Non-linear Matrix Equations: Equilibrium Analysis of Markov Chains

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1 No-lear Matrx Equatos: Equlbrum Aalyss of Marov Chas by Garmella Ramamurthy Accepted for publcato the Global Joural of Mathematcal Aalyss (publshed by Godavar Mathematcal Socety ) Report No: IIIT/TR/009/8 Cetre for Securty, Theory ad Algorthms Iteratoal Isttute of Iformato Techology Hyderabad , INDIA May 009

2 NON-LINEAR MATRIX EQUATIONS: EQUILIBRIUM ANALYSIS OF MARKOV CHAINS Garmella Rama Murthy, Assocate Professor, IIIT---Hyderabad, Gachbowl, HYDERABAD-50003, Adhra Pradesh, Ida.. Itroducto: I the research area of oe dmesoal stochastc processes, Marov chas acqure specal mportace due to large umber of applcatos. Oe of the smplest possble cotuous tme Marov chas, amely brth-ad-death process arses aturally may queueg models. Such a Marov cha has a effcet recursve soluto for the equlbrum probabltes. Specfcally the equlbrum probabltes form a geometrc sequece. The commo rato/recurso costat s the soluto of a quadratc equato. Evas ad Wallace cosdered a stochastc process called the Quas-Brth-ad-Death (QBD) process as a atural geeralzato of the brth-ad-death process. They showed that for a bloc-jacob geerator of cotuous parameter Marov processes (called QBD processes), the statoary probablty vector X may be parttoed to x vectors x, 0 whch are gve by x xo R for 0..(.) where the square matrx R s the mmal o-egatve soluto of a matrx-quadratc equato. A equlbrum probablty vector X, whch satsfes equato (.) wll be called a matrx-geometrc probablty vector. A atural questo whch arses s whether the matrx geometrc recursve soluto exsts for the equlbrum probabltes of a more geeral class of Marov processes. Marcel Neuts has show [Neu] that a matrx geometrc recursve soluto exsts for the equlbrum probabltes of a large class of processes, called G/M/-type Marov processes. Ths s fortuate sce these processes provde good stochastc models for varous problems arsg queueg ad vetory theores. The state space, E of a G/M/-type Marov process has the followg form: E {(, ) : 0, } (.) whch s fte but otherwse arbtrary.ths state space ca be clearly decomposed to levels by performg a lexcographc parttog o the frst state varable. For each level, a equlbrum probablty vector ca the be defed. It s gve by = [,,..., ] (.3) where, s the equlbrum probablty that the process s the state (,). The etre set of lmtg probabltes for the process s the specfed by the ftedmesoal vector

3 = [ 0 ]...(.4) I the case of a G/M/-type Marov process, the trastos betwee states varous levels are restrcted. Specfcally, upward trastos from level ca oly reach level (+); dowward trastos from level ca reach, oe trasto, ay level for <. Thus, wth the above parttog ad orderg of the state space to levels, the geerator matrx Q of a G/M/-type Marov process X t has the followg form: B0 C B A A B A A A Q B A3 A A A0...(.5) B4 A4 A3 A A A whch all matrces, except possbly some of the boudary matrces, are x. The equlbrum probablty vector s the uque soluto to Q = 0..(.6) Solvg the equato (.6) for drectly s tme cosumg ad very cumbersome. It s desrable to have a recursve method for computg. I other words, we would le to fd a expresso for terms of for <. as follows Neuts showed [Neu, Neu] that ca be foud terms of = R, (.7) Where the matrx R s the mmal oegatve soluto of The 0 X A 0..(.8) A matrces (.8) are the submatrces of the geerator Q (.5). Thus A matrx geometrc recurso exsts for the equlbrum probabltes. The rate matrx R ca be utlzed to express most of the steady state characterstcs of the G/M/ type Marov processes. Thus, effcet algorthms for the computato of ths matrx are therefore of cosderable terest appled probablty. Tradtoally, computato of rate matrx s approached utlzg teratve procedures. Sce the drect terato scheme requres large processg tme, a Pcard terato scheme ad other mproved schemes have bee proposed whch substatally decrease the umber of teratos [Neu]. Based o a expermetal study of varous schemes [Rama], a ew more effcet algorthm s proposed [Rama]. These teratve procedures for the computato of rate matrx trucate the power seres

4 (.8), thus effectvely mag a polyomal approxmato to t. Thus these teratve procedures are approxmate, tme cosumg ature. Also covergece problems could occur. I [LaR], a logarthmc reducto algorthm s proposed for QBD processes. I [RKC], a alteratve procedure to compute the egevalues, egevectors of R s proposed. Thus, the Jorda caocal form represetato ( Closed form Represetato ) of R s foud. Eve ths wor [RKC], the matrx power seres (.8) s trucated ad the soluto of a matrx polyomal equato s determed. But ths paper we cosder a rreducble, postve recurret G/M/- type Marov process, whose rate matrx satsfes a matrx power seres equato (ad ot a matrx polyomal equato). Method for computato of Jorda Caocal Form represetato of rate matrx s proposed. Ths research paper s orgazed as follows. I secto, the geeral problem of solvg matrx power seres equato Of the form (.8) s cosdered. Some terestg lemmas ad Theorems are proved. By explotg the propertes of A matrces (sub-matrces the geerator), Theorems related to the computato of egevalues ad left egevectors are proved Secto 3. Thus method of computato of rate matrx s dscussed Secto 4. It s brefly dscussed how the geeralzed egevectors ad thus the Jorda caocal form of rate matrx ca be computed. I Secto 4, by usg a smple trasformato, t s descrbed how the results developed for computato of R hold mutats mutads for the computato of matrx geometrc recurso matrx R arsg the equlbrum aalyss of dscrete tme Marov chas of G/M/-type. Furthermore, t s brefly descrbed how a smlar method ca be utlzed the equlbrum aalyss of M/G/-type Marov chas.. Matrx Power Seres Equatos: Soluto Techques: Algebrac Geometry: Cosder a matrx power seres equato of the followg form: 0 X D 0,..(.) where the coeffcet matrces { D ; 0 } are ow ad X s the uow matrx. For the sae of coveece, cosder the case where the elemets of D s are from the feld of real/complex umbers. The followg Lemma provdes a method of determg the egevalues of all possble solutos of (.). Lemma : Cosder a matrx Y whch satsfes (.) ad let ) D...(.) 0 H ( The H ( ) has the followg represetato 0 H ( ) ( I Y )( N )....(.3)

5 Where N Y D (.4). Thus the characterstc polyomal of every soluto of (.) dvdes the determetal trascedetal fucto 0 Det ( H ( ) ) = Det ( D )..(.5) Proof: It s easy to see that f s a egevalue of soluto Y, the t s a root of the determetal trascedetal fucto Det ( G( ) ). The more challegg part s the ecessty part. The Lemma s clam deals wth factorzato of matrx valued aalytc fuctos. A mportat research paper o ths topc was wrtte by M.L.J. Hautus [Hau] ettled, Operator Substtuto. Specfcally, the above Lemma follows from clam property.6 (Remader Theorem). From the pot of vew of symbolc algebra, the followg dervato provdes the requred factorzato. The approach orgated a dscusso wth Prof. Balasubramaa. 0 D = ( I Y ) D = ( I Y ) ( I Y... Y ) D N 0 = ( I Y) ( N ), where.(.6) Y D..(.7) QED The followg Theorem s a geeralzato of a result [Ga] for matrx polyomal equatos. Theorem : Cosder a matrx power seres equato of the form (.).e. 0 X D 0 Let the dmeso of X be. Let there be ftely may, say m ( m > ) roots of the trascedetal fucto Det ( 0 D ). The all possble solutos of (.) are m dvded to atmost equvalece classes ad soluto each class s determed As the soluto of a lear system of equatos.

6 Proof: From equato (.3); we ow that the characterstc polyomal of every soluto dvdes the trascedetal fucto Det ( the followg equato 0 D ). Ths ferece follows from Det( H ( )) Det ( I Y ) Det( 0 N ) (.8) Sce there are atmost m roots of Det ( 0 D ) ad sce each soluto of m (.) has roots, the solutos are dvded to classes. It s easy to see that the set of solutos each class costtute a equvalece class. Let the Jorda caocal form of a soluto th class be T E T. It should be oted that the set of egevalues determe D. Substtutg, the soluto (.), we have 0 T E T D 0 The above equato s equvalet to the followg oe. T [ E T D ] 0. 0 Sce T s o-sgular, we ecessarly have that [ 0 E T D ] 0...(.9) Sce egevalues are ow, E s ow. Thus we have to solve for T usg the above equato. It s clear that the above system of equatos are lear. QED. Lemma : Ra (Y) Ra ( D 0 ). Proof: Let f be a vector the left ull space of Y. f ( 0 Y D ) = 0, mples f D 0 = 0. Hece f les the left ull space of D 0. Thus, Ra ( D 0 ) Ra (Y). (.0) QED Remar: From the above Lemma, t s clear that f D 0 s osgular, the every soluto of (.) s osgular.

7 A. Reducto of a Matrx Power Seres Equato to a Matrx Polyomal Equato: Cosder a partcular soluto Y of (.). I the followg Lemma t s show that f the egevalues of Y are ow, the the matrx power seres equato (.) ca be reduced to a matrx polyomal equato of the followg form G s are matrx seres l 0 Y G G. 0, where Lemma 3: If Y satsfes (.) wth the matrx power seres beg absolutely coverget ad the characterstc polyomal of Y s ow, the 0 Y D Y D,..(.) 0 where the matrces D s are obtaed from the characterstc polyomal of Y. Proof: Let the characterstc polyomal of Y be By the Cayley-Hamlto Theorem, 0 I Y... Y 0. Equvaletly, 0 Y I Y... Y. N N Multplyg the above equato by Y ad substtutg for Y, a expresso for Y ca be obtaed terms of I, Y, Y,..., Y. Y where the coeffcets ca be repeated to fd Thus Y m ( ) ( ) ( ) 0 I Y... Y, 0 ( ) s are related to the m Y for arbtrary m terms of D s ad the coeffcets of the s. The process ( m) Y.(.) I, Y, Y,..., Y. Sce the matrx power seres o the left had sde of (.) s absolutely coverget, rearragg the terms wll ot affect the sum. Hece, substtutg the above m expresso for Y (.) ad regroupg the terms yelds the followg equato 0 Y D Y D QED 0 To llustrate the utlty of above results, a specal class of olear matrx equatos whch arse the equlbrum aalyss of G/M/-type Marov processes are cosdered. I the followg dscusso, we cosder G/M/-type

8 Marov processes whch are rreducble ad postve recurret.. System of No-Lear Matrx Equatos: Algebrac Geometry: Now let us cosder o-lear matrx equatos whch are of the followg form: Matrx Polyomal equatos.e. the uow as well as coeffcet matrces are compatble matrces e.g m m X Am X Am... X A A0 0.(.3) I the above equato X as well as A s are compatble matrces. Matrx Power Seres Equatos: 0.e uow X as well as coeffcets X A 0..(.4) A s are matrces. A. MATRIX POLYNOMIAL EQUATIONS: It s well ow that oe of the cetral goals of algebrac geometry s to determe the exstece, uqueess as well as cardalty of solutos of a system of mult-varate polyomal equatos. Bezout s Theorem s a mportat result ths drecto. It s ow show wth a example that the soluto of (.3) falls the doma of algebrac geometry. Let us cosder a specal case of (.3).e. a matrx quadratc equato [RaC] X A X A A 0, (.5) 0 where { X, A0, A, A } are matrces. For the sae of llustrato, let us cosder the smple case where these matrces are x matrces. Let the uows the matrx ( ) X be deoted as { x, x, x3, x4 }. Also, let { a ;, } be the etres of A for = 0,,. It s easy to see that the equato (.5) represets a system of 4 equatos uows { x, x, x3, x4 } of hghest degree. The four equatos are gve by () () () () (0) ( x x x3) a ( xx x x4 ) a ( x ) a ( x ) a a 0 () () () () (0) ( x x x3) a ( xx x x4 ) a ( x ) a ( x ) a a 0..(.6) () () () () (0) ( x 3x x4 x3) a ( x3x x4 ) a ( x3) a ( x4 ) a a 0 () () () () (0) ( x 3x x4 x3) a ( x3x x4 ) a ( x3) a ( x4 ) a a 0 By a atural geeralzato to matrx polyomal equatos, t s clear that we have a STRUCTURED system of mult-varate polyomal equatos. The soluto of a ARBITRARY matrx polyomal equato s dscussed [Ga]. The techques of LINEAR ALGEBRA are applcable to ths CLASS OF NON-LINEAR matrx equatos. Thus some STRUCTURED problems ALGEBRAIC GEOMETRY ca be solved usg LINEAR ALGEBRA techques.

9 CONJECTURE: It s coectured that ARBITRARY mult-varate polyomal equatos ca be mbedded PROPERLY CHOSEN TENSOR VARIATE POLYNOMIAL EQUATIONS. MULTI-VARIATE POWER SERIES EQUATIONS: As the case of matrx polyomal equatos, t s easy to see that matrx power seres equatos represet a STRUCTURED class of mult-varate power seres equatos. Thus, t s show ths research paper that such class of mult-varate equatos ca be solved usg the techques of lear algebra. CONJECTURE: It s coectured that ARBITRARY mult-varate power seres equatos ca be IMBEDDED a TENSOR VARIATE POWER SERIES EQUATIONS. 3 Computato of Egevalues ad Egevectors of Rate Matrx: The rate matrx R s the mmal oegatve soluto of the matrx power seres equato 0 R A 0.(3.) Where the A s are the submatrces the geerator matrx Q. O vog the Lemma, we realze that the egevalues of R are a subset of the roots of the determetal trascedetal fucto 0 K( ) Det ( A ) = Det ( G ( ) ) (3.) I geeral K( ) s a trascedetal fucto. The spectral radus of R, Sp(R), s strctly less tha oe, sce the rreducble Marov process X t s postve recurret. Hece, of the roots of K( ) le strctly sde the ut crcle the complex plae. But order to determe the egevalues of R uambguously from the roots of K ( ), t stll remas to localze the other roots to a dstct rego of the complex plae. Theorem () ths secto wll provde the desred result. Remar: Ivog results sub-secto (A) ( secto ), the matrx power seres equato (3.) ca be reduced to a matrx polyomal equato. The coeffcet matrces ths reduced equato are obtaed from those equato (3.) ad the coeffcets of characterstc polyomal of R If the coeffcet matrces the resultg, reduced matrx polyomal equato have the same propertes as those the equato [RKC], we ca drectly voe the results research paper [RKC] to arrve at more geeral coclusos (method for computato of egevalues ad egevectors of R). But t s ot clear how to deduce that the coeffcet matrces of the reduced equato have the same propertes as those of equato secto 3 of [RKC}. Thus we are aturally led to the followg dscusso. Relatoshp to the Method based o Complex Aalyss:

10 The results ths paper relatg to equlbrum aalyss of G/M/-type Marov chas are purely based o algebrac argumets. I the followg, these results are related to those obtaed usg the complex aalyss method. I the complex aalyss method, oe starts out to costruct a equlbrum probablty vector = [.. ] of the form a,..(3.3) where the (usually complex) umbers These ad the row vectors from the zeros of, le strctly sde the ut crcle. a are to be determed. ' s are ormally determed ( ) ( * * Det I A ( )) where ( ) A A, for, (3.4) 0 It should be oted that () s usually a trascedetal fucto. By a applcato of Rouche s Theorem ad the perturbato theory for matrces, t s the show that () has exactly zeros, satsfyg. Detaled argumets must the be employed to fd codtos uder whch all the zeros, are strctly wth the ut crcle. These codtos, foud for specfc models, are essece, specal cases of the ecessary ad suffcet codtos for the postve recurrece of the Marov cha derved [Neu]. Oce the zeros are determed, ther multplctes are examed to fd f the tral soluto (3.) eeds to be modfed case multple zeros are preset. For very specal cases such a examato s carred out [Ba], [BGK] ad [C]. It s evdet that whe the multplcty of the zeroes of () s oe or more, the matrx geometrc recurso for the equlbrum probablty vector s equvalet to the form (3.) ad ts modfcato. It s mmedate to see f, a are the egevalue ad left egevector of R respectvely, the * a A ( ) = 0. Lemma shows that the above codto s also suffcet. Now o utlzg the codto that the Marov cha s postve recurret f ad oly f SP( R ) < ad the Localzato Theorem prove the sequel, the egevalues of R ca be determed by computg the zeros of the trascedetal fucto Det ( K( ) ) whch are wth the ut crcle. Thus the Rouche s Theorem s ot voed to fer that there are zeroes of () whch are o or sde the ut crcle. Furthermore, o dagoalzablty costrat o R s mposed whch s ormally employed complex aalyss based methods. A characterzato of the geeralzed left egevectors of the rate matrx ca be derved alog the les of Lemma 5 [RKC]. I the followg Lemma, a ew proof of equalty (.7.) [Neu, p.33], s provded based o the results secto o matrx power seres equatos. Ths Lemma wll be utlzed the proof of Localzato Theorem. I [RKC], several results that are utlzed the proof of Localzato

11 Theorem are derved. The matrx power seres verso of the results ca easly be derved through a smple geeralzato of the argumets. Detals are avoded here for brevty. Oly mportat results are stated here wthout detaled proof. Utlzg the defto of K ( ) ad Lemma (.), we have K ( ) = Det ( I R) Det( N ()).(3.5) Theorem : N() s a dagoally domat matrx f the absolute value of s strctly less tha oe. Proof: The Theorem follows from a proof argumet very smlar to the oe utlzed [RKC]. The detaled argumet s avoded for brevty. QED. The followg Theorem s the ma result utlzed to determe the egevalues of R from the zeroes of the determetal trascedetal fucto K ( ). Theorem 3 (Localzato Theorem) : Gve a rreducble, postve recurret Marov process, suppose that A s rreducble. The s a egevalue of the rate matrx R f ad oly f K( ) = Det( G( )) = 0 ad <, where G( ) = 0 A. Proof: There are two approaches to provg the Theorem. The frst approach utlzes the same argumet as that utlzed complex aalyss based method. Specfcally Rouches Theorem s voed as [Ta], [Ba], [BGK] ad [C]. The secod approach s a geeralzato of the proof argumet utlzed [RKC]. Ths approach utlzes Theorem stated above. Detals are avoded for brevty. QED. The above Localzato Theorem provdes a method of fdg the egevalues of the rate matrx. I the followg Lemma t s show that the above Theorem also provdes a method of computg the left egevectors of the rate matrx. Lemma 3: The row vector u s a left egevector correspodg to a egevalue of R f ad oly f u ( 0 A ) = u ( G( )) = 0, (3.6) Proof: Suppose s a egevalue correspodg to the left egevector u. Necessary part of the above asserto follows from the defto of left egevector ad from (3.). Sce R satsfes (3.) ad (3.) s of the same form as (.), Lemma ca be appled. Hece 0 A = ( I R ) N( ). Now suppose that u satsfes ( ). O usg ( ) u ( I R) N( ) = 0. But the spectral radus of R s strctly less tha oe ad so the absolute value of

12 s strctly less tha oe. By the Localzato Theorem, N( ) s o-sgular f the absolute value of s strctly less tha oe. Therefore we have that u ( I R) = 0. Hece u s a left egevector of R. QED. I [RKC, Lemma 5], a alteratve characterzato of left egevectors of rate matrx R s provded. It ca be geeralzed to the matrx power seres equato case. Thus the geeralzed left egevectors ca be determed. 4 Computato of Rate Matrx of a G/M/-Type Marov Process: Cosder the case whe the rate matrx s dagoalzable. Oe suffcet case for dagoalzablty s that all the egevalues of R are dstct. Whe the rate matrx s dagoalzable, t ca be computed through the spectral represetato approach. The spectral represetato of R s of the followg form. R = T C T (4.) where C s a dagoal matrx wth the p o-zero egevalues of R o the dagoal ad the rows of the matrx T are the left egevectors of R. The egevalues of R ca be foud from the roots of the determetal trascedetal fucto K( ). Also, as show Lemma 3, a left egevector correspodg to a ozero egevalue ca be determed by fdg a vector the left ull space of the matrx 0 A. Thus the spectral represetato of rate matrx R ca be foud usg the results Secto 3. I [RKC], the problem of computato of rate matrx R as a lear programmg problem. Usg a smlar approach ca be utlzed for computato of R, the mmal o-egatve soluto of a matrx power seres equato. Whe the rate matrx R (a mmal o-egatve soluto of matrx polyomal equato) s ot dagoalzable, method for computato of all geeralzed left egevectors ad thus the Jorda caocal form represetato of R s dscussed [RKC]. Eve the matrx power seres case, smlar method ca be derved. Detals are avoded for brevty. 5 Computato of Rate Matrx of a G/M/-Type Marov Cha: Cosder a G/M/-type Marov cha X. The trasto probablty matrx, P s of the form

13 D0 C D C C D C C C P D C3 C C C0 (5.) D4 C4 C3 C C C The G/M/-type Marov cha X s assumed to be rreducble ad postve recurret. The equlbrum probabltes of the states o level, v are related to those at level +, through a matrx geometrc recurso ' v = v R..(5.) ' The rate matrx R s the mmal o-egatve soluto of Where C 0 R ' C R '..(5.3) C matrces are those the trasto probablty matrx P. Defg C for 0, ad C C I ad substtutg (5.3), we have 0 It ca be oted that each ' R C 0. (5.4) C for s oegatve ad that C has egatve dagoal elemets ad o-egatve off-dagoal elemets. Thus C s have the same propertes as A s. Sce (5.4) s also of the same form as (3.), the rate matrx ' R of G/M/-type Marov cha has may propertes whch are aalogous to those of R ad thus ca be computed usg smlar techques as the prevous secto. Detals are avoded for the sae of brevty. Now let us cosder M/G/-type Marov chas. I the aalyss of such Marov chas, a matrx power seres equato arses. As show [RKC}, computato of mmal o-egatve soluto of such matrx power seres equato ca be carred out usg the techques dscussed ( Sectos 3 ad 4) for G/M/-type Marov chas. Detals follow the same method as dscussed [RKC}. ACKNOWLEDGEMENTS The author would le to tha Prof. Paul Fuhrma for brgg to my atteto, the wor of Prof. M.L.J. Hautus. Also I would le to tha Prof. Balasubramaa for terestg dscussos related to Lemma.

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