Queueing Theory Analysis of Greedy Routing on Square Arrays. May 20, Abstract

Transcription

1 Queueng Theory Analyss of Greedy Routng on Square Arrays Mor Harchol æ Paul E. Black May 20, 1993 Abstract We apply queueng theory to derve the probablty dstrbuton on the queue buldup assocated wth greedy routng on an n æ n array of processors. We assume packets contnuously arrve at each node of the array wth Posson rate ç and have random destnatons. We assume an edge may be traversed by only one packet at a tme and the tme to traverse an edge s Posson dstrbuted wth mean 1. To analyze the queue sze n steady-state, we formulate the problem nto an equvalent Jackson queueng network model. It turns out that determnng the probablty dstrbuton on the queue sze at each node s then just a matter of solvng On 4 smultaneous lnear equatons whch determne the total arrval rate at each node and then pluggng these arrval rates nto a short formula for the probablty dstrbuton gven by the queueng theory. However, we even elmnate the need to solve these smultaneous equatons by dervng a very smple formula for the total arrval rates n the case of greedy routng. Lastly,we use ths smple formula to prove that the expected queue sze at a node of the næn array ncreases as the Eucldean dstance of the node from the center of the array decreases. æ Computer Scence Dvson, UC Berkeley, CA Supported by Natonal Physcal Scence Consortum NPSC Fellowshp. 0

2 1 Introducton æ PROBLEM JUSTIFICATION An array of processors s one of the most commonly used communcaton networks because t has avery smple layout whch uses an almost mnmal number of wres, and whch s also very easy to enlarge. The most common type of oblvous routng on arrays of processors s the greedy routng algorthm whch sends a packet ærst to ts correct column and only then to ts correct row. We nvestgate the problem of greedly routng packets whch arrve contnuously at the nodes of the array at Posson rate ç and have random destnatons. Ths problem s mportant snce t comprses the ærst half of any randomzed routng algorthm. Snce a wre edge of the array may be used by only one packet at a tme, packets naturally get delayed at the processor nodes of the array. It s therefore mportant n buldng arrays of processors that we create approprate szed buæers at each node of the array to hold packets whch are delayed n queue. In ths paper we determne the sze of these necessary buæers. æ PREVIOUS HISTORY Prevous work n ths area ncludes Leghton's work ëleghton,92ë whch examnes the same problem except that: æ Packets are prortzed at the queues n terms of Farthest Frst, rather than our method whch uses FCFS ærst come ærst serve. æ Chernoæ Bounds rather than queueng theory s the method of analyss æ Leghton only derves lmts on the tal end of the dstrbuton, rather than the whole dstrbuton. More closely related work s a paper by Stamouls and Tstskls ëstamouls, Tstskls,91ë. Ths work uses queueng theory, however the network analyzed s a hypercube rather than an array, and the authors are concerned more wth the problem of average delay of a sngle packet as t moves through the network, rather than wth an analyss of queue sze at each node of the network. æ SYNOPSIS OF PAPER In ths paper we solve the problem of determnng the queue sze at each nodeofthenæn array n steady-state by convertng the problem nto a Jackson Queueng Network Model whch can then be analyzed va queueng theory. The queueng theory analyss requres solvng On 4 smultaneous lnear equatons to determne the steady-state total arrval rate at each server, whch can then be plugged nto the queueng formula to determne a probablty dstrbuton on the queue sze at each server. One very nterestng observaton made n ths paper s that when the routng algorthm s greedy a very smple formula can be used to determne the total arrval rates, makng t unnecessary to solve all the smultaneous equatons. Ths greatly smplæes the process of determnng the probablty dstrbuton on the queue sze at each server. 1

3 Another consequence of ths smple formula s that t can be used to prove that the total arrval rate ncreases as we look at nodes closer and closer to the center of the n æ n array. Observng that the expected queue sze at a node s drectly proportonal to the total arrval rate at the node, we are then able to prove that the expected queue sze at a node of the array ncreases as we look at nodes closer to the center of the array. æ OUTLINE OF CHAPTERS Secton 2 contans a detaled problem deænton. In Secton 3 we provde background materal on the Jackson Queueng Network Model, and also provde the formula for the probablty dstrbuton on the queue sze at the servers of ths network model. In Secton 4 we ærstly show how to cast our orgnal problem nto the framework of the Jackson Queueng Network model. Havng formulated our problem n terms of a Jackson Queueng Network, we next llustrate the smultaneous equatons whch must be solved to determne the total arrval rate at each server whch n turn are used to determne the queue sze at the server. Secton 5 obvates the need to solve the smultaneous equatons of Secton 4 by developng a very smple formula for determnng the arrval rates. Secton 5 next proves that the total arrval rate at a node ncreases as we look at nodes closer to the center of the array, and furthermore, that total arrval rate s drectly related to expected queue sze. Secton 5 concludes wth a theorem statng that the expected queue sze at a node of the array ncreases as the node's Eucldean dstance from the center of the array decreases. Secton 6 llustrates the use of ths smple formula to mmedately determne the probablty dstrbuton on the queue szes of the 5 æ 5 array problem. It also llustrates the general propertes proved n Secton 5 for the 5 æ 5 array problem. In Secton 7, we summarze our results, and n Secton 8 we dscuss several alternatve useful analyses whch could be mplemented usng the same Jackson Queueng Network Theorem. In Appendx A, we gve a more formal dervaton for the arrval rate formula Secton 5, rather than the more ntutve proof provded n Secton 5. Appendx B looks at an alternatve way we could have set up the Jakcson Queueng Model for our routng problem on the array, n whch we use a dæerent way of classfyng the packets. Lastly, Appendx C llustrates the On 4 smultaneous equatons for the case of the 3 æ 3 array. 2 Problem Deænton Our network s an n æ n array of processors, as shown n Fgure 1. New packets arrve at processor P ;j at Posson rate ç, where 0 éçé1. Each packet s assgned a random destnaton n the array. A packet contans a destnaton æeld and a data æeld. 2

4 When a new packet arrves, t s routed to ts destnaton va the followng greedy routng algorthm: Frst, the packet s routed to ts correct column and next to ts correct row. If two packets requre the same edge, contenton s resolved va Frst-Come-Frst-Served FCFS. The tme t takes for a packet to move through an edge s Posson dstrbuted wth mean 1. Only one packet may be on an edge at a tme. Our goal s to compute the probablty dstrbuton on the queue sze at each processor, when the network s n steady-state. The ærst step n solvng the above problem s to convert t nto a common queueng network model. We next analyze the dstrbuton on the queue szes n the model, whch gves us the dstrbuton on the queue szes n our orgnal problem. In the next secton we descrbe the queueng network model we'll be usng. 3 Multple-Job-Class Open Jackson Queueng Network Model In ths secton we descrbe the Queueng Network Model we wll be usng n ths paper. The model we use allows packets to be assocated wth a class or type. There are smpler queueng models n whch the packets aren't typed, however, as we wll see n Secton 3 we wll need ths more extensve model to handle our routng problem of Secton 1. The Queueng Network Model we use ëbuzacott,shanthkumar,93ë assumes there are m servers wth one processor per server. There are r classes, ortypes of packets. Packets of class l arrve at server from outsde the network at a Posson rate r l. One of the nce features of ths model s that t allows packets to change ther class as they move from server to server. A packet of class l at server next moves to server j and becomes of class k type wth probablty p lk j. The queueng network model assumes a complete drected graph connectng the servers. We can model a network wth fewer edges, by smply makng some of the edge probabltes zero. A packet at server may also leave the network, wth some probablty, rather than contnung to another server. Lastly the servce rate at server s ç. We wll use the notaton n l to denote the number of packets at server of class l, and n P r l1 n l to denote the total numberofpackets at server. Theorem 1 Buzacott,Shanthkumar,93 When the queueng network s n steady state,! 0 1 p n 1 ;:::;n r n r l n Y l A 1, ç ç n ^ç n 1 ;:::;n r l1 1 where ç ^ç ^ç ç rx l1 ^ç l 3

5 l and ^ç r l + mx rx j1 k1 p kl j ^ç k j 2 The proof of ths theorem s gven n ëbuzacott,shanthkumar,93ë. The above theorem tells us how to compute the jont probablty ofhavng, for example, n 1 of type 1 at server and n 2 packets of type 2 at server and,..., and n r server. Itsays we must ærst solve m æ r smultaneous lnear equatons to obtan and 1;:::;m. Then we plug these ^ç l 's nto the above formula for p n 1 packets packets of type r at l ^ç ;:::;n r for l 1;:::;r l Note that ^ç represents the rate at whch packets of class l æow nto server, ncludng both those packets whch arrve from outsde as well as packets arrvng from other servers. Now, suppose we want toknow the probablty that there are n packets at server. By deænton, p n, the probablty that there are n packets at server, s the sum of Expresson 1 over all values of n 1 ;n 2 ;:::;n r such that n 1 + n 2 + :::+ n r n. Corollary 2 where ç s deæned asabove n Theorem 1. p n 1, ç ç n. Proof: Note that the expresson for p n 1 observe that by deænton P! ^çl r l1 1. ^ç ;:::;n r begns wth a multnomal probablty, and Lastly let N be a random varable representng the number of packets at server. Then, snce N has a dstrbuton whch s geometrc tmes a factor ç,wehave: EëN ë ç 1, ç varn ç 1, ç 2 4 Modelng the Routng Problem on an Array as a Jackson Queueng Network In ths secton we showhow to formulate the problem of greedy routng on an næn array n terms of the queueng network model ntroduced n Secton 2 so that we may apply Theorem 1 to determne the probablty dstrbuton on the queue szes. We buld up to the exact formulaton wth some dscusson. 4

6 A ærst attempt mght be to let each of the n 2 nodes of the array be a server, where the servers are connected only by those edges whch are n the array. All other edges between processors have 0 probabltes assocated wth them. Now the probablty that a packet moves from server ; j to server 0 ;j 0 s ether 1 or 0, dependng on the destnaton of the packet. If we now make the destnaton of the packet be the ëclass" assocated wth the packet, the probablty that a packet moves from server ; j to server 0 ;j 0 depends only on ; j, 0 ;j 0, and the class of the packet, as requred by the queueng network model of Secton 2. Note that f our queueng network model ddn't allow for classes, the process of movng from server to server would not be Markovan. Observe that n ths formulaton, the class of a packet does not change as the packet moves between servers. The above formulaton doesn't qute work, however. Suppose, for example we model the servce rate of each server as 1. Queung theory assumes one queue at each server. However, suppose that 2 packets arrve at the same node and want ext the node n dæerent drectons. The array allows both packets to leave the node snce they won't conæct wth each other. In the current queueng model formulaton, however, one packet would ærst have to wat for the other packet to ænsh. Settng the servce to 4, the number of outgong edges, ndcates that the node can send four packets out on one edge whch sn't rght, ether. We can æx ths problem by realzng that congeston n the array s an edge problem not a node problem. Therefore we assocate a server wth each outgong edge, rather than each node of the array, as shown n Fgure 2. Rows and columns are numbered from 0 to n, 1 wth 0; 0 beng n the upper, lefthand corner. We use the notaton P ;j;l to denote the left processor at row, column j. Lkewse P ;j;r P ;j;u P ;j;d respectvely denote the rght, up and down ; j processors. Lastly we let P ;j;c denote the center ; j processor. We wll refer to processors P ;j;l ;P ;j;r ;P ;j;u, and P ;j;d respectvely as the left, rght, up, and down petals of the æower ; j, and to P ;j;c as the center processor of æower ; j. Each petal processor has ts own queue and sts on an edge. Packets on queue at a petal processor may be thought ofaswatng to use that edge. Only one packet at a tme can use the edge, and the tme spent by a packet on an edge s on average one tme unt the servce rate of every petal processor s 1. When a packet orgnates at a node of the orgnal n æ n array, we model t as orgnatng on the petal of the correspondng æower whch corresponds to the ærst edge the packet must use to get to ts destnaton. The packet then moves from the petal of one æower to a petal of another æower, etc., untl t reaches ts destnaton, at whch pont tmoves to the center processor of ts destnaton æower and leaves the system. We set the servce rate for the center processors to be 1. The edges nto one processor n our queueng network are shown n Fgure 3. For clarty edges nto the central processor, edges from the petals to other nodes, and paths nto and out of the system are not shown. In the greedy algorthm a packet whch moves upward wll only move upward or leave the system subsequently. So the only edge nto the æower from the Up petal below goesto the Up petal of the mddle æower. Lkewse the only edge from the Down petal above stothe Down petal n the mddle. Packets travelng left or rght may contnue n the same drecton or may turn up or down or they 5

7 may leave the system not shown. So there are edges from the left and rght petals of the æowers on the sde to the Up, Down, and Left and Rght petals n the mddle. We are now ready to formulate the routng problem on an n æ n array as a Jackson Queueng Network. Gven an n æ n array of processors P ;j wth grd connectons, such that: æ new packets arrve atp ;j from outsde the system at rate ç, æ Each packet s assgned a random destnaton when t ærst arrves. æ The packet s routed to ts destnaton va the Greedy algorthm. æ The rate at whch a packet traverses an edge s Posson rate 1. æ Only one packet may traverse an edge at a tme. æ Edge contenton s resolved usng FCFS We analyze the queue szes at the nodes of the above array by lookng at the followng queueng network model: æ The number of servers, m, s5n 2, denoted by P ;j;r, P ;j;l, P ;j;u, P ;j;d, P ;j;c, for 0;:::;n, 1 and j 0;:::;n, 1. æ The number of classes, r, sn 2, one for each possble destnaton. æ Packets never change class. A packet of class d that s, for destnaton d at server P js next moves to server P 0 j 0 S0 wth probablty pd. It's value s 1 f the greedy algorthm routes js; 0 j 0 S 0 a packet at server ;j;s wth destnaton d to server 0 ;j 0 ;S0, and 0 otherwse. æ Packets of class d arrve at server ;j;s wth Posson rate r d ;j;s c æ q n 2, where q the number of possble destnatons a packet at server P ;j;s mght be headed for va the greedy algorthm. æ ç, the servce rate at server, s 1 for all petal servers and 1 for the center servers. For the above queueng network model, the system of lnear equatons specæed n Corollary 2 from Secton 2 become: r d + ^ç rx d1 mx j1 p d d j ^ç j 3 ç ^ç ç ^ç p n 1, ç ç n 6

8 Thus to calculate p n, the probablty ofhavng n packets at server, we need to ærst solve the d smultaneous equatons for all the ^ç 's and then sum them to obtan the ^ç 's. Snce the servce rate, ç s 1, the ^ç 's are the ç 's whch then gve usp n for each and any n wechoose. The expected numberofpackets queued at a node s the sum of the expected numberofpackets queued at each petal of the assocated æower. EëN rc ë X S2R;U;L;D EëN rcs ë The number of smultaneous lnear equatons generated by Equaton 3 s r æ m 5n 4. Solvng a system of 5n 4 lnear equatons n 5n 4 unknowns seems dauntng. For general networks wth feedback paths there are mutually dependent varables. However n the model of a greedy routng algorthms, no packet path has a loop, so no varables are mutually dependent. Ths, along wth other features of greedy routng makes a general analytc soluton of the system of equatons for arbtrary szed arrays feasble. As an example, Appendx C shows the smultaneous lnear equatons whch result for the case of a 3 æ 3 array. In the next secton, we propose, however, a far easer way to obtan the ^ç 's drectly whch doesn't d requre solvng smultaneous equatons for the ^ç 's. 5 A Smple Method For Determnng Queue Sze Recall from Secton 2 that ^ç represents the total rate at whch packets arrve at server from both outsde the system as well as from other servers. In the observaton below, we gve an equvalent nterpretaton to ^ç whch also allows t to be computed quckly. Observaton 3 The value ^ç has a smple, ntutve meanng: t s the number of paths through node weghted by the frequency of use of each path. If the frequences of use of all paths are the same, ^ç s just the number of paths tmes the frequency of use. For any oblvous routng scheme the number of paths through a node can be easly computed. Theorem 4 The total arrval rate of packets at petal node P r;c;s s ^ç Pr;c;R ç colp + 1n, colp, 1 n ^ç Pr;c;L ç n, colp colp n ^ç Pr;c;U ç n, rowp rowp n ^ç Pr;c;D ç rowp + 1n, rowp, 1 n Proof: Here we present an nformal proof of these equatons. A precse dervaton from the orgnal deænng equatons s gven n Appendx A. 7

9 Consder the number of paths through some rght petal P r;c;r. All paths through that petal must have a destnaton n a æower to the rght of t. There are nn, colp, 1 destnatons that paths mght have. See Fgure 4. Addtonally snce the algorthm routes packets to the correct column before changng rows, only packets whch arrve from outsde at the colp æowers to the left on the same row, plus those arrvng from the outsde at the æower P r;c, go through the petal. Thus there are a total of colp +1nn, colp, 1 paths through the petal. Snce the arrval rate at each æower from outsde s ç and each ofn 2 destnatons s equally lkely, the arrval rate at any rght petal P r;c;r s ç ncolp + 1n, colp, 1. The arguments for petals n other drectons s smlar. The above theorem gves us an easer way to compute all the ^ç 's and thereby the p n 's. Note that the above theorem assumes as we have assumed throughout ths paper that the arrval rate from outsde to the nodes n the orgnal n æ n array s the same for every node. If that s not the case, we can stll determne the probablty dstrbuton on the queue szes, however now we must solve the 5n 4 smultaneous equatons. Theorem 4 doesn't menton the arrval rate at the center server of each æower. Ths can safely be gnored. Snce any packet arrvng at the center node leaves the system, the arrval rate can't næuence the arrval rate at any other petal. Also snce packets leave the system mmedately, that s the servce rate ç PrcC s 1, no queue ever forms: and ç PrcC ^ç PrcC 1 0 EëN PrcC ë ç P rcc 1, ç PrcC 0 Theorem 5 The expected total number of packets n queue at a æower r;c, EëN r;c ë,decreases wth the æower's Eucldean dstance from the center of the array. Proof: The followng s a proof sketch. The center of the array sat n ; n. Let x c, n and y r, n, the x and y p oæsets of the node from the center of the array. We wll show that as 2 x2 + y 2, the dstance from the center of the array, ncreases, the expected queue sze decreases. From Theorem 4 we see that ^ç Pr;c;R ^ç Pr;c;L ç cn, c, and ^ç Pr;c;U ^ç Pr;c;D ç rn, r. Rewrtng n terms of x and y we have ^ç Pr;c;R ^ç Pr;c;L ç n + x n n2, x 2 2 4, x2, and ^ç Pr;c;U ^ç Pr;c;D ç n + y n n2, y y2. The expected queue sze at each petal s proportonal to the arrval rate, ^ç Pr;c;S, at the petal. The expected queue sze at the node s the sum of the expected queue szes at each petal. Therefore the expected queue sze at the node s drectly proportonal to the arrval rate at the petals. Eëqueue sze at node r;cë æ n2 2, x2 + y 2 æ,x 2 + y 2 8

10 Therefore the greater the value of x 2 + y 2, the smaller the expected queue at the node. Snce the Eucldean dstance from the center of the array s p x 2 + y 2, the queue sze at a node s nversely proportonal to t's Eucldean dstance. In the next secton we wll show an example of how the formulas derved n theorem 4 together wth Corollary 2 are used to quckly derve the exact probablty dstrbuton on the queue szes n the case of the 5 æ 5 array. We wll also observe the phenomenon of theorem 5, when analyzng the 5 æ 5 array. Before lookng at the example n the next secton, we menton that there are many other ways n whch greedy routng on an n æ n array can be formulated n terms of the queueng network model. One possblty s dscussed n Appendx B. The formulaton n Appendx B uses only 5 classes and one server per node, leadng to 5n 2 l smultaneous equatons for the ^ç 's rather than the 5n 4 smultaneous equatons we currently solve. In ths model, the class of a packet represents what drecton the packet arrved from left, rght, up, or down, or outsde. The probabltes p d j are now reals between 0 and 1 relatng to the average dstrbuton of packets from that node. Snce ths formulaton nduces loops, some varables are mutually dependent. 6 An Example: Numercal Results In ths secton we compute the probablty dstrbuton on the queue szes of the nodes ; j of the 5 æ 5 array of processors where 0;:::;4 and j 0;:::;4. To do ths we look at the assocated Jackson Queueng Network as descrbed n Secton 4, and compute the probablty dstrbuton on the queue sze of each petal processor n the Jackson Network. Then, for each æower, ; j, we sum up the queue sze of each of ts petals to obtan the queue sze for the æower, whch n turn s the queue sze of node ; j n the orgnal 5 æ 5 array of processors. To compute the probablty dstrbuton on the queue sze of each petal processor n the assocated Jackson Queueng Network, we smply use Corollary 2 as suggested by 3 see Secton 4, however rather than solvng smultaneous equatons, we derve the ^ç 's drectly by the formulae n Theorem 3. Fgure 5a shows the ^ç for each petal server as derved usng Theorem 4. We assume ç 5, 12 snce t makes the numbers nce. Snce ç 1 for all petal servers, ç ^ç. Corollary 2 then tells us that the probablty that there are n packets n queue at petal server, p n, s equal to 1, ç ç n. Wehave not drawn the probablty dstrbuton on the queue sze for each petal, however, t s clear that the dstrbuton s geometrc and therefore has an exponental shape. Note that snce the queue szes on the petals of a æower are not ndependent we can't smply combne these probablty dstrbutons, however we can sum ther expected values. In Fgure 5b, we show EëN ë, the expected number of packets n queue at, for each petal server. Lastly, n Fgure 5c, for each æower ; j, we total the expected numberofpackets n queue at each of ts petals, to obtan the expected number of packets n queue at node ; j of the orgnal 5 æ 5 array. Observe that Fgure 5c clearly llustrates the phenomenon descrbed n Theorem 5. 9

11 7 Concluson Ths paper combnes deas from the areas of communcaton networks, queueng theory, and combnatorcs to analyze the queue buldup at the nodes of an n æ n array durng greedy routng. The three man contrbutons of the paper are: æ Away to formulate the problem of greedy routng on an array as a Jackson Queueng Network model. æ A very smple method for computng the probablty dstrbuton on the queue sze n the Jackson Queueng Network wth greedy routng. æ A theorem showng that the expected queue sze s greater for nodes closest to the center of the array. 8 Future Extensons There are numerous possble extensons to the work n ths paper. One area of future work s to redo all the analyss n ths paper for the case of a torus, rather than an n æ n array, snce the torus has no dstngushed nodes. We conjecture ths wll result n all nodes havng the same sze queues. Another dea s to use the same Jackson Queueng Network Model, but examne only the queue buldup of packets whose destnaton s, say, the center node. Note that t s easy to look at the queues formed by just one class. The usefulness of such a study s n avodng bottlenecks when wrtng routng algorthms. Thrdly, the technques of ths paper may be appled to studyng other routng algorthms. Observe also that any concluson reached about queue sze can be used to derve conclusons about expected packet delay, snce packet delay and queue sze are related va Lttle's Formula. Lastly, n ths paper we have only dealt wth queue sze n steady state. The rate of queue buld up when the network s overloaded or the clearng of packets after an overload can be computed by castng the same balance equatons used to derve Theorem 1 as dæerental equatons and solvng them. For a start, see ëbuzacott,shanthkumar,93ë. 9 Acknowledgements We thank J. George Shanthkumar and Sheldon Ross for ther help and suggestons n developng these results. 10

12 A Dervaton of Theorem 4 In ths appendx we gve a rgorous dervaton of Theorem 4. For brevty we ndcate the row of processor P rcs by P r, the column by P c, and the sde whch t s on Rght, Left, Up, or Down by P S. To begn the dervaton, we deæne the probablty of a packet movng from one petal to another. p d j r;c;s r;c;s 8 é é: 8 f s R and d c é c and j s R and 8j r;c,1 r;c é j s R and j r;c,1 r;c,or f s U and d c c and d r é r and j s U and j r+1;c r;c,or é é: j s L and j r;c+1 r;c 1 f s L and d c é c and j s L and j8 r;c+1 r;c é j s R and j r;c,1 r;c,or f é: s D and d c c and d r é r and j s L and j r;c+1 r;c,or é: j s D and j r,1;c r;c 0 otherwse Of course, p d j s0fj r;c;s s outsde the array, that s, f j r é 0, j r ç n, j c é 0, or j c ç n. We gnore the probablty ofmovng to the center processor snce t has no eæect on queue szes. Applyng the above deæntons, the general system of lnear equatons specæed n Equaton 3 smplæes to P r;c;r r d P r;c;r + p d P r;c,1;r P r;c;r ^çd P r;c,1;r P r;c;u r d P r;c;u + p d P r;c,1;r P r;c;u ^çd P r;c,1;r + p d P r+1;c;u P r;c;u ^çd P r+1;c;u + p d P r;c+1;l P r;c;u ^çd P r;c+1;l P r;c;l r d P r;c;l + p d P r;c+1;l P r;c;l ^çd P r;c+1;l P r;c;d r d P r;c;d + p d P r;c,1;r P r;c;d ^çd P r;c,1;r + p d P r;c+1;l P r;c;d ^çd P r;c+1;l + p d P r,1;c;d P r;c;d ^çd P r,1;c;d Agan nodes ëbeyond" the edge and center nodes do not contrbute and are not counted. Ignorng the equatons at the edge nodes, we have only 4n 2 equatons to solve, each wth an average of 2 unknowns. The arrval rates at the petals from the outsde s r dr;c P r;c;s 8 é é: ç N 8 é é: f P S R and d c ép c,or f P S U and d c P c and d r ép r,or f P S L and d c ép c,or f P S D and d c P c and d r ép r 0 otherwse Let's begn by wrtng the equatons for Rght chans begnnng n the left æowers of the array. P r;0;r r d P r;0;r P r;c;r r d P r;c;r + p d P r;c,1;r P r;c;r ^çd P r;c,1;r 11

13 d d Snce each ^ç r;c;r only depends on the ^ç P r;c;r values to the left of t n the same row, we can easly compute ther values. To begn, we specalze the equatons for the destnaton P r;0;r P r;c;r ç f N d c é 0 0 otherwse ç N f d c ép c 0 otherwse Expandng ths for the next column ç P r;1;r f N d c é 1 0 otherwse ç f N d c é 1 0 otherwse 2 ç f N d c é 1 0 otherwse And n general + p d P r;c,1;r P r;c;r ^çd P r;c,1;r + p d P r;0;r P r;1;r ^çd P r;0;r + p d P r;0;r P r;1;r ç N f d c é 0 0 otherwse Smlarly P r;c;r Pc +1 ç N f d c ép c 0 otherwse P r;c;l P r;c;u P r;c;d n, Pc ç N f d c ép c 0 otherwse nn, P r ç f N d c P c and d r ép r 0 otherwse np r +1 ç f N d c P c and d r ép r 0 otherwse Next we need to sum over all the classes. The results are the equatons gven n Theorem 4 ^ç Pr;c;R ç n P c + 1n, P c, 1 ^ç Pr;c;L ç n n, P cp c ^ç Pr;c;U ç n n, P rp r ^ç Pr;c;D ç n P r + 1n, P r, 1 B Alternatve Queueng Network Model Desgn for Formulatng the Routng Problem on Array Wehave done all the work for ths appendx, but have left the wrte-up out due to tme constrants. Please see us f you have questons. 12

14 C Smultaneous Equatons for the 3 æ 3 Array Problem Although solvng an arbtrary system of 4 æ 81 lnear equatons n 4 æ 81 unknowns specæed by Equaton 3 s dauntng, the equatons derved from our model have a straght forward soluton. Snce no packet's path ever returns to the same node, there are no dependences loops among the varables n the system of lnear equatons. Recall that the general soluton gven n Theorem 4 allows one to ænd the ^ç's wthout wrtng out and solvng the system of lnear equatons. However these serve as a check that the results from the general soluton are, ndeed, correct. We use codng n the symbols to reduce the number of equatons whch need to be wrtten out. ë012ë The symbol ^ç 00R represents the destnaton classes 00, 10, and 20, that s ^ç 00R, ^ç 00R, and ^ç 00R. For nstance the ærst two statements represent nne equatons. ^ç ë012ë0 00R ^ç ë012ëë12ë 00R ^ç :ë12ë0 00D ^ç ë12ë0 00D ç ë12ë0 9 01L ^ç :ë012ë2 01R ^ç ë012ë2 01R ç ë012ë2 9 00R ^ç :ë012ë0 01L ^ç ë012ë0 01L ç ë012ë0 9 02L ^ç :ë12ë1 01D ^ç ë12ë1 01D ^ç ë012ë2 02L ^ç ë012ëë01ë 02L ^ç :ë12ë2 02D ^ç ë12ë2 02D ç ë12ë2 9 01R ^ç ë012ë0 10R ^ç ë012ëë12ë 10R ë12ë1 ë12ë1 00R 02L ^ç :00 10U ^ç 00 10U ç L 20U ^ç :20 10D ^ç 20 10D ç L 00D 13

15 ^ç :ë012ë2 11R ^ç ë012ë2 11R ç ë012ë2 9 10R ^ç :01 11U ^ç 01 11U ç R 21U 12L ^ç :ë012ë0 11L ^ç ë012ë0 11L ç ë012ë0 9 12L ^ç :21 11D ^ç 21 11D R 01D 12L ^ç :02 12U ^ç 02 12U ç R 22U ^ç ë012ë2 12L ^ç ë012ëë01ë 12L ^ç :22 12D ^ç 22 12D ^ç ë012ë0 20R ^ç ë012ëë12ë 20R R 02D ^ç :ë01ë0 20U ^ç ë01ë0 20U ç ë01ë0 9 21L ^ç :ë012ë2 21R ^ç ë012ë2 21R ç ë012ë2 9 20R ^ç :ë01ë1 21U ^ç ë01ë1 21U ë01ë1 ë01ë1 20R 22L ^ç :ë012ë0 21L ^ç ë012ë0 21L ç ë012ë0 9 22L ^ç :ë01ë2 22U ^ç ë01ë2 22U ç ë01ë2 9 21R ^ç ë012ë2 22L ^ç ë012ëë01ë 22L Agan n coded form, the soluton to all the varables s 14

16 ^ç ë012ëë12ë 00R ^ç ë12ë0 00D 3 ç 9 ^ç ë012ë2 01R 2 ç 9 ^ç ë012ë0 01L 2 ç 9 ^ç ë12ë1 01D 3 ç 9 ^ç ë012ëë01ë 02L ^ç ë12ë2 02D 3 ç 9 ^ç ë012ëë12ë 10R ^ç 00 10U 6 ç 9 ^ç 20 10D 6 ç 9 ^ç ë012ë2 11R 2 ç 9 ^ç 01 11U 6 ç 9 ^ç ë012ë0 11L 2 ç 9 ^ç 21 11D 6 ç 9 ^ç 02 12U 6 ç 9 ^ç ë012ëë01ë 12L ^ç 22 12D 6 ç 9 ^ç ë012ëë12ë 20R ^ç ë01ë0 20U 3 ç 9 ^ç ë012ë2 21R 2 ç 9 ^ç ë01ë1 21U 3 ç 9 ^ç ë012ë0 21L 2 ç 9 ^ç ë01ë2 22U 3 ç 9 ^ç ë012ëë01ë 22L For the 3 æ 3 array, the sum of EëN Pr;c;S ë are all EëN 00 ë 4ç EëN 01 ë 6ç EëN 02 ë 4ç EëN 10 ë 6ç EëN 11 ë 8ç EëN 12 ë 6ç EëN 20 ë 4ç EëN 21 ë 6ç EëN 22 ë 4ç 2ç. Example: Say ç 1. The expected queue lengths are 2 1 1:5 1 1:5 2 1:5 1 1:5 1 Example: Say ç 1. The expected queue lengths are 4 :4 :6 :4 :6 :8 :6 :4 :6 :4 d ^ç P r;c;s 's over all destnatons s the same: 2ç. The expected values 3 Note that the expected queue sze grows sgnæcantly toward the center of the array. Theorem 5, shows that ths s true for arrays of all szes. 15

17 References ëbuzacott,shanthkumar,93ë John A. Buzacott and J. George Shanthkumar. Stochastc Models of Manufacturng Systems. Prentce Hall, NJ, ëleghton,92ë F. Thomas Leghton. Introducton to Parallel Algorthms and Archtectures: Arrays- Trees-Hypercubes. Morgan Kaufmann Publshers, CA, ëross,83ë Sheldon M. Ross. Stochastc Processes. John Wley and Sons, NY,1983. ëstamouls,tstskls,91ë George D. Stamouls and John N. Tstskls. The Eæcency of Greedy Routng n Hypercubes and Butteræes. Journal of the ACM,