Runs and Patterns in a Sequence of Markov

American Open Journal of Statistics, 2011, 1, 115-127 doi:10.4236/ojs.2011.12014 Published Online July 2011 (http://www.SciRP.org/journal/ojs)

Runs and Patterns in a Sequence of Markov Dependent Bivariate Trials Kirtee Kiran Kamalja, Ramkrishna Lahu Shinde Department of Statistics, School of Mathematical Sciences, North Maharashtra University, Jalgaon, India E-mail: [email protected] Received May 13, 2011; revised June 2, 2011; accepted June 8, 2011

Abstract In this paper we consider a sequence of Markov dependent bivariate trials whose each component results in  X n    0   0  1  1   , n  0  of S    ,   ,   ,     0  1   0  1   Yn  

an outcome success (0) and failure (1) i.e. we have a sequence 

valued Markov dependent bivariate trials. By using the method of conditional probability generating functions (pgfs), we derive the pgf of joint distribution of X n0, k1 , X n1, k1 ; Yn0, k 2 , Yn1, k 2 where for i  0,1 , X ni , k1 de-



0

1

0

1



i

1 i

notes the number of occurrences of i-runs of length k in the first component and Yni, k 2 denotes the number i of occurrences of i-runs of length ki2 in the second component of Markov dependent bivariate trials. Further we consider two patterns 1 and  2 of lengths k1 and k2 respectively and obtain the pgf of joint distribution of  X n, 1 , Yn, 2  using method of conditional probability generating functions where X n, 1 Yn , 2  denotes the number of occurrences of pattern 1   2  of length k1  k2  in the first (second) n components of bivariate trials. An algorithm is developed to evaluate the exact probability distributions of the vector random variables from their derived probability generating functions. Further some waiting time distributions are studied using the joint distribution of runs. Keywords: Markov Dependent Bivariate Trials, Conditional Probability Generating Function, Joint Distribution

1. Introduction The distributions of several run statistics are used in various areas such as reliability theory, testing of statistical hypothesis, DNA sequencing, psychology [1], start up demonstration tests [2] etc. There are various counting schemes of runs. Some of the most popular counting schemes of runs are non-overlapping success runs of length k [3], overlapping success runs of length k [4], success runs of length at least k ,  - overlapping success runs of length k [5], success runs of exact length k [6]. The probability distribution of various run statistics associated with the above counting schemes have been studied extensively in the literature in different situations such as independent Bernoulli trials (BT), non-identical BT, Markov dependent BT (MBT), higher order MBT, binary sequence of order k , multi-state trials etc. But very little work is found on the distribution theory of run statistics in case of bivariate trials which has applications Copyright © 2011 SciRes.

in different areas such as start up demonstration tests with regard to simultaneous start ups of two equipment, reliability theory of two dimensional consecutive  k , r  out of  k  1, n  : F -Lattice system etc as specified by [7]. [7] have studied the distribution of sooner and later waiting time problems for runs in Markov dependent bivariate trials by giving system of linear equations of the conditional pgfs of the waiting times. The distribution of number of occurrences of runs in the two components of bivariate sequence of trials and their joint distributions are still unknown to the literature. Consider a sequence  X n , n  0 of S -valued trials where S is set of all possible outcomes of trials under study. The simple pattern  is composed of specified sequence of k states i.e.   a1a2  ak where a1 , a2 , ak  S . The number of occurrences of patterns can be counted according to the non-overlapping or overlapping counting scheme. The non-overlapping counting scheme starts recounting of the pattern immediately after the occurrence of the pattern while the overlapping OJS

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counting scheme of patterns allows an overlap of prespecified fixed length in the successive occurrences of patterns. Recently the study of distributions of different statistics based on patterns has become a focus area for many researchers due to its wide applicability area. Distribution of Wr , k ,  , the waiting time for the r th occurrence of pattern  of length k in the sequence of multistate trials is studied by [1,8,9]. [10] considered the sequence X 1 , X 2 , generated by Polya’s urn scheme and study the waiting time distribution of Wr , k ,  for r  1 . Joint distribution of number of occurrences of pattern 1 of length k1 and pattern  2 of length k2 in n Markov dependent multi-state trials is studied by [11]. [12] considered a sequence X i , i  1, 2, of m dimensional i.i.d. Random column vectors whose entries are 0,1 -valued i.i.d. random variables and obtain the waiting time distribution of two dimensional patterns with general shape. The general method, which is an extension of method of conditional pgfs, is used to study these distributions by [12]. Even though the distribution of waiting time of the pattern of general shape in the sequence of multi-variate trials with i.i.d. components has been done, the joint distribution of number of occurrences of patterns  i ,  i  1, 2, , m  in the sequence of i th component of the m -variate trials X 1 , X 2 , , X n is still unknown. Here we derive the pgf of joint distribution of number of occurrences of runs in both the components of the bivariate trials and generalize this study to the distribution of number of occurrences of patterns in both components of the bivariate trials.  X n   In this paper we consider the sequence   , n  0   Yn   of S -valued Markov dependent bivariate trials. In Section 2, we obtain the pgf of joint distribution of number of occurrences of i -runs of length ki1 in first components and i -runs of length ki2 in the second components of the bivariate trials  i  0,1 . We study this joint distribution of runs under the non-overlapping counting scheme of runs by using the method of conditional pgfs. Further in section 3, we study the joint distribution of number of occurrences of pattern 1 of length k1 in the first component and number of occurrences of pattern  2 of length k2 in the second component of bivariate trials. In Section 4, we develop an algorithm to evaluate the exact probability distributions of the random variables under study. As an application of the derived joint distributions, in Section 5, we obtain distributions of several waiting times associated with the runs and patterns in bivariate trials. In Section 6 we present some numerical work based on distribution of runs and patterns. Finally in Section 7, we discuss an application and Copyright © 2011 SciRes.

ET AL.

generalization of the studied distributions.

2. The Joint Distribution of Number of Occurrences of Runs  0   0   1   1  Let S    ,   ,   ,    . Consider the sequence  0   1   0   1   X n     , n  0  of S -valued Markov dependent bivari Yn   ate trials with the transition probabilities,  X   x   X  u   u   x  P   i      i 1       puv , xy    ,   S   Yi   y   Yi 1   v   v  y   i  1, 2, , n and the initial probabilities  X   x   x P   0       π xy for   S . Y y  y  0   

 

Let X ni , k1 Yni, k 2 i

 

be the number of i -runs ( i = 0,1) of

i

length ki1 ki2 in n trials associated with the first (second) component of bivariate trials (i.e. number of i -runs in X 0 , X 1 , , X n Y0 , Y1 , , Yn   . In this section we derive the joint distribution of



X

n , k01

, X n1, k1 ; Yn0, k 2 , Yn1, k 2 .

X

0 n , k01

, X n1, k1 ; Yn0, k 2 , Yn1, k 2 . Assume that for a non-negative

0

1

0

1

Let n  t0 , t1 ; s0 , s1  be the pgf of distribution of 1

0

1



integer c  n , we have observed until  n  c  trial X  (i.e. we have observed  i  , i  0,1, , n  c ). We de Yi  i ,m1i ; j , m2j    i ,m1i ; j ,m2j  t , t ; s , s as pgf of condifine c  0 1 0 1 c tional distribution of number of i -runs of length ki1 in X n  c 1 , , X n and number of j -runs of length k 2j in Yn  c 1 , , Yn given that we have observed  X 0   X1   X n c    ,   , ,   and currently have i -run of Y Y  0   1  Yn  c  th

length mi1 in first component and j -runs of length m 2j in the second component of bivariate trials is observed, mi1  1, 2, , ki1 , m 2j  1, 2, , k 2j , i, j  0,1. We define a c  1  a for a  0,1 . Now by assuming π 00  1 , we have,

n  t0 , t1 ; s0 , s1   n 0,0;0,0   t0 , t1 ; s0 , s1 

(2.1)

Also we have,

i , m1i ; j ,m2j  t , t ; s , s  1 for m1  1, 2,..., k 1 ,  0 1 0 1 i i

0

m 2j  1, 2,..., k 2j and i, j  0,1

(2.2)

Conditioning on the first trial we have the following system of recurrent relation of conditional pgfs for OJS

K. K. KAMALJA

i , m1i ; j , m2j   p  i , m1i 1; j ,1 t  p  ic ,1; j ,1 ij , ij c 1 i ij , i c j c 1

c  1, 2, , n .

c

n(0,0;0,0)  t0 , t1 ; s0 , s1 

i , m1i 1; jc ,1 t  p  ic ,1; jc ,1 i ij , i c j c c 1

 pij ,ij c c 1

 p00,00n(0,1;0,1)  p00,01n(0,1;1,1) 1 1  p00,10n(1,1;0,1)  p00,11n(1,1;1,1) 1 1

if mi1  ki1  1 , m 2j  k 2j

i , m1i ; j ,m2j   p  i ,1; j ,1  p  ic ,1; j ,1 ij ,ij c 1 ij , i c j c 1

Conditioning on the next trial from each stage, we have the following system of recurrent relations of conditional pgfs for c  1, 2, , n and i, j  0,1 .

c

i ,1; jc ,1  p

 pij ,ijc c 1

i ,m1i ; j ,m2j 

c

i , m1i 1; j ,m2j 1  p  ic ,1; j ,m2j 1  pij ,ijc 1 ij ,i c j c 1   pij ,ij c c 1

p

Thus we have

i ,1; j ,1 

i , m1i 1; j c ,1

c

c

ij , i c j c c 1

if mi1  1, 2, , ki1  2 , m 2j  1, 2, , k 2j  2 ,

i , m1i ; j , m2j 

c

i , m1i 1; j c ,1t  p  ic ,1; jc ,1 i ij , i c j c c 1



c

 pij ,ij c

  p  ij , ij c 1

p

i ,1; jc ,1  p  c 1



i



,1; j , m 2j 1

ij ,i c j c 1

ic ,1; jc ,1 

if mi1  ki1 , m 2j  1, 2, , k 2j  2

i , m1i ; j , m2j   p  i , m1i 1; j , m2j 1 s  p  ic ,1; j , m2j 1 s c ij , ij c 1 j j ij , i c j c 1 i , m1i 1; jc ,1  p

ic ,1; jc ,1

ij , i c j c

c 1

if mi1  1, 2, , ki1  2 , m 2j  k 2j  1 ,

i , m1i ; j , m2j   p  i , m1i 1; j , m2j 1 t s  p  ic ,1; j ,m2j 1 s ij , ij c 1 i j j ij , i c j c 1

c



t  p   i ij ,i c j c c 1

i , m1i 1; j c ,1

 pij ,ij c c 1



i c ,1; j c ,1

if mi1  ki1  1 , m 2j  k 2j  1 ,

i ,m1i ; j ,m2j   p  i ,1; j ,m2j 1 s  p  ic ,1; j ,m2j 1 s ij ,ij c 1 j j ij ,i c j c 1

c

i ,1; jc ,1  p

 pij ,ijc c 1

ic ,1; jc ,1

c c

ij ,i j

c 1

if mi1  ki1 , m 2j  k 2j  1 ,

i ,m1i ; j ,m2j   p  i ,m1i 1; j ,1  p  ic ,1; j ,1 ij ,ij c 1 ij ,i c j c 1

c



p

i , m1i 1; j c ,1

 pij ,ij c c 1







i c ,1; j c ,1

ij ,i c j c c 1

if mi1  1, 2, , ki1  2 , m2j  k 2j , Copyright © 2011 SciRes.



 k11 k02  k12



recurrent relations

of conditional pgfs for c  1, 2, , n  1 and these can be written as,

 c  t0 , t1 ; s0 , s1  

 c  c(0,0;0,0)

2

c(0,1;0,1)  c(0,1;0, k0 ) c(0,1;1,1) 

2

2

1

2

c(0,1;1, k1 )  c(1,1;1, k1 ) c(1, k1 ;1, k1 )

ij ,i c j c c 1

 pij ,ij c c 1

1 0

if mi1  ki1 , m2j  k 2j

where

if mi1  ki1  1 , m 2j  1, 2, , k 2j  2 , c

ij , i j

(2.3)

 pij ,ij c c 1

i ,1; j , m 2j 1

k

ic ,1; jc ,1

c 1

c c

1 1 1 1   1 2 12  A   Bi ti   B j s j    Bij ti s j   c 1  t0 , t1 ; s0 , s1  i 0 j 0 i 0 j 0  

i , m1i 1; j ,m2j 1 t  p  ic ,1; j , m2j 1  pij ,ijc 1 i ij , i c j c 1

i , m1i ; j , m 2j

117

ET AL.

1

1

i 0

j 0

1



1

and A   Bi1ti   B 2j s j    Bij12 ti s j is a square





i 0 j 0



matrix of size k01  k11 k02  k12 . Each row of this matrix corresponds to corresponding element of  c  t0 , t1 ; s0 , s1  and its elements are the coefficients of elements of  c 1  t0 , t1 ; s0 , s1  . For c  0 , we have,  0  t0 , t1 ; s0 , s1   1 where 1 is the column vector with all the elements equal to 1. Using (2.3) recurrently for c  1, 2, , n , we have,

 n  t0 , t1 ; s0 , s1  n

1 1 1 1     A   Bi1ti   B 2j s j    Bij12 ti s j   0  t0 , t1 ; s0 , s1  i 0 j 0 i 0 j 0  

(2.4) Hence from (2.1) and (2.4) we get the pgf of distribution of X n0, k1 , X n1, k1 ; Yn0, k 2 , Yn1, k 2 as,



0

1

n  t0 , t1 ; s0 , s1 

0

1



n

1 1 1 1    p  A   Bi1ti   B 2j s j    Bij12 ti s j  1 i 0 j 0 i 0 j 0  

(2.5)

OJS

K. K. KAMALJA

118

ET AL.

where p is the first column of identity matrix of order k01  k11 k02  k12 .

pattern of the length j of  2 in the second compoX   i  nent of bivariate trials is observed and  n  c      Yn c   b j 

3. The Joint Distribution of Number of Occurrences of Patterns

where i  0,1 and j  1, 2, , k2 . i, j Also let c, 1,  2  t  as pgf of conditional distribution of number of occurrences of pattern 1 in X n  c 1 , , X n and pattern  2 in Yn  c 1 , , Yn of bivariate trials given th that at  n  c  trial we have observed the sub-pattern of length i of 1 in the first component and a sub pattern of the length j of  2 in the second compo X   ai  nent of bivariate trials and  n  c     where  Yn  c   b j 







Let 1  a1a2  ak1 and  2  b1b2  bk2 be two patterns of lengths k1 and k2 respectively where ai , b j  0,1 , i  1, 2, , k1 ; j  1, 2, , k2 . We consider the non-overlapping counting scheme of patterns in which to count the number of occurrences of pattern  in a given sequence of n trials, one has to restart counting from scratch each time the pattern  occurs. Let X n , 1 denotes the number of non-overlapping occurrences of patterns 1 in the first component of n bivariate trials (i.e. in X 1 , X 2 , , X n ) and Yn ,  2 denotes the number of non-overlapping occurrences of patterns  2 in the second component of n bivariate trials (i.e. in Y1 , Y2 , , Yn ). Consider the random vector X n, 1 , Yn,  2 . In this section we obtain the pgf of joint distribution of X n, 1 , Yn,  2 using method of conditional pgfs. Let pgf of joint distribution of X n, 1 , Yn,  2 be n  t1 , t2   n  t  . Assume that for a non-negative integer th c  n , we have observed until  n  c  trial i.e.  X n c   X1   X 2   . For c  n , we define, the fol  ,   , ,   Y1   Y2   Yn  c 













lowing conditional pgfs. Let c( i , j )  t  be pgf of conditional distribution of number of occurrences of patterns 1 in X n  c 1 , , X n and number of occurrences of pattern  2 in Yn  c 1 , , Yn of bivariate trials given that currently (i.e. th at  n  c  trial) we have observed none of the subX  i patterns of 1 and  2 and  n  c     , where  Yn  c   j 

i  S .  j i, j Similarly let, c, 1  t  be pgf of conditional distribution of num b er of occurrences of patterns 1 in X n  c 1 , , X n and patterns  2 in Yn  c 1 , , Yn of th bivariate trials given that at  n  c  trial we have observed the sub-pattern of length i of 1 in the first component and none of the sub pattern of  2 is observed in the second component of bivariate trials and  X n  c   ai       where i  1, 2, , k1 and j  0,1 .  Yn  c   j  i, j Let c,  2  t  be pgf of conditional distribution of number of occurrences of patterns 1 in X n  c 1 , , X n and patterns  2 in Yn  c 1 , , Yn of bivariate trials th given that at  n  c  trial we have observed none of the sub-pattern of 1 in the first component and a sub Copyright © 2011 SciRes.

i  1, 2, , k1 and j  1, 2, , k2 . Let, aic  1  ai , i  1, 2, , k1 and bcj  1  b j , j  1, 2, , k2 .

For c  n , we assume that at

n  c

th

trial, the

sub-pattern of length i of 1 is observed in the first component of bivariate trials. If we condition on the next  X n  c 1   aic1  trial as   i.e. the sub-pattern of leng    Yn  c 1   yn  c 1  th i

of 1 observed at

 n  c  1

st

n  c

th

trial breaks at

trial then to check whether any sub-pattern

of 1 of length r  r  i  has occurred, we define the indicator function  ir1 as,

 ir1  I a1  ai  r  2 , a2  ai  r  3 ,..., ar 1  ai , ar  aic1 i  1, 2, , k1  1; r  i .

Similarly we have indicator function  2jr as,

 2jr  I b1  b j  r  2 , b2  b j  r  3 , ..., br 1  b j , br  bcj 1 j  1, 2, , k2  1; r  j .

Let



,

u1i  max  ir1 , u2i  max r  ir1  1, r  1, 2, , i 1 r  i

i  1, 2, , k1  1





v1j  max  2jr and v2j  max r  2jr  1, r  1, 2, , j 1 r  j

j  1, 2, , k2  1

Assuming π 00  1 , we have,

n  t1 , t2   n 0,0  t1 , t2 

(3.1)

We also have,

0 i , j   t   1

i for     S ;  j OJS

K. K. KAMALJA

0, i, j1  t   1

for i  1, 2, , k1 and j  0,1;

0, i, j2  t   1

for i  0,1 and j  1, 2, , k2 ;

119

ET AL.

c,i, j2  t   pib j , a1b j 1 c1,1,j 11,  2  t  t2  1, v2j  1,bcj1  j j  c 1, 1 ,  2  t  v1  c 1, 1  t  1  v1    a1c , v j   a1c ,bcj1  t 1  v j  pib , ac bc  c 1, 22  t  v1j  c 1  1 j 1 j 1   pib

0, i, j1,  2  t   1 for i  1, 2, , k1 and j  1, 2, , k2 . (3.2) Now for each 1  c  n ( c denotes the number of trials remaining to observe) we condition on the next trial to obtain the recurrent relations of conditional pgfs as follows.

 pib

c j , a1 b j 1

  1, b1c



a c ,b1c

 pij , a bc c 1, 1  t   pij , ac bc c 11 1 1

 t 

1 1

c , 1  t   pai j , ai1b1 c 1, 1 ,  2  t 

 pib

i

i 1





bc i 1 1

i

 pa j , a



 aic1 ,b1c  t 1 u i    u2i ,b1c  i  1   c 1, 1  t  u1  c 1  

 pa j , ac



i 1,b1c  c c 1, 1  t  b

if i  1, 2, , k1  2; j  0,1

c,i, j1  t   pai j , ai1b1 ci 1,1,11 , 2  t  t1

 aic1 ,1    u2i ,1  pa j , ac b  c 1,  2  t  u1i  c 1,  2  t  1  u1i  i 1 i 1    pa j , ac i

bc i 1 1



i , j 

c , 1

  u2i ,v2j  i j  c 1, 1 ,  2  t  u1 v1 

c c i j , ai 1b j 1

 

c

i

1 1

i

c 1

1 1

if i  k1 , j  0,1 . Similarly we have recurrent relations of the condii, j tional pgfs c,  2  t  for i  0,1 and j  1, 2, , k2 as follows.

c,i, j2  t   pib j , a1b j 1 c1,1,j 11,  2  t   1, v2j  1,bcj1  j j  c 1, 1 ,  2  t  v1  c 1, 1  t  1  v1    a1c ,v j   a1c ,bcj1  t 1  v j  pib , ac bc  c 1, 22  t  v1j  c 1  1 j 1 j 1 

 pib

c j , a1b j 1

 pib

c j , a1 b j 1

 a1c , j 1 t 

a

j c i 1 , v







 c 1,  22  t  1  u1i v1j

 aic1 ,bcj1  t 1  u i 1  v j     1  1  

c 1

 if i  1, 2, , k1  2; j  1, 2, , k2  2

 pa b

  i 1,v2j  i 1,bcj1  j j   c 1, 1 ,  2  t  v1  c 1, 1  t  1  v1  t1  

 pa b

 aic1 , j 1   u2i , j 1 i i   c 1, 1 ,  2  t  u1  c 1, 1  t  1  u1   

 pa b

  u2i ,v2j  i j  c 1, 1 ,  2  t  u1 v1 

c i j , ai 1b j 1

c i j , ai 1b j 1

c c i j , ai 1b j 1



 



 





if j  1, 2, , k2  2;





u2i ,bcj1  t u i 1  v j   1 1 

 c 1, 1

a

j c i 1 , v







 c 1,  22  t  1  u1i v1j

 aic1 ,bcj1  t 1  u i 1  v j     1  1  

 c 1

c 1,  2

Copyright © 2011 SciRes.



c,i, j1, 2  t   pai b j , ai1b j1 ci 1,1,j1, 12  t  t1

 a ,b   t 

 pa j , a bc c 1, 1  t   pa j , ac bc c 11





u2i ,bcj1  t u i 1  v j   1 1 



a c ,1  t   pai j , a1b1c 1,1 , 2  t   pai j ,acb1c11,2  t 



 c 1, 1

1,1

1

 a1c ,b1c  t 

c 1

 pa b

i  k1  1; j  0,1

1, b1c

c c j , a1 b1

 aic1 , j 1   u2i , j 1 i i   c 1, 1 ,  2  t  u1  c 1, 1  t  1  u1   

i 1,b1c   pa j , a bc c 1, 1  t  t1 i i 1 1

if

c 1, 1  t   pib

 pa b

 aic1 ,b1c  t 1  u i   (u2i ,b1c ) i  1   c 1,  2  t  u1  c 1  



1,b1c 

c j , a1b1

c 1, 2  t 

  i 1,v2j  i 1,bcj1  j j   c 1, 1 ,  2  t  v1  c 1, 1  t  1  v1   

c i j , ai 1b j 1



c j , a1 b1

 pa b

c i j , ai 1b j 1

i 1 1

i

  a1c ,1

if j  k2 The recurrent relations of the conditional pgfs c,i, j1,  2  t  for i  1, 2, , k1 and j  1, 2, , k2 are as follows. c,i, j1, 2  t   pai b j , ai1b j1 ci 1,1,j1, 12  t 

 aic1 ,1   u2i ,1 i i   c 1, 1 ,  2  t  u1  c 1,  2  t  1  u1   

b1

 

 a1c , j 1 t t 2

 c,i, j2  t   pib j , a1b1 c1,1 1, 1 ,  2  t   pib

 i 1,1

 pa j , ac



if j  k2  1;

if i, j  0,1 , i , j 

 

c 1,  2

 a1c ,1

 c i , j   t   pij , a1b1c1,1 1, 1 ,  2  t   pij , ac b c 1,  2  t  1 1



c j , a1b j 1



if i  k1  1; j  1, 2, , k2  2 OJS

K. K. KAMALJA

120

c,i, j1,  2  t   pai b j , a1b j 1 c1,1,j 11,  2  t 

ET AL.

c,i, j1,  2  t   pai b j , aii b1 ci 1,1,11 ,  2  t 

 pa b

 1,v2j  1,bcj1  j j   c 1, 1 ,  2  t  v1  c 1, 1  t  1  v1   

 pa b

 aic1 ,1   u2i ,1 i i   c 1, 1 ,  2  t  u1  c 1,  2  t  1  u1   

 pa b

c 1,  2

 pa b

c 1, 1  t 

 pa b

  u2i ,b1c   aic1 ,b1c  t 1  u i  i  1   c 1, 1  t  u1  c 1  

c i j , a1b j 1

c i j , a1 b j 1

 pa b i





 a1c , j 1 t 







c c i j , ai 1b1

c,i, j1,  2  t   pai b j , aii b1 ci 1,1,11 ,  2  t  t1

c,i, j1,  2  t   pai b j , ai1b j 1 ci 1,1,j1, 12  t  t2   i 1,v j  i 1,bcj1   pa b , a bc  c 1, 12,  2  t  v1j  c 1, 1  t  1  v1j i j i 1 j 1   aic1 , j 1   u2i , j 1  pa b , ac b  c 1, 1 ,  2  t  u1i  c 1, 1  t  1  u1i i j i 1 j 1 





   i j  c 1, 1 ,  2  t  u1 v1 

 

     t2 

u2i , v2j

c c i j , ai 1b j 1

( u2i , bcj 1 ) c 1, 1



 c 1

j 1

a 

i 1

j c i 1 , v2



c 1,  2

i 1

j 1





 pa b

 aic1 , j 1   u2i , j 1 i i   c 1, 1 ,  2  t  u1  c 1, 1  t  1  u1  t2  

 pa b

  u2i ,v2j  i j  c 1, 1 ,  2  t  u1 v1 





u2i ,bcj1  t u i 1  v j    aic1 ,v2j  t 1  u i v j   1  1  c 1,2    1  1

c 1, 1

 if i  k1  1; j  k2  1

c,i, j1,  2  t   pai b j , ai b j1c1,1,j 11,  2  t  t2

c c i j , a1 b j 1

 1,v2j  1,bcj1  j j   c 1, 1 ,  2  t  v1  c 1, 1  t  1  v1   





  a1c ,v2j   a1c ,bcj1  t 1  v j  j   c 1,  2  t  v1  c 1 1   





if i  k1 ; j  k2  1 Copyright © 2011 SciRes.

  u2i ,b1c   aic1 ,b1c  t 1  u i  i  1   c 1, 1  t  u1  c 1  





 pa b



 a1c ,1

c 1,  2  t 

a c , bc 1,b1c  t  p   a b ,acbc c11 1   t 

c c 1, 1 i j , a1b1

i j

1 1

if i  k1 ; j  k2

The above system of  k1  2    k2  2  recurrent rei, j lations of conditional pgfs c   t  ,  i, j  0,1 ;

c,i, j1  t  ,  i  0,1, , k1 ; j  0,1 ; c,i, j2  t  ,

 i  0,1; j  1, 2, , k2  and c,i, j ,  t   i  1, 2, , k1 ; j  1, 2, , k2  can be written as follows.  A  B1t1  B2 t2  B12 t1t2   c 1  t  if c  2,3, , n  c  t   if c  1  A  B1t1  B2 t2  B12 t1t2 1 1

2

  c

0,0 

  k1 ,1  0,1 c 0,1 c1,0 c1,1 c1,0 , 1 ...c , 1 c ,  2 

 1,2   k1 , k2  c1,, k22  c1,1 , 1 ,  2 c , 1 ,  2 c , 1 ,  2

 a1c , j 1  pa b , ac b c 1,  2  t  t2 i j 1 j 1  pa b

 pa b

i 1,bic 

c i j , ai 1b1

(3.3) where 1 is column vector with all its elements 1 and  n  t  is column vector with its elements as follows.

 aic1 ,bcj1  t 1  u i 1  v j   c 1    1  1  

c i j , a1b j 1

c 1, 1  t  t1



c i j , a1 b1

  i 1,v j   i 1,bcj1   pa b , a bc  c 1, 12,  2  t  v1j  c 1, 1  t  1  v1j  t1 i j i 1 j 1  

 pa b

 pa b



 c,i, j1,  2  t   pai b j , a1b1 c1,1 1, 1 ,  2  t   pa b

c,i, j1,  2  t   pai b j , ai1b j 1 ci 1,1,j1, 12  t  t1t2

c c i j , ai 1b j 1

 aic1 ,1   u2i ,1 i i   c 1, 1 ,  2  t  u1  c 1,  2  t  1  u1   

c i j , ai 1b1

if i  k1  1; j  k2

j 1

if i  1, 2, , k1  2; j  k2  1

c i j , ai 1b j 1

 pa b

c c i j , ai 1b1

 t  1  u  v 1  v   t  1  u 1  v  

t  u

( aic1 , bcj 1 )

i 1



if i  1, 2, , k1  2; j  k2

if i  k1 ; j  1, 2, , k2  2

 pa b



i 1,bic 

c i j , ai 1b1

  a1c ,v2j   a1c ,bcj1  t 1  v j  c 1,  2  t  v1j  c 1  c c  1  , a b j 1 j 1  



c i j , ai 1b1



From (3.1) we get,

n  t1 , t2   p ' n  t1 , t2  where p is first column of identity matrix of order  k1  2    k2  2  . The recurrent use of (3.3) gives the pgf of X n, 1 , Yn,  2 as follows.



n  t   p '  A  B1 t1  B2 t2  B12 t1 t2  1 n



(3.4)

OJS

K. K. KAMALJA

X

4. Exact distribution of

X

We note that the pgf of

X

case of pgf of

0 n , k01

n , 1

0 , X n1,k1 ;Yn0,k 2 , Yn1,k 2 n , k01 1 0 1

, Yn,  2





is a particular



, X n1, k1 ; Yn0, k 2 , Yn1, k 2 . Hence we de1

0

1

velop an algorithm to obtain the exact probability distri-

X

bution of

0 n , k01

, X n1, k1 ; Yn0, k 2 , Yn1, k 2 1

0

1



which can further

beapplied to obtain the exact probability distribution of

121

ET AL.

 

   x  e ; y  B I x  e

Cn x; y  Cn 1 x; y A 1

  Cn 1 i 0 1

1 i

i 1





i 1

 0





  Cn 1 x; y  e j 1 B 2j I y  e j 1  0 j 0 1

1









   Cn 1 x  ei 1 ; y  e j 1 Bij12 I x  ei 1  0; y  e j 1  0 i 0 j 0

  .   A   Bi1ti   B 2j s j    Bij12 ti s j  i 0 j 0 i 0 j 0  

(4.2) with C1  0;0   A , C1  ei 1 ; 0   Bi1 , C1  0; e j 1   B1j and C1  ei 1 ; e j 1   Bij12 for i, j  0,1 . Here ei is the i th row of the identity matrix of order 2. Proof Obviously for n  1 , we have, if x  0; y  0 A  1 if x  ei 1 ; y  0, i  0,1  Bi  2 C1 x; y   B j if x  0; y  e j 1 , j  0,1  2 if x  ei 1 ; y  ei 1 , i  0,1; j  0,1  Bij O otherwise 

Hence the joint probability distribution can be obtained by expanding the polynomial with respect to t0 , t1 , s0 and s1 . That is,

where O is the null matrix of same order as that of A. For n  2 , observe that C2 x, y satisfies (4.2). Assume that Equation (4.2) is true for some r  2 r  n  . Hence we have,

X

n , 1 , Yn ,  2



from its pgf.

Observe that the pgf of joint distribution of

X

0 n , k01

, X n1, k1 ; Yn0, k 2 , Yn1, k 2 1

0

1



as obtained in (2.5) in general

involves matrix polynomial Pn  t0 , t1 ; s0 , s1  in t0 , t1 , s0 and s1 of order n where Pn  t0 , t1 ; s0 , s1 

 

1



P X

1

 x0 , X

0 n , k01

1

 x1 ; Y

1 n , k11

0 n , k02

n

1

 y0 , Y

1 n , k12

 y1

r



1 1 1 1   x y 1 2 12  A  Bi ti   B j s j    Bij ti s j    Cr x, y t s     0 0 0 0 i j i j  , x y D     r Then

 coefficient of t0x0 t1x1 s0y0 s1y1 in n  t0 , t1 ; s0 , s1 

1 1 1 1   1 2 12  A   Bi ti   B j s j    Bij ti s j  i 0 j 0 i 0 j 0  

i.e.



 

 

P X n0, k1 , X n1, k1  x; Yn0, k 2 , Yn1, k 2  y 0

1

0

1

 coefficient of t s in n  t0 , t1 ; s0 , s1  x

(4.1)

y

Exact formula to obtain the coefficient matrix is quiet tedious since multiplication of matrices is not commutative operation. But the interesting recurrent relations are found between these coefficient matrices. Let

 

Cn  x0 , x1 ; y0 , y1   Cn x; y x

be the coefficient matrix of

y

t s in the expansion of matrix polynomial Pn  t0 ,t1 ; s0 , s1  and for n  1 , let Dn 

 x; y  x , y  0,1,, n for i  0,1 . The following i

i

Lemma gives the recurrent relations of the coefficient matrices of Cn x; y with Cn 1 x; y .

 

 

Lemma 4.1 Let Cn x; y x

of t s

y

 

 

recurrent relation. Copyright © 2011 SciRes.

satisfies the following

 

r 1

1 1 1 1    A   Bi1ti   B 2j s j    Bij12 ti s j i 0 j 0 i 0 j 0  1 1 1 1    A   Bi1ti   B 2j s j    Bij12 ti s j i 0 j 0 i 0 j 0 

r

     

  x y    Cr x , y t s   ( x , y )D  r  

 

1 1 1 1     A   Bi1ti   B 2j s j    Bij12 ti s j  i 0 j 0 i 0 j 0   1    Cr 1 x; y A  Cr 1 x  ei 1 ; y Bi1 I  x  ei 1  0 ( x , y )Dr 1  i 0

 







1







  Cr 1 x; y  e j 1 B 2j I y  e j 1  0

be the coefficient matrix

in the expansion of the matrix polynomial

Pn  t0 , t1 ; s0 , s1  Then Cn x; y

 

j 0

1

1



   Cr 1 x  ei 1 ; y  e j 1 i 0 j 0



 

Bij12 I x  ei 1  0; y  e j 1  0 t s x

y

OJS

K. K. KAMALJA

122

Hence we get the required proof of the lemma. Theorem 4.1 The exact probability distribution of

X

0

1

n , k01

0

1

, X n , k1 ; Yn , k 2 , Yn , k 2 1

0



1



is given by,

P ( X n , k1 , X n , k1 )  x;(Yn , k 2 , Yn , k 2 )  y 0

1

0

0

1

1

0

   x, y  is the

1

 p ' Cn x , y 1



0

1

0

1

,



y







12 12 12 E X n0, k1  p '  A  B11  B02  B12  B01  B10  B11 0

j 1



and



0 n 1, k01



5. Waiting Time Distributions Related to Runs and Patterns , X n1, k1 ; Yn0, k 2 , Yn1, k 2 1

0

1



from its pgf given in (2.5) can

be expressed as,



P ( X n0, k1 , X n1, k1 )  x ;(Yn0, k 2 , Yn1, k 2 )  y 0

1

0

1



1











1

1



   Cn 1 x  ei 1 ; y  e j 1 i 0 j 0

( X

 

(5.1) The components of the above expression can be interpreted as follows.

   

p ' Cn 1 x; y A 1



 P ( X n0, k1 , X n1, k1 )  x, (Yn0, k 2 , Yn1, k 2 )  y ;

( X

0

0 n 1, k01

1

0

1



, X n11, k1 )  x , (Yn01, k 2 , Yn11, k 2 )  y 1

Copyright © 2011 SciRes.

0

1



0

1

0 n 1, k01

0



1

.

, X n11, k1 )  x , (Yn01, k 2 , Yn11, k 2 )  y  ei 1 1

0

1



 

 

p ' Cn 1 x  ei 1 , y  e j 1 Bij12 1





 P ( X n0, k1 , X n1, k1 )  x , (Yn0, k 2 , Yn1, k 2 )  y ;

( X

0

1

0 n 1, k01

0

1

, X n11, k1 )  x  ei 1 , (Yn01, k 2 , Yn11, k 2 )  y  e j 1 1

0

1



(5.2) so that



0



1

0



1

1



n 1, k02

n 1, k12

 P ( X n0, k1 , X n1, k1 )  x, (Yn0, k 2 , Yn1, k 2 )  y ; 0

1

0

( X , X )  x , (Y , Y )  y   P ( X , X )  x , (Y , Y )  y ; ( X , X )  x  e , (Y , Y )  y   P ( X , X )  x , (Y , Y )  y ; ( X , X )  x , (Y , Y )  y  e     P ( X , X )  x , (Y , Y )  y ; ( X , X )  x  e , (Y , Y )  y  e  0

1

n 1, k01

n 1, k11

0 n , k01

i 0

j 0

0

1

n , k01

n , k11

0 n , k01

1 n 1, k11

1

0 n , k02

1 n 1, k11

0 n 1, k01

1

0

1 n , k11

0 n 1, k01

0 n 1, k01



1

Similarly for i, j  0,1 ,

i 0 j 0

 Bij12 I x  ei 1  0; y  e j 1  0 1



 



1



  Cn 1 x; y  e j 1 B 2j I y  e j 1  0 j 0

0

1

 P ( X n0, k1 , X n1, k1 )  x , (Yn0, k 2 , Yn1, k 2 )  y ;

1

1   p ' Cn 1 x; y A   Cn 1 x  ei 1 ; y Bi1 I  x  ei 1  0 i 0 

 

,

, X n11, k1 )  x  ei 1 , (Yn01, k 2 , Yn11, k 2 )  y

 

1

The exact probability distribution of 0



1

P ( X n0, k1 , X n1, k1 )  x ;(Yn0, k 2 , Yn1, k 2 )  y

(4.4)

n , k01

0

j 1

12  B01  B01 1

X

1

p ' Cn 1 x; y  ei 1 Bi2 1

in (2.5).



n

 

0

( X

(4.3)

Remark 4.1 The expected number of failure-runs of length k01 in first components of n Markov dependent bivariate trials is given by, d E X n0, k1    t0 ,1;1,1 t0 1 . 0 dt On simplifying this expression, we have,



 

p ' Cn 1 x  ei 1 ; y Bi1 1



x

X n0, k1 , X n1, k1 ; Yn0, k 2 , Yn1, k 2

For i  0,1 ,

 P ( X n0, k1 , X n1, k1 )  x , (Yn0, k 2 , Yn1, k 2 )  y ;

coefficient matrix of t s in where Cn expansion of matrix polynomial Pn  t0 , t1 ; s0 , s1  and it satisfies (4.2) Proof The proof follows by applying the Lemma 4.1 to matrix polynomial Pn  t0 , t1 ; s0 , s1  involved in the pgf of distribution of

ET AL.

1 n , k12

0 n 1, k02

i 1

0

n , k02

1 n 1, k11

1

n , k12

0 n 1, k02

1 n , k11

0 n , k02

i 1

1 n 1, k12

0 n 1, k02

1 n 1, k12

j 1

1 n , k12

1 n 1, k12

j 1

(5.3) The above interpretation is useful for deriving different waiting time distributions. Let F0i be the event that failure-run of length k0i occur for the first time in the i th component of the Markov dependent bivariate trials and F1i be the event that success-run of length k1i occur for the first time in the i th component of the Markov dependent bivariate trials i  1, 2 . Let WS be the waiting time for sooner occurring event between F01 , F02 F11 , F12 . To obtain the distriOJS

K. K. KAMALJA

bution of sooner waiting time we define the following random variables. WSi, j : The waiting time for sooner occurring event between F01 , F02 F11 , F12 given that F ji is the sooner event (i.e. j -run of length k ij of i th component of bivariate trials is the sooner event) i  1, 2 and j  0,1 . W ji : The waiting time for the first occurrence of j -r un of length k ij in the i th component of the  X   bivariate trials  n  , n  0  , i  1, 2 and j  0,1 .  Yn   12 Wij : The waiting time until the occurrence of both events Fi1 and F j2 simultaneously, i, j  0,1 (i.e. i run of length ki1 in the first component and j -run of length k 2j in the second component of the bivariate trials occur simultaneously). Then we have, 1



1





P WS  m    P WS1,i  m   P WS2, j  m i 0

1

1

j 0



   P Wij12  m i 0 j 0





ij

X

occurs for the first time at m trial given that none of the events F02 F11 , F12 has occurred until mth trial th



1 S ,0

P W



  m

m

can be written as follows.

X

0 m 1, k01

1

0

1





1

0

1



 p Cm 1  0, 0, 0, 0  B 1 '

Similarly the probability that 1-run of length k (i.e. event F11 ) occurs for the first time at mth trial given that none of the events F01 F02 , F12 has occurred until

Copyright © 2011 SciRes.



0

1



 p ' Am 1 B11 1

m  k11

Hence pgf of WS1,1 is, 1  S1 ,1  t   t p  I  A t  B111 .







Similarly P WS2,0  m and P WS2,1  m P WS2,0  m  p ' Am 1 B02 1

 P W

  m  p ' A

m 1

2 S ,1



is given by,

B12 1

and pgf of WS2, j is, Now the probability that both events Fi1 and Fj2 occurs simultaneously for the first time at mth trial

i.e. P W

12 i, j



m



can be written as follows.





P W0012  m  P ( X m0 , k1 , X m1 , k1 ; Ym0, k 2 , Ym1, k 2 )  1, 0,1, 0  ;

X

0

0 m 1, k01

,X

1

1 m 1, k11

0

0 m 1, k02

;Y

1

   0, 0, 0, 0

1 m 1, k12

,Y









P W0112  m  P X m0 , k1 , X m1 , k1 ; Ym0, k 2 , Ym1, k 2  1, 0, 0,1 ;

X

0 m 1, k01

0

1

0



1

, X m1 1, k1 ; Ym01, k 2 , Ym1 1, k 2   0, 0, 0, 0  1

0

1

 p 'Cm  0, 0, 0, 0  B 1  p ' A 12 01





m 1

12 01



B 1

0 m 1, k01

0

1

0

1





, X m1 1, k1 ; Ym01, k 2 , Ym1 1, k 2   0, 0, 0, 0  1

0

1

 p 'Cm  0, 0, 0, 0  B 1  p ' A





12 10



m 1



12 10

B 1

P W1112  m  P ( X m0 , k1 , X m1 , k1 ; Ym0, k 2 , Ym1, k 2 )   0,1, 0,1 ; 0 m 1, k01

0

1

0

1



, X m1 1, k1 ; Ym01, k 2 , Ym1 1, k 2   0, 0, 0, 0  1

0

m 1

1



12 11

B 1

In general, 1 1

m

1

 p 'Cm  0, 0, 0, 0  B 1  p ' A

1

1 S ,1



, X m1 1, k1 ; Ym01, k 2 , Ym1 1, k 2   0, 0, 0, 0 

12 11

 S1 ,0  t   t p '  I  At  B011 .

i.e. P W

1

 p 'Cm  0,1, 0, 0  B 1

X

m  k01

Hence pgf of WS1,0 can be given as,

mth trial

0

1 1

X

1 0

 p Am 1 B01 1

0 m 1, k01



, X m1 1, k1 ; Ym01, k 2 , Ym1 1, k 2   0, 0, 0, 0 

'

1

P W1012  m  P X m0 , k1 , X m1 , k1 ; Ym0, k 2 , Ym1, k 2   0,1,1, 0  ;

 P X m0 , k1 , X m1 , k1 ; Ym0, k 2 , Yn1, k 2  1, 0, 0, 0  ; 0

0

12 12 1  p ' Am 1 B00 1  p 'Cm  0, 0, 0, 0  B00

The probability that 0-run of length k01 (i.e. event F01 )

1 S ,0



 P X m0 , k1 , X m1 , k1 ; Ym0, k 2 , Ym1, k 2   0,1, 0, 0  ;

5.1. Sooner Waiting Time Distribution

i.e. P W



1

Let  S  t  ,   t  ,   t  and   t  be the pgf of WS , WSi, j , W and Wkj12 , i  1, 2 and k , j  0,1 respectively. In the next sub-section we obtain the sooner waiting time distribution. i j

i S, j i j



P WS1,1  m

 S2, j  t   t p '  I  At  B 2j 1, j  0,1 . . (5.4)



123

ET AL.

can be written as follows.





m 1 2 P Wi12 Bij 1, i, j  0,1 ,j  m  p'A

Also pgf of Wij12 is,

 ij  t   t p '  I  At  Bij12 1 1

Hence from (5.4), the exact probability distribution of

OJS

K. K. KAMALJA

124

random variable WS is given by, 1 1  2 1  P WS  m   p ' Am 1    B ij    Bij12  1 , i 0 j 0  i 1 j  0 



m  min k , k , k , k 1 0

1 1

2 0

2 1

.

ET AL.

i

0

(5.5)

fied form as follows.

 n1,0  t01   n  t0  t01 ,1;1,1  p '  A01  D01 t01  1 n

12 12 where A01  A  B11  B02  B12  B10  B11 and 1 1 12 12 D0  B0  B00  B01 . The exact probability distribution of X n1, k 1 can be 0 obtained from Lemma 4.1 as follows.



Let W be the waiting time for the ri occurrence of i -run of length ki j for the j th component of bivariate trials, i  0,1 ; j  1, 2 . We obtain the distribution of Wrij,i using distr ibution of X nj, k j number of occurth



0

1

0

1



as obtained in (2.5) is as

n  t0 , t1 ; s0 , s1  n

1 1 1 1   p '  A   Bi1ti   B 2j s j    Bij12 ti s j i 0 j 0 i 0 j 0  In particular pgf of marginal distribution of obtained by setting t1  s0  s1  1 in the pgf



  1  X n1, k 1 is



 



r

 A01 if i  0  1 with C1  i    D0 if i  1 O otherwise  where O is the null matrix of order same as matrix A01 and D01 . The probability in (5.8) can be written as follows.









P X n1, k 1  r  p ' Cn 1  r  A01  Cn 1  r  1 D01 1 0



The above components of P X n1, k 1  r 0 terpreted as follows.









can be in-

  r  1

p ' Cn 1  r  A01 1  P X n1, k 1  r ; X n1 1, k 1  r



0





0

p ' Cn 1  r  D01 1  P X n1, k 1  r ; X n11, k 1

Now we have,







0

0

(5.9)



P Wr10 ,0  m  P X m1 , k 1  r0 ; X m1 1, k 1  r0  1 0

0

0

1

0



1



and

is as follows. n  t0 ,1;1,1







P Wr10 ,0  m  p ' Cm 1  r0  1 D01 1

Particularly when r0  1 , we have,









1 P W1,0  m  p ' Am 1 D01 1

0

X n0, k 1 ,0 , X n1, k 1 ,1 ; Yn0, k 2 ,0 , Yn1, k 2 ,1

1 The pgf of W1,0 is given by,

  s   s p '  I  A01 s  D011 1

and formula for exact probability is n

1  12 12 12 12  t0  B01 t0  B10  p '  A  B01t0  B11   B 2j  B00  B11  1 j 0   (5.7)

Copyright © 2011 SciRes.

(5.8)

Using the pgf of X n1, k 1 and interpretations in (5.9) 0 we have,

follows.

n  t0 , t1 ; s0 , s1  o f

0

where Cn  r  is the coefficientn matrix of t01 in the matrix polynomial A01  D01t01 and in general for 2  m  n Cm  r  satisfies the recurrent relation Cm  r   Cm 1  r  A01  Cm 1  r  1 D01

i

rences of i -run of length ki j for the j th component of n bivariate trials i  0,1 ; j  1, 2 . The pgf n  t0 , t1 ; s0 , s1  of joint distribution of



P X n1, k 1  r  p 'Cn  r 1

5.2. Waiting Time Distribution for Runs

X n0, k 1 , X n1, k 1 ; X n0, k 2 , X n1, k 2

for i  0,1 ; j  1, 2 .

Then pgf of X n1, k 1 in (5.7) can be expressed in a simpli-

The pgf of WS is given by, 2 1 1 1  1   S  t   t p '  I  At     Bij    Bij12 1 (5.6) i 0 j 0  i 1 j  0  Here we note that it is quiet difficult to study the later waiting time distribution between F01 , F02 F11 , F12 , particularly when one is interested in studying the waiting time distribution for the later occurring event between more than two events. For the waiting time distribution of later occurring event between any two runs in the components of the bivariate trials or for the two patterns in the two components of bivariate trials the theory can be developed in general. For this we refer [13] who derive the waiting time distribution of later occurring event between the two events F0 and F1 where Fi  i  0,1 is the event that i -run of length ki occurs for the first time in the sequence of higher order Markov dependent BT. [13] use the joint distribution of number of occurrences of 1-runs (i.e. success-runs) of length k1 and the number of occurrences of 0-runs (i.e. failure runs) of length k0 .

j ri , i

 

Let pgf of X nj, k j be  nj,i ti j





 

1  m  p A01 P W1,0

m 1

D01 1 , m  k01 .

Similarly waiting time distributions of Wrij,i , i  0,1 and j  1, 2 can be obtained. OJS

K. K. KAMALJA

6. Numerical Study

Table 1. Distribution of

In this section we present the numerical study based on the joint distribution of patterns in the sequence of bivariate trials. We consider the sequence  X n   0   0   1   1     , n  0  of   ,   ,   ,    -valued Markov  0   1   0  1    Yn 

X 10,

1

Y10,

X

10,1



,Y10,2 .

2

Sum

0

1

2

3

0

0.00257

0.00666

0.00179

3.5E–05

0.01106

1

0.04765

0.08133

0.02006

0.00044

0.14949

2

0.16288

0.21551

0.04793

0.00113

0.42744

3

0.14927

0.15937

0.03184

0.00075

0.34122

p01,00  0.3 , p01,01  0.1 , p01,10  0.5 , p01,11  0.1 ,

4

0.03465

0.0294

0.00497

9.9E–05

0.06912

p10,00  0.1 , p10,01  0.4 , p10,10  0.3 , p10,11  0.2 ,

5

0.00101

0.00058

7.3E–05

1.2E–06

0.00167

p11,00  0.2 , p11,01  0.2 , p11,10  0.2 , p11,11  0.4

Sum

0.39803

0.49285

0.10666

0.00245

1.00000

dependent bivariate trials with π 00  1 with transition probabilities as follows. p00,00  0.2 , p00,01  0.3 , p00,10  0.4 , p00,11  0.1 ,

Let 1  01 and  2  110 . The joint distribution of X n, 1 , Yn,  2 is evaluated numerically for n  10 using the algorithm given in Theorem 4.1. The evaluated joint pmf of X n, 1 , Yn,  2 is described in the following Table 1.



125

ET AL.







7. Application and Extension [14] introduced the two-dimensional engineering system consisting of a grid of m  n components arranged in m -rows and n -columns. The system and its components can be either in working or failed state. The system fails if and only if a grid of size r  s components fails. Particularly, for m  r  1 , we can formulate the states of the components in this system as a sequence of n independent bivariate trials. We assume that the i th component in a column is in state 1 if it is in failed state and in state 0 otherwise. For i  1, 2, , n we define, 1 if first r components in i th column are in failed state Xi   0 otherwise and 1 if last r components in i th column are in failed state Yi   0 otherwise The reliability of consecutive-  r , s  -out-of-  r  1, n  : F-Lattice system can now be obtained simply by using the joint distribution of X n, 1 , Yn,  2 as, P(consecutive-  r , s  -out-of-  r  1, n  : F-Lattice system works) = P X n, 1  0, Yn,  2  0 , where 1 is 1-run of length s in the first component and  2 is 1-run of length s in the second component of the bivariate trials. Extending the concept of bivariate trials to multivariate trials, the joint distributions of number of occurrences of patterns  i of length ki  i  1, 2, , m  in the i th component of m -variate trials can be used to get the reliability of general two-dimensional consecutiver  s -out-of-  m, n  : F-Lattice system.





Copyright © 2011 SciRes.





The pgf n  t1 , t2 , , tm   n  t  of joint distribution of X ni , i , i  1, 2, , m , the number of occurrences of patterns  i of length ki in the i th component of m -variate trials obtained in general by using the method of conditional pgfs is of form, n

  n m n  t   p '  A   Bi ti   Bij ti t j  ...  B12...k t1t2 ...tk  1   i 1 i , j 1   i j   (7.1) Extending the Lemma 4.1 for the pgf in (7.1) exact joint probability distribution of X ni , i , i  1, 2, , m can be obtained. The above study of runs and patterns can be extended in another direction by generalizing the sequence of Markov dependent bivariate trials to the sequence of Markov dependent multivariate trials. In the following, we discuss briefly the method of deriving the joint dis  tribution of  Yn ,  2 , k , Yn ,  2 , k , , Yn ,  2 , k  where Yn,  2 , k , 2 2 r r  1 1 i i   i  1, 2, , r  denotes the number of occurrences of two dimensional patterns of rectangular shape  i2  i  1, 2, , r  in the sequence of Markov dependent multivariate trials.





Let Sm2  X   x1 , x2 , , xm  xi  0,1;i  1, 2, , m

 

'

so that # S  2 . Consider the sequence of m -variate Sm2 -valued Markov dependent trials  X i , i  0 . Let the transition probabilities of these Markov dependent trials  X i , i  0 be, 2 m



m



P X r  y X r 1  x  Px y , x, y  Sm2 , r  1 .

and initial probabilities be P  X 0  x    x  x  Sm2 . Consider the two dimensional pattern of rectangular shape,  i2  a i1 a i 2  a iki  i  1, 2, , r  where a i1 , a i 2 , , a iki  Sm2 . Let Yn,  2 , k ,  i  1, 2, , r  be the i number of occurrences of i patterns  2i in n trials OJS

K. K. KAMALJA

126

X 1 , X 2 , , X n . We obtain the joint distribution of    Yn ,  2 , k1 , Yn ,  22 , k2 , , Yn ,  r2 , kr  using the method for de1   riving joint distribution of number of occurrences of patterns  i  i  1, 2, , r  in the sequence of multi-state trials as done by [11]. For this consider the following transformation of sequence of m -variate sequence of Sm2 -valued Markov dependent trials  X i , i  0 to the univariate sequence of Markov dependent trials Z i , i  0 . Define the function g : Sm2  Sm10 such that



m

ET AL. Markov-Dependent Trials,” Journal of American Statistical Association, Vol. 78, No. 381, 1983, pp. 168-175. doi:10.2307/2287125

[2]

N. Balakrishnan and P. S. Chan, “Start-up Demonstration Tests with Rejection of Units upon Observing d Failures,” Annals of Institute of Statistical Mathematics, Vol. 52, No. 1, 2000, pp. 184-196. doi:10.1023/A:1004101402897

[3]

W. Feller, “An Introduction to Probability Theory and Its Applications,” 3rd Edition, John Wiley & Sons, Hoboken, 1968.

[4]

K. D. Ling, “On Binomial Distributions of Order k,” Statistics & Probability Letters, Vol. 6, No. 4, 1988, pp. 247-250. doi:10.1016/0167-7152(88)90069-7

[5]

S. Aki and K. Hirano, “Number of Success-Runs of Specified Length until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k,” Annals of Institute of Statistical Mathematics, Vol. 52, No. 4, 2000, pp. 767-777. doi:10.1023/A:1017585512412

[6]

A. M. Mood, “The Distribution Theory of Runs,” Annals of Mathematical Statistics, Vol. 11, No. 4, 1940, pp. 367-392. doi:10.1214/aoms/1177731825

[7]

S. Aki and K. Hirano, “Sooner and Later Waiting Time Problems for Runs in Markov Dependent Bivariate Trials,” Annals of Institute of Statistical Mathematics, Vol. 51, No. 1, 1999, pp. 17-29. doi:10.1023/A:1003874900507

[8]

M. Uchida, “On Generating Functions of Waiting Time Problems for Sequence Patterns of Discrete Random Variables,” Annals of Institute of Statistical Mathematics, Vol. 50, No. 4, 1998, pp. 655-671. doi:10.1023/A:1003756712643

[9]

D. L. Antzoulakos, “Waiting Times for Patterns in a Sequence of Multistate Trials,” Journal of Applied Probability, Vol. 38, No. 2, 2001, pp. 508-518. doi:10.1239/jap/996986759



g  x    2i 1 xi where Sm10  g  x  x  Sm2 . He nce



i 1



Sm10  0,1, 2, , 2m  1 . Each x in Sm2 can be treated as a m -digit binary number. The function g  x  converts this m -digit binary number into a unique equivalent decimal number in S m10 . Now corresponding to the m -variate sequence of Sm2 -valued Markov dependent trials  X i , i  0 , we have the univariate sequence of S m10 -valued Markov dependent trials Z i , i  0 where Z i  g  X i  . The transition probabilities for the sequence of trials  X i , i  0 and for the converted sequence of trials Z i , i  0 are related as follows.





P X r  y X r 1  x  Px y



 



 P Z r  g y Z r 1  g  x   pij x, y  S , r  1 2 m

 

where g  x   i and g y  j . Convert the pattern  i2  a i1 a i 2  a iki  i  1, 2, , r  into a pattern 10 i  bi1bi 2  biki where bij  g  a ij  for j  1, 2,  , ki . Now the original problem of studying the joint distribu  tion of  Yn ,  2 , k , Yn,  2 , k , , Yn,  2 , k  in the sequence of 2 2 r r  1 1  Sm2 -valued Markov dependent trials X 1 , X 2 , , X n reduces to the problem of studying distribution of    Yn , 10 , k1 , Yn , 102 , k2 , , Yn , 10r , kr  for the sequence of  1  S m10 -valued Markov dependent trials Z1 , Z 2 , , Z n . [15] obtain the joint distribution of number of occurrences of i -runs  i  0,1, 2, , m  of length ki , while [11] obtain the joint distribution of number of occurrences of patterns in the sequence of 0,1, , m -valued Markov dependent trials using the method of conditional pgfs. The related waiting time distributions can also be studied following the same process as in Section 5 in the case of Markov dependent multivariate trials.

7. References [1]

S. J. Schwager, “Run Probabilities in Sequences of

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[10] K. Inoue and S. Aki, “Generalized Waiting Time Problems Associated with Pattern in Polya’s Urn Scheme,” Annals of Institute of Statistical Mathematics, Vol. 54, No. 3, 2002, pp. 681-688. doi:10.1023/A:1022431631697 [11] K. S. Kotwal and R. L. Shinde, “Joint Distributions of Patterns in the Sequence of Markov Dependent MultiState Trials,” 2011 Submitted. [12] S. Aki and K. Hirano, “Waiting Time Problems for a Two-Dimensional Pattern,” Annals of Institute of Statistical Mathematics, Vol. 56, No. 1, 2004, 169-182. doi:10.1007/BF02530530 [13] K. S. Kotwal and R. L. Shinde, “Joint Distributions of Runs in a Sequence of Higher-Order Two-State Markov trials,” Annals of Institute of Statistical Mathematics, Vol. 58, No. 1, 2006, pp. 537-554. doi:10.1007/s10463-005-0024-6 [14] A. A. Salvia and W. C. Lasher, “2-Dimensional Consecutive k-out-of-n: F Models,” IEEE Transactions on Reliability, Vol. R-39, No. 3, 1990, pp. 382-385. doi:10.1109/24.103023 OJS

K. K. KAMALJA [15] R. L. Shinde and K. S. Kotwal, “On the Joint Distribution of Runs in the Sequence of Markov Dependent MultiState Trials,” Statistics & Probability Letters, Vol. 76, No.

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ET AL.

127

10, 2006, pp. 1065-1074. doi:10.1016/j.spl.2005.12.005

OJS