Simple procedures for finding mean first passage times in Markov chains

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1 Res. Lett. Inf. Math. Sci., 2005, Vol. 8, pp Availale online at Siple procedures for finding ean first passage ties in Markov chains JEFFREY J. HUNER Institute of Inforation and Matheatical Sciences Massey University at Alany, Auckland, New Zealand he derivation of ean first passage ties in Markov chains involves the solution of a faily of linear equations. By exploring the solution of a related set of equations, using suitale generalized inverses of the Markovian kernel I P, where P is the transition atrix of a finite irreducile Markov chain, we are ale to derive elegant new results for finding the ean first passage ties. As a y-product we derive the stationary distriution of the Markov chain without the necessity of any further coputational procedures. Standard techniques in the literature, using for exaple Keeny and Snell s fundaental atrix Z, require the initial derivation of the stationary distriution followed y the coputation of Z, the inverse I P + eπ where e = (1, 1,,1) and π is the stationary proaility vector. he procedures of this paper involve only the derivation of the inverse of a atrix of siple structure, ased upon known characteristics of the Markov chain together with siple eleentary vectors. No prior coputations are required. Various possile failies of atrices are explored leading to different related procedures. 1 Introduction In solving for ean first passage ties in irreducile discrete tie Markov chains typically the results are expressed in ters of the eleents of Z, Keney and Snell s fundaental atrix, ([7]), or A # the group inverse of I P, (Meyer, [8]) where P is the transition atrix of the Markov chain and I is the identity atrix. he coputation of Z = [I P + Π] -1 and A # = Z Π oth require the prior deterination of {π i }, the stationary distriution of the Markov chain. We explore the joint deterination of oth the stationary distriution and the ean first passage ties using appropriate generalized atrix inverses that do not require previous knowledge of the stationary distriution. In an earlier paper (Hunter [6]) the use of special classes of generalized atrix inverses was explored in order to deterine expressions for the stationary proailities and the ean first passage ties, the key properties of irreducile Markov chains. In this paper we consider instead a class of generalized inverses that are in fact atrix inverses to Eail address: j.hunter@assey.ac.nz

2 210 J.Hunter give alternative expressions for the stationary proailities and the ean first passage ties. We explore the structure of these atrix inverses in order to deterine if any special relationships exist to provide coputational checks upon any derivations of the key properties. 2 Generalized inverses of Markovian kernels Let P = [p ij ] e the transition atrix of a finite irreducile, -state Markov chain with state space S = {1, 2,, } and stationary proaility vector π = (π1, π 2,, π ). he following suary provides the key features of generalized inverses (g-inverses) of the Markovian kernel I P that we shall ake use of in developing our new results. he key results elow can e found in Hunter [2]. G is a g-inverse, or a Condition 1 g-inverse, of I P if and only if: (I P)G(I P) = I P. Let P e the transition atrix of a finite irreducile Markov chain with stationary proaility vector π Τ Τ. Let e = (1, 1,, 1) and t and u e any vectors. (a) I P + tu Τ is non-singular if and only if π t 0 and u e 0. () If π t 0 and u e 0 then [I P + tu ] 1 is a g-inverse of I P. All Condition 1 g-inverses of I P are of the for [ I P + tu ] 1 + ef + gπ for aritrary vectors f and g. G-inverses ay satisfy soe of the following additional conditions: Condition 2: G(I P)G = G, Condition 3: [(I P)G] Τ = (I P)G, Condition 4: [G(I P)] Τ = G(I P), Condition 5: (I P)G = (I P)G. If G is any g-inverse of I P, define A I (I P)G and B I G(I P), then (Hunter [5]) G = [I P + αβ Τ ] 1 + γ eπ, (2.1) where α = Ae, β Τ = π Τ B, γ + 1 = π Τ Gα = β Τ Ge = β Τ Gα (2.2) and π Τ α = 1, β Τ e = 1. (2.3) Further A = α π Τ (2.4) and B = e β Τ. (2.5)

3 Mean first passage ties in Markov chains 211 he paraeters α, β, and γ uniquely specify and characterize the g-inverse so that we can denote such a g-inverse as G(α, β, γ). In Hunter [5] it is shown that G(α, β, γ) satisfies condition 2 if and only if γ = 1, G(α, β, γ) satisfies condition 3 if and only if α = π/π Τ π, G(α, β, γ) satisfies condition 4 if and only if β = e/ e e, G(α, β, γ) satisfies condition 5 if and only if α = e and β = π. he Moore-Penrose g-inverse of I P is the unique atrix satisfying conditions 1, 2, 3 and 4 and has the for G = G(π/π π, e/e e, 1). (An equivalent for was originally derived y Paige, Styan and Wachter [10].) he group inverse of I P is the (unique) (1, 2, 5) g-inverse A # = G(e, π, 1), as derived y Meyer [8]. Keeney and Snell s fundaental atrix of finite irreducile Markov chains (see [7]) is Z = [I P + eπ ] 1 = G(e, π, 0), a (1, 5) g-inverse with γ = 0. he following results are easily estalished (see Hunter [2]) (a) u [I P + tu ] 1 = π /(π t). () [I P + tu ] 1 t = e /(u e). (2.6) (2.7) 3 Stationary distriutions here are a variety of techniques that can e used for the coputation of stationary distriutions involving the solution of the singular syste of linear equations, π (I P) = 0, suject to the oundary condition π e = 1. Since, as we shall see later, the derivation of ean first passage ties involves either the coputation of a atrix inverse or a atrix g-inverse, we consider only those techniques for solving the stationary distriutions that use g-inverses. his will assist us later to consider the joint coputation of the stationary distriutions and ean first passage ties with a inial set of coputations. We consider three specific classes of procedures - one using A = I (I P)G, one using B = I G (I P), and one using siply G. heore 3.1: ([2]) If G is any g-inverse of I P, A I (I P)G and v is any vector such that v Ae 0 then π = v A v Ae, (3.1)

4 212 J.Hunter Furtherore Ae 0 for all g-inverse of G so that it is always possile to find a suitale v. heore 3.1 utilizes the oservation that the atrix A has a very special structure. Fro (2.4) A = απ. Since, fro (2.3), π α = 1 it is clear that α 0 iplying Ae = α 0 and thus it is always possile to find a suitale v for heore 3.1. Knowledge of the conditions of the g-inverse usually leads to suitale choices of v that siplify v Ae. Corollary 3.1.1: ([6]) Let G e any g-inverse of I P, and A= I (I P)G. (a) For all such G, π = e A A e A Ae. () If G is (1, 3) g-inverse of I P, and e i is the i-th eleentary vector, If G is (1, 5) g-inverse of I P, A π = e A e Ae and, for any i = 1, 2,...,, π = e i e i Ae. π = e A e e and, for any i = 1, 2,...,, π = e i A. In certain cases the expression B = I G(I P) can also e used to find an expression for π. heore 3.2: ([6]) Let G e any g-inverse of I P that is not a (1, 2) g-inverse, B = I G (I P) and v any vector such that v e 0. hen π = v BG v BGe. Corollary 3.2.1: ([6]) Let G e any g-inverse of I P, and B = I G (I P). (a) For all G, except a (1, 2) g-inverse, π = e BG e BGe and, for any i = 1, 2,...,, π = e BG i e i BGe. () If G is a (1, 5) g-inverse of I P, then for any i = 1, 2,,, π = e i B. he aove theores and corollaries all require coputation of A or B, ased upon prior knowledge of G. If G is of special structure one can often find an expression for π in ters of G alone. heore 3.3: ([6]) If G is a (1, 4) g-inverse of I P, π = e G e Ge.

5 Mean first passage ties in Markov chains 213 Soe of the aove expressions are well known. heore 3.1 appears in Hunter [2], [3]. he first expression of Corollary () was originally derived y Decell and Odell [1]. Meyer [8] estalished the first expression of Corollary under the assuption that G is a (1, 2, 5) g-inverse (ut the 2-condition is not necessary). If v = e i, the i-th eleentary vector, then e Ae = e, which ust e non-zero i i α = α i for at least one such i. Since e i A consists of eleents of the i-th row of A, we can always find at least one row of A that does not contain a non-zero eleent. Furtherore, if there is at least one non-zero eleent in that row, all the eleents in that row ust e non-zero, since the rows of A are scaled versions of π. hus, if A = [aij ] then there is at least one i such a i1 0 in which case a ij 0 for j = 1,,. his leads to following result. heore 3.4: ([6]) Let G e any g-inverse of I P. Let A = I (I P)G [a ij ]. Let r e the sallest integer i (1 i ) such that a 0, then arj =, j = 1, 2,...,. a k= 1 rk k= 1 ik (3.2) p 1k g k1 In applying heore 3.4 one typically needs to first find a 11 ( = 1 g 11 + ). If a11 0 then the first row of A will suffice to find the stationary proailities. If not find a 21, a 31, and stop at the first non-zero a r1. For soe specific g-inverses we need only find the first row of A. For exaple MALAB uses the pseudo inverse routine pinv(i P), to generate the (1,2,3,4) g- inverse of I P. Corollary 3.4.1: ( [6]) If G is a (1, 3) or (1, 5) g-inverse of I P, and if A = I (I P)G [a ij ] then = a 1 j a 1k, j = 1, 2,...,. (3.3) Proof: If G satisfies condition 3, α=π/π π in which case α1 0. Siilarly if G satisfies condition 5, α = e in which case α 1 = 1. he non-zero for of α 1 ensures a G-inverse conditions 2 or 4 do not place any restrictions upon α and consequently the non-zero nature of a 11 cannot e guaranteed in these situations.

6 214 J.Hunter While (3.1), (3.2) and (3.3) are useful expressions for otaining the stationary proailities, the added coputation of A following the derivation of a g-inverse G is typically unnecessary, especially when additional special properties of G are given. Rather than classifying G as a specific ulti-condition g-inverse, we now focus on special class of g-inverses which are atrix inverses of the siple for [ I P + tu ] 1, where t and u are siple fors, selected to ensure that the inverse exists with π t 0 and u e 0. A general result for deriving an expression for using such a g-inverse is the following. heore 3.5: If G = and u e 0, then [I P + tu ] 1 where u and t are any vectors such that π t 0 π = u G u Ge. (3.4) Hence, if G = [g ij ] and u = (u 1, u 2,, u ), = r=1 u r u k g s=1 rs = π u k, j = 1, 2,...,. (3.5) g r. r =1 u r Proof: Using (2.6) it is easily seen that u [I P + tu ] 1 e = π e π t = 1 π t and (3.4) follows. he eleental expression (3.5) follows fro (3.4). he for for π aove has the added siplification that we need only deterine G (and not A or B as in heores 3.1 and 3.2 and their corollaries.) While it will e necessary to evaluate the inverse of the atrix I P + tu this ay either e the inverse of a atrix which has a siple special structure or the inverse itself ay e one that has a siple structure. Further, we also wish to use this inverse to assist in the deterination of the ean first passage ties (see Section 4). We consider special choices of t and u ased either upon the siple eleentary vectors e i, the unit vector e, the rows and/or coluns of the transition atrix P, and in one case a coination of such eleents. Let p e P denote the -th row of P. p a Pe a denote the a-th colun of P and ale 1 elow lists of a variety of special g-inverses with their specific paraeters. All these results follow fro the oservation that if G = [I P + tu ] 1 then, fro (2.2), the paraeters are given y α = t / π t, β = u / u e and γ + 1 = 1/{(π t)(u e)}. he special structure of the g-inverses given in ale 1 leads, in any cases, to very siple fors for the stationary proailities.

7 Mean first passage ties in Markov chains 215 In applying heore 3.5, oserve that π =u G if and only if u Ge = 1if and only if π t = 1. ale 1: Special g-inverses Identifier g-inverse Paraeters [I P + tu ] 1 α β γ G ee [I P + ee ] 1 e e / (1/) 1 (r ) -1 [I P + e p ] e (r ) p 0 G -1 e [I P + e e ] e e 0 G ae (c,r) G a G a [I P + p a e ] -1 p a / π a e / (1/π a ) 1 [I P + p -1 a p ] [I P + p -1 a e ] p a / π a p (r ) (1/π a ) 1 p a / π a e (1/π a ) 1 G ae [I P + e a e ] -1 e a /π a e / (1/π a ) 1 G a [I P + e ] a p -1 e a /π a (r ) p (1/π a ) 1-1 e a /π a e (1/π a ) 1 G a [I P + e ] a e G t -1 [I P + t e ] (t e e + p ) t e 0 Siple sufficient conditions for π t = 1 are t = e or t = α (cf. (2.3)). (his later condition is of use only if α does not explicitly involve any of the stationary proailities, as for ) G t G t is included in ale 1 as the update t e replaces the -th colun of I P y e. (See [10]). Corollary 3.5.1: If G = [I P + eu ] 1 where u Τ e 0, π = u G. (3.6) and hence if u Τ = (u 1, u 2,, u ) and G = [g ij ] then = u k g kj, j = 1, 2,...,. (3.7) In particular, we have the following special cases: (a) If u Τ = e Τ then G G ee = [ I P + ee ] 1 = [gij] and = g. j. (3.8)

8 216 J.Hunter () If u Τ = then G = [I P + ep ] -1 = [g ij ] and p (r ) = If u Τ -1 = e then G Ge = [I P + ee ] = [g ij ] and p k. (3.9) = g j. (3.10) Corollary 3.5.2: If G = [I P + te Τ ] -1 where π Τ t 0, π = e G e Ge, (3.11) and hence, if G = [g ij ], then = g kj g r =1 s=1 rs In particular, results (3.12) hold for G = G ae = g. j g.., j = 1, 2,...,. (3.12), Gee and G ae. In the special case of G ee, using (2.6) or (2.7), it follows that g.. = 1, and (3.12) reduces to (3.8). Corollary 3.5.3: If G = [I P + te ] -1 where π t 0, and hence, if G = [g ij ], then π = e G e Ge, (3.13) = g j s=1 g s In particular, results (3.14) hold for G = G In the special cases of and = g j g., j = 1, 2,...,. (3.14) a, Ga, and G t. G t, g. = 1 and (3.14) reduces to (3.10). Corollary 3.5.4: If G = [I P + tp ] -1 where π t 0, and hence, if G = [g ij ], then π = p G p Ge, (3.15)

9 Mean first passage ties in Markov chains 217 = i=1 p k s=1 p i g is, j = 1, 2,...,. (3.16) (c,r) G a G a In particular, results (3.16) hold for G =, and. In the special case of, the denoinator of (3.16) is 1 and (3.16) reduces to (3.9). hus we have een ale to find siple eleental expressions for the stationary proailities using any of the g-inverses in ale 1. In the special cases of G ee,, Ge and G t the denoinator of the expression given y equations (3.5) is always 1. (In each other case, oserve that.) u G = π / π denoinator of the expression u Ge is in fact 1/π, with We consider the g-inverses of ale 1 in ore detail in order to highlight their structure or special properties that ay provide either a coputational check or a reduction in the nuer of coputations required. Let g a = Ge a denote the a-th colun of G and g =e G denote the -th row of G. Fro the definition of G = [ I P + tu ] 1, pre- and post-ultiplication y I P + tu yields G PG + t u G = I, (3.17) G GP + Gt u = I. (3.18) Pre-ultiplication y π and post-ultiplication y e yields the expressions given y e a (2.6) and (2.7), i.e. u G = π /π t and Gt =e/u e. Relationships etween the rows, coluns and eleents of G follow fro (3.17) and (3.18) y pre- and postultiplication y and and the fact that g i. = g i e, g. j = e g j, g ij = e i Ge j. hese are suarised in the following theore. e heore 3.6: For any g- inverse of the for G = [ I P + tu ] 1, with π t 0 and u e 0, ( ) π, ( ) u, (a) (Row properties) g i p i G = e i t i π t g i g i P = e i 1 u e and hence g i. = p ik g k t i ( π k t k ). ( ) t, () (Colun properties) g j Pg i = e j π t

10 218 J.Hunter g j Gp i = e j ( u j u e) e, and hence g. j = p.k g kj + 1+ ( t k ) ( π k t k ), g. j = g.k p kj + 1+ u j ( u k ). (Eleent properties) g ij = p ik + δ ij t i ( π k t k ), g ij = g ik p kj + δ ij u j ( u k ). Let g rowsu = Ge = = [g 1., g 2.,...,g. ] denote the colun vector of row sus of G and g colsu = e G = g j = [g.1,g.2,...,g. ] the row vector of colun sus of G. g j=1 j j=1 ale 2 is constructed using results of (2.6), (2.7), heore 3.6 and the requisite definitions. A key oservation is that stationary distriution can e found in ters of just the eleents of the -th row of, G,,G and G his requires the a G a (a ) a. t deterination of just eleents of G. We exploit these particular atrices later. If the entire g-inverse has een coputed the stationary distriution can e found in ters of, the row vector of colun sus, in the case of G, and. In g colsu each of these cases there are siple constraints on and, possily reducing the nuer of coputations required, or at least providing a coputational check. (c,r) G a G a g a g rowsu ee G ae In the reaining cases of, and, the additional coputation of p G is required to lead to an expression for the stationary proailities. We can further explore inter-relationships etween soe of the g-inverses in ale 1 y utilizing the following result given y heore 3.3 of Hunter [4]. G ae heore 3.7: Let P e the transition atrix of a finite irreducile transition atrix of a Markov chain with stationary proaility vector π. Suppose that for i = 1, 2, π t i 0 and u e 0. hen i and hence that [I P + t 2 u 2 ] 1 = [I eu 2 u 2 e ][I P + t 1 u ] 1 [I t 2 π eπ ] + 1 π t 2 (π t 2 )(u 2 e ).

11 Mean first passage ties in Markov chains 219 [I P + t 2 u 2 ] 1 [I P + t 1 u 1 ] 1 = eu 2 u 2 e [I P + t 1 u 1 ] 1 t 2 π eu 2 π t 2 u 2 e [I P + t 1 u 1 ] 1 [I P + t 1 u 1 ] 1 t 2 π π t 2 + eπ (π t 2 )(u 2 e ). In particular, we wish to focus on the differences etween G, and. hese results are used in Section 4. aa G aa G aa G g-inverse ale 2: Row and colun properties of g-inverses t u g a a-th colun g colsu Colun su (r ) g -th row G ee e e π e g rowsu Row su Other properties G e e p e e p G = π e e π e G ae (c,r) G a p a p a e e a π π a Gp a = e p e a + (1 p a )e p G = π π a Gp a = e G aa (a = ) p a e e a π π a G a (a ) p a e e + e a π π a G ae e a e e π π a G aa (a = ) e (r ) a p e e a p G = π π G a (a ) e (r ) a p e e + π π p G = π π a G a e a e e π π a G t t e π e Gt = e

12 220 J.Hunter heore 3.8: (a) () Proof: (a) G aa G aa = e a πτ ee π a. (3.19) a G aa G aa = eπ ee π a = e( π e a π a ). (3.20) a G aa G aa = eπ π a e a π π a = (e e a ) π π a. (3.21) Using the results of heore 3.7, it is easily seen that ( c) ( c) ( c) ( r) eea ( r) pa π eea ( r) ( r) pa π eπ Gaa Gaa = Gaa G. ( c) aa Gaa + ( c) ( c) e e π p e e π p ( e e)( π p ) a a a a a a Using, e a e = 1, e a G aa = e a, p a = Pe a, π p a = π a, e a Pe a = p aa, equation (3.22) siplifies to G aa G aa = p aa eπ ee π a G Pe aa a π a π a Now oserve that, y the definition of G aa, (3.22) + eπ π a. (3.23) I = G aa G aa P + G aa e a p a. (3.24) Post-ultiplying (3.33) y e a yields e a = G aa e a G rr Pe a + G aa e a e a Pe a = e G aa Pe a + ep aa. (3.25) Sustitution of the expression for G fro (3.25) into (3.23) yields (3.19). aa Pe a () and hese results follow directly fro heore 3.7 and the row and colun properties of G aa and G aa, as given in ale 2. G aa G aa A close study of equation (3.19) shows that and differ only in the a-th row and a-th colun, with specific eleents in the a-th row and colun in each atrix as given in ale 2, and with all the other eleents identical. A foral proof follows fro (3.29), since for i a and j a, the (i,j)-th eleent of G G is given y aa aa e i (G aa G aa )e j = (e i e a )( π e j ) (e π i e)(e a e j ) = 0. a (A proof can e constructed via deterinants and cofactors defining the inverses and upon noting that in constructing I P + e a p a the only eleents of I G aa G aa P that are changed are in the a-th row where each eleent is zero apart fro the (a, a)- th eleent which is 1. Siilarly that in constructing I P + p a e a the only eleents of I P that are changed are in the a-th colun where each eleent is zero apart fro the (a, a)-th eleent which is 1.)

13 Mean first passage ties in Markov chains Mean first passage ties Let M = [ ij ] e the ean first passage tie atrix of a finite irreducile Markov chain with transition atrix P. All known general procedures for finding ean first passage ties involve the deterination of either atrix inverses or g-inverses. he following theore suarises the general deterination of M y solving the well known equations for the ij : ij = 1+ p ik kj, (4.1) k j using g-inverses to solve the atrix equation ( I P) M = E PM d, where E = ee = [1] and D = M d = (Π ) d -1 with Π = eπ. heore 4.1: (a) Let G e any g-inverse of I P, then M = [GΠ E(GΠ) d + I G + EG d ]D. (4.2) () Let H = G(I Π), then M = [EH d H + I]D. (4.3) Let C = I H, then M = [C EC d + E]D. (4.4) Proof: (a) Expression (4.2) appears in Hunter [3] as heore having initially appeared in the literature in Hunter [2]. () Expression (4.3) follows fro (4.2) upon sustitution. he technique was also used in a disguised for in Corollary of Hunter [6]. Expression (4.4) follows fro (4.3). It was first derived in Hunter [6]. he advantages of expressions (4.3) and (4.4) is that we can deduce siple eleental fors of ij direct fro these results. Corollary 4.1.1: Let G = [g ij ], H = [h ij ], and C = [c ij ] then 1 (a) ij = [ cij c jj + 1], for all i, j. (4.5) π () j 1, i= j, 1 π j ij = [ h jj hij + δij ] = π j 1 [ hjj hij ], i j. π j (4.6) 1 ij = [ g jj g ij + δij ] + [ g i. g j. ], for all i, j. π (4.7) j

14 222 J.Hunter Proof: (a) Result (4.5) follows directly fro (4.4) (correcting the results given in Hunter [6]). () Result (4.6) follows either fro (4.3) or (4.5) since h ij = δ ij c ij. Since H = G GΠ, h ij = g ij g ik = g ij g i., for all i, j. and result (4.7) follows fro (4.6). Note also that since C = I H c ij = δ ij g ij + g ik π k=1 j = δ ij g ij + g i., for all i, j. and hence result (4.7) follows alternatively fro (4.5). Note that expression (4.5) has the advantage that no special treatent of the i = j case is required. he following joint coputation procedure for and ij was given in Hunter [6], ased upon heore 3.4 and Corollary aove. (he version elow corrects soe inor errors given in the initial derivation.) heore 4.2: 1. Copute G = [g ij ], e any g-inverse of I P. 2. Copute sequentially rows 1, 2, r ( ) of A = I (I P)G [a ij ] until k=1 a. rk, (1 r ) is the first non-zero su. 3. Copute = a rj ark k=1, j = 1,...,. a k=1 rk, i= j, a rj 4. Copute ij = (g jj g ij ) a k=1 rk + (g k=1 a ik g jk ), i j. rj While this theore outlines a procedure for the joint coputation of all the and ij following the coputation of any g-inverse, the procedure contains the unnecessary additional coputation of the eleents of A. Oserve also that all the expressions of Corollary require knowledge of the stationary proailities. We consider instead first deriving expressions for ij. Let N = [n ij ] = [(1 δ ij ) ij ] so that N = (M M d )(M d ) -1. Note that n jj = 0 for all j. heore 4.3 follows directly fro (4.3) and (4.4), or y solving the atrix equation (I P)N =Π I, using g-inverse techniques.

15 Mean first passage ties in Markov chains 223 heore 4.3: N = [n ij ] = EH d H where H = G(I Π), so that n ij = (g jj g ij ) + (g i. g j. ), for all i, j. 1, i = j, Further, ij = (g jj g ij ) + (g i. g j. ), i j. Let us consider using the special g-inverses given in ale 1 and 2 to find expressions for all the and the ij. he results are suarised in ale 3. Note that siplification of the expressions for ij using,g and results fro the oservation that is in each case constant. he special case of deserves highlighting. g rowsu heore 4.4: If = [I P + e e ] 1 = [gij], then = g j, j = 1, 2,...,, G ee e (4.8) and ij = 1/ g j, i = j, (4.9) (g jj g ij ) g j, i j. his is one of the siplest coputational expressions for oth the stationary proailities and the ean first passage ties for a finite irreducile Markov chain. hese results do not appear to have een given any special attention in the literature. If the stationary proaility vector has already een coputed then the standard procedure is to copute either Keeny and Snell s fundaental atrix, ([7]), Z [I P + Π] -1, where Π = eπ, or Meyer s group inverse, ([8]), A # Z Π. Both of these atrices are in fact g-inverses of I P. he relevant results, which follow fro Corollary are as follows. heore 4.5: (a) If Z = [I P + eπ ] 1 = [z ij ] then M = [ij] = [I Z + EZ d ]D, and ij = 1, i = j; (z jj z ij ), i j. () If A # = [I P + eπ ] 1 eπ = [a # ij ] then M = [ij] = [I A # # + EA d ]D and (4.10) 1 π j, i = j; ij = (4.11) (a # jj a # ij ), i j. Proof: See Hunter, [3], Corollary 7.3.6C. hese are also special cases of (4.5) since Ze = e and A # e = 0, so that Σ j z ij = z i. = 1 for all i and Σ j a # ij = a # i. = 0 for all i.

16 224 J.Hunter Note the siilarity etween the expressions (4.9), (4.10) and (4.11), with (4.9) oviously the easiest of the three expressions to copute. ale 3: Joint coputation of { }and [ ij ] using special g-inverses g-inverse ij ij (i j) G g.j 1/ g.j (g jj g ij ) / g.j ee p k 1 p k k (g jj g ij ) p k g j 1/ g j (g jj g ij ) / g j k G g.j /g.. g.. / g. (g jj g ij ) g.. / g. + (g i. g j. ) ae (c,r) G a G a k i s p k p i g is i s k p i g is p k k i s (g jj g ij ) p i g is + (g p k g i. g j. ) kj k g j /g. g. / g j (g jj g ij ) g. / g j + (g i. g j. ) G ae g.j /g.. g.. / g. (g jj g ij ) g.. / g + (g i. g j. ) G a k i s p k p i g is i s k p i g is p k i s (g jj g ij ) p i g is + (g p k g i. g j. ) kj G a g j /g. g. / g j (g jj g ij ) g. / g j + (g i. g j. ) k G t g j 1/ g j (g jj g ij ) / g j + (δ i δ j.) If G = G t = [g ij ] then ij = g jj g ij + δ ij g j + δ i δ j, the eleental expressions of M, as given y Corollary 7.3.6D() of Hunter, [3]. It also appears, in the case =, in Meyer [9]. We have een exploring structural results. If one wished to find a coputationally efficient algorith for finding ased upon note that for π we need to solve the equations π = π P, or π (I P + ee ) = e. his reduces the prole to finding an efficient package for solving this syste of linear equations. Paige, Styan and Wachter [10] recoended solving for π using π (I P + eu ) = u with u = e (r ) j P = p j, using Gaussian eliination with pivoting. Other suggested choices included u = e j, the recoended algorith aove. We do not explore such coputational procedures in this paper. It is however interesting to oserve that the particular atrix inverse we suggest has een proposed in the past as the asis for a coputational procedure for solving for the stationary proailities. Mean first passage ties were not considered in

17 Mean first passage ties in Markov chains 225 [9] and techniques for finding the ij typically require the coputation of a atrix inverse. appears to e a suitale candidate. In deriving the ean first passage ties one is in effect solving the set of equations (4.1). If in this set of equations if we hold j fixed, (j = 1, 2,, ) and let j = (1j, 2j,, j ) then equation (4.1) yields j = [I P + p j e j ] 1 e = G jj e. (4.12) (his result appears in Hunter [3], as Corollary 7.3.3A). Note the appearance of one of the special g-inverses considered in this paper of the for of G with a = j. aa heore 4.6: For fixed j, 1 i, (a) = e G e. ij i jj (4.13) Further, if G jj = [g rs ] then ij = g i. δij () ij = ei G jj e + 1. (4.14) π j (r Further, if G ) jj = [g rs ] then ij = g i. + δ ij p 1 = jk g k., i = j, g i. 1, i j. ij 1 ij = i G δ e jje + π. (4.15) j Further, if G jj = [g rs ] then ij = g i = j, j. g i. g j. i j. Proof: Expressions (4.13), (4.14), and (4.15) follow, respectively, fro (4.12), (3.19) and (4.13), and (3.20) and (4.14) (or (3.21) and (4.13)). he eleental expressions for follow as the i-th coponent of the of G jj,g jj, and G jj. For case (), fro g rowsu ale 2 it follows that g j. = 1 and p jk g k. = p (r j G ) jj e = 1/. For case oserve that g j. = g j e = 1/. ij All of these results are consistent with equation (4.8). For exaple, for (4.12), with G jj = [gij], fro equation (3.14), π i = g ji /g j. for all i. Oserve that fro ale 2 that the G jj j-th row and colun of are, respectively, π / and ej, so that for fixed j, g jj = 1, and for i j, g ij = 0 and g ji =π i / with g j. = 1/. Sustitution in (4.7), for fixed j, yields jj = 1/ = g j. and for i j, ij = (g jj g ij ) g j. + (g i. g j. ) = g i., as given y (4.13).

18 226 J.Hunter he utilisation of special atrix inverses as g-inverses in the joint coputation of stationary distriutions and ean first passage ties leads to a significant siplification in that at ost a single atrix inverse needs to e coputed and often this involves a row or colun su with a very siple for, further reducing the necessary coputations. While no coputational exaples have een included in this paper, a variety of new procedures have een presented that warrant further exaination fro a coputational efficiency perspective. References 1. Decell, H.P., Jr., and Odell P. L. (1967). On the fixed point proaility vector of regular or ergodic transition atrices. Journal of the Aerican Statistical Association, 62, Hunter J.J. (1982). Generalized inverses and their application to applied proaility proles. Linear Algera Appl., 45, Hunter, J.J. (1983). Matheatical echniques of Applied Proaility, Volue 2, Discrete ie Models: echniques and Applications. Acadeic, New York. 4. Hunter J.J. (1988). Charaterisations of generalized inverses associated with Markovian kernels. Linear Algera Appl., 102, Hunter J.J. (1990). Paraetric fors for generalized inverses of Markovian kernels and their applications. Linear Algera Appl., 127, Hunter J.J. (1992). Stationary distriutions and ean first passage ties in Markov chains using generalized inverses. Asia-Pacific Journal of Operational Research, 9, Keeny J.G. and Snell, J.L. (1960). Finite Markov Chains. Van Nostrand, New York. 8. Meyer C.D. Jr. (1975). he role of the group generalized inverse in the theory of finite Markov chains. SIAM Rev., 17, Meyer C.D. Jr. (1978). An alternative expression for the ean first passage tie atrix. Linear Algera Appl., 22, Paige C.C., Styan G.P.H., and Wachter P.G. (1975). Coputation of the stationary distriution of a Markov chain. J. Statist. Coput. Siulation, 4,

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