Buffer-aided link selection with network coding in multihop networks

Transcription

1 Loughborough Unversty Insttutonal Repostory Buffer-aded lnk selecton wth network codng n multhop networks Ths tem was submtted to Loughborough Unversty's Insttutonal Repostory by the/an author. Ctaton: TIAN, Z....et al., Buffer-aded lnk selecton wth network codng n multhop networks. IEEE Transactons on Vehcular Technology, 65(9, pp Addtonal Informaton: c 2016 IEEE. Personal use of ths materal s permtted. Permsson from IEEE must be obtaned for all other uses, n any current or future meda, ncludng reprntng/republshng ths materal for advertsng or promotonal purposes, creatng new collectve works, for resale or redstrbuton to servers or lsts, or reuse of any copyrghted component of ths work n other works. Metadata Record: Verson: Accepted for publcaton Publsher: c IEEE Please cte the publshed verson.

2 Buffer-aded Lnk Selecton wth Network-codng n Mult-hop Networks Zhao Tan, Student Member, IEEE, Yu Gong, Gaoje Chen, Zh Chen, Member, IEEE and Jonathon Chambers, Fellow, IEEE Abstract Ths paper proposes a novel buffer-aded lnk selecton scheme based on network-codng n the multple hop relay network. Compared wth exstng approaches, the proposed scheme sgnfcantly ncreases the system throughput. Ths s acheved by applyng data buffers at the relays to decrease the outage probablty and usng network-codng to ncrease the data rate. The closed-form expressons of both the average throughput and packet delay are successfully derved. The proposed scheme has not only sgnfcantly hgher throughput than both the tradtonal and exstng buffer-aded max-lnk scheme, but also smaller average packet delay than the max-lnk scheme, makng t an attractve scheme n practce. Index Terms Mult-hop relay, lnk selecton, buffer-aded relay, network codng I. INTRODUCTION Relay network has been well nvestgated as an attractve scheme n wreless communcatons [1]. Current research manly focuses on the 2-hop relay network that every data packet takes two hops to transmt from the source to destnaton through a relay node [2] [4]. Relatvely less has been studed for relay networks wth more than two hops. The mult-hop relay network can be seen n many scenaros. A typcal example s the devce-to-devce (D2D communcatons n the cellular system, where some moble Z. Tan and Y. Gong are wth the Advanced Sgnal Processng Group, Loughborough Unversty, Loughborough, Lecestershre, UK, Emals: { z.tan,y.gong }@lboro.ac.uk. G. Chen s the correspondng author wth Insttute for Communcaton Systems (ICS, home of the 5G Innovaton Centre, Unversty of Surrey, Guldford, UK, E-mals: gaoje.chen@surrey.ac.uk. Z. Chen s wth the Natonal Key Laboratory of Scence and Technology on Communcatons, Unversty of Electronc Scence and Technology of Chna, Chengdu, Schuan , Chna, E-mal: chenzh@uestc.edu.cn. J. A. Chambers s wth the Communcatons, Sensors, Sgnal and Informaton Processng Group, Newcastle Unversty, Newcastle Upon Tyne, UK, E-mals: jonathon.chambers@ncl.ac.uk.

3 2 users may drectly communcate wth each other (D2D communcatons rather than through the base staton (cellular communcatons [5], [6]. Because the transmsson powers for the D2D moble users are usually strctly lmted to avod nterferng the base staton, mult-hop transmsson can be requred for D2D communcatons [7]. 1 Related work: Conventonally the mult-hop lnks are consecutvely selected for data transmsson. Recent research shows that applyng data buffers at the relays sgnfcantly mproves the transmsson performance. Due to the data buffers at the relays, when a data packet arrves at a node, t may not be mmedately forwarded to the next node. Instead, other lnks wth better sgnal-to-nose-rato (SNR may be selected for data transmsson. Ths so called adaptve lnk selecton ( [8], [9] s partcularly useful n the D2D cellular system, because the base staton often has the knowledge of the channel state nformaton (CSI of the D2D lnks to coordnate the nterference between the cellular and D2D communcatons. As a result, the base staton can always select the best lnk for data transmsson, rather than followng the conventonal hoppng sequence. Buffer-aded relay has attracted much attenton recently. Besde the aforementoned adaptve lnk selecton, buffer-aded relays have also been used n applcatons ncludng relay selecton [10] [13], cogntve rado networks [14] and physcal layer network securty [15]. Of partcular nterest s the maxlnk relay selecton scheme due to ts excellent outage performance [10]. In the max-lnk relay selecton, at any tme slot, the lnk wth the strongest channel SNR among all possble source-to-relay and relay-todestnaton lnks s always selected for data transmsson, leadng to the dversty order of 2N f the buffer sze s large enough (where N s the number of relays. The max-lnk scheme can be straghtforwardly used n the mult-hop lnk selecton, smply by selectng the lnk wth the hghest SNR among all possble mult-hop lnks at any tme slot. Of partcularly nterest s the average throughput of the mult-hop relay network whch s gven by η = R (1 P out, (1 where P out s the outage probablty of the system and R s the average data rate (wthout consderng the outage. It s known that the max-lnk scheme sgnfcantly reduces the outage probablty. However, the max-lnk scheme stll has the same data rate as the conventonal scheme, because n both schemes only one lnk s selected for data transmsson at any tme slot. On the other hand, snce the outage probablty tends to be zero when the SNR goes to nfnty, t s clear from (1 that the average throughput manly

4 3 depends on the average data rate at the hgh SNR range. Ths mples that the max-lnk selecton scheme manly mproves the system throughput at the low SNR range. On the other hand, t s well known that the physcal layer network codng can be used to ncrease the data rate of the two-way relay network, where two source nodes exchange data packets through a sngle relay node [16] [18]. To be specfc, n the physcal layer network codng scheme, the two sources can transmt packets to, or receve packets from, the relay node smultaneously. Thus the data rate can reach 1 packet per tme slot, rather than 1/2 n the conventonal approach. Ths encourages us to apply the physcal layer network codng n the mult-hop relay selecton to ncrease the data rate. Ths can be acheved by smultaneously selectng two or more lnks for data transmsson. As a result, the throughput of the mult-hop network at the hgh SNR range can be mproved. 2 Contrbuton: In ths paper, we propose a novel mult-hop lnk selecton scheme whch seamlessly ntegrates the max-lnk selecton and physcal layer network codng so that the average throughput s sgnfcantly mproved at both low and hgh SNR ranges. The man contrbutons of ths paper are lsted as follows: Proposng a novel buffer-aded network-codng lnk selecton scheme for the mult-hop relay network. The proposed scheme has sgnfcantly hgher throughput than exstng buffer-aded max-lnk scheme. Descrbng a new analyss tool to obtan the average throughput of the proposed scheme. Both the outage probablty and average data rate are successfully derved to obtan the average throughput of the proposed scheme. Frst, the outage probablty analyss s based on the Markov chan of the buffer states, whch s much more dffcult than those n exstng approaches (e.g. [10] due to the complcated lnk selecton rules. Partcularly, we descrbe a trells dagram to derve the transton probabltes between buffer states, based on whch the outage probablty s obtaned. Secondly, n the proposed mult-hop scheme, due to the smultaneous lnk transmsson, the calculaton of the average data rate s far from straghtforward. In ths paper, a trells dagram s descrbed to successfully obtan the average data rate. The analyss not only shows deep nsght n understandng the mult-hop relay network, but also provdes gudance n analyzng smlar systems. Dervng the closed-form expresson of the average packet delay of the proposed scheme. The average packet delay s an mportant ssue n buffer-aded schemes. The analyss shows that the proposed

5 4 scheme not only has larger throughput, but also shorter packet delay, than the max-lnk scheme, makng t an attractve scheme n practce. The remander of the paper s organzed as follows: Secton II shows the system model of the N-hop relay network; Secton III proposes the buffer-aded network-codng lnk selecton scheme; Secton IV and V analyze the outage probablty and average data rate of the proposed scheme respectvely; Secton VI analyzes the average packet delay; Secton VII shows smulaton results to verfy the proposed scheme; fnally Secton VIII concludes the paper. II. N -HOP RELAY NETWORK The system model of the N-hop relay network s shown n Fg. 1, where there are one source node (S, one destnaton node (D and (N 1 number of relay nodes (R 1,, R N 1. We assume that there are no drect lnks between two nodes separated by two hops or more, and all relays apply the decode-and-forward (DF protocol and operate n the half-duplex mode. Fg. 1. The system model of the N-hop relay network. For later use, the hoppng lnks are consecutvely named as lnk 1, lnk 2,, lnk N respectvely, as s shown n Fg. 1. The channel coeffcent and gan for lnk at tme slot t s denoted as h (t and γ (t = h (t 2 respectvely. We assume that all channel lnks are ndependent and dentcally dstrbuted (..d. Raylegh fadng 1, so that the channel gans γ (t are exponentally dstrbuted wth the same average gan as γ = E h (t 2 for all = 1,, N. We also assume wthout losng generalty that transmsson powers and all nose varances are normalzed to unty. The most straghtforward way to transmt packets through the N-hop network s to let lnk 1, lnk 2,, lnk N be consecutvely used for data transmsson. Ths so-called consecutve-hoppng scheme s used as a baselne to compare wth other schemes n the paper. If the transmsson rates at all nodes are the same as r t, the average data rate of the consecutve-hoppng scheme s gven by R (con hoppng = 1 N r t. (2 1 Whle the analyss n ths paper s based on the..d. channel assumpton, t can be generalzed to the case that every lnk has dfferent average channel gan.

6 5 In ths paper, we assume the channels are quas-statc so that the coeffcents reman unchanged durng one hop nterval but ndependently vary from one hop to another. We also assume that, when a lnk becomes outage, the packet wll be re-sent by the transmsson node correspondng to the lnk (rather than the source node S. For Raylegh fadng channels, the probablty that lnk becomes outage s gven by P out, = P (C < r t = 1 e γ (3 where = 2 rt 1, C = log(1 + γ whch s the nstantaneous capacty for lnk. Or the probablty that a packet takes k tme slots to successfully pass lnk s (P out, k 1 (1 P out,. Thus the average number of tme slots for a packet passng through lnk s gven by T = = k (P out, k 1 (1 P out, k=1 1 = 1 1 P out, e γ Because all channels are..d., the average number of slots for a packet passng through the overall N-hop network s N T. Then the average throughput for the consecutve hoppng scheme s obtaned as (4 η (con hoppng = Comparng (1, (2 and (5, we can have the outage probablty as r t = r t N T N e γ. (5 P (con hoppng out = 1 e γ. (6 III. BUFFER-AIDED LINK SELECTION BASED ON NETWORK-CODING In ths secton, we wll frst apply the buffers at the relays to reduce the outage probablty and use the physcal layer network codng to ncrease the data rate. We then propose a novel lnk selecton scheme for the mult-hop relay network by ntegratng the buffer-aded and network codng approaches. A. Decrease the outage probablty wth buffers at the relays The max-lnk relay selecton scheme descrbed n [10] can be straghtforwardly appled n the multhop lnk selecton. To be specfc, n the buffer-aded lnk selecton, every relay s equpped wth a data

7 6 buffer of the sze L. We assume that the relay R has buffer Q, where = 1,, N 1. At any tme slot, when a data packet arrves at a relay node, t s stored n the buffer. At the next tme slot, unlke the tradtonal scheme, the stored data packet s not necessarly forwarded to the next node. Instead the lnk wth the hghest SNR among all of the avalable lnks s selected for data transmsson. A lnk s consdered avalable f the buffers of the correspondng transmttng and recevng nodes are not empty and full respectvely. Thus n the max-lnk scheme, the lnk for data transmsson s selected as lnk = arg max lnk A {γ }, (7 where A s the set contanng all avalable lnks, and recall that γ s the nstantaneous channel SNR for lnk. Wthout losng generalty, we assume that the source S always have data to transmt and the buffer sze s n the unt of packet. Because one packet s transmtted at one tme slot at fxed rate, f an avalable lnk s selected, there must be a packet avalable for transmsson and the buffer at the recevng node s large enough to store the packet. In the max-lnk scheme, because only one lnk s selected for data transmsson at any tme, the average data rate s stll the same as that n the tradtonal scheme whch s gven by R (max lnk = 1 N r t, (8 Then the average throughput of the max-lnk scheme s gven by where P (max lnk out smlar analyss as those n [10]. η (max lnk = 1 ( N 1 P (max lnk out r t, (9 s the outage probablty of the max-lnk scheme whch can be obtaned by followng Because P (max lnk out < P (con hoppng out, the throughput of the buffer-aded max-lnk scheme s hgher than that of the tradtonal scheme. One the other hand, because the max-lnk and tradtonal schemes have the same data rate, and further notng that P (max lnk out 0 when SNR, the two schemes have smlar throughput when the SNR s hgh enough. Ths ndcates that the buffer-aded lnk selecton manly mproves the throughput at the low SNR range.

8 7 B. Increase the data rate wth network codng We suppose at one tme slot, all odd numbered lnks transmt data at the same tme, and at the next tme slot all even numbered lnks transmt smultaneously. Thus a relay node may receve data from both the prevous and next nodes. Wthout losng generalty, at tme t, we assume that node R receves data from ts prevous node a and next node b smultaneously. Then the receved sgnal at relay R at tme slot t s gven by y (t = h (t x a + h b, (t x b + n (t, (10 where x a and x b are the data packets transmtted from nodes a and b respectvely, h b, (t s the channel coeffcent for the b R lnk, and n (t s the nose at node R. It s clear from (10 that h b, (t x b forms the nter-relay nterference. Because x b s transmtted from R to node b prevously, t can be stored at R. Wth the prncple of physcal layer network codng ( [18], the nter-relay nterference can be completely removed from (10, so that the receved sgnal at R becomes y (t = h (t x a + n (t. (11 Therefore, wth the physcal layer network codng, all odd (or even numbered lnks can be used for data transmsson smultaneously wthout causng any nter-relay nterference. As an example, the network codng based transmsson scheme for the 4-hop relay network s shown n Fg. 2. Fg. 2. Network-codng based 4-hop relay transmsson. As s llustrated n Fg. 2, on average t only takes two hops to transmt one data packet from S to D, no matter how many hops there are n the relay network. Thus the data rate for the network-codng based scheme s gven by R (net codng = 1 2 r t. (12

9 8 On the other hand, when the odd-numbered lnks are used for data transmsson, the outage occurs when mn odd {C } < r t. Smlar to (4, the average number of tme slots for a packet passng through odd-numbered lnks can be obtaned as 1 T odd =, (13 1 P out,odd ( where P out,odd = P mn {C } < r t = (1 e N o γ, N o = N/2 whch s the number of oddnumbered lnks and. rounds up the embraced value to the nearest nteger. Smlarly, the average odd number of tme slots for a packet passng through even-numbered lnks can be obtaned as 1 T even =, (14 1 P out,even ( where P out,even = P mn {C } < r t = (1 e Ne γ, N e = N/2 whch s the number of oddnumbered lnks and. rounds down the embraced value to the nearest nteger. Thus the average even throughput of the network-codng based scheme can be obtaned as η (net codng = r t (1 P out,odd (1 P out,even = r t T odd + T even (1 P out,odd + (1 P out,even = r t e No γ e Ne γ e N o γ + e N e γ (15 Comparng (1, (12 and (15, we can have the outage probablty for the network-codng based scheme as P (net codng out No = e γ + e N e γ e N o γ 2e N o γ e N e γ + e N e γ. (16 From (16, and notng that ether N o = N e or N o = N e + 1, P (net codng out s bounded as 1 e N e γ P (net codng out 1 e N o γ (17 Comparng (6 and (17 clearly shows that both upper and lower bounds of P (net codng out P (con hoppng out so that P (net codng out are larger than P (con hoppng out (18 Therefore, whle the network-codng scheme has hgher data rate than the consecutve-hoppng scheme, ts outage performance s however worse than the latter. To be more specfc, because the outage probablty

10 9 P out 0 when the SNR, the average throughput s manly determned by the data rate when the SNR s large enough. Thus at the hgh SNR range, the average throughput of the N-hop network wth the network-codng scheme s always about r t /2. On the other hand, when the SNR, the outage probablty P out 1 so that the throughput s more determned by the outage probablty than by the data rate. Ths mples that, when the SNR s very small, the network-codng based scheme has lower throughput than the tradtonal scheme. Therefore, the network-codng scheme mproves the throughput at the hgh SNR range. C. Buffer-aded network-codng lnk selecton In order to ncrease the average throughput over all SNR ranges, we propose a novel lnk selecton scheme by ntegratng the buffer-aded max-lnk and network-codng approaches. Ths s acheved by addng smultaneous lnk transmsson n the buffer-aded lnk selecton rules. Generalzng from the network-codng scheme, we understand that any lnks separated by two hops or more can be smultaneously selected for data transmsson. We denote N s as the number of smultaneously transmttng lnks at one tme slot. For the N-hop relay network, we have 1 N s N/2 (19 For any N s, there exst D(N s possble lnk selectons, whch s represented by the selecton vector as lnk (Ns = [lnk (Ns (1,, lnk (Ns (D(N s ], (20 where lnk (N s ( s the th lnk selecton for N s smultaneous lnk transmsson. For later use, we denote lnk n as the smultaneous transmsson of lnk 1,, lnk n. For example, n the 4-hop relay network, we have 1 N s 2, and lnk (N s=1 = [lnk 1, lnk 2, lnk 3, lnk 4 ] lnk (N s=2 = [lnk 1+3, lnk 1+4, lnk 2+4 ] (21 The prncple of the proposed scheme s to let as many lnks for smultaneous transmsson as possble. To be specfc, at tme slot t, the lnk(s for transmsson s/are selected followng the rules below: Step 1: Frst, let N s = N/2, and fnd the selecton vector lnk (Ns, or lst all possble lnk selectons for N s smultaneous lnk transmssons.

11 10 If none of the lnk selectons n lnk (N s s avalable, then go to Step 2. Otherwse, use the max-mn to choose the best N s smultaneous lnk transmsson among all avalable lnks n lnk (N s, as { lnk (N s b = arg max lnk (Ns ( A where A s the set contanng all avalable lnks. Check whether the lnk selecton lnk (N s b If lnk (N s b and go to Step 2. } mn {γ }, (22 lnk lnk (Ns ( s n outage or not. s n outage, then no N s smultaneous lnk transmsson s possble at tme t Otherwse select lnk (Ns b for data transmsson at tme t. Step 2: Let N s (N s 1 and repeat Step 1 untl N s = 1. In order to better understand the proposed lnk selecton rule, we consder the 4-hop relay network as an example. We suppose at tme slot t, all lnks are avalable except lnk 3. Then the selecton vectors for avalable lnks are obtaned by removng all selectons contanng lnk 3 n (21, so that we have lnk (N s=1 = [lnk 1, lnk 2, lnk 4 ] lnk (N s=2 = [lnk 1+4, lnk 2+4 ] (23 Then the lnks are selected as followng. Step 1: Let N s = 2, and fnd the best selecton of 2 smultaneous lnk transmsson as lnk (Ns=2 b = arg max {mn{γ 1, γ 4 }, mn{γ 2, γ 4 }} (24 We assume that soluton from (24 s lnk (N s=2 b = lnk 1+4. Then we check whether mn{c 1, C 4 } < r t or not If no, lnk 1+4 s not n outage and s selected for data transmsson at tme slot t. Otherwse, go to Step 2. Step 2: Let N s = 1, and fnd the best selecton of sngle lnk transmsson as We assume that soluton from (25 s lnk (Ns=1 b lnk (N s=1 b = arg max {γ 1, γ 2, γ 4 } (25 = lnk 2. Then we check whether mn{c 2 } < r t

12 11 or not If no, then choose lnk 2 for data transmsson. Otherwse, outage occurs. The proposed buffer-aded network-codng scheme takes advantages of both the network-codng and max-lnk schemes. On the one hand, because hgher lnk selecton prorty s gven to smultaneous transmsson, the average data rate s hgher than that of the tradtonal scheme. Partcularly, when SNR, we have P out 0 so that the average throughput of the proposed scheme s r t /2, whch s the same as that for the network-codng scheme. On the other hand, n the proposed scheme, the outage occurs only f all avalable lnks are n outage. Ths s smlar to the max-lnk scheme. Therefore, the outage performance of the proposed and max-lnk scheme are smlar. From (1, the average throughput of the proposed scheme s gven by η (buffer code = R (buffer code (1 P (buffer code out, (26 where P (buffer code out and R (buffer code are the outage probablty and average data rate of the proposed scheme, whch are gven by (40 and (51 obtaned n the followng two sectons respectvely. IV. OUTAGE PROBABILITY At any tme, the numbers of data packets n the relay buffers form a state. Because each buffer has sze L, there are (L + 1 N 1 states n total, where the -th state vector s defned as s = [Ψ (Q 1, Ψ (Q 2,..., Ψ (Q N 1 ], = 1,..., (L + 1 N 1, (27 where 0 Ψ (Q k L for all k = 1,..., N 1 whch s the buffer length (or the number of data packets n the buffer of Q k at state s. At every tme, dependng on whch lnk(s s/are selected for transmsson, the state may move to several possble states at the next tme, formng a Markov chan. Consderng all possble states, the outage probablty of the buffer-aded network-codng scheme can be obtaned as (L+1 N 1 P (buffer code out = π p s out, (28 =1 where π and p s out are the statonary probablty and outage probablty for state s respectvely. In the followng two subsectons, we derve p s out and π respectvely.

13 12 A. p s out: outage probablty for state s Accordng to the lnk selecton rules of the proposed buffer-aded network-codng scheme, at state s, outage occurs only f all avalable lnks are n outage. Recallng that a lnk s avalable when the buffers of the correspondng transmsson and recevng nodes are not empty and full respectvely, we defne the avalable-lnk vector for the state s n the N-hop network as a = [a (1, a (2,..., a (N] (29 where a (n can only be 1 or 0, ndcatng that the correspondng lnk n s avalable or not avalable at state s respectvely. For nstance, n the 4-hop example n Secton III-C where the buffers are at the state that all lnks except lnk 3 are avalable, we have a = [ ]. Because all channels are..d., the outage probablty for state s s gven by p s out = (P (C < r t a +, (30 where P (C < r t s the probablty that a sngle lnk becomes outage, and a + s the total number of avalable lnks at state s whch s the number of 1 -s n a. Because C = log(1 + γ and the SNR γ s exponentally dstrbuted, we have P (C < r t = F γ ( = ( 1 e γ, (31 where F γ (. s the cumulatve dstrbuton functon (CDF of γ and = 2 rt 1. Substtutng (31 nto (30 gves ( p s out = F a + γ = 1 e γ a +, (32 where s gnored n F γ ( wthout causng any confuson. B. π : the statonary probablty of state s In order to obtan the statonary probablty π for every state, frst we need to calculate the state transton matrx A whch s an (L + 1 N 1 by (L + 1 N 1 matrx, where the entry A j, = P (X t+1 = s j X t = s s the transton probablty that the state moves from s at tme t to s j at tme (t + 1. We suppose that the buffer state s s at tme slot t. If the outage occurs, the buffer state remans at s at the tme slot (t + 1. Otherwse, s may move to several possble states at (t + 1, whch are denoted

14 13 as s j1,, s jq respectvely. The state transton from s to s jq (j q {j 1,, j Q } s the result of one partcular lnk selecton whch s represented by the selecton vector defned as sel (jq = [sel (jq (1,, sel (jq (N], (33 where sel (jq (n can only take values of 1 or 0, ndcatng the correspondng lnk n s selected or not respectvely. For example, n the 4-hop network, sel = [ ] represents the lnk selecton of lnk 1+4. Wth these observatons, we have p s out, ( A j, = P sel (j j =, j {j 1,, j Q } 0, otherwse ( where P sel (j s the probablty to choose the lnk selecton sel (j ( (32, below we calculate P. sel (j (34 at state s. Whle p s out s gven by Accordng to the proposed lnk selecton rules, the lnk selecton at state s depends on the outage events at every avalable lnks, where the prorty s gven to as many smultaneously lnk transmsson as possble. Only when a lnk s both avalable and not n outage, may t be used for data transmsson. We defne the good-lnk vector to ndcate whether the lnks are good or not for data transmsson at state s as g = [g (1, g (2,..., g (N] (35 where g (n can only take values of 1, 1 or 0, g (n = 1 ndcates that the correspondng lnk n s not only avalable but also not n outage, g (n = 1 ndcates that lnk n s avalable but n outage, and g (n = 0 ndcates that lnk n s not avalable. Comparng (29 and (35 shows that, for every state s, t corresponds to one avalable-lnk vector a, whch agan corresponds a set of good-lnk vectors ncludng all possble lnk outages of the avalable lnks. Because the state s has a + avalable lnks, there are G = ( a ( a + a 1 good-lnk + vectors for s, denotng as g (1,, g (G respectvely, where ( a + n s the (combnaton probablty that n lnks become outage among all a + avalable lnks. The probablty of the k-th good-lnk vector s obtaned as ( P g (k = g(k F + g γ F (k γ, k = 1,, G (36

15 14 where g (k + and g (k gve the number of 1 -s and 1 -s n g (k s the probablty that a sngle lnk s not n outage. respectvely, and F γ = 1 F γ whch From the proposed lnk selecton rules, for every good-lnk vector, t may lead to several possble lnk selectons, dependng on the channel gans at the current tme slot. On the other hand, one lnk selecton may also correspond to several good-lnk vectors. As a result, we can form a 2 stage trells-lke dagram for the state s, as s llustrated n Fg. 4 for the 4-hop network. At the frst stage, there are G nodes, where each node corresponds to one good-lnk vector g. At the second stage, there are Q nodes, each correspondng to one lnk selecton vector sel. We assume that the k-th node at stage 1, g (k, leads to N k nodes at stage 2, denotng as sel (n 1,, sel (n N k node g (n respectvely. Because the channels are..d., the probabltes for the pathes from to any of these N k nodes at stage 2 are the same, or we have P ( g (k sel (j = ( P g (k 1 N k, j {n 1,, n Nk } 0, otherwse (37 Then further from (34, the transton probablty from s to s j s the summaton of the probabltes of all pathes that ends at the node sel (j, whch s gven by ( A j, = P sel (j G = P k=1 ( g (k sel (j, j {j 1,, j Q } (38 Substtutng (38 nto (34, and applyng t on all states, we can obtan the state transton matrx A. Because the transton matrx A s column stochastc, rreducble and aperodc 2, the statonary state probablty vector s obtaned as (see [20] and [21] π = (A I + B 1 b, (39 where π = [π 1,, π (L+1 N 1] T, b = [1,, 1] T, I s the dentty matrx and B n,l s an n l all one matrx. 2 Column stochastc means all entres n any column sum up to one, rreducble means that s s possble to move from any state to any state, and aperodc means that t s possble to return to the same state at any steps [19], [20]

16 15 Fnally, substtutng (32 and (39 nto (28 gves the outage probablty of the overall system as (L+1 N 1 P (buffer code out = π p s out = dag(a π =1 = dag(a (A I + B 1 b, (40 where dag(a s the vector consstng of all dagonal elements of A. 1 Illustraton - the 4-hop relay network: In order to better understand the above analyss, we gve an example of the 4-hop relay network wth buffer sze of L = 4. As an llustraton, we consder the state transton for the state s = [2 0 2], or the buffer lengthes at nodes R 1, R 2 and R 3 are 2, 0 and 2 respectvely. Ths s actually the same example n Secton III-C where the selecton rules are explaned. As s shown n Fg. 3, there are 5 possble states that s can move to at tme (t + 1, denotng as s j1,, s j5 respectvely, and each s jq corresponds to one lnk selecton. Fg. 3. State transton dagram for the state s = [2 0 2] n the 4-hop relay network wth buffer sze of L = 4. At state s = [2 0 2], all lnks except lnk 3 are avalable, so that the avalable-lnk vector s gven by a = [ ] (41 The trells dagram for the transton probablty of state s s shown n Fg. 4, where there are 7 nodes (good-lnk vectors at stage 1, and 5 nodes (lnk selecton vectors at stage 2. The probabltes for every lnk selecton can be obtaned from Fg. 4. For example, for lnk 2+4 whch s hghlghted n red, we have A j1, = P (s s j1 = P (sel (j 1 ( = P g (1 = [ ] 1 ( 2 + P g (4 = [ ] 1 (42 = 1 2 F 3 γ + F γ F 2 γ

17 16 Fg. 4. Trells dagram for the transton probablty for the state s = [2 0 2] n the 4-hop relay network. V. AVERAGE DATA RATE In ths secton, we frst ntroduce the concept of effectve hops and then use a trells dagram to obtan the average data rate. A. Effectve hops In the proposed N-hop lnk selecton scheme, although every data packet needs to go through the N hops consecutvely to reach the destnaton, at some tme slots, several packets may be smultaneously transmtted at dfferent lnks. Thus by average, t takes fewer than N tme slots to delver one packet to the destnaton, or the number of effectve hops to transmt one packet s fewer than N. To be specfc, at one tme slot, f several packets are transmtted smultaneously, ths tme slot s only counted as one effectve hop for one of the packets. In order to better understand the nfluence of the smultaneous transmsson on the effectve hop number, we look at the example of the 4-hop network as s shown n Fg. 2. Specfcally, for data packet x(2, we have the followng observatons: At tme slot t = 1, data packets x(2 and x(1 are smultaneously transmtted at lnk 1 and lnk 3 respectvely. We assume that the tme slot s counted as one effectve hop only for the packet at the lnk wth the lowest number. At t = 1, the lowest numbered lnk s lnk 1. Thus t = 1 contrbutes one effectve hop only for x(2 transmsson, but not for x(1 transmsson. Smlarly, t = 2 contrbutes one effectve hop for x(2 transmsson, but not for x(1.

18 17 At t = 3, x(3 and x(2 are smultaneously transmtted at lnk 1 and lnk 3 respectvely. Because the lowest numbered lnk s lnk 1, t = 3 contrbutes one effectve hop only for x(3 transmsson, but not for x(2. Smlarly t = 4 s counted as one effectve hoppng tme for x(3, but not for x(2. Therefore, although x(2 goes through all 4 hops to reach the destnaton, only t = 1 and t = 2 are counted as ts effectve hoppng tmes, or the number of effectve hops for x(2 transmsson s 2. Ths leads to the followng rule as: Effectve hoppng rule: at any tme slot t, f multple data packets are transmtted smultaneously, the tme slot s only counted as one effectve hoppng tme for the packet transmtted at the lowest numbered lnk. In the proposed buffer-aded network codng scheme, because dfferent smultaneous lnk transmssons may be selected at dfferent tme slots, dfferent data packets have dfferent numbers of effectve hops and the average data rate s obtaned as R (buffer code = 1 n r t, (43 where n s the average number of effectve hops to transmt one data packet. B. Trells dagram to obtan n Below we use the trells dagram to analyze average number of effectve hops n. In the proposed N- hop relay scheme, for any data packet, t must go through all lnks (lnk 1,, lnk N consecutvely to reach the destnaton. We suppose that at tme slot t, a packet needs to go through lnk n. There exst several possble lnk selectons to make ths happen: ether only lnk n s selected, or lnk n s selected smultaneously wth other lnks. On the other hand, the lnk selectons only depend on the buffer states at tme slot t, but not on other packet transmssons. Wth ths observaton, we descrbe an N-stage trells dagram to represent all possble lnk selectons for one packet transmsson, as s llustrated n Fg. 5 for the 4-hop network. Every stage contans a set of nodes, where every node corresponds to one possble lnk selecton to pass through the correspondng lnk. Trells nodes at adjacent stages are nter-connected, formng paths from stage 1 to N. The total number of paths s gven by N p = N (1 t N (N t (44

19 18 where N (n t s the number of trells nodes at stage n. Every path corresponds to one combnaton of lnk selectons for a packet passng through the network. Supposng that the kth path conssts of n k -th trells node at the n-th stage, the k-th path s represented as path k = { } sel (1k,, sel (N k, k = 1,, N p, (45 where sel (nk s defned n (33 whch s the n k -th lnk selecton for a packet passng through lnk n. ( In order to obtan the number of effectve hops for the k-th path, we defne a bnary functon H sel (n k. ( If H sel (n k = 1, then the correspondng transmsson at stage contrbutes one effectve hop; otherwse ( f H sel (n k = 0, no effectve hop s contrbuted at ths stage. From the effectve hoppng rule, we understand that a tme slot s counted as one effectve hoppng tme for a packet, only f there are no other packets are transmttng smultaneously at lower numbered lnks. Thus we have ( H sel (n k = ( 1, L sel (n k ( 0, L sel (n k ( where L sel (n k gves the ndex of the frst 1 n the selecton vector sel (nk. < = (46 Then number of effectve hops for path k s then gven by N e (path k = N n=1 ( H sel (n k (47 On the other hand, the probablty to choose path k s gven by ( where P sel (n k P (path k = N P n=1 s the probablty to select sel (n k whch s gven by ( sel (n k, (48 ( where P sel (n k ( (L+1 N 1 ( P sel (n k = π P =1 sel (n k s the probablty to select sel (nk at state s whch s gven by (38. Then from (47 and (48, the average number of effectve hops s obtaned by averagng over all pathes (49 n the trells as N p n = N e (path k P (path k (50 k=1

20 19 Substtutng (50 nto (43 gves the average data rate as R (buffer code = 1 Np k=1 N e(path k P (path k r t, (51 C. An llustraton of the hoppng trells dagram for the 4-hop relay network Fg. 5 shows the hoppng trells dagram for the 4-hop relay network, where there are 4 stages (or columns correspondng to a packet passng through lnk 1 to lnk 4 respectvely. At stage 1, there are 3 nodes correspondng to 3 selecton vectors, namely [ ], [ ] and [ ] respectvely. We note that the frst element of all of the three vectors at stage 1 s 1. Therefore, for a packet to go through lnk 1, t must correspond to one of these selecton vectors. Smlarly, there are 2, 2 and 3 trells nodes at stage 2, 3 and 4 respectvely. Fg. 5. Hoppng trells dagram for the 4-hop relay network. It s clear n Fg. 5 that there are = 36 paths for the 4-hop relay network, where each path corresponds to one combnaton of lnk selectons for a packet passng through the network. For example, the k-th path, whch s hghlghted wth red, s represented as { } sel (1k, sel (2k, sel (3k, sel (4 k = { [ ], [ ], [ ], [ ] }, (52 whch corresponds to the combnaton of selectons as lnk 1+4, lnk 2, lnk 1+3 and lnk 1+4 consecutvely. Table I lsts the effectve hops at every stage for the path n (52. Partcularly, at stage 1, sel (1 k = [ ] whch s the lnk selecton for the frst hop for ths path. It s clear that the ndex of the ( frst 1 s L sel (1 k = 1 whch s equal to the hop ndex (or the frst hop. Thus sel (1k contrbutes ( one effectve hop for ths path, or we have H sel (1 k = 1. On the other hand, at stage 4, although sel (4 k = sel (1 k = [ ], sel (4 k does not contrbute one effectve hop for ths path. Ths s because that, for sel (4k, the hop ndex s now 4 whch s not equal to the ndex of the frst 1 (whch s stll 1. Then from (47, the number of effectve hops for the path n (52 s gven by ( N n=1 H sel (n k =

21 20 ( L H sel (n k TABLE I EFFECTIVE HOPS FOR THE PATH IN (52 [ ] [ ] [ ] [ ] n sel (n k ( sel (n k = 2. The probablty to choose ths packet s gven by P (path k = P ([ ] P ([ ] P ([ ] P ([ ] (53 VI. AVERAGE PACKET DELAY The delay of a packet n the N-hop network s defned as the duraton between the tme when the packet leaves the source and the tme when t arrves the destnaton. In the non-buffer-aded schemes (e.g. the tradtonal or network-codng based scheme, when a packet reaches one node, t wll be mmedately forwarded to the next node at the followng tme slot, so that the delay for every packet s N tme slots. On the other hand, n the buffer-aded scheme, because the data packets may queue at the relay nodes, the packet delay also ncludes the queung tme. We partcularly note that the packet delay s dfferent from the number of effectve hops, where the latter does not take nto account of the queueng tmes at the relays. Because t takes one tme slot to transmt a packet from the source to R 1, the average packet delay n the network s gven by D (buffer code = 1 + N 1 k=1 D k, (54 where D k s the average delay at relay R k. Usng Lttle s law [22], the average delay at node can be obtaned as D k = L k η k, k = 1,, N (55 where L k and η k are the average queung length and average throughput at node R k respectvely. Because all nodes are connected n seres, the average throughput at every node s the same, whch s equal to the system average throughput as η k = η (buffer code, k = 1,, N (56

22 21 where η (buffer code s gven by (26. On the other hand, the average queung length at relay R k s obtaned by averagng the buffer lengths over all buffer states as L k = (L+1 N 1 l=1 π l Ψ l (Q k, k = 1,, N (57 where we recall that Ψ l (Q k gves the number of packets (or the buffer length of buffer Q k at state s l. Substtutng (56 and (57 nto (55, and further nto (54, gves the proposed average packet delay n the buffer-aded network-codng scheme as D (buffer code = 1 + (L+1 N 1 l=1 N 1 k=1 π lψ l (Q k η (buffer code. (58 It s nterestng to compare the average packet delays of the two buffer-aded schemes: the max-lnk and proposed schemes respectvely. On the one hand, the proposed scheme has hgher throughput than the max-lnk scheme, or η (buffer code > η (max lnk. On the other hand, because of the smultaneous data transmsson n the proposed scheme, the data packets move more quckly through the system, resultng n shorter queung lengthes at the relays, than the max-lnk scheme. From the Lttle law (as s shown n (55, the average packet delay of the proposed scheme s sgnfcantly smaller than that of the max-lnk scheme. VII. SIMULATION In ths secton, numercal results are shown to verfy the proposed scheme n ths paper. In all smulatons, the transmsson powers and the nose powers are normalzed to unty, the transmsson rates are set as r t = 1, all channels are..d. Raylegh fadng, and the channel coeffcents remans unchanged durng one hoppng tme slot but vary ndependently from one tme slot to another. Both the smulaton and theoretcal results are shown, where the smulaton results are obtaned by averagng over 100, 000 ndependent runs. Other parameters ncludng the buffer sze and number of hops are set ndvdually for every smulaton. A. Average system throughput Fg. 6 (a and (b shows outage probablty and average data rate for consecutve-hoppng, max-lnk, network-codng, and buffer-aded network-codng schemes n the 5-hop relay network respectvely. Frst, for the proposed scheme, the smulatons well match the theoretcal results for both the outage probablty

23 22 and data rate, whch verfes the analyss n ths paper. It s nterestng to observe n Fg. 6 (a that the proposed and tradtonal max-lnk schemes have smlar outage performance. Ths s not surprsng because both proposed and max-lnk are buffer-aded schemes, where the outage occurs only when all of the avalable lnks are n outage. Fg. 6 (a also shows that buffer-aded schemes (ncludng both the proposed and tradtonal max-lnk schemes have the best outage performance, whle the network-codng scheme has even has worse outage probablty than the consecutve-hoppng scheme. On the other hand, t s shown n Fg. 6 (b that the network-codng scheme has the hghest data rate (0.5 packet/tme-slot, whle both the consecutve-hoppng and max-lnk schemes have the lowest data rate (0.2 packet/tme-slot. It s nterestng to observe that the proposed scheme has smlar data rate as the consecutve-hoppng scheme at low SNR range. But when the SNR s hgh enough, the data rate of the proposed scheme approaches to that of the network-codng scheme. Combnng Fg. 6 (a and (b, t s well expected that the proposed scheme must have the hghest throughput among all schemes. Ths wll be verfed n the followng smulaton. outage probablty network codng consecutve max lnk L=4 proposed wth L=4 (Smulaton proposed wth L=4 (Theory channel SNR [db] (a Outage probablty. average data rate network codng proposed wth L=4 (Smulaton proposed wth L=4 (Theory max lnk wth L=4 consecutve channel SNR [db] (b Average data rate. Fg. 6. Outage probablty and average data rate for consecutve-hoppng, max-lnk, network-codng, and buffer-aded network-codng schemes n the 5-hop relay network. Fg. 7 (a and (b shows the average throughput for dfferent schemes n the 5-hop and 3-hop relay networks respectvely, where n both the max-lnk and proposed schemes, the buffer szes for the 3-hop and 5-hop network are set as L = 3 and L = 4 respectvely. In Fg. 7, the smulatons also perfect matches the theoretcal results for the proposed scheme. As s expected, n both 3-/5- hop networks, the network-codng scheme can acheve the maxmum throughput of 1/2 at hgh SNRs (e.g. SNR >20 db, but t has lower throughput than the consecutve-hoppng scheme for small SNRs. The reason s shown n Fg. 6 that, compared wth the consecutve-hoppng scheme wth data rate of R = 1/N, though the network-codng scheme ncreases the data rate to R = 1/2, t also ncreases the outage probablty. For

24 23 the max-lnk scheme, t sgnfcantly ncreases the throughput at low SNRs, but has the same throughput of 1/N as the tradtonal scheme at hgh SNRs, where the reason s also shown n Fg. 6. Ths verfes the our expectaton that the network-codng and max-lnk schemes mprove the throughput at hgh SNRs and low SNRs respectvely. On the other hand, t s clearly shown n Fg. 7 that the proposed buffer-aded network-codng scheme takes advantage of both network-codng and max-lnk schemes, leadng to sgnfcantly mprovement n throughput at all SNR ranges. Partcularly, when the SNR s large enough, the proposed scheme has the same maxmum throughput of 1/2 as the network-codng scheme proposed wth L=4 (Smulaton proposed wth L=4 (Theory network codng max lnk wth L=4 consecutve proposed wth L=3 (Theory proposed wth L=3 (Smulaton network codng max lnk L=3 consecutve average throughput average throughput channel SNR [db] (a 5-hop channel SNR [db] (b 3-hop. Fg. 7. Throughput comparson among tradtonal, network-codng, max-lnk and buffer-aded network-codng schemes. Fg. 8 compares the average throughput vs the buffer sze L between the max-lnk and proposed schemes for the 3-hop relay network. It s clearly shown that, for every buffer sze L, the proposed scheme always has hgher throughput than the max-lnk scheme, where the former can reach the date rate of 1/2 and the latter can only reach 1/3 when the SNR s very large. In both schemes, the average throughput becomes hgher wth larger buffer sze, but the mprovement becomes less sgnfcant when the buffer sze s larger. For example, the throughput dfference between those for L = 10 and L = 5 s trval n both schemes. B. Average packet delay Ths smulaton nvestgates the average packet delay. The unt of the delays s tme slot, where one tme slot s used for a packet transmttng from one node to the next. Table II compares the theoretcal analyss (based on (58 and smulaton results of the proposed buffer-aded network-codng scheme for both 3-hop and 5-hop network, where the channel SNR s set as 15 db. It s clearly shown that, n both networks, the theoretcal analyss very well matches the smulaton results. Together wth the results n

25 24 average throughput proposed wth L=10 proposed wth L=5 proposed wth L=3 proposed wth L=1 max lnk wth L=10 max lnk wth L=5 max lnk wth L=3 max lnk wth L=1 consecutve channel SNR [db] Fg. 8. Throughput vs buffer length L, for the max-lnk and buffer-aded network-codng schemes n the 3-hop relay network. Fg. 8, we obtan that t s not necessary to have a very large buffer sze L as otherwse t not only has lttle mprovement n throughput but also unnecessarly ncreases the average packet delay. TABLE II AVERAGE PACKET DELAYS OF THE BUFFER-AIDED NETWORK-CODING SCHEME 3-hop Average Delay 5-hop Average Delay Buffer sze Smulaton Theory Smulaton Theory L= L= L= Table III compares average packet delays between the max-lnk and proposed schemes for the 3-hop relay network. It s clearly shown that, whle both schemes have larger average packet delays wth larger buffer sze L, the max-lnk scheme has approxmately 50% larger average packet delay than the proposed scheme. Ths well matches our expectaton n Secton VI. TABLE III AVERAGE PACKET DELAYS COMPARISON BETWEEN THE MAX-LINK AND PROPOSED SCHEME IN THE 3-HOP NETWORK. Channel SNR 3-hop Schemes L=5 L=10 L=20 10 db Proposed Max-lnk db Proposed Max-lnk VIII. CONCLUSION In ths paper, we proposed a novel buffer-aded network-codng lnk selecton scheme for the N-hop relay network. The proposed scheme appled buffers at the relays to decrease the outage, and used network-

26 25 codng to ncrease the data rate. As a result, the throughput at all SNR ranges s ncreased. We descrbed new analyss tools to analyze the outage probablty and average data rate, based on whch the average throughput of the proposed scheme was successfully obtaned. We also analyzed the average packet delay. The analyss shows that, the proposed scheme not only has hgher throughput, but also lower average packet delay, than the exstng buffer-aded max-lnk scheme, makng t an attractve approach n the mult-hop network. ACKNOWLEDGEMENT The authors would lke to thank the edtors and anonymous revewers for ther constructve comments to mprove the qualty of ths manuscrpt. Partcularly the dervaton of the average throughput for the tradtonal consecutve hoppng scheme s based on the suggeston from one of the revewers. REFERENCES [1] W. J. Huang, Y. W. Hong, and C. C. J. Kuo, Lfetme Maxmzaton for Amplfy-and-Forward Cooperatve Networks, IEEE Trans. Wreless Commun., vol. 7, no. 5, pp , May [2] A. Papadoganns, H. Bang, D. Gesbert, and E. Hardoun, Effcent Selectve Feedback Desgn for Multcell Cooperatve Networks, IEEE Trans. Veh. Technology, vol. 60, no. 1, pp , Jan [3] A. Nosratna, T. E. Hunter, and A. Hedayat, Cooperatve Communcatons n Wreless Networks, IEEE Communcatons Magazne, pp , [4] J. N. Laneman, D. N. C. Tes, and G. W. Wornell, Cooperatve Dversty n Wreless Networks: efffent Protocols and Outage behavor, IEEE Trans. Inf. Theory, vol. 50, no. 12, pp , Dec [5] P. Pahlevan, M. Hundebll, M. V. Pedersen, D. E. Lucan, H. Charaf, F. H. P. Ftzek, H. Bagher, and M. Katz, Novel Concepts for Devce-to-Devce Communcaton Usng Network Codng, IEEE Commun. Mag., vol. 52, no. 4, pp , Apr [6] C. Yu, K. Doppler, C. B. Rbero, and O. Trkkonen, Resource Sharng Optmzaton for Devce-to-Devce Communcaton Underlayng Cellular Networks, IEEE Trans. Wreless Commun., vol. 10, no. 8, pp , Jun [7] L. Le, Z. Zhong, C. Ln, and X. Shen, Operator Controlled Devce-to-Devce Communcatons n LTE-advanced Networks, IEEE Trans. Wreless Commun., vol. 19, no. 3, pp , Jun [8] N. Zlatanov, R. Schober, and P. Popovsk, Buffer-Aded Relayng wth Adaptve Lnk Selecton, IEEE J. Sel. Areas Commun., vol. 31, no. 8, pp , Aug [9] N. Zlatanov, R. Schober, and P. Popovsk, Buffer-Aded Relayng wth Adaptve Lnk Selecton-Fxed and Mxed Rate Transmsson, IEEE Trans. Inform. Theory, vol. 59, no. 5, pp , May [10] I. Krkds, T. Charalambous, and J. S. Thompson, Buffer-Aded Relay Selecton for Cooperatve Dversty Systems wthout Delay Constrants, IEEE Trans. Wreless Commun., vol. 11, no. 5, pp , May [11] A. Ikhlef, D. S. Mchalopoulos, and R. Schober, Buffers Improve the Performance of Relay Selecton, n Proc IEEE Global Commun. Conf, Houston, Texas, USA, pp. 1 6, Dec [12] Z. Tan, G. Chen, Z. Chen, Y. Gong, and J. A. Chambers, Buffer-aded Max-lnk Relay Selecton n Amplfy-and-Forward Cooperatve Networks, IEEE Trans. Veh. Technology, vol. 64, no. 2, pp , May 2014.

27 26 [13] B. Xa, Y. Fan, J. Thompson, and H. V. Poor, Bufferng n a Three-Node Relay Network, IEEE Trans. Wreless Commun., vol. 7, no. 11, pp , Nov [14] G. Chen, Z. Tan, Y. Gong, and J. A. Chambers, Decode-and-Forward Buffer-Aded Relay Selecton n Cogntve Relay Networks, IEEE Trans. Veh. Technology, vol. 63, no. 9, pp , Mar [15] G. Chen, Z. Tan, Y. Gong, Z. Chen, and J. A. Chambers, Max-Rato Relay Selecton n Secure Buffer-Aded Cooperatve Wreless Networks, IEEE Trans. nform. Forenscs and Securty., vol. 9, no. 4, pp , Apr [16] H. Yang, W. Meng, B. L, and G. Wang, Physcal Layer Implementaton of Network Codng n Two-way Relay Networks, n Proc IEEE Int. Conf. Commun., Ottawa, Canada, pp , Jun [17] W. Lang, Y. Chen, K. Xu, H. Tan, and Y. Xu, Jont LDPC and Physcal-layer Network Codng for Two-way Relay Channels wth Dfferent Carrer Frequency Offsets, n Proc Internatonal Conference on Wreless Communcatons & Sgnal Processng, Huangshan, Chna, pp. 1 4, Oct [18] R. H. Y. Loue, Y. L, and B. Vucetc, Practcal Physcal Layer Network Codng for Two-way Relay Channels: Performance Analyss and Comparson, IEEE Trans. Wreless Commun., vol. 9, no. 2, pp , Feb [19] C. M. Grnstead and J. L. Snell, Introducton to Probablty: Second Revsed Edton, Amercan Mathematcal Socety, [20] J. R. Norrs, Markov Chans, Cambrdge Unversty Press, [21] A. Berman and R. J. Plemmons, Nonnegatve Matrces n the Mathematcal Scences, Socety of ndustral and appled mathematcs, [22] J. D. C. Lttle and S. C. Graves, Lttle s Law, Internatonal Seres n Operatons Research & Management Scence, vol. 115, pp , 2008.