Fast Node Cardinality Estiation and Cognitive MAC Protocol Design for Heterogeneous MM Networks Sachin Kada, Chaitanya S. Raut and Gaurav S. Kasbekar Abstract Machine-to-Machine MM networks are an eerging technology with applications in nuerous areas including sart grids, sart cities, vehicular teleatics, healthcare, security and public safety. In this paper, we design a ediu access control MAC protocol that supports ulti-channel operation for a heterogeneous MM network, with three types of MM devices e.g., those that send eergency, periodic and noral type data, operating as a secondary network using Cognitive Radio technology. Also, we design an estiation protocol for rapidly obtaining separate estiates of the nuber of active nodes of each traffic type, and use these estiates to find the optial contention probabilities to be used in the Cognitive MAC protocol. We copute a closed for expression for the expected nuber of tie slots required by our estiation protocol to execute as well as a siple upper bound on it, which shows that the expected nuber of tie slots required by our protocol to obtain the above estiates is sall. Also, we atheatically analyze the perforance of the Cognitive MAC protocol and obtain expressions for the expected nuber of successful contentions and the expected aount of energy consued per frae. Finally we evaluate the perforance, in ters of average throughput and average delay, of our MAC protocol using siulations. I. Introduction Machine-to-Machine MM counications is an eerging technology, in which data generation, processing and transission is done with inial huan intervention [1]. MM networks have applications in nuerous areas including sart grids, sart cities, vehicular teleatics, healthcare, industrial autoation, security and public safety [1] [3]. It is challenging to design ediu access control MAC protocols for MM networks due to their unique characteristics such as liited access to energy sources ost MM devices are battery operated, need to provide network access to a very large nuber of devices, the fact that the Quality of Service QoS requireents of MM devices differ fro those of Huanto-Huan HH counications and are also different for different MM devices 1 etc [1], [3], [4]. Several wireless technologies such as Bluetooth, Wi-Fi, ZigBee and cellular networks including LTE-Advanced and 8.16 are potential candidates for enabling MM counications; however, these technologies have soe shortco- The authors are with the Departent of Electrical Engineering, Indian Institute of Technology IIT Bobay, Mubai 476, India. Their eail addresses are {sachink, csraut, gskasbekar}@ee.iitb.ac.in. 1For exaple, soe MM nodes need to transit data e.g., sart eter readings periodically, soe need it for sending eergency or alar essages e.g., in healthcare and security applications, soe need it for transission of noral data traffic and soe need it for reliable transission of data packets e.g., in reote payent gateway systes [1], [3], [4]. ings [5]. Cognitive Radio technology is a proising alternative to the above wireless technologies for enabling MM counications []. Cognitive Radio Networks CRNs have eerged as a proising solution to alleviate the artificial spectru scarcity wherein ost of the usable radio spectru is allocated, but underutilized caused by the traditional spectru regulation policy of assigning exclusive licenses to users to operate their networks in different geographical regions [6]. In CRNs, there are two types of spectru users priary users PUs, which have prioritized access to channels, and secondary users SUs that detect and use spectru holes, i.e., chunks of spectru that are currently not in use by the PUs [6]. Operating an MM network as a secondary network using Cognitive Radio technology has the advantage that a large aount of spectru, which is allocated to other users, but underutilized, becoes available for MM counications [5]. However, this requires the design of efficient Cognitive MAC protocols in order to provide channel access to an extreely large nuber of MM devices, while satisfying the unique service requireents of MM applications described in the first paragraph of this section, as well as ensuring avoidance of interference to PUs. The design of a Cognitive MAC protocol that supports ulti-channel operation involves addressing additional challenges [7] including achieving coordination aong nodes 3, overcoing the ulti-channel hidden terinal proble [8], and balancing the traffic load of the secondary MM nodes over the free channels in real-tie [9]. In this paper, we design a Cognitive MAC protocol for MM networks that overcoes the above challenges. Now, consider an MM network in which a large nuber of MM devices interittently transit soe inforation e.g., sart eter readings, inforation collected by sensors to a base station BS. In any given tie frae, the BS is unaware of the nuber of active nodes, i.e., those that need to transit soe data to the BS in the current frae. There is a need to rapidly estiate the nuber of active nodes since this estiate can be used to deterine the optial values of various paraeters of the MAC protocol such as contention probabilities and the aounts of tie to be used for contention and for data transission in the current frae. For exaple, recall that for the Slotted ALOHA protocol, the optial contention probability is the reciprocal of the nuber of active nodes [1]. Also, in [11] [13], cardinality estiation Specifically, Wi-Fi has high power consuption, due to which it is not suitable for battery operated MM devices, and Bluetooth has high latency when the nuber of devices is large, as is the case in MM networks [5]. ZigBee operates on unlicensed bands and is prone to interference fro Wi-Fi networks and other equipent e.g., icrowave ovens that use those bands [], [5]. Due to the high deand for HH counication services such as voice, video, eails etc, only a liited aount of radio spectru is available with cellular operators to support MM counications []. 3Note that for two nodes to be able to exchange data, both ust have their wireless transceiver tuned to a coon channel at a tie.
is perfored and using the estiates obtained, the contention probabilities that axiize the throughput of their respective MAC protocols for MM networks are deterined. In a heterogeneous MM network, i.e., one in which different types of nodes are present e.g., those that send eergency, periodic and noral type data, we need to obtain separate estiates of the nuber of active nodes of each traffic type. In prior work, several protocols have been designed [11] [13] to estiate the nuber of active nodes of a hoogeneous MM network see Section II. However, to the best of our knowledge, so far no estiation protocol has been designed for obtaining separate estiates of the nuber of active nodes of each traffic type in a heterogeneous MM network. Note that executing a cardinality estiation protocol for a hoogeneous MM network ultiple ties to do this is inefficient. In this paper, we consider an MM network with three types of nodes, which we refer to as Type 1, Type and Type 3 nodes; e.g., these ay be eergency, periodic and noral data type nodes. We design an estiation protocol to rapidly obtain separate estiates of the nuber of active nodes of each traffic type see Section III. We copute a closed for expression for the expected nuber of tie slots required by our estiation protocol to execute as well as a siple upper bound on it, which shows that the expected nuber of tie slots required by our protocol to obtain the above estiates is sall. Next, we use our estiation protocol as part of a Cognitive MAC protocol that we design for heterogeneous MM networks see Section IV. In the proposed MAC protocol, tie is divided into fraes of equal duration, with each frae containing an estiation window, a contention window CW and a data transission window DTW. Whenever a node succeeds in contention on a given channel during the CW, the BS reserves the requested nuber of tie slots for data transission by that node in the DTW. Slotted ALOHA [1] is used for contention in the CW, and the contention probability used by each node is the reciprocal of the estiated nuber of contending nodes on the channel; thus, the estiates obtained using our estiation protocol are used for optiizing the contention probabilities. We atheatically analyze the perforance of the proposed MAC protocol and obtain expressions for the expected nuber of successful contentions and the expected aount of energy consued per frae see Section V. Finally, using siulations, we evaluate the perforance, in ters of average throughput and average delay, of our MAC protocol and copare it with that of a hypothetical ideal protocol, which is assued to know the exact nuber of active nodes at any tie see Section VI. II. Related Work A schee to estiate the nuber of active nodes in MM networks is proposed in [1]. In the proposed ethod every device selects a slot uniforly at rando fro a set of slots and transits a Power Save-poll essage. The access point AP estiates the nuber of nodes by using the nuber of epty slots with the axiu likelihood ML estiation ethod. In [11] an iterative ethod is proposed for estiation of nodes and is obtained using drift analysis on the access results of consecutive epty and collision slots of the past slots. In [13] estiation of nodes is carried out in two phases. i Coarse phase: In this phase every node sends a busy tone with probability 1/ i in slot i. This process continues till all nodes stop sending busy tones in soe slot j. Average length of this phase is log N where N is nuber of active nodes. ii Refine phase: In this phase each node sends a busy tone with probability used by node to send last busy tone in the coarse phase. Average length of this phase depends on accuracy requireent. In this estiation schee, it uses only one control channel for node estiation due to which all other channels reain unused during estiation phase. In the proposed estiation schee, we use all the available channels for node estiation due to which the efficiency of the schee is iproved. Tag cardinality estiation ethods are extensively discussed in literature of Radio-frequency identification RFID systes. Most of the tag cardinality estiation schees ai to achieve the following accuracy requireent. Pr{ ˆn n εn} 1 δ, 1 where n, ˆn are actual and estiated nuber of tags respectively, ε is confidence interval and δ is error probability. The schee proposed in [14] tags select a slot uniforly at rando and transit a packet in the selected slot. The nuber of epty slots and collision slots are counted by the reader server and these values are used to obtain the zero estiator ZE and collision estiator CE respectively. More accurate aong ZE and CE is chosen as the estiator of tags and it is called as unified probabilistic estiator UPE. Liitations of this schee are all tags ust be readable in any given iteration and approxiate nuber of tags need to infor the server. These liitations are addressed in [15], which uses only epty slots for the cardinality estiation process and estiator is called as enhanced zero based estiator EZB. In both the described estiation algoriths require large nuber of slots. Energy efficient cardinality estiation process is proposed in [16]. At every polling server sends a request packet with contention probability p along with frae size f. Any user can poll in one of the slots with contention probability p. Polling stops when the accuracy requireent described in 1 gets fulfilled. Three different cardinality estiation algoriths; Maxiu Likelihood Estiation Algorith MLEA, Average Su Estiation Algorith ASEA and Enhanced MLEA are proposed in [16]. In MLEA optiised contention probability p i = 1.594/ ˆn i 1 and in ASEA p i = 1/ ˆn i 1, where ˆn i is estiated nuber of tags in i th iteration, are used. Basic cardinality estiation algorith presented in [17] require Olog n slots. Optiized version of the sae algorith require Olog log n slots which also satisfy accuracy requireent of 1. In [18] RFID tag cardinality estiation schee is based on new distinct eleent counting ethod described in [19]. Every tag has a counter corresponding to a rando nuber. After every tie slot counter decreases by one and tag transits when counter becoes zero. Here server need not scan the entire estiation window. Server infors frae length f to every node. It awaits response fro every node. If k be nuber of waiting slots then k + log f are required nuber of slots to achieve the accuracy described in 1. Reader observes the position of epty and non epty 1 slots. First Non-Epty slots Based FNEB estiator is used to estiate the nuber of tags. 7 ties faster estiation schee than UPE and EZB is proposed in [] and it is called Average Run based Tag ART cardinality estiation ethod which uses slotted ALOHA
protocol. Reader sends a packet to all tags indicating frae size. Every tag picks a slot randoly to respond. Epty and non epty 1 responses are collected by reader, cardinality estiation is done based on the average run size of 1s in the obtained binary sequence. However, all the node cardinality estiation schees studied in the above papers are for a hoogeneous network, wherein all nodes are alike. In contrast, in this paper, we propose an estiation schee for a heterogeneous network with three different types of nodes, which efficiently coputes separate estiates of the nuber of active nodes of each type. Extensive surveys related to MAC protocol design for MM networks are provided in [3], [1]. In [], a hybrid MAC protocol that uses contention-based channel access CSMA/ CA when the network load is low and reservation-based access when the load is high is proposed. In [3], a hybrid MAC protocol, in which each tie frae consists of a contention period followed by a transission period, is proposed. The devices that successfully contend in the contention period are assigned a tie slot for data transission in the transission period. Extension of protocol proposed in [3] is extended to heterogeneous MM networks in [4], where service requireents of different devices are different along with different priorities. In [8], the MAC protocol is proposed for ultichannel ad hoc networks and this is odified in [13] to suite for MM networks. Tie is divided into fraes and they are further divided into 3 phases naely estiation phase, contention phase and data transission phase. Nuber of active users are estiated in estiation phase. In the contention phase, all active users tune to a coon control channel and contend for channel access with contention probabilities which are obtained with the help of nuber of estiated nodes. The nodes which are successful in contention transit their data packets in data transission phase siultaneously on different channels. In the protocol proposed in [5], tie is divided into slots, and in each slot, nodes contend with contention probability which is the statistical estiate of present traffic load, using a Request to Send RTS packet and it is responded with Clear to Send CTS packet, followed by transission of a data packet. In [1], the 8.11ah MAC protocol is odified for MM counications as follows: first estiation of the nuber of active MM devices is done and this estiation is used to adapt the length of the Restricted Access Window, in which only MM devices are allowed to contend. In [11], a odified version of the Slotted-ALOHA schee is presented, in which results of the previous slots are considered to estiate the transission attept probability of the current slot which results in axiising the throughput. In [6], an overload control echanis is presented for MM counication in LTE-Advanced networks, in which based on the traffic load on the rando access channel RACH base station adjusts the nuber of RACH resources. However, to the best of our knowledge, our Cognitive MAC protocol is the first to eploy separate estiates of the nubers of active nodes of different types for selecting the optial contention probabilities in a heterogeneous MM network. III. Fast Node Cardinality Estiation Schee In this section, we present our fast node cardinality estiation schee for heterogeneous MM networks. The estiation Type 1 nodes Base Station Type nodes Type 3 nodes Figure 1. The figure shows a base station and three types of nodes within its range the area inside the circle. proble is defined in Section III-A. In Section III-B, the Lottery Frae LoF based protocol [7], [8], which is a cardinality estiation schee for hoogeneous networks, and which we extend to estiate cardinalities in heterogeneous MM networks, is briefly described. Our proposed fast estiation schee is described in Section III-C. A closed for expression for the expected nuber of tie slots required by our estiation protocol to execute is coputed in Section III-D and a siple upper bound on it is established in Section III-E. A. The Estiation Proble Consider a heterogenous MM network consisting of a base station BS and three different types, say Type 1, Type and Type 3, of MM devices nodes in its range as shown in Figure 1. We denote the sets of nodes of Type 1, Type and Type 3 as N 1, N and N 3 respectively; let 4 N b = n b, b {1,,3}. Tie is divided into fraes of equal duration, and in each frae only a subset of the nodes of each type are active, i.e., have data to send to the BS. Also, each frae is divided into tie slots of equal durations. Let n b be the nuber of active nodes of Type b, b {1,,3}, in a given frae. Our objective is to design an estiation protocol to estiate the values of n 1, n and n 3 rapidly, i.e., using a sall nuber of tie slots. B. Review of the LoF Based Protocol The LoF based estiation protocol was designed in [7] and uses the probabilistic bitap counting technique proposed in [8] for tag cardinality estiation in RFID systes. The LoF based estiation protocol is designed for a hoogeneous network. Our proposed protocol extends the LoF based protocol to a heterogeneous network with three types of nodes. So we provide a brief review of the LoF based protocol in this subsection. Every tag equivalent to a node in MM networks has a unique binary identification ID nuber that is l bits in length. The hash value h of any tag is defined as the position 4 A denotes the cardinality of set A.
S 1 S S 1 1 S 1 S t 1 1 S t 1 BP B B1 Bt 1 1 K 1 R 1 st Phase nd Phase 3 rd Phase Figure. The figure shows the structure of the Estiation Window. BP denotes a Broadcast Packet. of the least significant zero bit in its ID. For exaple, h111 = 1 and h11111 = 4, where hi denotes the hash value corresponding to ID I. So if h is the hash value of a rando tag, then assuing that each of the l bits of the corresponding ID independently equal or 1 with probability 1/ each, Ph = i = 1/ i+1, i =,1,,...l 1 and 5 Ph = l = 1/ l. Now, tie is divided into slots of equal duration. During the estiation process, each active tag with hash value h transits a packet in the h th tie slot, for h =,1,,...,l. A corresponding bitap BM of s and 1s is generated by the RFID syste reader equivalent to the BS in MM networks based on the slot results; the h th bit of the BM is if the h th tie slot is epty i.e., one in which no node transits and 1 if the h th tie slot is non-epty i.e., one in which one or ore nodes transit. Let ρ = in{h BMh = }; then the estiated value of n the actual nuber of active tags is ˆn = 1.897 ρ [7]. It is also proved in [7] that the LoF based protocol executes within log n all slots, where n all is the total nuber of all possible binary IDs. Note that if the LoF based protocol is executed thrice to separately estiate n 1, n and n 3 in the network odel for MM networks described in Section III-A, then log n 1,all n,all n 3,all slots are required, where n b,all is the total nuber of all possible binary IDs of the b th type of nodes. To reduce the nuber of slots, we propose a fast estiation schee, which is described in the following subsection. C. Proposed Estiation Schee The estiation process is carried out in three phases, which we describe in Sections III-C1, III-C and III-C3. We refer to the set of slots used during the estiation process as the Estiation Window EW. The structure of a typical EW is shown in Figure. At the end of the estiation process, separate estiates, say ˆn 1, ˆn and ˆn 3, of the nuber of active nodes of the three types, n 1,n and n 3 see Section III-A, are obtained. For each b {1,,3}, the estiate ˆn b is equal to and hence, as accurate as the estiate of n b that would have been obtained if the LoF protocol [7], [8] were used for the estiation. However, note that the total nuber of tie slots used in our estiation schee is uch saller than the nuber of tie slots that would have been required if the LoF protocol were separately executed thrice to estiate n 1,n and n 3. At a high level, our estiation schee operates as follows. Let t = log axn 1,all,n,all,n 3,all. Also, for b {1,,3} and i {,1,...,t 1}, let B p b,i be 1 respectively, if the i th slot would have been non-epty respectively, epty if the LoF protocol were used to estiate the nuber of active nodes of Type b. Fro Section III-B, it is clear that if the bit patterns B p b,i, b {1,,3}, i {,1,...,t 1}, are known, then the 5If all the bits of the ID are 1, then its hash value is defined to be l. BP LoF estiates ˆn 1, ˆn and ˆn 3, of n 1,n and n 3 respectively, can be deduced. In our estiation schee, the bit patterns B p b,i, b {1,,3}, for ost values of i are found in the first phase; abiguity about the rest reains, which is resolved in the second and third phases. 1 First Phase: In the first phase, t slots are used. Every two consecutive slots constitute a block see Figure ; let B i denote the i th block. In block B i, i {,1,...,t 1}, active nodes fro N 1 whose hash value is i send a packet containing the sybol α in both the slots of B i. Also, in block B i, active nodes fro N respectively, N 3 whose hash value is i send a packet containing the sybol β only in the first slot respectively, only in the second slot of B i. So every slot has four possible outcoes, which are as follows: i Epty E if no node transits in the slot, ii Collision C if two or ore nodes transit, iii α if exactly one node of Type 1 transits, iv β if exactly one node of Type or Type 3 transits. The possible outcoes in a block are shown in the first two coluns of Table I 6. Note that for b {1,,3}, i {,1,...,t 1}, B p b,i equals 1 respectively, if and only if atleast one node respectively, no node of Type b transits in block B i. The bit patterns B p b,i, b {1,,3} corresponding to each possible block outcoe are shown in the last three coluns of Table I. For exaple if Slot1 results Table I. Outcoe in Block i Bit patterns Slot1 Slot B p 1,i B p,i B p 3,i E E E C 1 E β 1 C E 1 C C C α 1 1 C β 1 1 α C 1 1 α α 1 β E 1 β C 1 1 β β 1 1 E, C and denote Epty, Collision and abiguous result respectively. in C and Slot results in α, then it iplies that exactly one node fro N 1, at least one node fro N and none fro N 3 have transitted. Siilarly if both the slots result in β, then it iplies that exactly one node each fro N and N 3, and none fro N 1 have transitted. The outcoe C, C in which collisions occur in both the slots ay be due to transissions by atleast two nodes fro N 1, or by one node fro N 1 and at least one node each fro N and N 3, or by atleast two nodes each fro N and N 3. Due to the above abiguity, if the outcoe C, C occurs in the i th block, then the second phase is used to find the bit patterns B p 1,i, B p,i and B p 3,i. Let C I be the set of block nubers i in which the abiguous outcoe C, C has occurred. A broadcast packet BP is sent by the BS after the first phase see Figure, which contains a list of the block nubers in set C I. Second Phase: In the second phase, only the nodes fro N 1 whose hash value belongs to the set C I participate. Specifically, for each j = 1,,..., C I, in the j th slot of the second phase, the nodes fro N 1 whose hash value equals the block nuber of the j th block whose outcoe was C, C 6Note that the block results E, α, α, E, α, β and β, α cannot occur under the above protocol.
in the first phase transit. Nodes fro N and N 3 do not transit in the second phase. Now, consider the slot in the second phase corresponding to the i th block in the first phase, where i C I. If the slot result is epty, then it follows that B p 1,i =, B p,i = 1 and B p 3,i = 1; also, if the slot result is one packet transission, then B p 1,i = 1, B p,i = 1 and B p 3,i = 1. If the slot result is C, then B p 1,i = 1; however, abiguity about the values of B p,i and B p 3,i still reains and it is resolved in the third phase. Let C II C I be the set of block nubers i for which a collision occurred in the second phase. A BP is sent by the BS after the second phase see Figure, which contains a list of the block nubers in set C II. 3 Third Phase: In this phase, only those active nodes fro N and N 3 participate, whose corresponding blocks in the first phase and corresponding slots in the second phase resulted in collisions. That is, the active nodes fro N and N 3 whose hash value belongs to C II participate. The odd respectively, even nubered slots of the third phase are used by nodes fro N respectively, N 3. Specifically, for each j = 1,,..., C II, in slot j 1 respectively, j of the third phase, the active nodes fro N respectively, N 3 whose hash value equals the first phase block nuber, say i, of the j th eleent of C II transit. If slot j 1 is epty, then B p,i =, else B p,i = 1. Siilarly, if slot j is epty, then B p 3,i =, else B p 3,i = 1. Also, since B p 1,i = 1, the above abiguity is resolved in the third phase. D. Deterination of Expected Nuber of Tie Slots Required by Estiation Protocol to Execute Recall that t slots are required in the first phase. Let K respectively, R be the nuber of slots required in the second phase respectively, third phase. 1 Deterination of E[K]: Note that K t. Let S i 1 respectively, S i represent the result of the first respectively, second slot of B i. Also, let I F denote the indicator rando variable corresponding to event F, i.e., { 1, if F occurs, I F =, else. Clearly, K = t 1 I {S i So: 1 =C,Si =C}. t 1 E[K] = PS i 1 = C,Si = C. The conditions under which collisions occur in both the slots of B i are as follows: 1 At least two nodes fro N 1 transit in B i, Exactly one node fro N 1 and at least one node each fro N and N 3 transit in B i, 3 At least two nodes each fro N, N 3 and none fro N 1 transit in B i. Let Q 1 i, Q i and Q 3 i denote the probabilities of the events in 1, and 3 respectively. Then: PS i 1 = C,Si = C = Q 1i + Q i + Q 3 i. 3 It is easy to show that Q 1 i = 1 un 1 vn 1, Q i = vn 1 1 un 1 un 3 and Q 3 i = un 1 1 un vn 1 un 3 vn 3, where un = 1 1/ i+1 n and vn = n1/ i+1 1 1/ i+1 n 1. By and 3, the expected nuber of slots required in the second phase is: t 1 E[K] = {Q 1 i + Q i + Q 3 i}. 4 Deterination of E[R]: Note that R K. It is easy to show that: t 1 E[R] = Q 1 i. 5 The expected total nuber of slots required by the estiation protocol to execute is t + + E[K] + E[R] see Figure, where E[K] and E[R] are given by 4 and 5 respectively. E. Upper Bound on Expected Nuber of Tie Slots Required by Estiation Protocol to Execute Although the expressions derived in Section III-D are exact, they are coplicated. So to gain insight, in this subsection, we provide siple upper bounds on E[K] and E[R] and use the to obtain an upper bound on the expected total nuber of slots required by the estiation protocol to execute. Let n = axn 1,n,n 3, x = the sallest integer greater than or equal to x, and l y = log y. Theore 3.1: E[K] l n 1 + n 1 3n [ 1 + n n 3 + 1n n 3 5n 1 n 7n 1 n ]. Theore 3.: E[R] l n 1 + n 1. 3n The proofs of Theores 3.1 and 3. are provided Appendix. As an exaple, consider the case where n 1 = n = n 3 = n say and n 1,all = n,all = n 3,all. Then the expected total nuber of slots required by the estiation protocol to execute is bounded by t + + E[K] + E[R] t + + l n + 1.76 + l n.333 = t + 3l n +.41. Hence, the nuber of tie slots saved copared with the case where the LoF protocol is executed thrice to separately estiate n 1, n and n 3 is at least 3t t + 3l n +.41 = t 3l n.41. IV. Cognitive MAC Protocol for Multi-channel MM Networks A. Overview For concreteness, we henceforth assue that Type 1, Type and Type 3 nodes are eergency, periodic and noral data nodes respectively. Tie is divided into fraes of equal durations. Let M T be the total nuber of channels and q i be the probability that a priary user PU is present on channel i {1,...,M T } in any given frae 7. Also, in a frae, suppose there are M f free channels, say {a 1,a,...,a Mf }; then out of these, we use the M fe channels with the lowest values of q i for eergency data traffic, the M fp channels with the next lowest values of q i for periodic data traffic and the rest for noral data traffic, for soe M fe,m fp. We assue that all MM devices are in the range of the base station BS see Figure 1. In each frae, only the BS senses the M T channels 7We assue that the probabilities q i are known to the base station; for exaple, they can be estiated using past observations of PU occupancies on different channels.
a1 UL DL DL CW1 DTW1 a1 UL DL DL CW DTW a UL DL DL CW3 CWMf DTW3 DTWMf amf UL DL DL SW BW1 EW BW CDTW Figure 3. The figure shows the structure of a frae. Only the free channels are shown. a 1 1 M f + 1..... a M f +.................... R s 1....... R s.............. a Mf M f M f.... Figure 4. The figure shows the schee used for nubering the reserved R s slots in the EW. The first slot of channel a 1 is nubered 1, the first slot of channel a is nubered,..., the first slot of channel a Mf is nubered M f, the second slot of channel a 1 is nubered M f + 1 and so on. to check for the presence of PUs 8. Figure 3 shows the structure of a frae. The BS senses every channel in the sensing window SW to check for the presence of PUs. In the first broadcast window BW 1, all the active nodes receive the list of channels that are free in the current frae fro the BS see Section IV-B. The fast node cardinality estiation schee described in Section III is executed in the estiation window EW to estiate the nuber of active nodes of each type see Section IV-C. In the second broadcast window BW, the list of channels assigned to each type of node and the optial contention probabilities which are coputed using the estiates obtained in the EW are broadcast by the BS see Section IV-D. In the Contention and Data Transission Window CDTW, active nodes contend on the channels assigned to the using Slotted ALOHA [1]; upon each successful contention, the BS reserves the requested nuber of slots for data transission by the node in the DTW see Section IV-E. Periodic nodes require channels for periodically transitting data. In particular, when a periodic node r with T r data packets contends successfully, the BS reserves one slot each in T r successive fraes for data transissions by node r. Node r does not participate again in the contention process in these T r fraes. B. First Broadcast Window BW 1 The BS and every node store the list of all channels, sorted in increasing order of q i. In BW 1, the BS repeatedly broadcasts a packet on the first free channel say f of the above list; this packet contains the list of channels that are free in the current frae. Each active node tunes to channels in increasing order of q i, listening for one tie slot on each channel, until it tunes to channel f and receives the list broadcast by the BS. C. Estiation Window EW Recall that the fast node cardinality estiation schee described in Section III requires t ++ C I + C II = R s say slots to execute. R s slots are reserved 9 in the EW for the estiation process. In the EW, all the M f free channels in the frae are utilized for the estiation. The schee used for nubering the reserved R s slots in the ulti-channel environent is shown in Figure 4. 8Since MM devices are low-cost and battery-operated devices, our protocol does not require the to have sensing capability. 9Note that although the value of R s is not known in advance, after the first respectively, second phase of the estiation schee, the BS can find the value of C I respectively, C II see Sections III-C1 and III-C. So the inforation required to reserve R s slots is available with the network. D. Second Broadcast Window BW After the EW, the BS knows the estiated nubers of active nodes with eergency ˆn e, periodic ˆn p and noral ˆn n data packets. Based on the values of ˆn e, ˆn p and ˆn n, out of the M f free channels, M fe, M fp and M fn channels are assigned to eergency, periodic and noral data nodes respectively, where M f = M fe + M fp + M fn ; the BS broadcasts the lists of channels assigned to each type of node in BW. For instance, let w e,w p and w n be weights positive real nubers associated with the eergency, periodic and noral data classes respectively. Then M fe,m fp and M fn ay be selected to be approxiately ˆn e w e M f ˆn e w e + ˆn p w p + ˆn n w n, ˆn p w p M f ˆn e w e + ˆn p w p + ˆn n w n ˆn and n w n M f ˆn e w e + ˆn p w p + ˆn n w n respectively. We use w e w p w n to ensure that eergency respectively, periodic data is provided a higher priority than periodic respectively, noral data. To balance the load across the assigned channels, each eergency respectively, periodic, noral node selects one channel fro the M fe respectively, M fp, M fn channels at rando and tunes to it in the CDTW. Now, recall that if n nodes contend using Slotted ALOHA, then the value of the contention probability p that axiizes the throughput is p = 1/n [1]. So the BS sets the probabilities of contention for eergency, periodic and noral data nodes to ˆp e = inm fe / ˆn e,1, ˆp p = inm fp / ˆn p,1 and ˆp n = inm fn / ˆn n,1 respectively and broadcasts the values of ˆp e, ˆp p and ˆp n in BW. Finally, there ay be soe periodic data nodes with tie slots in the DTW of the current frae reserved in past fraes; a packet containing a list of such reserved slots is also broadcast by the BS in BW. E. Contention and Data Transission Window CDTW After BW, all nodes switch to their respective selected channels for contention and data transission. Every channel in this window is divided into a Contention Window CW and a Data Transission Window DTW of variable lengths see Figure 3. Each active node fro N 1 contends using Slotted ALOHA [1] with contention probability ˆp e in the first slot of the CW on its channel, which is an uplink UL slot, and listens to the channel for an acknowledgent ACK packet fro the BS in the second slot, which is a downlink DL slot. If a node successfully contends in the UL slot, then it is allotted the requested nuber of slots in the DTW by the BS, starting fro the rightost available slot of the current frae; the BS includes the list of allotted slots in the ACK packet that it broadcasts in the following DL slot. The node then switches to idle sleep state to save energy and wakes up to transit in its allotted slots in the DTW. Now, since the
UL DL UL DL UL DL 1 3 4 Figure 5. Contention Window T dm T d d d M M 1 1 Data Transission Window The figure shows the CDTW used in the analysis in Section V-A. nuber of contending nodes has reduced by 1, the BS odifies ˆp e to in1/ ˆn e /M fe 1,1 and broadcasts this value in the DL slot. In case of an unsuccessful contention collision or epty slot, the BS does not send any ACK. This process continues until the CW and DTW on that channel are separated by a single slot; then, the BS transits a broadcast packet inforing the reaining contending nodes to switch to idle state to save energy for the rest of the frae. However, if three successive UL slots are epty, then it is taken by the BS to be an indication that with a high probability all the active nodes on the channel have already successfully contended; in this case, the BS can allot the reaining free slots of the channel to active nodes of other channels. A siilar procedure is followed by active nodes fro N and N 3 on their selected channels with paraeter sets ˆp p, ˆn p and ˆp n, ˆn n respectively. V. Perforance Analysis In this section, we obtain closed-for expressions for the expected nuber of successful contentions and the expected aount of energy consued per frae under the Cognitive MAC protocol described in Section IV. A. Expected Nuber of Successful Contentions Here, we focus on only one channel and hence only nodes of a single type contend on it. Assue that n nodes of this type are active on the channel at the start of a given frae and let ˆn be the estiated value of n. Let the length of the CDTW of the frae be T slots. Let M be the nuber of successful contentions in the given frae. Recall that contentions occur only in UL slots. For tractability, we assue that upon every successful contention, the BS reserves a constant nuber, say d, of slots for the successful node fro the last available slot in that frae as shown in Figure 5. If no successful contentions take place in the frae, then M = and if all contentions are successful, then M = T / + d since M + dm = T. So M T / + d. For each x, let T x =.5T xd. By definition: T /+d EM = = PM = 6 Let S x be the event that a successful contention occurs when x nodes contend. Let PS x = r x and p x be the contention probability used when x nodes contend. In the proposed protocol, p n j = in1/ ˆn j,1, j =,1,... see Section IV-E. We now find the distribution of M. There are M = successful contentions if and only if for soe integers k 1,k,...,k, the first k 1 1 contention attepts are unsuccessful with n contending nodes and the k th 1 attept is successful, k 1 + 1 th to k 1 th attepts are unsuccessful with n 1 contending nodes and k th attept is successful,..., and k + 1 th to T th attepts are unsuccessful with n contending nodes. So: PM = = T +1 k 1 =1 T + k =k 1 +1 T + j+1... k j+1 =k j +1... T k =k 1 +1 1 r n k 1 1 r n 1 r n 1 k k 1 1 r n 1... 1 r n j k j+1 k j 1 r n j...1 r n T k. Note that r n j = n jp n j 1 p n j n j 1. EM can be obtained fro 6 and 7. B. Expected Aount of Energy Consued per Frae Let γ I,γ T,γ R be the energy spent by a node per slot in the idle state, transission state and reception state respectively. Let us classify the slots in the given frae into uplink slots, downlink slots and data transission slots; let the total energy spent by all the active nodes in the be E UL,E DL and E DT respectively. So the total expected aount of energy spent per frae is EE UL + EE DL + EE DT. We copute EE UL, EE DL and EE DT in Sections V-B1, V-B and V-B3 respectively. 1 Energy Spent in UL Slots: Note that there are a total of T M uplink slots in the frae. In each of these slots, soe of the active nodes are in transission state and the rest are in idle state. So, E UL = T M i=1 L i γ T + n L i γ I, where L i is the nuber of nodes that transit in UL slot i, which depends on N i the nuber of contending nodes in UL slot i and p Ni. Taking expectations and conditioning on the values taken by M: EE UL = T /+d = + T EL i /M = γ T i=1 n EL i /M = So EL i /M = can be calculated as, EL i /M = = j= 7 γ I PM =. 8 n j l i PL i = l i /M =,N i = n j l i = PN i = n j/m = 9 Distribution of N i follows fro the faous Gabler s ruin proble [9], i.e., PN i = n j/m = = PN i 1 = n j/m = 1 P i S n j /M = + PN i 1 = n j + 1/M = P i S n j+1 /M = 1 where P i S x /M = = Probability of success with x nodes given M = in slot i and j =,1,...i 1. Now, P i S n j /M = = P im = /S n j P i S n j, 11 PM =
P i M = /S n j = T +1 k 1 =1 T + k =k 1 +1 T + j+1... k j+1 =k j +1 1 r n k 1 1 r n 1 r n 1 k k 1 1 r n 1... 1 r n j k j+1 k j 1...1...1 r n T k.... T k =k 1 +1 1 Average Throughput.4.3..1 Average Throughput v/s Eergency Nodes Noral Nodes Periodic Nodes Average Throughput.6.5.4.3..1 Average Throughput v/s Eergency Nodes Noral Nodes Periodic Nodes Where P i M = /S n j = for j. Now distribution of L i is given by, PL i = l = N i l p l Ni 1 p Ni Ni l. Let L i,n j = The nuber of nodes that transit in UL slot i given n j nodes, j =,1,,...,. So conditional distribution is given by, PL i = l i /M =,N i = n j = 1 PL i,n j = l i T PL i,n j = l i 13 By using 7, 8, 9, 1, 11, 1, 13, we get EE UL. Energy Spent in DL Slots: In these slots, contending nodes are in reception state and the rest are in idle state. So, E DL = T M i=1 N i γ R + n N i γ I. Taking expectations and conditioning on the values taken by M, we get: EE DL = T /+d = + T EN i /M = γ R i=1 n EN i /M = γ I PM =. 14 By definition of conditional expectation, EN i /M = = j= n jpn i = n j/m =. EE DL can then be found using 7, 1 and 14. 3 Energy Spent in DT Slots: Since only one node is transitting in these slots, all other nodes are in the idle state. So, E DT = dm γ T + n 1γ I and EE DT = dem γ T + n 1γ I. 15 Using 6 and 15, we can calculate EE DT. C. Expectation of the Efficiency with the Proposed Estiation Schee Suppose there are n active nodes of a class in a given frae and let ˆn be the estiated value of n obtained using the schee described in Section III-C. Also, suppose free channels are allocated to the class in the CDTW in the frae. Consider one of these channels and n n nodes select this channel. Recall that on this channel, each of the n nodes contends using slotted ALOHA with contention probability / ˆn. Fro [8], we know that ˆn = c ρ, where c = 1.897 and ρ is defined in Section III-B as the position of right ost zero in BITMAP. Fro [8], we get the distribution of ρ as, Pρ = k = Pρ k Pρ k + 1, 16 n where Pρ k = k j= 1v j 1 j and v j indicates k the nuber of one bits in the binary representation of j. We..4.6.8 1..4.6.8 1 Figure 6. The following paraeters are used in these plots: M T = 3, N = 5, k e = k p = k n = 1. In the left plot we use w e = w p = w n = 1 whereas in the right plot we use w e = 3, w p =, w n = 1. now find the expected value of efficiency, say η A, which is defined to be the probability of successful contention in the first UL slot of the frae. Note that η A is a easure of accuracy of the proposed estiation schee. So, η A = n ˆn 1 / ˆnn 1. 1 Eη A = n E = n r n 1 c ρ, c ρ 1 1 c ρ 1 n 1 c ρ By using 16 and 17, we can find Eη A. VI. Siulations Pρ = r. 17 In this section, we evaluate the perforance of the proposed Cognitive MAC protocol, in ters of average throughput and average delay, via siulations. Also, we copare the perforance of the proposed protocol with a hypothetical ideal protocol to find out how accurate the proposed estiation schee is. The ideal protocol is siilar to the proposed protocol, with the difference being that it is assued to know the exact nuber of active nodes at any tie 1. Let M T,M f, ˆn e, ˆn p, ˆn n,w e,w p,w n and q i be as defined in Section IV. At the beginning of each frae, data packets arrive at rando at each node; the nuber of packets that arrive at a node belonging to the eergency respectively, periodic, noral data class is a Poisson rando variable with ean e respectively, p, n. Also, each frae is divided into 5 slots and transission of a data packet takes 1 slot. An eergency respectively, noral node which has successfully contended for access during the CW can reserve at ost k e respectively, k n consecutive slots in the DTW. A periodic node can reserve one slot per frae for at ost k p consecutive fraes. The liits k e, k n and k p are iposed to ensure short-ter fairness in the transission opportunities that different nodes get. We consider a balanced load condition wherein there are an equal nuber, say N, of nodes of each type in the network and e = p = n = say. Figure 6 shows the average throughput per node versus for the three classes with different paraeter values. It can be seen that initially the average throughput for any given class equals the arrival rate, but after a particular value of, the average throughput saturates, which is the unstable 1Note that the ideal protocol is not practically ipleentable and is considered only for coparison with the proposed protocol.
region of operation. The left plot in Figure 6 shows that when w e = w p = w n = 1 and k e = k p = k n = 1, the average throughput curves of all three classes roughly coincide; this is because they are treated alike by the protocol. In contrast, the right plot in Figure 6 shows that in the unstable region, eergency respectively, periodic nodes achieve a higher average throughput than periodic respectively, noral nodes when w e = 3, w p =, w n = 1; this is because a higher weight results in ore channels being allocated to a class. Figure 7 shows the average throughput under the proposed protocol and the ideal protocol versus for the eergency and noral classes. These plots show that the perforance of the proposed protocol is close to that of the ideal protocol for both classes: in particular, in the unstable region of operation, on average, the proposed protocol achieves 87.5% respectively, 68% of the average throughput under the ideal protocol for the eergency respectively, noral class. Average Throughput Average Throughput for Eergency Nodes v/s 5 4 3 1 Proposed Protocol Ideal Protocol Arrival Rate 1 3 4 5 Average Throughput 5 4 3 1 Average Throughput for Noral Nodes v/s Proposed Protocol Ideal Protocol Arrival Rate 1 3 4 5 Figure 7. The following paraeters are used in these plots: M T = 3, N = 5, w e = 3, w p =, w n = 1, k e = k p = k n = 5. Figure 8 shows the average packet delay versus for the eergency and noral classes with different paraeter values. The left plot of Figure 8 shows that when w e = w p = w n = 1 and k e = k p = k n = 5, the average delay curves of the two classes roughly coincide; on the other hand, when the weights w e = 3, w p =, w n = 1 are used see the right plot of Figure 8, the average delay for eergency nodes is uch lower than that of noral nodes, which is because ore channels are allocated to eergency nodes. Average Delay 5 15 1 5 Average Delay v/s Eergency Nodes Noral Nodes.5 1 1.5.5 3 Average Delay 7 6 5 4 3 1 Average Delay v/s Eergency Nodes Noral Nodes.5 1 1.5.5 3 Figure 8. The following paraeters are used in these plots: M T = 3, N = 5, k e = k p = k n = 5. In the left figure we use w e = w p = w n = 1 whereas in the right figure we use w e = 3, w p =, w n = 1. VII. Conclusions and Future Work We designed a Cognitive MAC protocol for a heterogeneous MM network with three types of nodes. 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