U J.T. (: 33- (pr. 0 Optmzaton of Local Routng for Connected odes wth Sngle Output Ports - Part I: Theory Dobr tanassov Batovsk Faculty of Scence and Technology ssumpton Unversty Bangkok Thaland E-mal: <dbatovsk@au.edu> bstract The optmzaton of packet flows n a set of cooperatve nodes wth sngle output ports s consdered. sngle output port relays a packet to a sngle connected node at a tme. The dfferent servce tme dstrbutons to dstnct connected nodes are consdered n terms of multclass queung wth a sngle frst-come frst-serve queue and a sngle server n each node. The analytc model s appled to cases of two three and four connected nodes wth M/M/ queues relayng packets n a chosen drecton. nalytcal solutons for two connected nodes are obtaned. The nfluence of other arbtrary packet flows s consdered as background traffc. Drected lnks are used for local connectvty wthn the set of cooperatve nodes. Keywords: Local routng cooperatve node sngle output port multple servce rates background traffc drected lnks. Introducton The mplementaton of mult-hop routng n moble ad hoc networks (MET 0 reles on the exstng technologcal constrants whch restrct the number of wreless channels whch can be used by a moble user to relay nformaton (IEEE 80. 007; IEEE 80.b 999; IEEE 80.g 003; IEEE 80.n 009; IEEE-S 00. lthough multple-nput multple-output (MIMO technology s used n the physcal layer of the latest wreless standards the ncreased bandwdth s utlzed for a drected transmsson towards a sngle destnaton n most cases. Power consumpton frequency reuse topology control securty ssues etc. are factors whch lmt the number of output ports n a moble node. lso the exstng moble unts whch communcate drectly only wth an access pont or a base staton are desgned to operate wth a sngle output port and connect to a sngle destnaton n a centralzed network. The modfcaton of the exstng hardware confguratons to operate n a decentralzed network would use an upgrade of the exstng technology and the utlzaton of a sngle output port whch can swtch consecutvely between dfferent connectons wth dfferent servce dstrbutons s a realstc scenaro. The packets to be relayed to dfferent connected nodes wth dfferent servce tmes can be consdered as belongng to dfferent classes or dfferent types of customers for the sngle server at the output port. The sngle queue whch stores packets arrvng ndependently from several nput ports wll contan several classes of packets. The packets of a gven class whch could be servced faster than other packets wll have to wat for the slower servcng of packets to be delvered n channels wth low sgnal-to-nose ratos. The performance of such mult-class sngle-server systems for moble communcatons s obvously outperformed by alternatve prospectve systems contanng several ndependent output ports whch could relay packets to several dstnct destnatons at the same tme. s the advent of such advanced mult-port systems s delayed due to bandwdth lmtatons t s worth studyng to what extent the sngle output port systems can be optmzed to relay packets n a partcular drecton (by usng underutlzed lnks wth shorter servce Regular Paper 33
U J.T. (: 33- (pr. 0 tmes and reduce the average delay n small sets consstng of several cooperatve nodes. Such local routng may prove benefcal n congested parts of the network (hot spots where some local paths are utlzed more frequently that others by the global routng algorthms. The decentralzed networks mplement best effort algorthms whch result n overlappng paths and the absence of global routng control requres local optmzaton of packet flows. The arbtrary nter-arrval tme dstrbutons at the nput ports and the multclass servce tme dstrbutons at the sngle output port of each local moble node result n a mult-varable optmzaton n locally reroutng portons of the global packet flows among adacent neghbors by applyng a local decson makng. nalytc Model The deal optmzaton of packet flows s dffcult to acheve n practce because the number of varables s a functon of the local connectvty and even a node degree of ust two or three lnks poses challenges for real-tme estmaton. The engneerng approach s to reduce the complexty by separatng the problem nto smaller ones whch can be treated ndependently from each other n frst approxmaton n order to acheve sub-optmal performance. One can optmze the traffc n partcular drecton by consderng all other traffc as a background traffc and partton the general problem nto smaller solvable cases. In a set of cooperatve nodes the arrval traffc n one of the detected general drectons s descrbed by the nter-arrval tme dstrbutons characterzed by the average arrval rates and the squared coeffcents n of tme varaton c of sad dstrbutons = n where s the number of local cooperatve nodes. The mult-class servce tme dstrbutons are known by the average servce rates out and the squared coeffcents of tme varaton = of departng packets and c out also the average servce rates squared coeffcents of tme varaton and the c = of packets crculatng among the cooperatve nodes. The unknown average departure rates and the correspondng unknown squared out coeffcents of tme varaton c = out of departng packet flows are to be determned by optmzng the local packet exchange wth the unknown rates and the correspondng unknown squared coeffcents of tme varaton c = for assumng that f 0 then 0. ll average rates are measured n packets/sec and ther recprocal values the average tme ntervals of the tme dstrbutons are measured n seconds/packet. The range of the ndexes and = goes wthout sayng n the formulae that follow. t a local node the total average arrval rate s gven by n ( the utlzatons of a node due to packets of dstnct classes are out out and ( out (3 and the total utlzaton of a node s out. ( The mean servce rate of a node for all packet classes (customers s as follows:. (5 out The followng constrant apples to all average rates assumng ntally that there are no packet drop rates: n out k (6 where and are the numbers of Regular Paper 3
U J.T. (: 33- (pr. 0 actve lnks used n each drecton and and the cumulatve average rate for all cooperatve nodes s. (7 total The unknown squared coeffcents of varaton c and c can be obtaned as out descrbed by Belch et al. (998. In the prvate case of M/M/ frst comefrst serve (FCFS queues (Klenrock 975 the number of watng packets of a gven class s estmated by the expresson (Puolle and Wu 986; Belch et al. 998: Q M/M/. (8 The number of watng packets for GI/G/ queues: Q f c c Q (9 GI/G/ ( S M/M/ s estmated usng a chosen approxmaton f ( c c where c and c S S represent the squared coeffcents of varaton of arrval and servce tme dstrbutons nvolved n the method of decomposton for open non-product-form networks (llen 990; Whtt 983ab 993 99; Belch et al. 998; Batovsk 008. The correspondng average delay of watng n a queue for a gven class of packets s obtaned from the average number of watng packets for the sad class wth the use of the Lttle s Law (Lttle 96. The sum of average ( Total ( ( out ( watng tme and average servce tme (whch s recprocal to the average servce rate gves the average delay n a node for the gven class of packets. The average delay for a set of ndependent nodes s obtaned from the average rate-delay product (Inthawadee and Batovsk 008; Batovsk 009 whch n a general form s shown as: D out Dout. (0 Total The cooperaton among the set of local nodes results n a modfed expresson: D out Dout D ( Total where out n k 0. ( umercal solutons for the mnmzaton of the average delay D have been obtaned for multple ndependent output ports and M/M/ queues n each port wth the method of Lagrange multplers (Inthawadee and Batovsk 008; Inthawadee 009. The formula of Theorem by Inthawadee and Batovsk (008 for M/M/ queues n cooperatve nodes formng a dstrbuted gateway wth multple output ports can be modfed to nclude the background traffc n each node: ( where = s the utlzaton of node due to background packet flows. The method of Lagrange multplers results n systems of non-lnear equatons for nodes wth sngle ports. n alternatve approach has been used (Batovsk 009; Inthawadee 009 n equalzng (whenever possble the rate-delay products among connected nodes so that 0 (3 D D... ( out The level of complexty ncreases for nodes wth a sngle output port because the delays D out and D n Eq. ( depend on a multtude of local arrval rates whch are ncluded n n Eq. (. The advantage of the method of decomposton for open networks s that the Regular Paper 35
U J.T. (: 33- (pr. 0 average delay of ndvdual flows can be estmated separately from the rest of the traffc. The analytc approach for the optmzaton of the average delay of the set of cooperatve nodes can be summarzed as follows: - For a gven set of connected nodes the drecton of nternal flows s chosen so that the underutlzed nodes receve and relay more traffc n a partcular drecton. - The method of decomposton for open networks allows one to estmate separately the average number of watng packets of every packet flow. - The Lttle s Law s used to estmate the average delay of every packet flow gven the average number of watng packets of sad flow and the average source rate. Internal Lnk Drectvty s stated by Inthawadee and Batovsk (008 the nternal packet flows have chosen drectons pontng to the underutlzed nodes whch may share the load of congested neghbors. It s assumed that a packet may perform only one hop wthn the set before leavng t. set of two connected nodes forms a cooperatve par. set of three nodes forms a trangular confguraton of three pars n a closed loop. set of four nodes forms a tetrahedral confguraton of four pars n four nterconnected trangular closed loops. However the drectvty of chosen nternal lnks reduces the number of pars partcpatng n the optmzaton problem. For two nodes ( and there s a sngle par so that the followng drected lnks could be establshed n two dfferent scenaros: or For three nodes ( and 3 there s one node sendng traffc to two neghbors or one node recevng traffc from two neghbors or two pars so that the followng drected lnks could be establshed n two dfferent scenaros: 3 or 3. For four nodes ( 3 and there s one node sendng traffc to three neghbors or recevng traffc from three neghbors so that the followng drected lnks could be establshed n two dfferent scenaros: 3 or 3. Other scenaros could splt the trangular or tetrahedral confguraton nto ndvdual pars to be optmzed ndependently because (accordng to the two assumptons: only one nternal hop s allowed and a chosen drectvty apples to the selected lnks a node s not allowed to receve and send nternal traffc at the same tme for a gven optmzaton. ote that the drectvty of the lnks (for a porton of the traffc to be optmzed locally n a partcular drecton s a logcal concept used by the optmzaton algorthm and t does not apply to the background traffc related to other packet flows whch may flow n both drectons for a gven connecton. M/M/ Queues smplfcaton of Eq. ( s obtaned for M/M/ queues snce all squared coeffcents of varaton are equal to and the soluton depends only of the average source and servce rates. Two nodes: Let the known average rates be denoted as n out and and the correspondng unknown rates be denoted as out and =. In the absence of arbtrary background traffc the equalzaton of rate-delay products as gven by Eq. ( for the scenaro for M/M/ queues can be wrtten as follows: n n ( n ( Regular Paper 36
U J.T. (: 33- (pr. 0 ( n n ( (5 where: out n (6 out n (7 n and (8 n. (9 Equaton (5 s to be solved for the unknown nternal rate and t s a quadratc equaton for whch can be smplfed as follows: n ( n out ( (( n n ( (0 The analytcal soluton of Eq. (5 s gven by the followng analytc expresson: [( 8 ( ( n n n n ] ( From the two possble solutons of the quadratc Eq. (5 the postve real soluton s chosen f t does exst for: 8 ( ( n n n 0. ( In the presence of arbtrary background traffc the equalzaton of rate-delay products s based on the followng equaton:. n n ( n ( n (3 n out ( where and are the average utlzatons of background traffc n nodes and correspondngly. Equaton (3 s also to be solved for the unknown nternal rate and smlarly to Eq. (5 can be smplfed as follows: n ( n out ( (( n n ( ( The analytcal soluton of Eq. (3 s gven by the followng analytc expresson: [( 8 ( ( ( ( ( ( n n ( n ( (. ( ( ( n ( (5 From the two possble solutons of the quadratc Eq. (3 the postve real soluton s chosen f t does exst for: 8 ( ( ( ( ( ( ( ( n ( n ( ( ( ( n 0. (6 ] Regular Paper 37
U J.T. (: 33- (pr. 0 Obvously Eq. ( can be obtaned from Eq. (5 for = 0 and = 0. The soluton for scenaro for M/M/ queues can be wrtten by smply exchangng the places of ndexes and n the above equatons. Three nodes: In the presence of arbtrary background traffc the equalzaton of ratedelay products as gven by Eq. ( for the scenaro 3 for M/M/ queues can be wrtten as follows: ( n n ( 3 3 n ( 3 n3 3 3 3( 3 ( n n ( 3 ( n3 3 (7 n3 3 3 out 3( 3 where: out n 3 (8 out n (9 out 3 n 3 3 (30 (3 3 n 3 out and (3 n out. (33 n3 3 3 3 3 out 3 3 and and 3 are the average utlzatons of background traffc n nodes and 3 correspondngly. For the scenaro 3 the equalzaton of rate-delay products for M/M/ queues can be wrtten as follows: ( n ( ( n 3 3 3 n n ( 3 3 n 3 3 ( ( n3 n3 ( 3 3 3 3 n 3 3 3( (3 where: out n 3 (35 out n (36 out 3 n 3 3 (37 (38 n 3 3 out 3 and (39 n out. (0 n3 3 3 3 out 3 and and 3 are the average utlzatons of background traffc n nodes and 3 correspondngly. Four nodes: In the presence of arbtrary background traffc the equalzaton of ratedelay products as gven by Eq. ( for the scenaro 3 for M/M/ queues can be wrtten as follows: 3 Regular Paper 38
U J.T. (: 33- (pr. 0 n n 3 ( n ( 3 n3 3 3 3( 3 n ( out out n 3 3 n ( n3 n3 3 3 ( 3 3 3 n n ( out ( where: out n 3 ( out n (3 out 3 n 3 3 ( out n (5 (6 n 3 out (7 n out and (8 n3 3 3 3 3 out 3 3 (9 n out and 3 and are the average utlzatons of background traffc n nodes 3 and correspondngly. For the scenaro 3 the equalzaton of rate-delay products for M/M/ queues can be wrtten as follows: n n B out n 3 n ( n B out n3 3 n3 3 out 3( 3 3 n 3 B out n n out ( out n (50 where: out n 3 (5 out n (5 Regular Paper 39
U J.T. (: 33- (pr. 0 out 3 n 3 3 (53 out n (5 n out n out (55 (56 and (57 n3 3 3 3 out 3 (58 n out and 3 and are the average utlzatons of background traffc n nodes 3 and correspondngly. Dscusson The equatons derved for three and four nodes can be solved usng standard numercal methods. Rate-delay equalzaton usually can take place n cases when the mean values and the coeffcents of varaton of the servce tme dstrbutons of lnks between the partcpatng nodes do not dffer sgnfcantly. If there s a node wth ncreased servce tmes ts capablty n sharng traffc s rather lmted and there s no guarantee that a soluton for rate-delay equalzaton wth other adacent neghbors does exst. Therefore the selecton of partcpatng nodes depends on the channel condtons n each node. The extenson of the analytc model for GI/G/ queues can be made usng the method of decomposton for open non-product-form networks (Puolle and Wu 986; Belch et al. 998; Batovsk 008 to estmate the delay wth Eq. (9. summary of the sad method of mergng-flow-splttng s provded below. The followng ft for the functon f ( c c n Eq. (9 covers a wde S range of squared coeffcents of varaton (Whtt 993: c cs f ( c cs (59 wth (here the ndexes and go wthout sayng (Whtt 993: ( c cs cs ; c cs c 3c S c 3c S (60 cs c cs 3c 3 ; c cs ( c cs ( c cs ; c c cs where c (6 (c ; 0 c (6 (63 ( 3 exp 3 (6 3 mn (65 ( ( m ( 5m mn 0. (66 6m where: s the overall node utlzaton c s the squared coeffcent of varaton (scv of the nter-arrval tme dstrbuton of a gven packet flow; and c s the scv of the servce tme S dstrbuton of sad packet flow. For the case of a sngle server consdered n ths contrbuton m =. The mergng process determnes the overall scv coeffcent of a node (Belch et al. 998: c [( c c ] (67 where the addtonal terms c and are the arrval scv and the average arrval rate of background traffc. The overall scv coeffcent of the servce tme of a node s obtaned from: c S ( c S ( c B S (68 where the addtonal terms c S and are the servce scv and the average servce rate of background traffc. Knowng c and c one can estmate the scv of the flow process (Whtt 983ab: S Regular Paper 0
U J.T. (: 33- (pr. 0 ( cs cd ( ( c (69 m where the number of servers m =. lternatve estmatons of the flow process can be used nstead (Belch et al. 998. The splttng process determnes the scv coeffcent of a departng process (Belch et al. 998: c ( c (70 D where s the overall average arrval rate n node ncludng the background traffc. The rate-delay equalzaton for nodes wth for GI/G/ queues s qute complex due to the non-lnear analytcal expressons nvolved n the estmaton of the scv coeffcents. The scv coeffcents depend on the unknown rates of traffc sharng among the nodes. The smpler llen-cunneen approxmaton (llen 990 can also be used nstead of Eq. (59: c cs f ( c cs. (7 Computatonal experments wth the use of the method of decomposton for cooperatve nodes consstng of two three and four nodes wth GI/G/ queues and dfferent levels of background traffc are ncluded n the second part of ths contrbuton. Concluson The local routng depends on a sgnfcant number of statstcal parameters and the level of complexty ncreases rapdly wth the ncluson of more cooperatve nodes. The obtaned analytcal solutons for two connected nodes wth M/M/ queues can be used as ntal approxmatons n obtanng solutons for the more complex problems wth three and four nodes and GI/G/ queues. Statstcally pars of only two nodes appear more frequently n arbtrary topologcal confguratons. The sets of three and four nodes have an ncreased capablty to share traffc whenever the local topology has such nodes n a close proxmty to each other. References llen.o. 990. Probablty statstcs and queueng theory wth computer scence applcatons. cademc Press Professonal Inc. San Dego C US. Batovsk D.. 008. Sem-analytc evaluaton of qualty of servce parameters n multhop networks. ssumpton Unversty Journal of Technology (U J.T. (: 5- prl. Batovsk D.. 009. Local routng n multhop networks. ssumpton Unversty Journal of Technology (U J.T. (: 7-6 prl. Belch G.; Grener S.; de Meer H.; and Trved K.S. 998. Queueng networks and Markov chans: Modelng and performance evaluaton wth computer scence applcatons. Wley-Interscence John Wley & Sons Inc. ew York Y US. IEEE 80.. 007. IEEE 80.: Wreless L medum access control (MC and physcal layer (PHY specfcatons. Insttute of Electrcal and Electroncs Engneers Standards ssocaton (IEEE-S ew York Y US. valable: http://standards.eee.org/geteee80/downloa d/80.-007.pdf. IEEE 80.b. 999. 80.b-999: Hgher speed physcal layer extenson n the. GHz band. Insttute of Electrcal and Electroncs Engneers (IEEE ew York Y US. valable: http://standards.eee.org/geteee80/downloa d/80.b-999.pdf. IEEE 80.g. 003. IEEE 80.g-003: Further hgher data rate extenson n the. GHz band. Insttute of Electrcal and Electroncs Engneers (IEEE ew York Y US. valable: http://standards.eee.org/geteee80/downloa d/80.g-003.pdf. IEEE 80.n. 009. IEEE 80.n-009 - mendment 5: Enhancements for hgher throughput. Insttute of Electrcal and Electroncs Engneers Standards ssocaton (IEEE-S ew York Y US. valable: http://dx.do.org/0.09%fieeestd.00 9.53073. Regular Paper
U J.T. (: 33- (pr. 0 IEEE-S. 00. IEEE 80. L/M wreless Ls. Insttute of Electrcal and Electroncs Engneers Standards ssocaton (IEEE-S ew York Y US. valable: http://standards.eee.org/geteee80/80.. html. Inthawadee S. 009. Routng technques n three-dmensonal ad hoc networks. Ph.D. Dssertaton n Telecommuncatons Scence Department of Telecommuncatons Scence Faculty of Scence and Technology ssumpton Unversty Bangkok Thaland May 009. Inthawadee S.; and Batovsk D.. 008. Flow control n dstrbuted gateways. nnals of Telecommuncatons 63(9-0: 53-7 September-October. Klenrock L. 975. Queueng systems Volume : Theory Wley Interscence ew York Y US. Lttle J.D.C. 96. proof of the queueng formula L = λ W. Operatons Research 9: 383-7 May-June. MET. 0. MET - Moble ad-hoc networks. IETF - The Internet Engneerng Task Force Fremont C US. valable: http://www.etf.org/html.charters/manetcharter.html. Puolle G. and Wu. 986. soluton for multserver and multclass open queueng networks. Informaton Systems and Operatons Research (3: -30. Whtt W. 983a. The queueng network analyzer. Bell System Techncal Journal 6(9: 779-85 ovember. Whtt W. 983b. Performance of the queueng network analyzer. Bell System Techncal Journal 6(9: 87-3 ovember. Whtt W. 993. pproxmatons for the GI/G/m queue. Producton and Operatons Management (: -6 Sprng. Whtt W. 99. Towards better mult-class parametrc-decomposton approxmatons for open queueng networks. nnals of Operatons Research 8: -8. Regular Paper