Introduction In this paper, we consider the probem of optimay scheduing a set of jobs on a singe machine which is susceptibe to occasiona breakdowns.

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1 Stochastic Scheduing on a Repairabe Machine with Erang Uptime Distribution Wei Li, Chinese Academy of Sciences W. John Braun y, University of Winnipeg Yiqiang Q. Zhao y, University of Winnipeg Abstract A set of jobs is to be processed on a machine which is subject to breakdown and repair. When the processing of a job is interrupted by a machine breakdown, the processing ater resumes at the point at which the breakdown occurred. We assume that the machine uptime is Erang distributed and that processing and repair times foow genera distributions. Simpe permutation poicies on both machine parameters and the processing distributions are given which minimize the weighted number of tardy jobs, weighted ow times, and the weighted sum of the job deays. Key words: Stochastic scheduing, breakdowns, Erang uptimes, Nonpreemptive resume mode. Institute of Appied Mathematics, Chinese Academy of Sciences, Beijing, 8, China and aso with the Department of Mathematics and Statistics, University of Winnipeg, Winnipeg, Manitoba, Canada, R3B E9. y Department of Mathematics and Statistics, University of Winnipeg, Winnipeg, Manitoba, Canada, R3B E9.

2 Introduction In this paper, we consider the probem of optimay scheduing a set of jobs on a singe machine which is susceptibe to occasiona breakdowns. When a breakdown occurs, the job currenty being processed is hated unti the machine is repaired; then the job is resumed at the point at which it was interrupted. Scheduing probems of this sort have been studied thoroughy in the case of reiabe machines and in cases where job processing times and due dates are known in advance. However, such assumptions are not very reaistic; machines are often subject to unpredictabe breakdowns and possiby engthy repairs. Furthermore, processing times and deadines may be more adequatey modeed as random variabes. Singe-machine stochastic scheduing modes incorporating breakdowns and repairs were rst studied by Gazebrook [4]. Subsequenty, many authors have investigated stochastic probems on unreiabe or repairabe machines (see references []-[7], []-[5]). However, amost a of the resuts are based on one or more strong assumptions. Aahverdi and Mittentha [3, 4, 5], and Birge et a. [6] obtained some powerfu resuts in the case of deterministic processing times. In case of exponentia uptimes, it is aways possibe to transform a scheduing probem on an unreiabe machine to a probem on a reiabe machine by extending the processing time of a particuar job to the tota time that the job occupies the machine. For some exampes, see Gazebrook [4], Pinedo and Rammouz [3], Aahverdi and Mittentha [3], and Li and Cao [8]. Under the assumption of exponentia processing times, some simpe poicies have been obtained (see, for exampe, Pinedo and Rammouz [3], Chang et a. [7], Du and Pinedo [], and Li and Cao [7]), but these are usuay based on some stronger conditions, and the poicies obtained are not reated to the machine parameters. Few resuts have been obtained when specic distributiona assumptions have not been made about the machine uptime. In stochastic scheduing probems where specic distributiona assumptions have not been made about processing requirements, or machine uptimes and downtimes, optima processing strategies coud depend upon both the eapsed time since the ast repair period and the eapsed processing time of the current job. Therefore, it woud seem very dicut to nd optima index poicies which depend on uptime and downtime distributions as we as the processing time distribution. In view of this, we investigate a singe machine scheduing mode in which the machine is subject to breakdowns with Erang uptimes and genera repair times. Each job's processing time is aso a random variabe with a genera distribution. This mode woud seem to be as genera as can be

3 reaisticay considered, if one seeks a schedue which depends on the machine parameters as we as the processing distribution. Li and Gazebrook [9] have considered the competey genera case, but their resuts are not reated to the machine parameters. In order to proceed, we restrict our attention to the cass of simpe recourse strategies. Under such a strategy, the sequence of job competion times in a schedue is xed, but the competion times may be postponed as a resut of the machine breakdowns (see Aahverdi and Mittentha [4, 5] or Frenk []). Further assumptions and notation are as foows: A set of jobs J = f; ; ; Ng (N < ) is to be processed on a singe machine which is subject to breakdown and repair. Job j has weight h j, which is basicay a priority factor, denoting the importance of job j reative to the other jobs in the system. For exampe, this weight may represent the actua cost of keeping the job in the system. This cost coud be a hoding or inventory cost; it coud aso be the amount of vaue aready added to the job. A jobs have a common due date d which is a random variabe with an exponentia distribution and rate r; this index represents the committed shipping or competion date (the date the jobs are promised to the customer). The competion of a job after its due date is aowed, but a penaty is incurred. The jth job has a random processing requirement j with distribution function F j (), having hazard rate function j () and nite expectation = j. The machine is avaiabe for processing from time unti the rst breakdown occurs at time U. The machine then takes time D to be repaired, whie no processing takes pace. The repair having been competed, the machine is again avaiabe for processing from time U + D unti time U + D + U, and so on. The machine up times U ; U ; ; are independent random variabes with an Erang distribution function U() having m phases and where each phase is exponentiay distributed with parameter. That is, U(t) =? e?t m? k= (t) k : k! The repair times D ; D ; are aso independent and identicay distributed but with genera distribution function D(), having hazard rate () and nite expectation =. A of the above random variabes are assumed to be mutuay independent. Under the above assumptions, we sha consider the so-caed non-preemptive resume mode; that is, if a job is being processed when a breakdown occurs, the processing time is cumuative after the machine is repaired. Our goa is to nd schedues, depending on both machine parameters and the processing distributions, for the jobs which minimize one of severa possibe objectives. The objectives we consider are the weighted ow time, the weighted numbers of tardy jobs, and the weighted sum of job tardiness. Genera resuts are obtained which can be simpied for certain 3

4 specia cases aready in the iteratures [8], [9] and [3]. In addition, some new resuts are obtained for specia cases not yet considered in the iterature. A Reation Between Competion Time and System Parameters Competion times are important random variabes in scheduing theory. Since our goa is to optimize performance characteristics reated to job competion time, it is the purpose of this section to nd a usefu reationship between the competion time of job n and the parameters of the system. We rst set out some notation which wi be used throughout this paper. Let EY represent the expectation of a random variabe Y. C (n) represent the competion time of the nth job (n = ; ; ; N) under the scheduing poicy ; is a permutation of the vector (; ; ; N). p! i = exp? (i = ; ; ; m), the mth roots of unity. i m i (s) = + s?! i m p?sd repair time distribution.?i i (s) = s (i = ; ; ; m), where D is a random variabe having the (i = ; ; ; m? ) and m (s) = h?m s + (??sd ) i. e = (; ; ; ), e i = (; ; ; ; ; ; ) and e represent the transpose of e. (n;i;j) denote a scheduing poicy in which job i and job j have been assigned to the nth and (n + )st positions, respectivey, (n = ; ; ; N? ; i; j = ; ; ; N). (n;j;i) is the poicy which is identica to (n;i;j) except for a switch in the nth and (n + )st positions. For the purpose of considering the competion time of the nth job (n = ; ; ; N) in any given poicy, we dene S n (t) to be the state of the system at time t before the nth job has been competed. This stochastic process can be in any of the foowing states [ fg f(i; k) : i = ; ; ; ; m; k = ; ; ; ng where denotes the absorbing state corresponding to the competion of the nth job, and (i; k) denotes the state in which the machine is processing the kth job whie the uptime is in phase i (i = ; ; ; m); phase corresponds to machine downtime. That is, state (; k) corresponds to the machine being repaired after having broken down whie the kth job is being processed. 4

5 Ceary, the process S n (t) is not Markovian, but it can be extended to a Markov process with the use of suppementary variabes (e.g. Cox[9], Chaudhry and Tempeton[8]). To do this, we denote the eapsed processing time of the kth job at t by V k (t), and the eapsed repair time at t by V (t). Then f(s n (t); V k (t); V (t)); t ; k = ; ; ; ng is a Markov process. Set p i;k (t; x)dx = P fs n (t) = (i; k); x V k (t) < x + dxjs n () = (; ); V () = g p ;k (t; x; y)dy = P fs n (t) = (; k); V k (t) = x; y V (t) < y + dyjs n () = (; ); V () = g where i = ; ; ; m and k = ; ; ; n: Notice that the kth job in the schedue is the job whose processing time has hazard function (k). Lemma. The system is governed by the foowing (k)(x) p ;k (t; x) = p ;k (t; x; y)(y)dy; k = ; ; ; @x + + (k)(x) p i;k (t; x) = p i?;k (t; x) i = ; ; m; k = ; ; n () with boundary conditions and @y + (y) p ;k (t; x; y) = k = ; ; ; n (3) p i; (t; ) = (i = ; ; ; n) (4) p i;k (t; ) = Z p i;k? (t; x) (k?) (x)dx; i = ; ; ; m; k = ; ; n (5) p ;k (t; x; ) = p m;k (t; x); k = ; ; ; n (6) p i;k (; x) = i; k; (x); i = ; ; ; m; k = ; ; ; n (7) p ;k (; x; y) = k = ; ; ; n (8) where (x) denotes the Dirac distribution and i;k denotes the Kronecker deta. Some detais of the proof are suppied in the appendix to this paper. The previous emma aows us to nd a reation between the Lapace transform of the competion times and the Lapace transform of the processing times and downtimes, a resut which is crucia to most of the scheduing poicies provided ater. 5

6 Lemma. For n = ; ; ; N; and s,?sc (n) =? m m m i= The proof is suppied in the appendix. i (s)!?i k k (s) 4? E (s) n j= (j) A As a specia case, we can obtain the foowing resut, previousy obtained in [3]. 3 5 Coroary.3 When the uptime is exponentiay distributed with rate, then for n = ; ; ; N; and s,?sc (n) = E exp@?[s + (??sd )] n j= (j) A Proof: This foows directy from Lemma., since when m =, (s) = (s) = s + (??sd ). Remark. We note that when the uptime has a more genera form it is not aways possibe to obtain the Lapace transform of C (n) in terms of system parameters. Even if the Lapace transform of the competion times can be found, it is not aways convenient for obtaining optima scheduing poicies. In the case discussed, as we wi see in ater sections, the above resut gives an exceedingy usefu reation. 3 The Optima Schedue for the Weighted Fow Time In this section, we sha nd schedues which minimize the expectation of the tota weighted competion times, P k h (k) C (k). This quantity gives an indication of the tota hoding or inventory costs incurred by a schedue. The sum of the competion times is often referred to as the ow time whie the tota weighted competion time is referred to as the weighted ow time. Some simpe permutation poicies have been found, under the assumption that the machine's uptime is exponentiay distributed (see [3], [8] and [], for exampe). However, as far as we know, resuts depending on parameters of both processing and the machine for the case of nonexponentia uptime have not been reported. Here, we sha consider this often-studied scheduing probem in the sense of expectation. Frequenty, however, the poicies that minimize the objective in expectation minimize the objective stochasticay as we (see pp. 8, []). We have the foowing necessary and sucient condition: 6

7 Theorem 3. The scheduing poicy (n;i;j) is better than the poicy (n;j;i) if and ony if ( j h j + m + m? i! m = m? i h i ( + m + j m = [?? i ]E exp! [?? j ]E exp where = () = (?! ) ( = ; ; ; m? ). "? "? (k) + j!#)!#) (k) + i Proof: According to Lemma., we know that for any poicy the expected competion time of the nth job is: EC (n) = P where d in = n i? E exp? i N E =?h i ( + m h (n;i;j) (k) C (n;i;j) (k)!? + j m + n m i= + m? (i) m i=! i d in (k). Therefore, by direct cacuation, we obtain: m? = ( +h j + + m? m i m Thus, the Theorem is proved. = N h (n;j;i) (k) C (n;j;i) (k)!! [?? j ]E exp! [?? i ]E exp "? "? (k) + i!#)!#) (k) + j It shoud be pointed out that the above necessary and sucient condition is dependent on the sequence before the nth job in a schedue, so it is impossibe to get an index optima poicy reated to both the parameters of the processing times and the machine. However, since the above condition has a nice structure, some optima schedue poicies can sti be found as foows: Theorem 3. The scheduing poicy = (; ; ; N) is optima if for i < j and t where and the quantity H C (t; i; j) = h i i + j m h C (x) = me?x H C (t; i; j) H C (t; j; i) Z h C (x + t)p ( i x i + j )dx (x) km? (km? )! ; x is ony dependent on the parameters of the uptime distribution of the machine. 7

8 Proof: Let EW F T [] represent the expected sum of the weighted ow times under poicy. We sha prove that EW F T [ ] EW F T [] for every poicy. Without oss of generaity, suppose there exists a poicy and an n as we as i < j satisfying (n) = j and (n + ) = i. We next show that EW F T [ (n;i;j) ] EW F T [ (n;j;i) ]. In fact, by the identity: m? =! e? x =? + me?x (x) km? (km? )! and ( h i +!#) + m?! [?? j ]E exp "? m j m (k) + i = 8 < = h i : + Z! 9 = h C (x)p j m (k) + i x (k) + i + j dx ; = i j Z H C (t; i; j)dp (k) t! as we as the resut in Theorem 3., we see that EW F T [ (n;i;j) ] EW F T [ (n;j;i) ]. Thus, the theorem foows from a pairwise interchange argument (see [6]). The foowing consequence was previousy found in [3]. Coroary 3.3 The expected weighted ow time is minimized by the schedue where the jobs are arranged in decreasing order of h i i, provided one of the foowing conditions hods:. m = ;. the job processing times are a exponentiay distributed. Proof:. This foows directy from Theorem 3., since H C (t; i; j) = h i i + if m =.. This foows directy from Theorem 3.. However, we oer an aternative proof to show how to use the concusion in Theorem 3. to obtain the resut for this specia case. Note that, in this case, ( i h i + m + j m = i h i 8 < : + m Z " m = m? =!! e? x [?? j ]E exp # dp "? (k) + i + j x (k) + i!#)!9 = ; 8

9 using the corresponding equation with j and i interchanged together with m! e?x = me?x (x) km? (km? )! = We see that the condition in Theorem 3. hods if j h j i h i. A pairwise interchange argument competes the proof. If we are interested ony in optima poicies which are independent of the machine parameters we can get the foowing resut previousy obtained in [9]. Coroary 3.4 The poicy = (; ; ; N) minimizes the expected weighted ow time if hods for i < j and t. h j P ( j t i + j ) h i P ( i t i + j ) Proof: This is a straightforward appication of Theorem 3., since the above condition impies h i i h j j. 4 The Optima Schedue for the Weighted Number of Tardy Jobs In this section, we sha consider the probem of minimizing the expectation of the weighted number of tardy jobs, P k h (k)u (k). Here, U (k) = ( ; if C(k) > d ; if C (k) d where d, the common due date of the jobs, is an exponentia random variabe with rate r. This quantity is an often-used objective in practice since it is a measure that can be recorded very easiy. It is a more genera cost function than the one studied in Section 3. Now, the cost is discounted at a rate of r per unit time. That is, if the jth job is not competed by time t, an additiona expected cost h j re?rt dt is incurred over the period [t; t + dt]. If job j is competed at time t, the tota expected cost incurred over the period [; t] is h j (? e?rt ). In the present paper, we sha nd optima schedues which minimize the expected vaue of this objective. Some simpe permutation poicies have been found previousy in the case of exponentia uptime (see [3] and [8], for exampe), but in the nonexponentia case, no resuts have been reported. 9

10 Since EU (n;i;j) (k) =? E exp[?rc (n;i;j) (k) ] hods for k = ; ; ; N, a direct cacuation using Lemma. gives the foowing necessary and sucient condition: Theorem 4. The scheduing poicy (n;i;j) is better than the poicy (n;j;i) if and ony if h j m = h i m = m p= m p=!?p p (r)?? (r) i E exp (r)!?p p (r)?? (r) j E exp (r) " "? (r)? (r) (k) + j!#!# (k) + i (9) We note here that it is not possibe to obtain a genera index optima poicy since the above necessary and sucient condition depends on the sequence before the nth job. However, one optima scheduing poicy can sti be found : Theorem 4. The poicy = (; ; ; N) is optima, if for i < j and t, where H U (t; i; j) = h i R H U (t; i; j) H U (t; j; i) h U (x + t)p ( i x i + j )dx and the quantity h U (x) = me?(+r)x m p= k= is ony dependent on the parameters of the machine. x mp?rd km+p? p (r) (km + p? )! ; x Proof: The eft-hand-side of (9) can be written as h j m = = h j = Z m!?p p= Z 4 m m = p= H U (t; j; i)dp p (r) E (r) " p (r)!?p exp? (r)( j + e? (r)x (k) t 3 (k) )! 5 P (k) + j x!? exp? (r)( (k) + j + i ) (k) + i + j! dx The resut now foows from a pairwise interchange argument and Theorem 4.. The foowing specia cases can be found in [3].!#

11 Coroary 4.3 When m =, the schedue which minimizes the expected weighted number of tardy jobs is in decreasing order of h i? i?? i, where = r + (??rd ). Proof: This foows from Theorem 4. upon noting that H U (t; i; j) = e?t h i? i?? j or, more simpy, by using Theorem 4. directy and a pairwise interchange argument. Coroary 4.4 If the processing times are a exponentia, the schedue which minimizes the expected weighted number of tardy jobs is in decreasing order of h i i Proof: This is a straightforward appication of Theorem 4.. Aternativey, this resut can be obtained by using the same idea as in Coroary 3.3 and m =!?p e? (r)x ; x Theorem 4. yieds a simpe optima poicy reated ony to the parameters of the processing time distribution as in [9]. Coroary 4.5 The poicy = (; ; ; N) minimizes the expected weighted number of tardy jobs if h j P ( j t i + j ) h i P ( i t i + j ) hods for any i < j and t. 5 The Optima Schedue for the Sum of Weighted Tardiness In this section, we sha consider schedues which minimize the expectation of the sum of weighted tardiness, P k h (k) T (k), where T (k) = maxf; C (k)? dg. Many papers have deat with this scheduing probem in the deterministic case, but few have considered the stochastic case. Pinedo [] was the rst to consider this scheduing probem. He assumed that the processing time of job i is an exponentia random variabe with rate i, the weight associated with job i is h i and job i is due at time d i, which is a random variabe. By using a compatibiity condition, i.e., i h i k h k impies d i st d k, he proved that the poicy

12 that minimizes this objective is the one where the jobs are arranged in decreasing order of h i i. However, when the machine is unreiabe, the question is more compicated. The case in which uptimes are arbitrary has been considered by [3] in case of exponentia processing times. We now consider the case of arbitrary processing times with Erang uptimes. We have the foowing necessary and sucient condition: Theorem 5. The scheduing poicy (n;i;j) is better than the poicy (n;j;i) if and ony if ( h j + + m?! [?? i ]E exp "? m i m? mr m m = p= h i ( + m? mr m m = p=!?p p (r) (r) j + m!?p p (r) (r) =?? (r) i E exp m? =! " [?? j ]E exp?? (r) j E exp "? (r) (k) + j!#!#9 = (k) + j ;!# (k) + i "?? (r)!#9 = (k) + i ; Proof: The argument foows the same ines as the proofs of Theorems 3. and Theorem 4., after one notes that hods for k = ; ; ; N. ET (n;i;j) (k) = EC (n;i;j) (k)? r h??rc (n;i;j) (k) i Theorem 5. The poicy = (; ; ; N) is optima if, for i < j and t, where H T (t; i; j) = h i R h T (x) = + e?x H T (t; i; j) H T (t; j; i) h T (x + t)p ( i x i + j )dx and the quantity (x) km? (km? )!? r e?(+r)x m k= p= (x ) is ony dependent on the parameters of the machine. x mp?rd km+p? p (r) (km + p? )! Proof: This foows from Theorem 5. and + m m = + e?x! k e? kx? mr m m p= (x) km? (km? )!? r e?(+r)x p!?p k e? k(r)x m k= p= x mp?rd km+p? p (r) (km + p? )!

13 Theorem 5. aso yieds the foowing resut Coroary 5.3 When m =, the optima schedue arranges the jobs in the order (; ; ; N) if the compatibiity conditions h j j h i i and h j? j?? j h i?i?? i hod for i < j, where = r + (??rd ). Theorem 5. yieds three additiona resuts, of which the rst is new, and the others were ontained in [3] and [9]. Coroary 5.4 When m =, the schedue which minimizes the expected sum of the weighted tardiness of the jobs arranges the jobs in the order (; ; ; N) if hods for i < j and t. i h i + j h j + r? e?t j? i r? e?t i? j?? j?? i Proof: Here, we note that H T (t; i; j) = h i + r? e?t j? i?? j r j Coroary 5.5 If the processing times are a exponentia, the schedue which minimizes the expected sum of the weighted tardiness of the jobs arranges the jobs in decreasing order of h i i. Coroary 5.6 The poicy = (; ; ; N) minimizes the expected sum of the weighted tardiness if h j P ( j t i + j ) h i P ( i t i + j ) hods for any i < j and t. 3

14 6 Concusions and Further Research In this paper, we have studied a cass of singe unreiabe machine stochastic scheduing probems, with genera processing time, Erang uptime and genera downtime distributions. We have obtained some genera optima scheduing poicies which are not index poicies and which depend on both processing parameters and machine parameters, under the nonpreemptive resume mode assumption. Some of the specia cases provided are aso new to the iterature. The method of suppementary variabes has been used successfuy here to nd usefu representations for the competion times. It shoud be noted that, because objective functions are often functions of competion times, much research focuses on competion times, a probem for which the method of suppementary variabes seems we-suited. It shoud be emphasized that nding optima schedues which depend on machine parameters has proved dicut in cases where the uptime distribution is nonexponentia. Thus, the extension to Erang uptimes given here represents a signicant step forward. We aso wish to point out here that the optimaity conditions found in this paper are both necessary and sucient. This is in contrast to other papers in the ed which usuay provide ony sucient conditions. When conditions sucient for optima poicies of simpe structure fai, it is benecia to understand the cost impications of impementing simpe poicies nevertheess. Li and Gazebrook [9] estabished an upper bound on the oss incurred when a processing poicy is adopted under the simpifying assumption of an exponentia processing requirement. Severa suboptimaity bounds for the nonexponentia case sha be given in another paper. One question for future research that was pointed out by the referee concerns the case of deterministic uptime. By reaceing by m and etting m become innitey arge, we can obtain the specia cases of a our resuts for constant uptime. Obtaining simpe poicies wi require some eort and wi be the subject of a subsequent paper in which numerica impementation of our poicies wi aso be considered. We have restricted our attention to the nonpreemptive resume mode, but it woud be interesting to see if simiar resuts can be found for the nonpreemptive repeat mode. In that case, the makespan woud seem to be an important objective to study rst. So far, we have found that if the uptime has an Erang distribution, the Lapace transform of the competion time of the nth job in any 4

15 given poicy possesses a nice structure, but optimaity resuts are sti sought. Acknowedgements The authors woud ike to thank the referee for a carefu reading of the origina manuscript. His/Her comments and suggestions have ed to a much better presentation of the paper. The authors acknowedge that this work was supported by research grants from the Natura Sciences and Engineering Research Counci of Canada (NSERC). Dr. Wei Li aso acknowedges that this work was supported by research grants from the Nationa Natura Science Foundation of China (NNSFC) and from the University of Winnipeg, Manitoba, Canada. 5

16 Appendix Proof of Lemma.: The proof of these equations foows a standard probabiistic argument (for exampe, see [8] or [9]). Since these equations are the starting point for a other deveopments in the paper, we provide some detais of the proof for () and (7). The arguments for (), (3), (4), (5) and (6) are simiar to that for (), and equation (8) is immediate from the denition. To obtain equation (), consider the state of the system at t and at t + t. In order that S n (t + t) = (; k) and V k (t + t) = x + t (x > ), it is necessary that (a) at time t, S n (t) = (; k) and V k (t) = x. Within (t; t + t), the uptime of the machine is sti in phase and the kth job has not been competed; or (b) at time t, S n (t) = (; k), and V k (t) = x, V (t) = y for any y >. Within (t; t + t), the repair of the machine has been competed. Using a conditioning argument, we have Thus, Z p ;k (t + t; x + t) = p ;k (t; x)[? t? (k) (x)t] + p ;k (t; x; y)(y)tdy + o(t) p ;k (t + t; x + t)? p ;k (t; x + t) + p ;k (t; x + t)? p ;k (t; x) =?p ;k (t; x)[t + (k) (x)t] + Dividing by t and etting t! gives equation (). Z p ;k (t; x; y)(y)tdy + o(t) Most of the equations given by (7) foow immediatey from the mode assuptions. The ony exception needing more proof is p ; (; x) = (x). That is, we must show p ; (; x) = for a x >, whie Z p ; (; x)dx = for any " > : Z " p ; (; x)dx = The rst reation is obvious, because it is impossibe for the machine to have processed the rst job for a non-zero amount of time at time t=. Next, by the denition of p ; (; x), it foows that Z p ; (; x)dx = Z P fs n () = (; ); x V () < x + dxjs n () = (; ); V () = g = Thus, together with the rst reation, we can get the second reation. 6

17 R Remark: p ;(; x)dx = can be aso directy obtained from the genera reation in the equation (7) used in the foowing proof of the Lemma.. Proof of Lemma.: Soving the Lapace transformed version of (3) with the boundary condition (6) gives p ;k(s; x; y) = e?sy [? D(y)]p m;k(s; x); k = ; ; ; n () where p (s) denotes the Lapace transform of a function p(t). Using (), (), (7) and the corresponding denitions, we p k(s; x) = Ap k(s; x)? (k) (x)p k(s; x); + k (x)e (k = ; ; ; n) () where p k (t; x) = (p ;k (t; x); p ;k (t; x); ; p m;k (t; x)), and A = Noting that? (s);? (s); ;? m (s) are the m dierent eigenvaues of matrix A, we can write?i i? A = T f? i (s)g T? () C A mm where fx i g = diagfx i g, T = m?..... C. A and T? = m!?m!?m! m?m m?!! m?......! m! m m? C A Substituting () into () and soving, we have p (s; x) = [? F () (x)]?i T f? i (s)xgt? i? U(x)e (3) and p k(s; x) = [? F (k) (x)]?i T f? i (s)xgt? i? L k (4) where L k (k = ; ; n) is a coumn vector to be determined, and U(x) is one if x > and zero otherwise. Using boundary conditions (4), (5) and the above two resuts and dening L = e, vectors L k (k = ; ; n) can be recursivey determined as L k =?i T fe exp[? i (s) (k?) ]gt? i? L k? 7

18 = =?i T? T i? 8 < : Then substituting (5) into (4) gives us Since p k(s; x) = [? F (k) (x)] T? P (C (n) > t) = i? n 4 m e? i(s)x E exp 4?i (s) k? j= 9 3 = (j) 5 ; e (5)?i Z i= T 8 < : e? i(s)x E exp 4?i (s) k? j= 3 (j) 5 e (k = ; 3; ; n) (6) p i;k (t; x)dx + Z Z the resuts in equations (), (3) and (6) can be combined to give = = s Z e?st P (C (n) > t)dt = n 4 m Z i= e + n Z s (??sd )e m p k(s; x)dx h i se + (? E?sD ) T? i? = s e f i(s)g T :? i (s) = sm m m i= 8 < i (s)!?i k k (s) e?i p i;k(s; x)dx + T 8 < 4? E (s) 4? E (s) Z Z :? i (s) n j= (j) A (j) A 3 9 = ; p ;k (t; x; y)dxdy5 (7) p ;k(t; x; y)dxdy 3 5 4? E (s) 39 = 5 The ast equation above foows since it is a specia case of the formua n j= efy i gt fd i gt? fx i ge = m d k y i!?i k m i= when x = and x i = (i = ; 3; ; m). Now, by using the reationship: E[e?sC (n)] =? s Z 8 m 3 ; T? n j= i? (j) A e 39 = 5 ; 5 (8) e?st P (C (n) > m j= x j! j k A

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