BMAP/M/C Bulk Service Queue with Randomly Varying Environment

Transcription

1 IOSR Journal of Engneerng (IOSRJEN) ISSN (e): , ISSN (p): Vol. 05, Issue 03 (March. 2015), V1 PP BMAP/M/C Bulk Servce Queue wth Randomly Varyng Envronment Rama.G, Ramshankar.R, Sandhya.R 3, Sundar.V 4, Ramanarayanan.R Independent Researcher B. Tech, Vellore Insttute of Technology, Vellore, Inda Independent Researcher MS9EC0, Unversty of Massachusetts, Amherst, MA, USA Independent Researcher MSPM, School of Busness, George Washngton Unversty, Washngton.D.C, USA Senor Testng Engneer, ANSYS Inc., 2600, Drve, Canonsburg, PA 15317, USA Professor of Mathematcs, (Retred), Vel Tech Dr. RR & Dr.SR Techncal Unversty, Chenna. Abstract: Ths paper studes two stochastc BMAP arrval and bulk servce C server queues (A) and (B) wth k varyng envronments. The arrvals to the queue are governed by a batch Markovan arrval process of verson and the bulk servce tmes are exponental wth parameter μ n the envronment for 1 k* respectvely. When the envronment changes from to j, changes occur for arrval and servce as follows: the arrval BMAP representaton changes from the verson to the j verson, the resdual arrval tme starts wth the statonary probablty vector of the j verson BMAP, t becomes the ntal j verson upon arrval of customers and the exponental servce tme parameter changes from μ to μ j for 1, j k*. The system has nfnte storng capacty and the servce bulk szes are fnte valued random varables. Matrx parttonng method s used to study the models. In Model (A) the maxmum of the arrval szes M n all the envronments s greater than the maxmum of the servce szes N n all the envronments, (M > N), and the nfntesmal generator s parttoned as blocks of the sum of the number of BMAP phases of all envronments tmes the maxmum of the arrval szes for analyss. In Model (B) the maxmum of the arrval szes M n all the envronments s less than the maxmum of the servce szes N n all the envronments, (M < N), where the nfntesmal generator s parttoned usng blocks of the sum of the number of BMAP phases of all envronment tmes the maxmum of the servce szes for analyss. Fve dfferent cases assocated wth C, M and N due to parttons are treated. They are namely, (A1) M >N C, (A2) M C >N (A3) C >M >N, whch come up n Model (A); (B1) N C and (B2) C >N, whch come up n Model (B) respectvely. For the cases when C M or N Matrx Geometrc results are obtaned and for the cases when C > both M and N Modfed Matrx Geometrc results are presented. The basc system generator s seen as a block crculant matrx n all the cases. The statonary queue length probabltes, ts expected values, ts varances and probabltes of empty queue levels are derved for the models usng Matrx Methods. Numercal examples are presented for llustraton Keywords: Block Crculant, BMAP Arrval, Bulk Servce, C servers, Infntesmal Generator, Matrx methods. I. INTRODUCTION In ths paper two mult server queues wth batch Markovan arrval process (BMAP) and bulk servce have been studed wth random envronments usng matrx geometrc methods. For M/M/1 bulk queues wth random envronment models one may refer to Rama Ganesan, Ramshankar and Ramanarayanan [1] and M/M/C bulk queues wth random envronment models are of nterest n Sandhya, Sundar, Rama, Ramshankar and Ramanarayanan [2]. PH/PH/1 bulk queues wthout varaton of envronments have been treated by Ramshankar, Rama Ganesan and Ramanarayanan [3] and the same type of queues wth random varaton of envronments are studed by Ramshankar, Rama, Sandhya, Sundar and Ramanarayanan [4]. Bn, Latouche and Men [5] have studed numercal methods for Markov chans. Chakravarthy and Neuts [6] have dscussed n depth a multserver queue model. Gaver, Jacobs and Latouche [7] have treated brth and death models wth random envronment. Latouche and Ramaswam [8] have analyzed Analytc methods. For matrx geometrc methods and models one may refer Neuts [9]. Batch Markovan arrval processes are presented by Lucantony [10] and are analyzed also by Cordero and Khroufch [11]. The models consdered n ths paper are general compared to exstng queue models n lterature snce a BMAP arrval and mult server bulk servce queue wth random envronment has not been studed at any depth so far. The number of servers ncreases wth the arrval of number of customers tll t becomes C. Usually the parttons of the bulk arrval models have M/G/1 upper- Hesenberg block matrx structure wth zeros below the frst sub dagonal. The decomposton of a Toepltz sub matrx of the nfntesmal generator s requred to fnd the statonary probablty vector. In ths paper the parttonng of the matrx s carred out n a way that the statonary probablty vectors have a Matrx Geometrc soluton or a Modfed Matrx Geometrc soluton for nfnte capacty C server bulk arrval and bulk servce queues wth randomly varyng envronments. 33 P a g e

2 Two models (A) and (B) on BMAP/M/C bulk queue systems under k* varyng envronments wth nfnte storage space for customers are studed here usng the block parttonng method. The M/PH/1 and PH/M/C queues wth random envronments have been studed by Usha n [12] and [13] wthout bulk arrvals and bulk servces. It has been notced by Usha n [12, 13] that when the envronment changes the remanng arrval and servce tmes are to be completed n the new envronment. The resdual arrval tme and the resdual servce tme dstrbutons n the new envronment are to be consdered at an arbtrary epoch snce the spent arrval tme and the spent servce tme have been n the prevous envronment wth dstnct szes of PH phase. Further new arrval tme and new servce tme from the start usng ntal PH dstrbutons of the new envronment cannot be consdered snce the arrval and the servce have been partly completed n the prevous envronment ndcatng the statonary versons of the arrval and servce dstrbutons n the new envronments are to be used for the completons of the resdual arrval and servce tmes n the new envronment and on completon of the same the next arrval and servce onwards they have ntal versons of the PH dstrbutons of the new envronment. The statonary verson of the dstrbuton for resdual tme has been well explaned n Q-Mng He [14] where t s named as equlbrum PH dstrbuton. Ramshankar, Rama, Sandhya, Sundar, Ramanarayanan n [4] have studed PH/PH/1 queue models wth bulk arrval, bulk servce wth random envronment ntroducng the statonary verson for the resdual tmes. In ths paper the statonary probablty startng vector of the new verson s used when the envronment changes for the resdual arrval tme and t becomes the ntal new verson of BMAP dstrbuton after the arrval. Model (A) presents the case when M, the maxmum of all the maxmum arrval szes n the envronments s bgger than N, the maxmum of all the maxmum servce szes n all the envronments. In Model (B), ts dual, N s bgger than M, s treated. In general n Queue models, the state space of the system has the frst co-ordnate ndcatng the number of customers n the system but here the customers n the system are grouped and consdered as members of M szed blocks of customers (when M >N) or N szed blocks of customers (when N > M) for fndng the rate matrx. For the C server system under consderaton, Model (A) gves three cases namely (A1) M > N C, (A2) M C > N and (A3) C > M > N and Model (B) gves two cases namely (B1) N C, and (B2) C > N. The case M=N wth varous C values can be treated usng Model (A) or Model (B). The matrces appearng as the basc system generators n these models due to block parttons are seen as block crculant matrces. The statonary probablty of the number of customers watng for servce, the expected queue length, the varance and the probablty of empty queue are derved for these models. Numercal cases are presented to llustrate ther applcatons. The paper s organzed n the followng manner. In secton II and secton III the BMAP/M/C bulk servce queues wth randomly varyng envronment n whch maxmum arrval sze M s greater than maxmum servce sze N and the maxmum arrval sze M less than the maxmum servce sze N are studed respectvely wth ther sub cases. In secton IV numercal cases are presented. II. MODEL (A). MAXIMUM ARRIVAL SIZE M GREATER THAN MAXIMUM SERVICE SIZE N 2.1Assumptons for M > N. () There are k* envronments. The envronment changes as per changes n a contnuous tme Markov chan wth nfntesmal generator Q 1 of order k* wth statonary probablty vector ϕ. () In the envronment for 1 k*, the batch arrvals occur n accordance wth Batch Markovan Arrval Process wth matrx representaton for the rates of batches of sze m M gven by the fnte sequence {D m, 0 m M } wth phase order k where D 0 has negatve dagonal elements and ts other elements are non-negatve; D m has non-negatve elements for 1 m M. Let D M = m=0 D m and φ be the statonary probablty vector of the generator matrx D wth φ D = 0 and φ e = 1 for 1 k*. () When the envronment changes from to j for 1, j k*, the arrval process BMAP of the j verson starts as per statonary (equlbrum) probablty vector of the j verson of the arrval process for the completon of the j resdual arrval tme there after the arrvals occur as per BMAP of the j verson, namely, { D m 0 m M j }. (v)customers are served n batches of dfferent bulk szes. There are s servers to serve when s customers are present n the system for 1 s C. When C or more than C customers are present n the system the number of servers to serve customers s C. In the envronment for1 k*, the tme between consecutve bulk servces has exponental dstrbuton wth parameter sμ when s customers (s servers ) are n the system for 1 s C and wth parameter Cμ when C or more than C customers (C servers )are present where μ s the parameter of sngle server exponental servce tme dstrbuton. At each servce epoch n the envronment, ψ customers are served wth probabltes gven by P (ψ = j) = q j for 1 j N when more than N customers are watng for 34 P a g e

3 N q j BMAP/M/C Bulk Servce Queue wth Randomly Varyng Envronment servce where j =1 =1. When n customers n < N are n the system, then j customers are served wth N probablty, q j for 1 j n-1 and n customers are served wth probablty j =n q j for 1 k*. (v) When the envronment changes from to j, the exponental servce tme parameter of sngle server changes from μ to μ j, the bulk servce sze ψ changes to ψ j and the maxmum servce sze N changes to N j. (v) The maxmum batch arrval sze of all BMAPs, M= max 1 M s greater than the maxmum servce sze N= max 1 N 2.2.Analyss There are three sub cases for ths model namely (A1) M > N C, (A2) M C >N and (A3) C > M >N. Sub Cases (A1) and (A2) admt Matrx Geometrc solutons and they are treated n sub secton (2.2.1). Modfed Matrx Geometrc soluton s presented for Sub Case (A3) whch s studed n sub secton (2.2.2). The state of the system of the contnuous tme Markov chan X (t) under consderaton s presented as follows. X (t) = {(n, m,, j): for 0 m M-1; 1 k*, 1 j k and n 0}(1) The chan s n the state (n, m,, j) when the number of customers n the system s n M + m, for 0 m M-1, 0 n <, the envronment s for 1 k* and the arrval phase s j for 1 j k. When the number of customers n the system s r, then r s dentfed wth (n, m) where r on dvson by M gves n as the quotent and m as the remander.. Let the survvor probabltes of servces ψ be respectvely for the envronment state for 1 k*. P(ψ >m)= Q m m =1- n=1 q n, for 1 m N -1 (2) Q m =0 for m N and Q 0 = 1. (3) Sub Cases: (A1) M > N C and (A2) M C > N When M > N C or M C > N, the BMAP/M/C bulk queue admts matrx geometrc soluton as follows. The chan X (t) descrbng them, has the nfntesmal generator Q A,2.1 of nfnte order whch can be presented n block parttoned form gven below. B 1 A A 2 A 1 A Q A,2.1 = 0 A 2 A 1 A A 2 A 1 A 0 0. (4) A 2 A 1 A 0 0 In (4) the states of the matrces are lsted lexcographcally as 0, 1, 2, 3,. n,. Here the vector n s of type 1 x M =1 k and n = ( (n, 0, 1, 1),(n, 0, 1, 2) (n, 0, 1, k 1 ), (n, 0, 2, 1),(n, 0, 2, 2) (n, 0, 2, k 2 ),,(n, 0, k*, 1),(n, 0, k*, 2) (n, 0, k*, k ), (n, 1, 1, 1),(n, 1, 1, 2) (n, 1, 1, k 1 ), (n, 1, 2, 1),(n, 1, 2, 2) (n, 1, 2, k 2 ),,(n, 1, k*, 1),(n, 1, k*, 2) (n, 1, k*, k ),, (n, M-1, 1, 1),(n, M-1, 1, 2) (n, M-1, 1, k 1 ), (n, M-1, 2, 1),(n, M-1, 2, 2) (n, M-1, 2, k 2 ),,(n, M-1, k*, 1),(n, M-1, k*, 2) (n, M-1, k*, k ) ) for n 0. The matrces B 1 and A 1 have negatve dagonal elements, they are of order M =1 k and ther off dagonal elements are non- negatve. The matrces A 0, anda 2 have nonnegatve elements and are of order M =1 k and they are gven below. Let Q = D 0 + ( Cμ +(Q 1 ), )I k for 1 k* (5) where I ndcates the dentty matrx of order gven n the suffx, Q s of order k. Consderng the change of envronment swtches on statonary verson of BMAP arrval n the new envronment, the followng matrx Ω of order =1 k s defned whch s concerned wth change of envronment durng arrval tme and servce tme. Q 1 Ω 1,2 Ω 1,3 Ω 1, Ω 2,1 Q 2 Ω 2,3 Ω 2, Ω= Ω 3,1 Ω 3,2 Q 3 Ω 3, (6) Ω,1 Ω,2 Ω,3 Q where Ω,j s a rectangular matrx of type k x k j whose all rows are equal to (Q 1 ),j φ j for j, 1, j k*. In the envronment, for 1 k*, the matrx of arrval rates of n customers correspondng to the arrval n BMAP s D n whch s a matrx wth non-negatve elements for 1 n M and D n = 0 matrx for n > M (7) and the rate wth whch n customers are served by a sngle server for 1 n N s gven by S,n =μ q n and S,n = 0 f n > N. (8) 35 P a g e

4 1 D n D n 0 0 Let Λ n = D n 0 for 1 n M (9) D n In (9) Λ n s a square matrx of order =1 k ; D j n s a square matrx of order k j for 1 j k* and 0 appearng as (,j) component of (9) s a block zero rectangular matrx of type k x k j. S 1,n I k S 2,n I k2 0 0 Let U n = 0 0 S 3,n I k3 0 for 1 n N (10) S,n I k In (10) U n s a square matrx of order =1 k ; S j,n I kj s a square matrx of order k j for 1 j k* and 0 appearng as (, j) component of (10) s a block zero rectangular matrx of type k x k j. The matrx A for = 0, 1, 2 are as follows. A 0 = Λ M Λ M 1 Λ M Λ M 2 Λ M Λ M 3 Λ M Λ 3 Λ 4 Λ M 0 0 Λ 2 Λ 3 Λ M 1 Λ M 0 Λ 1 Λ 2 Λ M 2 Λ M 1 Λ M (11) A 2 = 0 0 CU N CU N 1 CU 2 CU CU N CU 3 CU CU N CU N CU N (12) Ω Λ 1 Λ 2 Λ M N 2 Λ M N 1 Λ M N Λ M 2 Λ M 1 CU 1 Ω Λ 1 Λ M N 3 Λ M N 2 Λ M N 1 Λ M 3 Λ M 2 CU 2 CU 1 Ω Λ M N 4 Λ M N 3 Λ M N 2 Λ M 4 Λ M 3 CU A 1 = N CU N 1 CU N 2 Ω Λ 1 Λ 2 0 CU N CU N 1 CU 1 Ω Λ 1 Λ M N 2 Λ M N 3 Λ M N 1 Λ M N 2 (13) 0 0 CU N CU 2 CU 1 Ω Λ M N 4 Λ M N CU N CU N 1 CU N 2 Ω Λ CU N CU N 1 CU 1 Ω For defnng the matrces B 1 the followng component matrces are requred Usng (2) and (3) let V,n = μ Q n I k for 1 n N -1 whch s a matrx of order k for 1 k*and let V 1,n V n = 0 V 2,n 0 0 for 1 n N-1. (14) V,n Ths s a matrx of order =1 k and 0 appearng n the (, j) component s a 0 matrx of type k x k j for 1, j k*. Let U = μ 1 I k μ 2 I k μ I k1 (15) 36 P a g e

5 BMAP/M/C Bulk Servce Queue wth Randomly Varyng Envronment In (15), U s matrx of order =1 k and 0 appearng n the (, j) component s a rectangular 0 matrx of type k x k j for 1, j k*. To wrte B 1 the block for 0 s to be consdered whch has queue length L= 0, 1, 2 M-1. When L = 0 there s only arrval process wthout servce. The change n the envronment from to j swtches on BMAP j verson as per statonary (equlbrum) probablty vector n the new envronment j whenever t occurs for 1, j, k*. In the empty queue (L=0) when an arrval occurs n the envronment both the arrval tme and the servce tme start. In block 0 when L =1, 2,,M-1 all the processes arrval, servce and envronment are actve as n other blocks n for n > 0. Consderng the change of envronment swtches on BMAP arrval process n the new envronment through the statonary (equlbrum) probablty vector when the queue s empty, the followng matrx Ω of order =1 k s defned whch s concerned wth the change of envronment durng arrval tme and s smlar to Ω defned n (6). T 1 Ω 1,2 Ω 1,3 Ω 1, Ω 2,1 T 2 Ω 2,3 Ω 2, Ω = Ω 3,1 Ω 3,2 T 3 Ω 3, (16) Ω,1 Ω,2 Ω,3 T Here T = D 0 + dag(q 1 ), and Ω,j s a rectangular matrx of type k x k j whose all rows are equal to (Q 1 ),j φ j presentng the rates of changng to phases n the new envronment for j and 1, j k*. The matrx B 1 for Sub Case (A1) where N > C and Sub Case (A2) where C > N are gven below n (17) and (18) respectvely. For the case when C=N, the matrxb 1 may be wrtten by placng C n place of N n the N-th block row n (18) and there after the multpler of U j s C. Let Q 1,j = Ω ju for 0 j C and Q 1,C =Ω For the case when M = C, the multpler C does not appear as a multpler for the U j matrces n the matrx B 1 n (18) n the 0 block of (4) and C appears as a multpler for all U j matrces n the matrces of A 1 and A 2 from row block 1 onwards. The basc generator of the bulk queue whch s concerned wth only the arrval and servce s a matrx of order [ M =1 k ] gven below n (21) where Q A =A 0 + A 1 + A 2 (19) Its probablty vector w gves, wq A =0 and w. e = 1 (20) It s well known that a square matrx n whch each row (after the frst) has the elements of the prevous row shfted cyclcally one place rght, s called a crculant matrx. It s very nterestng to note that the matrx Q A s a block crculant matrx where each block matrx s rotated one block to the rght relatve to the precedng block partton. 37 P a g e

6 In (21), the frst block-row of type [ =1 k ] x[ M =1 k ] s, W = (Ω + Λ M, Λ 1, Λ 2,, Λ M N 2, Λ M N 1, Λ M N + CU N,, Λ M 2 + CU 2, Λ M 1 + CU 1 ) whch gves as the sum of the blocks Ω + Λ M + Λ 1 + Λ 2 + +Λ M N 2 + Λ M N 1 + Λ M N + CU N + +Λ M 2 + CU 2 + Λ M 1 + CU 1 = Ω whch s the matrx gven by Q 1 Ω 1,2 Ω 1,3 Ω 1, Ω 2,1 Q 2 Ω 2,3 Ω 2, Ω = Ω 3,1 Ω 3,2 Q 3 Ω 3, (22) Ω,1 Ω,2 Ω,3 Q where usng (5) and (6), Q = D + dag (Q 1 ), for 1 k*. The statonary probablty vector of the basc generator gven n (21) s requred to get the stablty condton. Consder the vector w = (ϕ 1 φ 1,ϕ 2 φ 2,, ϕ φ ) (23) where ϕ = (ϕ 1, ϕ 2,, ϕ ) s the statonary probablty vector of the envronment, φ = ( φ,j ) s the statonary probablty vector of the arrval BMAP D for 1 k*. It may be noted φ φ D =0. Ths gves φ φ Q = φ (Q 1 ), φ for 1 k*. Now the frst column of the matrx multplcaton of wω s φ 1 (Q 1 ) 1,1 φ 1,1 + φ 2 (Q 1 ) 2,1 φ 11 [φ 2 e] φ (Q 1 ),1 φ 11 [φ e] = 0 snce (φ )e = 1 and ϕq 1 =0. In a smlar manner t can be seen that the frst column block of the matrx multplcaton of wω s ϕ 1 (Q 1 ) 1,1 φ 1 + ϕ 2 (Q 1 ) 2,1 φ 1 [(φ 2 )e] ϕ (Q 1 ),1 φ 1 [(φ )e] = 0 and -th column block s ϕ 1 (Q 1 ) 1, φ [(φ 1 )e] +ϕ 2 (Q 1 ) 2, φ [(φ 2 )e] +...+ϕ (Q 1 ), φ + +ϕ (Q 1 ), φ [(φ )e]= 0. Ths shows that w Ω + Λ M + wλ 1 + wλ 2 + +wλ M N 2 + wλ M N 1 + wλ M N + wcu N + +wλ M 2 + wcu 2 + wλ M 1 + wcu 1 = w Ω =0. So (w, w,,w).w= 0 = (w, w,.w) W where W s the transpose W. Ths shows (w, w...w) s the left egen vector of Q A and the correspondng probablty vector s w w =, w, w,.., w (24) M M M M where w s gven by (23). Neuts [5], gves the stablty condton as, w A 0 e < w A 2 e where w s gven by (24). Takng the sum cross dagonally n the A 0 and A 2 matrces, t can be seen usng (9), (10), (11) and (12) that w A 0 e= 1 w M nλ M n=1 n e= 1 M n M n=1 =1 φ (φ D n )e = 1 ( φ M M =1 n=1 n (φ D n )e = 1 ( φ M M =1 φ ( n=1 n D n )e <w A 2 e= 1 w N ncu N M n n=1 =1 n φ φ μ q n e) = C M n=1 e= C M N =1 φ μ n=1 n q n ) = C ( φ M =1 μ E(ψ ). Ths gves the stablty condton as M =1 φ φ ( n=1 n D n )e < C =1 φ μ E (ψ ) (25) Ths s the stablty condton for the BMAP/M/C bulk servce queue wth random envronment for Sub Case (A1) M > N C and Sub Case (A2) M C > N. When (25) s satsfed, the statonary dstrbuton exsts as proved n Neuts [9]. Let π (n, m,, j), for 0 m M-1, 1 k*, 1 j k and 0 n < be the =1 k wth π n = (π(n, 0, 1, 1), π(n, statonary probablty of the states n (1) and π n be the vector of type 1xM 0, 1, 2) π(n, 0, 1, k 1 ), π(n, 0, 2, 1), π(n, 0, 2, 2), π(n, 0,2, k 2 ) π(n, 0, k*, 1), π(n, 0, k*, 2), π(n, 0,k*, k ) π(n, M-1, 1, 1), π(n, M-1, 1, 2) π(n, M-1, 1, k 1 ), π(n, M-1, 2, 1), π(n, M-1, 2, 2), π(n, M-1,2, k 2 ) π(n, M-1, k*, 1), π(n, M-1, k*, 2), π(n, M-1,k*, k )for n 0. The statonary probablty vector π = (π 0, π 1, π 3, ) satsfes πq A,2.1 =0 and πe=1. (26) From (26), t can be seen π 0 B 1 + π 1 A 2 =0. (27) π n 1 A 0 +π n A 1 +π n+1 A 2 = 0, for n 1. (28) Introducng the rate matrx R as the mnmal non-negatve soluton of the non-lnear matrx equaton A 0 +RA 1 +R 2 A 2 =0, (29) t can be proved (Neuts [9]) that π n satsfes π n = π 0 R n for n 1. (30) Usng (27) and (30), π 0 satsfes π 0 [B 1 + RA 2 ] =0 (31) The vector π 0 can be calculated up to multplcatve constant by (31). From (26) and (30) 38 P a g e

7 π 0 I R 1 e=1. (32) Replacng the frst column of the matrx multpler of π 0 n equaton (31) by the column vector multpler of π 0 n (32), a matrx whch s nvertble may be obtaned. The frst row of the nverse of that same matrx s π 0 and ths gves along wth (30) all the statonary probabltes. The matrx R gven n (29) s computed usng recurrence relaton 1 R 0 = 0; R(n + 1) = A 0 A 1 R 2 1 (n)a 2 A 1, n 0. (33) The teraton may be termnated to get a soluton of R at an approxmate level where R n + 1 R(n ) < ε Sub Case: (A3) C > M > N When C > M > N, the BMAP/M/C bulk queue admts a modfed matrx geometrc soluton as follows. The chan X (t) descrbng ths Sub Case (A3), can be defned as n (1) presented for Sub Cases (A1) and (A2). It has the nfntesmal generator Q A,2.2 of nfnte order whch can be presented n block parttoned form gven below. When C > M, let C = m* M + n* where m* s postve nteger and n* s nonnegatve nteger wth 0 n* M-1. B 1 A A 2,1 A 1,1 A A 2,2 A 1,2 A Q A,2.2 = 0 0 A 2,3 A 1,3 A (34) A 2,m A 1,m A A 2 A 1 A A 2 A 1 In (34) the states of the matrces are lsted lexcographcally as 0, 1, 2, 3,. n,. Here the vector n s of type 1xM =1 k and n = ( (n, 0, 1, 1),(n, 0, 1, 2) (n, 0, 1, k 1 ), (n, 0, 2, 1),(n, 0, 2, 2) (n, 0, 2, k 2 ),,(n, 0, k*, 1),(n, 0, k*, 2) (n, 0, k*, k ), (n, 1, 1, 1),(n, 1, 1, 2) (n, 1, 1, k 1 ), (n, 1, 2, 1),(n, 1, 2, 2) (n, 1, 2, k 2 ),,(n, 1, k*, 1),(n, 1, k*, 2) (n, 1, k*, k ),, (n, M-1, 1, 1),(n, M-1, 1, 2) (n, M-1, 1, k 1 ), (n, M-1, 2, 1),(n, M-1, 2, 2) (n, M-1, 2, k 2 ),,(n, M-1, k*, 1),(n, M-1, k*, 2) (n, M-1, k*, k ) ) for n 0.The matrces B 1, A 1,j for 1 j < m and A 1 have negatve dagonal elements, they are of order M =1 k and ther off dagonal elements are non- negatve. The matrces A 0 anda 2 have nonnegatve elements and are of order M =1 k and the matrces A 0, A 1 and A 2 are same as defned earler for Sub Cases (A1) and (A2) n equatons (11), (12) and (13). Snce C > M the number of servers n the system s equals the number of customers n the system L up to customer length becomes C. Once number of customers becomes L C, the number of servers n the system remans C. When the number of customers becomes less than C, the number of servers reduces and equals the number of customers. The matrx A 2,j for 1 j < m*-1 s gven below. 0 0 jmu N jmu N 1 jmu 2 jmu (jm + 1)U N (jm + 1)U 3 (jm + 1)U (jm + N 2)U A 2,j = N (jm + N 2)U N (jm + N 1)U N (35) The matrx A 2,m s as follows gven n (36) when C = m*m + n* and n* s such that 0 n* N-1. Here the multpler of U j n the row block ncreases by one tll the multpler becomes C = m*m + n* and there after the multpler s C for U j for all blocks. When N n* M-1, A 2,m s same as n (35) for j = m* 0 0 (Mm )U N (Mm )U N 1. (Mm )U 2 (Mm )U (Mm +1)U N. (Mm +1)U 3 (Mm +1)U CU N CU n +2 CU n +1 A 2,m = CU N CU N 1 (36) CU N P a g e

8 The matrx A 1,m s as follows when C = m*m + n* and n* s such that 0 n* N-1. The multpler of U j ncreases by one tll t becomes C = m*m + n* and thereafter n all the blocks the multpler of U j s C. When n* = N or n* > N then, n the matrx A 1,m, there s slght change n the elements. When n* = N, n the N+1 block row and thereafter C appears as multpler of U j, and when n* > N wth n* = N + r for 1 r M-N-1, n the n*+1 block row U N appears n the r + 1 column block. C appears as multpler for t and as the multpler of U j thereafter n all row blocks respectvely. The basc system generator for ths Sub Case s same as (21) wth probablty vector as gven n (24). The stablty condton s as presented n (25). Once the stablty condton s satsfed the statonary probablty vector exsts by Neuts [9]. As n the prevous Sub Cases, πq A,2.2 =0 and πe=1. (40) The followng may be noted. π n A 0 +π n+1 A 1 +π n+2 A 2 = 0, for n m*, the rate matrx R s same as n prevous Sub Cases wth same teratve method for solvng the same and π n satsfes π n = π m R n m for n m*. (41) The set of equatons avalable from (40) are π 0 B 1 +π 1 A 2,1 = 0, (42) π A 0 +π +1 A 1,+1 +π +2 A 2,+2 = 0, for 0 m*-2 (43) and π m 1 A 0 +π m A 1,m +π m +1 A 2 = 0. (44) m 1 The equaton πe=1 n (40) gves =0 π e + π m (I-R) 1 e = 1 (45) Usng π m +1 =π m R and equatons (42), (43), (44) and (45) the followng matrx equatons can be seen where Q A,2.2 s gven by (48). (π 0, π 1, π 3, π m )Q A,2.2 =0 (46) e (π 0, π 1, π 3, π m ) (I R) 1 e =1 (47) 40 P a g e

9 B 1 A A 2,1 A 1,1 A Q A,2.2 = 0 A 2,2 A 1,2 A A 2,3 A 1,3 A (48) A 2,m RA 2 + A 1,m Equatons (46) and (47) may be used for fndng (π 0, π 1, π 3, π m ). Replacng the frst column of the frst column- block n the matrx gven by (48) by the column vector multpler n (47) a matrx whch s nvertble can be obtaned. The frst row of the nverse matrx gves (π 0, π 1, π 3, π m ).Ths together wth equaton (41) gve all the probablty vectors for ths Sub Case Performance Measures (1) The probablty P(L = r), of the queue length L = r, can be seen as follows. Let n 0 and j for 0 m M-1 be non-negatve ntegers such that r = n M + m. Then t s noted that P (L=r) = =1 k π n, m,, j, where r = M n + m. j =1 k (2) P (Queue length s 0) = P (L=0) = =1 j =1 π 0, 0,, j. (3)The expected queue level E(L), can be calculated as follows. For Sub Cases (A1) and (A2) t may be seen as follows. Snce π n, m,, j = P [L = M n + m, and envronment state =, arrval BMAP phase=j], for n 0, 0 m M-1, 1 j k and 1 k*, M 1 k E(L) = n=0 m=0 =1 j =1 π n, m,, j Mn + m = n=0 π n. (Mn Mn, Mn+1 Mn+1, Mn+2 Mn+2 Mn+M-1 Mn+M-1) where n the multpler vector Mn appears =1 k tmes; Mn+1 appears =1 k tmes; and so on and fnally Mn+M-1appears =1 k tmes. So E(L) =M n=0 nπ n e +π 0 ( I R) 1 ξ. Here ξ s a (M =1 k ) x1 type column vector ξ= 0, 0,1,,1,2,,2,, M 1,, M 1 where 0,1, 2, M-1 appear =1 k tmes n order.ths gves E (L) = π 0 ( I R) 1 ξ + Mπ 0 (I R ) 2 Re. (49) M 1 k For Sub Case (A3), E(L) = n=0 m=0 =1 j =1 π n, m,, j Mn + m = M n=0 nπ n e + n=0 π n ξ = m 1 M n=0 nπ n e+ =0 π ξ + π m (I-R) 1 ξ. Lettng the generatng functon of probablty vector Φ(s) = =0 π s, t can be seen, Φ(s) = m 1 =0 π s +π m s m (I-Rs) 1 m 1 and n=0 nπ n e = Φ (1)e = =0 π e+π m m*(i-r) 1 e + π m (I-R) 2 Re. Usng ths, t s noted that m 1 E(L) = M [ =0 π e + π m m*(i-r) 1 e + π m (I-R) 2 m 1 Re] + =0 π ξ + π m (I-R) 1 ξ (50) (4) Varance of queue level can be seen usng Var (L) = E (L 2 ) E(L) 2. Let η be column vector η=[0,..0, 1 2, ,.. 2 2, M 1) 2, (M 1) 2 of type (M =1 k ) x1 where 0,1, 2, M-1 appear =1 k tmes n order. Then t can be seen that the second moment, for Sub Cases (A1) and (A2) E (L 2 M 1 k ) = n=0 m =0 =1 j =1 π n, m,, j [Mn + m] =M 2 n=1 n n 1 π n e + n=0 nπ n e + n=0 π n η + 2M n=0 n π n ξ. So, E(L 2 )=M 2 [π 0 (I R) 3 2R 2 e + π 0 (I R) 2 Re]+π 0 (I R) 1 η + 2M π 0 (I R) 2 Rξ (51) Usng (49) and (51) the varance can be wrtten for Sub Cases (A1) and (A2). For the Sub Case (A3) the second moment can be seen as follows. E (L 2 M 1 k ) = n=0 m=0 =1 j =1 π n, m,, j [Mn + m] = M 2 n=1 n n 1 π n e + n=0 nπ n e + n=0 π n η + 2M n=0 n π n ξ = M 2 m 1 [Φ (1)e + =0 π e+π m m*(i-r) 1 e + π m (I-R) 2 m 1 Re] + =0 π η + π m (I-R) 1 m 1 η + 2M [ =0 π ξ+π m m*(i-r) 1 ξ + π m (I-R) 2 R ξ]. Ths gves E (L 2 ) = M 2 m 1 [ =1 1 π e + m*(m*-1)π m (I R) 1 e +2m*π m (I-R) 2 Re +2π m (I-R) 3 R 2 e m 1 + =0 π e+π m m*(i-r) 1 e + π m (I-R) 2 m 1 Re] + =0 π η + π m (I-R) 1 m 1 η +2M [ =0 π ξ+π m m*(i- R) 1 ξ + π m (I-R) 2 R ξ]. (52) (52) Usng (50) and (52) the varance can be wrtten for Sub Case (A3). III. MODEL (B). MAXIMUM ARRIVAL SIZE M LESS THAN MAXIMUM SERVICE SIZE N In ths Model (B) the dual case of Model (A), namely the case, M < N s treated. Here the parttonng =1 k and the customers are consdered as members of N blocks. M plays no role n the matrces are of order N partton where as t played the major role n Model (A). Two Sub Cases namely (B1) N C and (B2) C > N come up n the Model (B). (When M =N and for varous values of C greater than them, or less than them or equal to them, both Models (A) and (B) are applcable and one can use any one of them.) The assumpton (v) of Model (A) s modfed wthout changng others. 41 P a g e

10 3.1Assumpton. (v) The maxmum batch arrval sze of all BMAPs, M= max 1 M s greater than the maxmum servce sze N=max 1 N. 3.2.Analyss Snce ths model s dual, the analyss s smlar to that of Model (A). The dfferences are noted below. The state space of the chan s as follows defned n a smlar way presented for Model (A). X (t) = {(n, m,, j): for 0 m N-1, for 1 k*, for 1 j k and 0 n < }. (53) The chan s n the state (n, m,, j) when the number of customers n the queue s, n N + m, the envronment state s and the BMAP arrval phase s j for 0 m N-1, for 1 k*,for 1 j k and 0 n <. When the customers n the system s r then r s dentfed wth (n, m) where r on dvson by N gves n as the quotent and m as the remander Sub Case: (B1) N C The nfntesmal generator Q B,3.1 of the Sub Case (B1) of Model (B) has the same block parttoned structure gven n (4) for the Sub Cases (A1) and (A2) of Model (A) but the nner matrces are of dfferent orders and elements. B" 1 A" A" 2 A" 1 A" A" Q B,3.1 = 2 A" 1 A" A" 2 A" 1 A" 0 0. (54) A" 2 A" 1 A" 0 0 In (54) the states of the matrces are lsted lexcographcally as 0, 1, 2, 3,. n,. Here the vector n s of type 1 x N =1 k and n = ( (n, 0, 1, 1),(n, 0, 1, 2) (n, 0, 1, k 1 ), (n, 0, 2, 1),(n, 0, 2, 2) (n, 0, 2, k 2 ),,(n, 0, k*, 1),(n, 0, k*, 2) (n, 0, k*, k ), (n, 1, 1, 1),(n, 1, 1, 2) (n, 1, 1, k 1 ), (n, 1, 2, 1),(n, 1, 2, 2) (n, 1, 2, k 2 ),,(n, 1, k*, 1),(n, 1, k*, 2) (n, 1, k*, k ),, (n, N-1, 1, 1),(n, N-1, 1, 2) (n, N-1, 1, k 1 ), (n, N-1, 2, 1),(n, N-1, 2, 2) (n, N-1, 2, k 2 ),,(n, N-1, k*, 1),(n, N-1, k*, 2) (n, N-1, k*, k ) ) for n 0. The matrces, B 1, A 0, A 1 and A 2 are all of order N =1 k. The matrces B 1 and A 1 have negatve dagonal elements and ther off dagonal elements are non- negatve. The matrces A 0 and A 2 have nonnegatve elements. They are all gven below. Usng the same matrces presented n model (A), for Ω, Λ j, U j, V j, U, Ω and Q 1,j n (6), (9), (10), (14) to (17) the parttonng matrces are defned below A Λ 0 = M Λ M 1 Λ M (55) Λ 2 Λ 3 Λ M Λ 1 Λ 2 Λ M 1 CU N CU N 1 CU N 2 Λ M 0 CU 2 0 CU 1 0 CU N CU N 1 CU 3 CU CU N CU 4 CU 3 A CU 5 CU 4 (56) CU N 1 CU N CU N CU N CU N A 1 = Ω Λ 1 Λ 2 Λ M CU 1 Ω Λ 1 Λ M 1 Λ M CU 2 CU 1 Ω Λ M 2 Λ M 1 Λ M 0 0 CU N M 1 CU N M 2 CU N M 3 Ω Λ 1 Λ 2 Λ M 1 Λ M CU N M CU N M 1 CU N M 2 CU 1 Ω Λ 1 Λ M 2 Λ M 1 CU N M+1 CU N M CU N M 1 CU 2 CU 1 Ω Λ M 3 Λ M 2 CU N 2 CU N 3 CU N 4 CU N M 2 CU N M 3 CU N M 2 Ω Λ 1 CU N 1 CU N 2 CU N 3 CU N M 1 CU N M 2 CU N M 1 CU 1 Ω (57) 42 P a g e

11 IOSR Journal of Engneerng (IOSRJEN) ISSN (e): , ISSN (p): Vol. 05, Issue 03 (March. 2015), V1 PP In (58) the case N > C has been presented. When C=N, V j and U j n B 1 do not get C as multpler n (58) and C appears as a multpler of U j n A 2 and A 1 n (56) and (57). The multpler of matrces U j and V j concernng the servces ncreases by one n each row block from thrd row block as the row number ncreases by one, up to the row C+1 and t remans C n row blocks after that as gven above. The basc generator (59) whch s concerned wth only the arrval and servce s Q B = A 0 + A 1 + A 2. Ths s also block crculant. Usng smlar arguments gven for Model (A) t can be seen that ts probablty vector s w w =, w, w,.., w where w s as seen n Model (A), where w= (ϕ N N N N 1 φ 1, ϕ 2 φ 2,, ϕ φ ) and the stablty condton remans the same as n Model (A). Followng the arguments gven for Sub Cases (A1) and (A2) of Model (A), one can fnd the statonary probablty vector for Sub Case (B1) of Model (B) also n matrx geometrc form. All performance measures n secton 2.3 ncludng the expectaton of customers watng for servce and ts varance for Sub Cases (A1) and (A2) of Model (A) are vald for Sub Case (B1) of Model (B) wth M s replaced by N. It can also be seen that when N = C the system admts Matrx Geometrc soluton as n Model (A) Sub Case: (B2) C > N The nfntesmal generator Q B,3.2 of the Sub Case (B2) of Model (B) has the same block parttoned structure gven n (34) for Sub Case (A3) of Model (A) but the nner matrces are of dfferent orders and elements. When C > N > M, the BMAP/M/C bulk queue admts a modfed matrx geometrc soluton as follows. The chan X (t) descrbng ths Sub Case (B2), can be defned as n the Sub Case (B1). It has the nfntesmal generator Q B,3.2 of nfnte order whch can be presented n block parttoned form gven below. When C > N, let C = m* N + n* where m* s postve nteger and n* s nonnegatve nteger wth 0 n* N-1. B 1 A A 2,1 A 1,1 A A 2,2 A 1,2 A A 2,3 A 1,3 A Q B,3.2 = (60) A 2,m A 1,m A A 2 A 1 A A 2 A 1 In (60) the states of the matrces are lsted lexcographcally as 0, 1, 2, 3,. n,. Here the vector n s of type 1 x N =1 k and n = ( (n, 0, 1, 1),(n, 0, 1, 2) (n, 0, 1, k 1 ), (n, 0, 2, 1),(n, 0, 2, 2) (n, 0, 2, k 2 ),,(n, 0, k*, 1),(n, 0, k*, 2) (n, 0, k*, k ), (n, 1, 1, 1),(n, 1, 1, 2) (n, 1, 1, k 1 ), (n, 1, 2, 1),(n, 1, 2, 2) (n, 1, 2, 43 P a g e

12 k 2 ),,(n, 1, k*, 1),(n, 1, k*, 2) (n, 1, k*, k ),, (n, N-1, 1, 1),(n, N-1, 1, 2) (n, N-1, 1, k 1 ), (n, N-1, 2, 1),(n, N-1, 2, 2) (n, N-1, 2, k 2 ),,(n, N-1, k*, 1),(n, N-1, k*, 2) (n, N-1, k*, k ) ) for n 0. The matrces B 1, A 1j for 1 j m* and A 1 have negatve dagonal elements, they are of order N =1 k and ther off dagonal elements are non- negatve. The matrces A 0, A 2,j and A 2 for 1 j m* have nonnegatve elements and are of order N =1 k and the matrces A 0, A 1 and A 2 are same as defned earler for Sub Case (B1) n equatons (55), (56) and (57). Snce C > N the number of servers n the system s equals the number of customers n the system L up to customer length becomes C= m* N + n*. Once number of customers L C, the number of servers n the system remans C. When the number of customers becomes less than C, the number of servers agan falls and equals the number of customers. Usng the same matrces presented n model (A), for Ω, Λ j, U j, V j U, Ω and Q 1,j n (6), (9), (10), (14) to (17) the parttonng matrces are defned below. The matrx A 2,j s gven for 1 j < m*-1, as jnu N jnu N 1 jnu 2 jnu 1 0 (jn + 1)U N (jn + 1)U 3 (jn + 1)U 2 A 2,j = (61) 0 0 (jn + N 2)U N (jn + N 2)U N (jn + N 1)U N The matrx A 2,m s as follows gven n (62) when C = m*n + n* where 0 n* N-1. (Nm )U N (Nm )U N 1. (Nm )U 2 (Nm )U 1 0 (Nm +1)U N. (Nm +1)U 3 (Nm +1)U 2 A 2,m = 0 0 CU N CU n +2 CU n +1 (62) CU N CU N CU N Let Q 1,j = Ω ju for 0 j C and Q 1,C =Ω as n Sub Cases (A1) and(a2). Then B 1, s defned as follows. The matrx A 1,m s n (65) when C = m*n + n* and 0 n* N-1. From row block n*+1, the multpler of U j s C. A 1,m = Q 1,Nm Λ 1 Λ 2 Λ M (Nm +1)U 1 Q 1,Nm +1 Λ 1 Λ M 1 Λ M (Nm +2)U 2 (Nm +2)U 1 Q 1,Nm +2 Λ M 2 Λ M 1 Λ M 0 0 CU n CU n 1 CU n 2 Q 1,C Λ 1 Λ 2.. (65) CU n +1 CU n CU n 1 CU 1 Q 1C Λ 1.. CU N 2 CU N 3 CU N 4 CU N M 2 CU N M 3 CU N M 2 Q 1C Λ 1 CU N 1 CU N 2 CU N 3 CU N M 1 CU N M 2 CU N M 1 CU 1 Q 1C The basc generator for ths model s also same as (59) whch s concerned wth only the arrval and servce. Q B = A 0 + A 1 + A 2. Ths s also block crculant. Usng smlar arguments gven for Model (A) t can be 44 P a g e

13 seen that ts probablty vector s w = w, w, w,, w, where w = (ϕ N N N N 1φ 1, ϕ 2 φ 2,, ϕ φ ) and the stablty condton remans the same. Followng the arguments gven for Sub Case (A3) n secton of Model (A), one can fnd the statonary probablty vector for Sub Case (B2) of Model (B) also n modfed matrx geometrc form. All the performance measures gven n secton 2.3 ncludng the expectaton of customers watng for servce and ts varance for Sub Case (A3) are vald for Sub Case (B2) of Model (B) except M s replaced by N. IV. NUMERICAL ILLUSTRATION For the BMAP/M/C bulk models, the varyng envronment s consdered to be governed by the Matrx Q 1 = 5 5. Nne examples three for each are studed for the cases M = N =3; M = 3, N= 2 and M = 2, N = wth the number of servers n each case as C = 2, 3 and 4. Matrx geometrc results are seen for C = 2 and C = 3 M or N. Modfed Matrx Geometrc results are seen when C = 4 > M and N. The servce tme parameters of exponental dstrbutons are respectvely fxed n the two envronments E1 and E2 as μ 1 = 5 and μ 2 =.5 for sngle server respectvely For the case M=3, BMAP, the batch Markovan arrval process for E1 s gven by D 0 = 2 3, D 1 1 = , D = , D = and BMAP for the envronment E2 s gven by D = , D = , D = , D 3 1 = For the case M=2, D for =0 and =1 and D for = 0, 1, 2, 3 are as gven above for the case M = 3 but t s assumed that D = and D 3 1 = The bulk sze servce probabltes are gven n table 1for the case when M = N = 3 for the two envronment. For the case M =3, N=2 the probabltes of bulk servce sze 2 n E1 s fxed as.5 and of bulk sze 3 n E1 s fxed as 0; and other probabltes are unchanged. Table 1: Servce probabltes Envronment 1 P(sze 1) P(sze 2) P(sze 3) Envronment 2 P(sze 1) P(sze 2) P(sze 3) Servce Servce Thrty teratons are performed for all the models to terate the rate Matrx R and the norms of convergence are recorded. Queue length probabltes and block sze probabltes are calculated. Expected queue length and Standard devaton are presented. They show sgnfcant varatons when M, N and C are changed. The probabltes of queue lengths and block szes are presented n fgures 1 and 2 for all the nne examples. Table2: Results Obtaned For Sx Matrx Geometrc Models wth Servers C=2, 3 and Three Modfed Matrx Geometrc Models wth Servers C=4. 45 P a g e

14 M=3=N M=3,N=2 M=2,N=3 M=3=N M=3,N=2 M=2,N=3 M=3=N M=3,N=2 M=2,N=3 BMAP/M/C Bulk Servce Queue wth Randomly Varyng Envronment C=2 C=2 C=2 C=3 C=3 C=3 C=4 C=4 C=4 Fgure 1: Probabltes of Queue lengths P(L=0) P(L=1) P(L=2) P(L=3) P(L=4) M=3=N M=3,N=2M=2,N=3 M=3=N M=3,N=2M=2,N=3 M=3=N M=3,N=2M=2,N=3 C=2 C=2 C=2 C=3 C=3 C=3 C=4 C=4 C=4 Fgure 2: Probabltes of Block Szes π0e π1e π2e π3e π(n>3)e V. CONCLUSION Two BMAP/M/C bulk servce queues and ther sub cases wth randomly varyng envronments have been studed. The envronment changes the batch Markovan arrval processes, the servce rates, and the probabltes of bulk servces. Matrx geometrc ( modfed matrx geometrc) results have been obtaned by sutably parttonng the nfntesmal generator by groupng of customers, envronments, BMAP and PH phases together respectvely when the number of servers s not greater than ( greater than) the maxmum of the maxmum arrval and maxmum servce szes. The basc system generators of the queues are block crculant matrces whch are explctly presentng the stablty condton n standard form. Numercal results for varous bulk queue models are presented and dscussed. Effects of varaton of rates on expected queue length and on probabltes of queue lengths are exhbted. The decrease n arrval rates (so also ncrease n servce rates) makes the convergence of R matrx faster whch can be seen n the decrease of norm values. Bulk BMAP/PH/C queue wth randomly varyng envronments causng changes n szes of the PH phases may produce further results f studed snce BMAP/PH/C queue s a most general form almost equvalent to G/G/C queue. VI. ACKNOWLEDGEMENT The fourth author thanks ANSYS Inc., USA, for provdng facltes. The contents of the artcle publshed are the responsbltes of the authors. REFERENCES [1]. Rama Ganesan, Ramshankar.R and Ramanarayanan. R (2014) M/M/1 Bulk Arrval and Bulk Servce Queue wth Randomly Varyng Envronment, IOSR-JM, Vol.10,Issue 6, Ver III, Nov-Dec, [2]. Sandhya.R, Sundar.V, Rama.G, Ramshankar.R and Ramanarayanan.R, (2015) M/M//C Bulk Arrval And Bulk Servce Queue Wth Randomly Varyng Envronment,IOSR-JEN,Vol.05,Issue02, V1,pp [3]. Ramshankar.R, Rama Ganesan and Ramanarayanan.R (2015) PH/PH/1 Bulk Arrval and Bulk Servce Queue,IJCA, Vol. 109,.3,27-33 [4]. Ramshankar.r, Rama.G, Sandhya.R, Sundar.V, Ramanarayanan.R, (2015), PH/PH/1 Bulk Arrval and Bulk Servce Queue wth Randomly Varyng Envronment, IOSRJEN, Vol.05,Issue.02, V4 pp [5]. Bn.D, Latouche.G and Men.B (2005). Numercal methods for structured Markov chans, Oxford Unv. Press, Oxford. 46 P a g e

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