fig. 5 shows queue length distribution and access delay when p ¼ 0:8. Among all the cases shown in Fig. 5a, the number-limited case with ¼ 0:2 tends to result in the worst packet dropping probability. In this case, the access delay performance may be unacceptable for delay-sensitive applications (Fig. 5b).

7.3.3 Effect of the Probability of Arrival (Traffic Load) Figs. 6 and 7 demonstrate the effect of packet arrival probability (i.e., traffic load) on the system performance. For both these figures, the value of the wakeup probability is set to be the same as the successful packet transmission probability (i.e., channel-dependent wakeup). We study two different traffic loads, ¼ 0:1 and ¼ 0:4, which are the same for both streams. Moreover, we set the setup time according to the channel condition (i.e., the probability of successful packet transmission). Under a low traffic load condition (e.g., 1 ¼ 2 ¼ 0:1) and higher probability of success, both the standard and the number-limited strategies achieve similar gain in the energy saving factor. However, the performance of the standard exhaustive strategy degrades when the probability of transmission error increases (Fig. 6a). For a high traffic load condition (e.g., 1 ¼ 2 ¼ 0:4), the difference in the energy saving performance with these two strategies is quite significant. For the number-limited case, we observe that, with relatively high traffic intensity, the energy saving factor is almost independent of the channel condition


391

Fig. 6. Effect of the probability of success on the energy saving factor and packet dropping for channel-dependent wakeup.

Fig. 7. Effect of the probability of success on queue length and delay channel-dependent wakeup (for p ¼ 0:8, c ¼ p ¼ 0:8).

(Fig. 6a). While the energy saving performance is better in the number-limited case compared to that in the standard case, degradation in the QoS performance is more significant in the former case (Figs. 6 and 7). Using the analytical model, the trade-off between the QoS and energy savings can be analyzed quantitatively under different channel conditions. In Fig. 8, the energy saving factor is plotted as a function of probability of arrival and probability of success. Here, the probability of service is c ¼ 0:8 for both types of arrivals. As expected, the energy saving performance improves with an increasing probability of successful transmission and/or decreasing probability of arrival. Given a particular required energy saving performance and a given channel condition, the packet arrival rate at a node can be controlled so that the desired performance objectives can be achieved.

7.3.4 Effect of Number of Neighboring Nodes For the predictive p-persistent CSMA protocol, we investigate the effect of the number of neighboring nodes on queue length distribution and average queue length when

the size of each of the queues at the tagged node is 30. While the probability mass function for the high and low node density scenarios varies monotonically (which is intuitive), it follows a “convex” variation in a medium node density scenario (Fig. 9a, for 1 ¼ 2 ¼ ¼ 0:15). This is due to the adaptive variation in px;i in the predictive p-persistent CSMA protocol. Note that, given a particular node density, the probability mass function can be utilized for traffic shaping (to adjust the traffic arrival rate) when the queue length (and, hence, the access delay) is required to be statistically bounded. Fig. 9b shows variation in average queue length when the node density varies. In this case, we consider two different arrival rates (a ¼ 2 ¼ ¼ 0:1 and 1 ¼ 2 ¼ ¼ 0:4), which correspond to low and high traffic cases, respectively. As expected, the average queue-length increases with increasing traffic arrival rate.

7.3.5 Effect of Network Perturbations Variation in the system dynamics due to three different phenomena, namely, variation in the number of neighbor

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Fig. 8. Effect of the probability of arrival and probability of success on energy savings: (a) for wakeup probability, ¼ 0:2 and (b) for wakeup probability ¼ 0:8.

Fig. 9. Effect of neighbor nodes on (a) queue length distribution and (b) average queue length ðpi ¼ 0:9; ¼ 0:8Þ.

nodes (or effect of mobility), variation in the network topology (e.g., due to power saving mechanism), and variation in traffic arrival, can be modeled in the proposed framework. For example, in the p-persistent CSMA case, mobility of the nodes causes variations in the number of neighbor nodes, which affects N and, thereafter, affects px;i . The parameter pw;j can incorporate the effect of the power saving mechanism and consequent changes in the network topology. This parameter can also represent the selfish behavior of some nodes (e.g., which are unwilling to transmit the relayed data packets). Also, due to the sleep and wakeup mechanism, when there is no packet arrival in the tagged node, it will not attempt any transmission. In this way, it automatically adapts to the variation in traffic arrival.

7.4 Optimal Parameter Setting The proposed analytical framework can be used to set the system parameters optimally. For example, the optimum wakeup probability and the optimum traffic load can be

found simultaneously under constrained packet dropping probability performance. We can formulate an optimization problem as follows: minimize :

jpd ð; Þ ptar d j;

ð4Þ

where the decision variables are and ; note that similar formulations can be developed under constraints on energy or other QoS factors. We use the Hooke-Jeeves direct search method [38] to solve this optimization problem. This heuristic direct search technique is used to find the minimum of the cost function using the smallest number of iterations. This is done by using a succession of simple moves known as exploratory searches and pattern moves [39]. Fig. 10 shows typical variations in the optimum value of and under constraints on the packet dropping probability and energy saving factor, respectively. In these cases, we set p ¼ 0:5, c ¼ 0:8, and p ¼ 0:6 for both types of arrivals. We observe that (in Figs. 10a and 10b), for a lower packet dropping probability, the probability of wakeup needs to be


393

Fig. 10. (a) Optimum wakeup probability for the target packet dropping probability, (b) optimum arrival probability for the target packet dropping probability, and (c) optimum arrival probability for the target energy saving factor (for p ¼ 0:5, p ¼ 0:6, c ¼ 0:8).

increased. Meanwhile, a lower traffic arrival rate is required to meet the target packet dropping probability requirement. As is evident from Fig. 10c, traffic load needs to be reduced when we need more energy savings in a node. Results for both the standard and the number-limited cases are shown in Fig. 10 from where the relative performances of these two strategies become evident. The frequency of state switching (between the active and the sleep mode) is higher for the number-limited scheme since the server is not allowed to serve more than a packet during each cycle. However, the traffic load that can be served under a constrained packet loss probability is lower compared to that for the standard strategy. On the other hand, regarding the energy saving performance, the optimum traffic load in the number-limited case is higher than that in the standard case (Fig. 10c) under different values of packet dropping probability.

8

CONCLUSION

AND

FUTURE WORK

A novel queuing analytical framework has been presented for performance evaluation of distributed and energyaware wireless MAC protocols under nonsaturation condition and with differentiated services between two types of services. The presented analytical framework is very general and comprehensive in that it considers the Markovian arrival process, phase-type distribution for service time, and phase-type distribution for vacation period with two types of service disciplines, namely, the standard exhaustive and the number-limited exhaustive both in multiple vacation. In this general analytical framework, the MAC layer access delay (e.g., time required for channel sensing and contention resolution) is taken into account through setup time, the distribution of which is phase-type. Impact of nonideal wireless channel condition has been also taken into account. The trade-off between the QoS performances for the high-priority packets (i.e., queue length distribution, distribution of access delay, packet dropping probability and throughput) and the energy saving performance at a node has been analyzed under different system parameter settings. Although the analyses presented in this paper are for the high-priority packets, similar analyses can be also performed for the low-priority packets.

We have assumed that packet arrival process remains unaffected when a node goes into the sleep mode. In a practical scenario, the node may turn off the transceiver circuitry so that there is neither transmission nor reception activity in the node. The presented model can be extended to consider this case as well. In the presented model, we have assumed that the node switches from the active mode to sleep mode when the queue becomes empty and it switches from sleep mode to active mode as soon as the queue becomes nonempty. However, this may not be an efficient strategy when the transition/switching overheads are taken into account. By using a threshold-based mechanism for mode switching, the transition overhead can be reduced. For example, a node may decide to wakeup only when at least a threshold number of packets are available in the queues. This situation can be modeled by following the procedures used to model the return policies, namely, N-policy or NT-policy in a queuing system [40]. In addition, a more realistic wireless channel model, such as the Nakagami fading channel, and also other physical layer aspects, such as adaptive modulation and coding, can be integrated into the model.

ACKNOWLEDGMENTS This work was supported by a grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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[32] J. Misic and V.B. Misic, “Duty Cycle Management in Sensor Networks Based on 802.15.4 Beacon Enabled MAC,” Ad Hoc and Sensor Wireless Networks J., vol. 1, no. 1, 2005. [33] N. Akar, N.C. Oguz, and K. Sohraby, “Matrix-Geometric Solutions of M/G/1-Type Markov Chains: A Unifying Generalized StateSpace Approach,” IEEE J. Selected Areas in Comm., vol. 16, no. 5, June 1998. [34] U. Vornefeld, “Analytical Performance Evaluation of Mobile Internet Access via GPRS Networks,” Proc. European Wireless Conf., http://docenti.ing.unipi.it/ew2002/proceedings/108.pdf, Feb. 2002, [35] M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins Univ. Press, 1981. [36] C. Xiaoming and H. Geok-Soon, “A Simulation Study of the Predictive p-Persistent CSMA Protocol,” Proc. IEEE 35th Ann. Simulation Symp., 2002. [37] S. Rakshit and R.K. Guha, “Fair Bandwidth Sharing in Distributed Systems: A Game-Theoretic Approach,” IEEE Trans. Computers, vol. 54, no. 11, pp. 1384-1393, Nov. 2005. [38] R. Hooke and T.A. Jeeves, “Direct Search Solution of Numerical and Statistical Problems,” J. ACM, vol. 8, pp. 212-229, Apr. 1961. [39] E.K.P. Chong and S.H. Zak, An Introduction to Optimization, second ed. John Wiley and Sons, 2001. [40] A.S. Alfa and I. Frigui, “Discrete NT-Policy Single Server Queue with Markovian Arrival Process and Phase Type Service,” European J. Operational Research, vol. 88, no. 3, pp. 599-613, 1996. Afshin Fallahi (S’04) received the BSc degree in electrical engineering from the University of Tehran, Iran, in 1996. In 1999, he received the MSc degree in telecommunications engineering from Tarbiat Modarres University (TMU), Iran. Currently, he is a PhD student in the Electrical and Computer Engineering Department at the University of Manitoba, Winnipeg, Canada. His main research interests are in the areas of modeling, analysis, and optimization for wireless networks. He is a student member of the IEEE. Ekram Hossain (S’98-M’01-SM’06) received the PhD degree in electrical engineering from the University of Victoria, Canada, in 2000. He is currently an associate professor in the Department of Electrical and Computer Engineering at the University of Manitoba, Winnipeg, Canada. He was a University of Victoria Fellow and also a recipient of the British Columbia Advanced Systems Institute (ASI) graduate student award. Dr. Hossain’s research interests include radio link control, transport layer protocol design, and cross-layer optimization issues for wireless networks, cognitive radio networks, and mobile computing. Currently, he serves as an editor for the IEEE Transactions on Wireless Communications, the IEEE Transactions on Vehicular Technology, IEEE Wireless Communications, the IEEE/KICS Journal of Communications and Networks, Wireless Communications and Mobile Computing (Wiley InterScience), and the International Journal of Sensor Networks (Inderscience Publishers). He served as a technical program committee member for the IEEE WCNC ’07, ICC ’07, ICC ’06, ICC ’05, WCNC ’05, WCNC ’04, Globecom ’04, Globecom ’03, and IFIP Networking ’05, and as the technical program cochair for the Symposium on Next Generation Mobile Networks at the ACM International Wireless Communications and Mobile Computing Conference (IWCMC ’06), Vancouver, Canada, 3-6 July 2006. One of the papers coauthored by Dr. Hossain won the Best Student Paper Award at IWCMC ’06. He was a recipient of the Lucent Technologies, Inc. research award for his contribution to the IEEE International Conference on Personal Wireless Communications (ICPWC), 1997. He is a senior member of the IEEE.