Journal of Quality and Reliability Engineering

Volume 2013 (2013), Article ID 494976, 9 pages

http://dx.doi.org/10.1155/2013/494976

Research Article

Distributions of Patterns of Pair of Successes Separated by Failure Runs of Length at Least and at Most Involving Markov Dependent Trials: GERT Approach

Kanwar Sen,¹ Pooja Mohan,² and Manju Lata Agarwal³

¹Department of Statistics, University of Delhi, Delhi 7, India

²RMS India, A-7, Sector 16, Noida 201 301, India

³The Institute for Innovation and Inventions with Mathematics and IT (IIIMIT), Shiv Nadar University, Greater Noida 203207, India

Received 25 June 2012; Revised 10 December 2012; Accepted 11 December 2012

Academic Editor: Tadashi Dohi

Copyright © 2013 Kanwar Sen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We use the Graphical Evaluation and Review Technique (GERT) to obtain probability generating functions of the waiting time distributions of 1st, and th nonoverlapping and overlapping occurrences of the pattern , involving homogenous Markov dependent trials. GERT besides providing visual picture of the system helps to analyze the system in a less inductive manner. Mean and variance of the waiting times of the occurrence of the patterns have also been obtained. Some earlier results existing in literature have been shown to be particular cases of these results.

1. Introduction

Probability generating functions of waiting time distributions of runs and patterns have been studied and utilized in various areas of statistics and applied probability, with applications to statistical quality control, ecology, epidemiology, quality management in health care sector, and biological science to name a few. A considerable amount of literature treating waiting time distributions have been generated, see Fu and Koutras [1], Aki et al. [2], Koutras [3], Antzoulakos [4], Aki and Hirano [5], Han and Hirano [6], Fu and Lou [7], and so forth. The books by Godbole and Papastavridis [8], Balakrishnan and Koutras [9], Fu and Lou [10] provide excellent information on past and current developments in this area.

The probability generating function is very important for studying the properties of waiting time distributions of runs and patterns. Once a potentially problem-specific statistic involving runs and patterns has been defined, the task of deriving its distribution can be very complex and nontrivial. Traditionally, combinatorial methods were used to find the exact distributions for the numbers of runs and patterns. By using the theory of recurrent events, Feller [11] obtained the probability generating function for waiting time of a success run of size in a sequence of Bernoulli trials. Fu and Chang [12] developed general method based on the finite Markov chain imbedding technique for finding the mean and probability generating functions of waiting time distributions of compound patterns in a sequence of i.i.d. or Markov dependent multistate trials. Ge and Wang [13] studied the consecutive--out-of-: system involving Markov Dependence.

Graphical Evaluation and Review Technique (GERT) has been a well-established technique applied in several areas. However, application of GERT in reliability studies has not been reported much. It is only recently that Cheng [14] analyzed reliability of fuzzy consecutive--out-of-: system using GERT. Agarwal et al. [15, 16], Agarwal & Mohan [17] and Mohan et al. [18] have also studied reliability of -consecutive--out-of-: system and its various generalizations using GERT, and illustrated the efficiency of GERT in reliability analysis. Mohan et al. [19] studied waiting time distributions of 1st, and th nonoverlapping and overlapping occurrences of the pattern , involving Markov dependent trials, using GERT. In this paper, probability generating functions of the waiting time distributions of 1st, and th nonoverlapping and overlapping occurrences of a pattern involving pair of successes separated by a run of failures of length at least and at most , , involving Homogenous Markov Dependence, that is, probability that component fails depends only upon the state of component and not upon the state of the other components, (Ge and Wang [13]) have been studied using GERT. Mean and variance of the time of their occurrences can then be obtained easily. Some earlier results existing in literature have been shown to be particular cases.

2. Notations and Assumptions

  : pair of successes separated by a run of failures of length at least and at most , . : indicator random variable for state of trial , or 1 according as trial is success or failure. : ;  : , probability that trial is a success given that preceding trial is also a success, for . : , probability that trial is a success given that preceding trial is a failure, for .

3. Brief Description of GERT and Definitions

GERT is a procedure for the analysis of stochastic networks having logical nodes (or events) and directed branches (or activities). It combines the disciplines of flow graph theory, MGF (Moment Generating Function), and PERT (Project Evaluation and Review Technique) to obtain a solution to stochastic networks having logical nodes and directed branches. The nodes can be interpreted as the states of the system and directed branches represent transitions from one state to another. A branch has the probability that the activity associated with it will be performed. Other parameters describe the activities represented by the branches.

A GERT network in general contains one of the following two types of logical nodes (Figure 1):

(a)nodes with Exclusive-Or input function and Deterministic output function and(b)nodes with Exclusive-Or input function and Probabilistic output function.

(a)   Deterministic output

(b) Probabilistic output

Figure 1: Type of GERT nodes.

Exclusive-Or Input. The node is realized when any arc leading into it is realized. However, one and only one of the arcs can be realized at a given time.

Deterministic Output. All arcs emanating from the node are taken if the node is realized.

Probabilistic Output. Exactly one arc emanating from the node is taken if the node is realized.

In this paper type (b) nodes are used.

The transmittance of an arc in a GERT network, that is, the generating function of the waiting time for the occurrence of required system state is the corresponding -function. It is used to obtain the information of a relationship, which exists between the nodes.

If we define , as the conditional function associated with a network when the branches tagged with a are taken times, then the equivalent generating function can be written as follows:

The function is the generating function of the waiting time for the network realization.

Mason’s Rule (Whitehouse [20], pp. 168–172). In an open flow graph, write down the product of transmittances along each path from the independent to the dependent variable. Multiply its transmittance by the sum of the nontouching loops to that path. Sum these modified path transmittances and divide by the sum of all the loops in the open flow graph yielding transmittance as follows:

where

For more necessary details about GERT, one can see Whitehouse [20], Cheng [14] and Agarwal et al. [15].

4. Waiting Time Distribution of the Pattern

Theorem 1. , the probability generating function for the waiting time distribution of the 1st occurrence of the pattern involving Homogenous Markov Dependence is given by

Then,

The GERT network for this pattern is represented by Figure 2 where each node represents a specific state as described below:

 : initial node : a trial resulting in failure : a trial resulting in success corresponding to beginning of the th occurrence of the pattern ,  1: a component is in failed state preceded by working component : th contiguous failed trial preceded by a success trial,  : th occurrence of the required pattern, .

Figure 2: GERT network representing 1st occurrence of the pattern .

There are in all nodes designated as and 0¹ representing specific states of the system, that is, a sequence of homogenous Markov dependent trials. GERT network can be summarized as follows. If the first trial results in failure then state is reached from with conditional probability otherwise state with conditional probability . Further, if the preceding trial is failure (state ) followed by a contiguous success trial then state is reached from state with conditional probability otherwise it continues to move in state with conditional probability . State represents first occurrence of a success trial (may or may not be preceded by failed trials) and if next contiguous trial(s) is (are) also success then it continues to move in state with conditional probability otherwise it moves in state 1 with conditional probability . Again if the next contiguous trial results in failure then system state 2 occurs with conditional probability otherwise state with conditional probability . Similar procedure is followed till contiguous failed trials occur preceded by a success trial. Now, if the next contiguous trial is failure then state is reached with conditional probability otherwise state 0¹, first occurrence of required pattern. However, if the system is in state and next contiguous trial is failure system moves to state with conditional probability . Again a similar procedure is followed till node . However, if there occur contiguous failed trials after state then the system moves to node and continues to move in that state until a success trial occurs at which it moves to state and again a similar procedure is followed for the remaining trials till state 0¹, first occurrence of the required pattern. Details on the derivation of the theorem are given in the appendix.

Theorem 2. , the probability generating function for the waiting time distribution of the th nonoverlapping occurrence of the pattern involving Homogenous Markov Dependence is given by

Then,

and can be obtained by applying (6).

The GERT network for (say) nonoverlapping occurrence of the pattern is represented by Figure 3. Each node represents a specific state as described earlier in Figure 2.

Figure 3: GERT network representing the th (, here) nonoverlapping occurrence of the pattern .

Theorem 3. , the probability generating function for the waiting time distribution of the mth overlapping occurrence of the pattern involving Homogenous Markov Dependence is given by

Then,

and can be obtained by applying  (6).

The GERT network for th (, say), overlapping occurrence of the pattern is represented by Figure 4. Each node represents a specific state as described earlier in Figure 2.

Figure 4: GERT network representing the th (, here) overlapping occurrence of the pattern .

Particular Cases. (i) For , , and , in the pattern , that is, for a run of at most failures bounded by successes, the probability generating function becomes

verifying the results of Koutras [3, Theorem 3.2].

(ii) If , , that is, pair of successes are separated by a run of failures of length at least one and at most , that is, then (4), (7), and (9), respectively, become

(iii) If and , then it reduces to the pattern , that is,

(iv) If ,

then (4), (7), (9) reduce to the results of Mohan et al. [19], that is,

5. Conclusion

In this paper we proposed Graphical Evaluation and Review Technique (GERT) to study probability generating functions of the waiting time distributions of , and nonoverlapping and overlapping occurrences of the pattern ), involving homogenous Markov dependent trials. We have also demonstrated the flexibility and usefulness of our approach by validating some earlier results existing in literature as particular cases of these results.

Appendix

Proof of Theorems

From the GERT network (Figure 2), it can be observed that there are paths to reach state 0¹ from the starting node S, see Table 1.

Table 1

No.
Paths
Value

1
to 1A to 0₁ to 1 to 2⋯to to 0¹

2
to 1A to 0₁ to 1 to 2⋯to to to 0¹

3
to 1A to 01?> to 1 to 2⋯to to to to 0¹

to 1A to 0₁ to 1 to 2⋯to to…to to 0¹

to 1A to 0₁ to 1 to 2⋯to to…to to to 0¹

to 0₁ to 1 to 2⋯to to 0¹

to 0₁ to 1 to 2⋯to to to 0¹

to 0₁ to 1 to 2⋯to to to to 0¹

to 0₁ to 1 to 2⋯to to…to to 0¹

to 0₁ to 1 to 2⋯to to…to to to 0¹

Now, to apply Mason’s rule, we must also locate all the loops. However only first and second order loops exist. First order loops are given in Table 2.

Table 2

No.
First order loops
Value

1
1A to 1A

2
0₁ to 0₁

3
0₁ to 1 to 0₁

4
0₁ to 1 to 2 to 01>

0₁ to 1 to 2 to…to to 0₁

0₁ to 1 to 2 to…to to to to 0₁

to

However, paths at Nos. , , ⋯, 2( also contain first order nontouching loop ( to ), whose value is given by . Further, first order loop No. forms first order nontouching loop to both paths, whose value is given by .

Also second order loops corresponding to first order loops from No. 2 to No. are given (since first order loop no. 1, that is, to forms nontouching loop with each of the other mentioned first order loops and can be taken separately, Whitehouse [20] pp. 257) in Table 3.

Table 3

Second order loops
Value

No. 2 and No.

No. 3 and No.

No. 4 and No.

No. and No.

Thus, by applying Mason’s rule we obtain the following generating function of the waiting time for the occurrence of the pattern as follows:

which yields (4).

Similarly, for the nonoverlapping occurrence of the pattern (Theorem 2) once when state 0¹ is reached, that is, the first occurrence of required pattern, if the next contiguous trial results in success then state is directly reached with conditional probability otherwise state with conditional probability and continues to move in that state with conditional probability until a success occurs at which it moves to state with conditional probability . Thus, there are two paths to reach node from node 0¹. Now, on reaching node again the same procedure is followed for the second occurrence of the required pattern, represented by node 0². Thus, for the second occurrence of the pattern node 0¹ acts as a starting node following a similar procedure as for the first occurrence.

It can be observed from the GERT network that there are paths (in general for any there are paths) for reaching state 0² () from the starting node . Thus, proceeding as in Theorem 1 and applying Mason’s rule we obtain the following generating function of the waiting time for the 2nd nonoverlapping occurrence of the required pattern as follows:

Proceeding similarly as above, we can obtain generating function as given by (7).

For th overlapping occurrence of the pattern (Theorem 3), proceeding as in Theorem 1, once when state 0¹ is reached, that is, the first occurrence of the required pattern, if the next contiguous trial results in failure then state 1 is directly reached with conditional probability otherwise it continues to move in state 0¹ with conditional probability until a failed trial occurs resulting in occurrence of state 1 with conditional probability . Now, on reaching node 1 if the next contiguous trial results in failure then state 2 occurs with conditional probability otherwise state 0¹ with conditional probability . Again similar procedure is followed for the second occurrence of the required pattern. Thus, for the second occurrence, node 0¹ acts as a starting node following a similar procedure as for the first occurrence (except that there is only one path to reach node 1 from node 0¹).

It can be observed that in the GERT network for 2nd overlapping occurrence of the required pattern (Figure 4), there are (in general for any , there are ) paths for reaching state 0² from the starting node .

Now pairs (, are identical. Thus, proceeding as in the Theorem 1 and by applying Mason’s rule the generating function of the waiting time for the 2nd overlapping occurrence of the required pattern is given by

Similarly, proceeding as above we can obtain generating function as given by (9).

Acknowledgments

The authors would like to express their sincere thanks to the editor and referees for their valuable suggestions and comments which were quite helpful to improve the paper.

References

J. C. Fu and M. V. Koutras, “Distribution theory of runs: a Markov chain approach,” Journal of the American Statistical Association, vol. 89, no. 427, pp. 1050–1058, 1994.

S. Aki, N. Balakrlshnan, and S. G. Mohanty, “Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials,” Annals of the Institute of Statistical Mathematics, vol. 48, no. 4, pp. 773–787, 1996.

M. V. Koutras, “Waiting time distributions associated with runs of fixed length in two-state Markov chains,” Annals of the Institute of Statistical Mathematics, vol. 49, no. 1, pp. 123–139, 1997.

D. L. Antzoulakos, “On waiting time problems associated with runs in Markov dependent trials,” Annals of the Institute of Statistical Mathematics, vol. 51, no. 2, pp. 323–330, 1999.

S. Aki and K. Hirano, “Sooner and later waiting time problems for runs in markov dependent bivariate trials,” Annals of the Institute of Statistical Mathematics, vol. 51, no. 1, pp. 17–29, 1999.

Q. Han and K. Hirano, “Sooner and later waiting time problems for patterns in Markov dependent trials,” Journal of Applied Probability, vol. 40, no. 1, pp. 73–86, 2003.

J. C. Fu and W. Y. W. Lou, “Waiting time distributions of simple and compound patterns in a sequence of rth order markov dependent multi-state trials,” Annals of the Institute of Statistical Mathematics, vol. 58, no. 2, pp. 291–310, 2006.

A. P. Godbole and S. G. Papastavridis, Runs and Patterns in Probability: Selected Papers, Kluwer Academic Publishers, Amsterdam, The Netherlands, 1994.

N. Balakrishnan and M. V. Koutras, Runs and Scans With Applications, John Wiley and Sons, New York, NY, USA, 2002.

J. C. Fu and W. Y. Lou, Distribution Theory of Runs and Patterns and its Applications: A Finite Markov Chain Imbedding Approach, World Scientific Publishing Co, River Edge, NJ, USA, 2003.

W. Feller, An Introduction to Probability Theory and its Applications, vol. 1, John Wiley and Sons, New York, NY, USA, 3rd edition, 1968.

J. C. Fu and Y. M. Chang, “On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials,” Journal of Applied Probability, vol. 39, no. 1, pp. 70–80, 2002.

G. Ge and L. Wang, “Exact reliability formula for consecutive-k-out-of-n:F systems with homogeneous Markov dependence,” IEEE Transactions on Reliability, vol. 39, no. 5, pp. 600–602, 1990.

C. H. Cheng, “Fuzzy consecutive-k-out-of-n:F system reliability,” Microelectronics Reliability, vol. 34, no. 12, pp. 1909–1922, 1994.

M. Agarwal, K. Sen, and P. Mohan, “GERT analysis of m-consecutive-k-out-of-n systems,” IEEE Transactions on Reliability, vol. 56, no. 1, pp. 26–34, 2007.

M. Agarwal, P. Mohan, and K. Sen, “GERT analysis of m-consecutive-k-out-of-n: F System with dependence,” International Journal for Quality and Reliability (EQC), vol. 22, pp. 141–157, 2007.

M. Agarwal and P. Mohan, “GERT analysis of m-consecutive-k-out-of-n:F system with overlapping runs and ()-step Markov dependence,” International Journal of Operational Research, vol. 3, no. 1-2, pp. 36–51, 2008.

P. Mohan, M. Agarwal, and K. Sen, “Combined m-consecutive-k-out-of-n: F & consecutive kc-out-of-n: F systems,” IEEE Transactions on Reliability, vol. 58, no. 2, pp. 328–337, 2009.

P. Mohan, K. Sen, and M. Agarwal, “Waiting time distributions of patterns involving homogenous Markov dependent trials: GERT approach,” in Proceedings of the 7th International Conference on Mathematical Methods in Reliability: Theory, Methods, Applications (MMR '11), L. Cui and X. Zhao, Eds., pp. 935–941, Beijing, China, June 2011.

G. E. Whitehouse, Systems Analysis and Design Using Network Techniques, Prentice-Hall, Englewood Cliffs, NJ, USA, 1973.