Delay Performance of Different MAC Schemes for Multihop Wireless Networks

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1 Delay Performance of Dfferent MAC Schemes for Multhop Wreless Networks Mn Xe and Martn Haengg Department of Electrcal Engneerng Unversty of Notre Dame Notre Dame, IN 46556, USA Emal: Abstract Ths paper studes the end-to-end (e2e) delay performance of a multhop wreless network fed wth a sngle constant bt rate (CBR) source. Two MAC schemes are nvestgated, m- phase TDMA and probablstc slotted ALOHA. A delay model s used to analyze the resultng tandem queueng system and derve tght upper bounds on the delay mean. The e2e delay lnearly ncreases wth the route length and ts dstrbuton converges to a Gaussan dstrbuton. For channels wth recepton probablty greater than.5, TDMA sgnfcantly outperforms ALOHA. For unrelable channels, the stuaton s less clear. I. INTRODUCTION Nowadays many wreless applcatons demand delay guaranteed servces, whch s more challengng than n the wred network because of the random delay ncurred by the errorprone wreless channel. In order to provde such QoS guarantees, t s mportant to know the system performance. The subject of ths paper s the analyss of the delay performance of multhop wreless networks fed wth a sngle CBR source. Regardng the e2e delay, we take nto account both the transmsson delay and queueng delay []. The multhop topology makes the e2e delay jontly affected by several factors, ncludng the routng algorthm, theschedulng polcy, and the MAC scheme. We concentrate on the mpact of MAC schemes. The assumpton of a sngle CBR flow makes a smple FIFO dscplne suffcent for packet schedulng. Moreover, the multhop network contans only one actve route such that the topology s essentally reduced to one dmenson (-D). The resultng performance provdes an upper bound for 2-D networks wth multple actve routes. Two MAC schemes are consdered, ) determnstc m-phase TDMA, n whch a node s allocated to transmt once n m tme slots and nodes m hops apart can transmt smultaneously; and ) probablstc slotted ALOHA, n whch every node ndependently decdes to transmt wth probablty /m f t has packets buffered. From a queueng perspectve, the -D network, referred to as lne network, s a tandem system. So, ths paper focuses on the wreless MAC scheme and the tandem queueng system. Many analyses of MAC schemes are based on a snglehop topology and the nfnte populaton model [2] [6]. The nfnte populaton model assgns a new node to a new arrvng packet, and the transmsson attempts n the network are Posson dstrbuted wth the spatal densty of nodes. Ths model excludes the queueng delay and accounts only for the transmsson delay and access delay. In contrast, the analyss of tandem queueng systems focuses on the queueng delay [7] []. The channel s generally set to be error-free so that the access and transmsson delay ncurred by MAC-dependent collson and nterference can be gnored. Our man contrbuton s to jontly analyze the tandem system and the MAC scheme n the wreless envronment. Both the queueng delay and transmsson/access delay are taken nto account. An upper bound on the e2e delay s derved. Wth a sngle CBR source, the delays experenced at a sngle node converge to a geometrc dstrbuton, and the e2e delay converges to a Gaussan dstrbuton. The delay means and varances are lnear n the route length, and we also conclude that the delay s very senstve to the channel recepton probablty. II. ANALYSIS OF THE M-PHASE TDMA SYSTEM The lne network s a chan of N + sngle-server nodes usng the FIFO servce dscplne. Node s the source node that generates fxed-length packets at a constant rate /r, r N,.e., one packet n r tme slots. Node N + s the destnaton. All remanng nodes are pure relays. The channel s slotted to one packet duraton. Transmsson attempts are made at the slot boundares. As n [5], [6], the channel error s characterzed by the recepton probablty, whch depends on the receved sgnal-to-nose-and-nterference rato (SNIR) and the predetermned SNIR threshold for a successful recepton. Faled packets wll be retransmtted untl successfully receved. The m-phase TDMA scheduler lets nodes, m +, 2m +,... ( m) transmt smultaneously at tmes, m +, 2m +,... and so on. A transmsson can be ether a transmsson of new packet or retransmsson of a faled packet. The tme s dvded nto frames of m tme slots, where m < r for stablty. Consderng a heavy traffc load, we assume m<r<2m. For a node, the begnnng of a frame s the begnnng of the tme slot allocated to ths node. If we

2 observe the system at the frame level, the transmsson tme s geometrc wth. For smplcty, we denote the geometrc dstrbuton wth parameter by G pr.thetraffc ntensty s defned by m/( r) <. We start the analyss wth the source node. A. Delay dstrbuton of the frst node At the frame level, the servce tme s G pr.form<r< 2m, the nterarrval tme r/m s not an nteger n terms of frame. The number of arrvals n one frame jumps between and and depends on the arrvals of all prevous r frames, whch makes t dffcult to use the conventonal approach of establshng a Markov chan to keep track of the buffer sze. To take advantage of the constant nterarrval tme, we resort toadelaymodelthatwasfrst used n []. The system state s denoted by the current delay of the Head-of-Lne (HOL) packet n terms of tme slots although the state transtons occur at the frame boundares. The transton probablty from state to state j s:, j, : r m P j p r, j + m, () <, j + m. The absolute value of the negatve state ndcates the remanng tme pror to the arrval of the next packet. Surely, the server s dle when the system state s negatve. Gven r<2m, the server dle tme does not exceed one frame. At frame t, let the HOL packet be packet k and ts delay be d k (t). Packet k s transmtted at the begnnng of frame t. If the transmsson s successful, packet k departs and packet k + becomes the HOL packet at frame t +. The delay of packet k + at frame t s d k+ (t) d k (t) r. It ncreases by m up to d k+ (t +) d k (t) r + m d k (t) at frame t +. That s, the system state transts from d k (t) to j d k (t) wth probablty.ifj <, packetk s the last queued packet and the buffer becomes empty after packet k s departure. Snce the server dle tme s bounded by one frame, a new packet wll arrve n the mddle of frame t +. Ths new packet wll wat m j > slots before t s able to access the channel at the next frame. Then the negatve state j transts to a postve state m j wth probablty. If the transmsson s faled (wth probablty ), the HOL packet remans at the buffer and s retransmtted at frame t+, wth ts delay ncreased by m to d k (t +)d k (t)+m. Denote the steady-state probabltes by {π }.From (), we derve the generatng functon (z-transform) G(z) P h G(z) π hz h z z m. (2) p z m+ Based on G(), we can calculate the average server dle tme P I P π as follows: P I G(z) z lm z z p z m z m+. (3) The average server busy tme s P B. Besdes, the average departure rate P B m/r s consstent wth the expected arrval rate n terms of frames. If the transmsson s successful, the HOL packet departs the node wth a delay as the sum of the system state and one slot for transmsson. The delay dstrbuton {d () d () P π j π j } s π P j π. (4) j Specfcally, for, () leads to π + <m π π + + π m π + +( )π m m. These equatons also hold f π s replaced by d () +.Thenthe delay dstrbuton can be recursvely derved wth respect to d (). The generatng functon of {d()} s (z m ) G D (z) z ( p)z m+, (6) resultng n a delay mean D (5) 2( ). (7) Fg. compares the smulaton result wth the analytcal delay dstrbuton gven by () and (4) for a system wth m 3,r 4,. Our queueng analyss accurately characterzes the delay performance of the source node. Fg smulaton analyss delay D d Delay probabltes of the source node n the TDMA system B. Delay dstrbuton of the relay nodes The output of node s the nput to node 2. We characterze ths output process by the nterdeparture tme T ().Duetothe frame structure, the packet departs only at the frame boundary, whch allows us to observe and measure the nterdeparture tme at the frame level. The packet transmsson tme s S () G pr n terms of frames. Observng at the packet departng nstant, f the queue s non-empty (probablty ep B ), the nterdeparture tme s S () ; otherwse, the nterdeparture tme s S () plus the server dle

3 tme of one frame. Note that P B and ep B are the server busy probablty observed at any frame and at the packet departng frame, respectvely. Snce the arrval process s not memoryless, ep B 6 P B. Usng the generatng functon of T (), we derve the nterdeparture tme dstrbuton {t () }: t () ( P eb ( ) 2 ( ep B ) > Snce the average nterdeparture tme s equal to the nterarrval tme r/m, wehave ep B + m. (9) The dstrbuton (8) corresponds to a correlated on-off process, whch s characterzed by a transton matrx (see [, Eqn.()]). In our case, the transton probabltes are a and a /m. Correlaton and burstness are nduced n the output process even though the channel errors are ndependent and the nput process s smooth. Node 2 (the frst relay node) s characterzed as a queueng system wth a correlated on-off source and a geometrc server. A smlar system s analyzed n [], n whch the maxmum delay s bounded. We extend the result to the case of nfnte delay and conclude that the delay of node 2 s geometrcally dstrbuted n terms of frames. In terms of tme slots, the delay dstrbuton s ½ d (2) ( x)x j f jm +,j otherwse where m( ) x m( ) 2 + p 2 <, r and the delay mean s D 2 +m C. End-to-end delay (8) (). () Smulaton results (Fg. 3(a)) confrm that the relay-node delay converges to a geometrc dstrbuton and the e2e delay converges to a Gaussan dstrbuton. Ths observaton allows us to approxmate the delay of the ndvdual relay nodes by the geometrc dstrbuton gven n (). The end-to-end delay s the sum of the ndvdual node delays,.e., D P N D. Approxmately, D (>2) can be upper bounded by D 2, the local delay at the frst relay node. Therefore, the delay mean s uppoer bounded by D D +(N )D 2 +(N )( + m ). (2) 2( ) Fg. 4(a) s a comparson of ths upper bound (2) wth the smulaton result. III. ANALYSIS OF THE PROBABILISTIC SLOTTED ALOHA MAC SCHEME m-phase TDMA ncurs a substantal amount of overhead to establsh the frame structure. Therefore, we consder the probablstc slotted ALOHA, n whch every node ndependently and probablstcally determnes whether to transmt. To compare wth the TDMA scheme, we set the transmt probablty to be /m. The traffc and channel model reman unchanged. Agan, we start wth the source node. A. Delay dstrbuton of the frst node Unlke the TDMA system, the ALOHA system s observed at the slot level. Wth the channel recepton probablty,a packet s correctly receved f and only f the node attempts to transmt and the transmsson s successful, wth probablty s /m (gven the assumpton that the arrval and transmsson are ndependent). Otherwse, the transmsson fals. The transmsson tme s G s. We employ the delay model agan and denote the system state by the current delay of the HOL packet. The state transton probabltes are s, j + P j s, j (r ) (3) <, j +. Snce the maxmum nterval between a packet departure and the next packet arrval s (r ) slots, the mnmum negatve state s (r ). Rewrtng the balance equatons, we obtan r + X π s π j, (r ), k max{,}, (4) jk From (4), we deduce the system busy probablty P B P π /(sr), whch s dentcal to the busy probablty of the TDMA system. A possble soluton to (4) s the geometrc dstrbuton π α π for all. Snce P π and P B /sr, α sapostverealrootofthe polynomal y r y s +. (5) s Based on Descartes Sgn Rule and /s >, there are exactly two postve real roots rrespectve of r. The desred root α les between and. The delay dstrbuton {d () } s expressed n terms of {π } usng (4), whch leads to d () ( α)α ( ). Fg. 2 shows the smulated delay dstrbuton for a system wth m 3,r 4,.8, n whch the parameter α 569 s very close to the soluton α 57 of (5). We also derve that the delay mean D /( α) can be approxmated as D m 2( ). (6)

4 Fg smulaton analyss D d Delay probabltes of the source node n the ALOHA system B. Delay dstrbuton of the relay nodes The analyss of the output process of the source node s more complcated for ALOHA snce the server dle tme ranges between and (r ) slots. We establsh a new queueng model to characterze the nterdeparture tme. Instead of observng the system at every tme slot, we observe the moment when the HOL packet departs. The system state s denoted by the delay of the next HOL packet. The state transton probabltes are: P out j s( s) r k+j, k max{,}. (7) The balance equatons for all (r ) are π out +r X s πj out. (8) Smlar to (4), a soluton to (8) s the geometrc dstrbuton π out ( β)β r+. After some manpulatons, t can be establshed that β α. Then, the nterdeparture tme T () s ½ + S wth probablty π T () out S wth probablty PB out, (9) where S G s,andp out B busy rate. The dstrbuton {t () s t () j P πout } s α r s the server ³sα r ( s) +( α)α r k ( s) k α( s), (2) where k mn{r, }. In order to compare wth the TDMA system, we approxmate (2) by a correlated on-off process. Gven t () (s + α )/α and the condton that the mean T r, we obtan a transton matrx, where a ( s)/α and a ( s)/((r )α). Then, the delay dstrbuton of node 2 s approxmated by a geometrc dstrbuton wth x ( s)/(sa +( s)a ). The delay mean s D 2 +m α. (2) C. End-to-end delay As shown n Fg. 3(b), the e2e delay converges to a Gaussan dstrbuton. We approxmate the delay of a relay node by the geometrc dstrbuton of node 2. Agan, the e2e delay s upper bounded by the sum of all node delays. Wth an approxmaton of (6), the delay mean s upper bounded by D m 2 α +(N )( + m α) +(N )(m ). (22) If N s large, D s manly affected by the delay mean of the relay nodes. Recallng () and (2), we are able to compare the delay mean between two MAC schemes, D TDMA D ALOHA α, (23) where the second approxmaton holds when the traffc load s hgh and thus α. IV. SIMULATION RESULTS Ths secton presents a set of smulaton results to confrm the analyses n Secton II and III. We frst consder the e2e delay dstrbuton (from the source node to dfferent relay nodes) of a lne network wth m 3,r 4,,N 8 (shown n Fg. 3). For both MAC schemes, the delay dstrbuton converges to a Gaussan dstrbuton node m3,r4,,n8 node delay D d (a) TDMA Fg node m3,r4,,n8 node delay D d (b) ALOHA Probablty dstrbutons of the e2e delay Fg. 4 compares the analytcal upper bounds (2) and (22) wth the smulaton result, showng that the bounds are comparatvely tght. The e2e delay mean ncreases lnearly wth the number of nodes. The devaton from strct lnearty s caused by the non-geometrc output process of the frst node. Further, the delay mean (D 292) of the ALOHA system s about 4 tmes that of the TDMA system (D 85) n agreement wth our analyss (23): /( )4for. Inthe ALOHA network, the drastc ncrease n the delay s caused by the lack of central control and the resultng reducton of spatal reuse. To dscuss the mpact of, Fgs. 5 and 6 exhbt the delay mean and varance for dfferent n a network wth m

5 delay mean smulaton upper bound (a) TDMA Fg. 4. delay mean smulaton upper bound (b) ALOHA Comparson of the e2e delay mean 4,r 7,N 2. When s closer to the traffc load.6, both the mean and varance ncrease substantally. Mean delay mean (a) Delay mean Fg (a) Delay mean Fg. 6. delay varance p r (b) Delay varance Comparson of delay performance of TDMA Delay varance p r (b) Delay varance Comparson of delay performance of ALOHA V. CONCLUSION For both MAC schemes, the delay performance s senstve to the recepton probablty, partcularly when s close to the normalzed traffc load. Besdes, the e2e delay mean and varance lnearly ncrease wth the number of nodes. Such lnearty allows us to regard these nodes as almost ndependent and analyze them ndvdually. The relay-node delay dstrbuton converges to a geometrc dstrbuton. Thus, f the path s long, the e2e delay dstrbuton s well approxmated by a Gaussan dstrbuton. We derved an upper bound on the e2e delay mean for both MAC schemes. When the traffc load s heavy, the delay mean of the ALOHA network s approxmately /( ) tmes than that of the TDMA network. Therefore, for a channel suffcently good, say >.5, TDMA outperforms ALOHA n both throughput and delay. However, f the channel s not as good as expected, sayng <.5, ALOHA s more effcent to take advantage of the random channel errors and acheves a smaller delay than TDMA. Note that n practce, due to nterference, of the TDMA system s hgher than that of the ALOHA system f both systems are optmzed [2], whch enhances the benefts of usng TDMA. Our future work s to analyze the delay performance of MAC schemes n assocaton wth the fadng channel characterstcs. ACKNOWLEDGMENT The authors would lke to thank the support of the Center for Appled Mathematcs (CAM) Fellowshp of the Unversty of Notre Dame and the partal support of NSF (grant ECS ). REFERENCES [] C.-K. Toh, Mnar Delwar, and Donald Allen, Evaluatng the Communcaton Performance of an Ad Hoc Wreless Network, IEEE Transactons on Wreless Communcatons, vol., no. 3, pp , July 22. [2] Jens C. Arnbak and Wm Van Bltterswjk, Capacty of Slotted ALOHA n Raylegh-Fadng Channels, IEEE Journal on Selected Areas n Communcatons, vol. 5, no. 2, pp , Feb [3] Malcolm C. H. Peh, Stephen V. Hanly, and Phlp Whtng, Random Access wth Multpacket Recepton over Fadng Channels, n Australan Communcatons Theory Workshop 23, Feb. 23. [4] Flamno Borgonovo and Mchele Zorz, Slotted ALOHA and CDPA: A Comparson of Channel Access Performance n Cellular Systems, ACM Wreless Networks, vol. 3, no., pp. 43 5, Mar [5] Yang Yang and Tak-Shng Peter Yum, Delay Dstrbutons of Slotted ALOHA and CSMA, IEEE Transactons on Communcatons, vol. 5, no., pp , Nov. 23. [6] Mchele Zorz and Slvano Pupoln, Slotted ALOHA for Hgh-Capacty Voce Cellular Communcatons, IEEE Transactons on Vehcular Technology, vol. 43, no. 4, pp. 2, Nov [7] John A. Morrson, Two Dscrete-Tme Queues n Tandem, IEEE Transactons on Communcatons, vol. 27, no. 3, pp , Mar [8] J. Hsu and P. J. Burke, Behavor of Tandem Buffers wth Geometrc Input and Markovan Output, IEEE Transactons on Communcatons, vol. 24, no. 3, pp , Mar [9] Moshe Sd, Tandem Packet-Rado Queueng Systems, IEEE Transactons on Communcatons, vol. 35, no. 2, pp , Feb [] Mchael J. Neely, Exact Queueng Analyss of Dscrete Tme Tandems wth Arbtrary Arrval Processes, n ICC, 23, vol. 4, pp [] Kelvn K. Lee and Samuel T. Chanson, Packet Loss Probablty for Bursty Wreless Real-Tme Traffc Through Delay Model, IEEE Transactons on Vehcular Technology, vol. 53, no. 3, pp , May 24, correspondence. [2] Martn Haengg, On the Local Throughput of Large Interference- Lmted Wreless Networks, n 39th Annual Conference on Informaton Scences and Systems (CISS 5), Baltmore, MD, Mar. 25.

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