Distributions of Patterns of Pair of Successes Separated by Failure Runs of Length at Least 𝑘_𝟏 and at Most 𝑘_𝟐 Involving Markov Dependent Trials: GERT Approach

Abstract

We use the Graphical Evaluation and Review Technique (GERT) to obtain probability generating functions of the waiting time distributions of 1st, and 𝑚th nonoverlapping and overlapping occurrences of the pattern Λ^𝑘1,𝑘2_𝑓=𝑆𝐹𝐹⋯𝐹|{z}_{𝑘1≤𝑘𝑓≤𝑘2}𝑆(𝑘₁>0), 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 𝐴=𝑆𝑆⋯𝑆|{z}_𝑘 of size 𝑘 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-𝑘-out-of-𝑛: 𝐹 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-𝑘-out-of-𝑛: 𝐹 system using GERT. Agarwal et al. [15, 16], Agarwal & Mohan [17] and Mohan et al. [18] have also studied reliability of 𝑚-consecutive-𝑘-out-of-𝑛: 𝐹 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 𝑚th nonoverlapping and overlapping occurrences of the pattern Λ^𝑘=𝑆𝐹𝐹⋯𝐹|{z}_𝑘𝑆, involving Markov dependent trials, using GERT. In this paper, probability generating functions of the waiting time distributions of 1st, and 𝑚th nonoverlapping and overlapping occurrences of a pattern involving pair of successes separated by a run of failures of length at least 𝑘₁ and at most 𝑘₂, Λ^𝑘1,𝑘2_𝑓=𝑆𝐹𝐹⋯𝐹|{z}_{𝑘1≤𝑘𝑓≤𝑘2}𝑆(𝑘₁>0), involving Homogenous Markov Dependence, that is, probability that component 𝑖 fails depends only upon the state of component (𝑖−1) 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

  Λ^𝑘1,𝑘2_𝑓=𝑆𝐹𝐹⋯𝐹|{z}_{𝑘1≤𝑘𝑓≤𝑘2}𝑆: pair of successes separated by a run of failures of length at least 𝑘₁ and at most 𝑘₂, 𝑘₁>0. 𝑋_𝑖: indicator random variable for state of trial 𝑖, 𝑋_𝑖=0 or 1 according as trial 𝑖 is success or failure. 𝑝₀,𝑞₀: Pr{𝑋₁=0}; 𝑞₀=1−𝑝₀ 𝑝₁,𝑞₁: Pr{𝑋_𝑖=0∣𝑋_𝑖−1=0}, probability that trial 𝑖 is a success given that preceding trial (𝑖−1) is also a success, for 𝑖=2,3,…;𝑞₁=1−𝑝₁. 𝑝₂,𝑞₂: Pr{𝑋_𝑖=0∣𝑋_𝑖−1=1}, probability that trial 𝑖 is a success given that preceding trial (𝑖−1) is a failure, for 𝑖=2,3,…;𝑞₂=1−𝑝₂.

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 𝑊-function. It is used to obtain the information of a relationship, which exists between the nodes.

If we define 𝑊(𝑠∣𝑟), as the conditional 𝑊 function associated with a network when the branches tagged with a 𝑧 are taken 𝑟 times, then the equivalent 𝑊 generating function can be written as follows: 𝑊(𝑠,𝑧)=^∞_𝑟=0𝑊(𝑠∣𝑟)𝑧^𝑟. (1) The function 𝑊(0,𝑧) 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 𝑇 as follows: ∑𝑇=path∗∑nontouchingloops∑loops, (2) where loops=1−firstorderloops+secondorderloops−⋯nontouchingloops=1−firstordernontouchingloops+secondordernontouchingloops−thirdordernontouchingloops+⋯ (3)

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 Λ^𝑘1,𝑘2_𝑓

Theorem 1. 𝑊_𝑆𝐹𝐹⋯𝐹|{z}_{𝑘1𝑓2≤𝑘≤𝑘}^𝑆(0,𝑧), the probability generating function for the waiting time distribution of the 1st occurrence of the pattern Λ^𝑘1,𝑘2_𝑓 involving Homogenous Markov Dependence is given by 𝑊_𝑆𝐹𝐹⋯𝐹|{z}_{𝑘1𝑓2≤𝑘≤𝑘}^𝑆()=𝑞0,𝑧₀𝑝₂𝑧²+𝑝₀𝑧1−𝑞₂𝑧×𝑞₁𝑝₂𝑞₂^𝑘1−1𝑧^𝑘1+11−𝑞₂𝑧^{𝑘2−𝑘1+1}×1−𝑞₂𝑧1−𝑞₂𝑧−𝑝₁𝑧−𝑞₁𝑝₂𝑧²+𝑞₂𝑝₁𝑧²−𝑞₁𝑝₂𝑞₂^𝑘2𝑧^𝑘2+2+𝑞₁𝑝₂𝑞₂^𝑘1−1𝑧^𝑘1+1−1. (4) Then, 𝐸𝑊Λ^𝑘1,𝑘2_𝑓=𝐸𝑚𝑖𝑛𝑖𝑚𝑢𝑚𝑛𝑢𝑚𝑏𝑒𝑟𝑜𝑓𝑡𝑟𝑖𝑎𝑙𝑠𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑𝑡𝑜𝑜𝑏𝑡𝑎𝑖𝑛𝑡ℎ𝑒𝑝𝑎𝑡𝑡𝑒𝑟𝑛Λ^𝑘1,𝑘2_𝑓=𝑞₂𝑞₁+𝑝₂+𝑞₁𝑞^𝑘1₂𝑞₀+𝑝₂1−𝑞^{𝑘2−𝑘1}₂⁺¹𝑞₁𝑝₂𝑞^𝑘1₂1−𝑞^{𝑘2−𝑘1}₂⁺¹,Var𝑊Λ^𝑘1,𝑘2_𝑓=𝑑²W_S𝐹𝐹⋯𝐹|{z}_{𝑘1𝑓2≤𝑘≤𝑘}^𝑆(0,𝑧)𝑑𝑧²_𝑧=1+𝐸𝑊Λ^𝑘1,𝑘2_𝑓−𝐸𝑊Λ^𝑘1,𝑘2_𝑓². (5)(6)

The GERT network for this pattern is represented by Figure 2 where each node represents a specific state as described below: 𝑆: initial node 1^𝐴: a trial resulting in failure 0_𝑖: a trial resulting in success corresponding to beginning of the 𝑖th occurrence of the pattern Λ^𝑘1,𝑘2_𝑓, 𝑖=1,2,…𝑚 1: a component is in failed state preceded by working component 𝑗: 𝑗th contiguous failed trial preceded by a success trial, 𝑗=2,…,𝑘₁−1,𝑘₁,𝑘₁+1,…,𝑘₂,𝑘₂+1 0^𝑖: 𝑖th occurrence of the required pattern, 𝑖=1,2,…,𝑚.

Figure 2 

GERT network representing 1st occurrence of the pattern Λ𝑘1,𝑘2𝑓=𝑆𝐹𝐹⋯𝐹|{z}𝑘1≤𝑘𝑓≤𝑘2𝑆.

There are in all 𝑘₂+5 nodes designated as 𝑆,1^𝐴,0₁,1,2,…,𝑘₁−1,𝑘₁,𝑘₁+1,…,𝑘₂−1,𝑘₂,𝑘₂+1 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 1^𝐴 is reached from 𝑆 with conditional probability 𝑞₀ otherwise state 0₁ with conditional probability 𝑝₀. Further, if the preceding trial is failure (state 1^𝐴) followed by a contiguous success trial then state 0₁ is reached from state 1^𝐴 with conditional probability 𝑝₂ otherwise it continues to move in state 1^𝐴 with conditional probability 𝑞₂. State 0₁ 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 0₁ with conditional probability 𝑝₁ otherwise it moves in state 1 with conditional probability 𝑞₁. Again if the next contiguous trial results in failure then system state 2 occurs with conditional probability 𝑞₂ otherwise state 0₁ with conditional probability 𝑝₂. Similar procedure is followed till 𝑘₁ contiguous failed trials occur preceded by a success trial. Now, if the next contiguous trial is failure then state 𝑘₁+1 is reached with conditional probability 𝑞₂ otherwise state 0¹, first occurrence of required pattern. However, if the system is in state 𝑘₁+1 and next contiguous trial is failure system moves to state 𝑘₁+2 with conditional probability 𝑞₂. Again a similar procedure is followed till node 𝑘₂. However, if there occur 𝑘₂−𝑘₁+1 contiguous failed trials after state 𝑘₁ then the system moves to node 𝑘₂+1 and continues to move in that state until a success trial occurs at which it moves to state 0₁ 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. 𝑊𝐹_𝑆^𝑛𝑜𝐹⋯𝐹|{z}_{𝑘1𝑓2≤𝑘≤𝑘}^𝑆(0,𝑧)_𝑚, the probability generating function for the waiting time distribution of the 𝑚th nonoverlapping occurrence of the pattern Λ^𝑘1,𝑘2_𝑓 involving Homogenous Markov Dependence is given by 𝑊𝐹_𝑆^𝑛𝑜𝐹⋯𝐹|{z}_{𝑘1𝑓2≤𝑘≤𝑘}^𝑆(0,𝑧)_𝑚=𝑞₀𝑝₂𝑧²+𝑝₀𝑧1−𝑞₂𝑧×𝑝₂𝑞₁𝑞₂^𝑘1−1𝑧^𝑘1+11−𝑞₂𝑧^{𝑘2−𝑘1+1𝑚}×𝑝₁𝑧1−𝑞₂𝑧+𝑞₁𝑝₂𝑧^2𝑚−11−𝑞₂𝑧^−𝑚×1−𝑞₂𝑧−𝑝₁𝑧+𝑞₂𝑝₁𝑧²−𝑞₁𝑝₂𝑧²+𝑞₁𝑝₂𝑞^𝑘1₂⁻¹𝑧^𝑘1+1−𝑞₁𝑝₂𝑞₂^𝑘2𝑧^{𝑘2+2−𝑚}. (7) Then, 𝐸𝑊^𝑛𝑜Λ^𝑘1,𝑘2_𝑓=𝐸𝑚𝑖𝑛𝑖𝑚𝑢𝑚𝑛𝑢𝑚𝑏𝑒𝑟𝑜𝑓𝑡𝑟𝑖𝑎𝑙𝑠𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑𝑡𝑜𝑜𝑏𝑡𝑎𝑖𝑛𝑡ℎ𝑒𝑚𝑡ℎ𝑛𝑜𝑛𝑜𝑣𝑒𝑟𝑙𝑎𝑝𝑝𝑖𝑛𝑔𝑜𝑐𝑐𝑢𝑟𝑟𝑒𝑛𝑐𝑒𝑜𝑓𝑝𝑎𝑡𝑡𝑒𝑟𝑛Λ^𝑘1,𝑘2_𝑓=𝑞₁𝑞^𝑘1₂1−𝑞^{𝑘2−𝑘1}₂⁺¹×𝑝₁−𝑝₀+𝑚𝑞₁+𝑝₂+𝑚𝑞₂𝑞₁+𝑝₂×𝑞₁𝑝₂𝑞^𝑘1₂1−𝑞^{𝑘2−𝑘1}₂⁺¹⁻¹, (8) and Var[𝑊^𝑛𝑜(Λ^𝑘1,𝑘2_𝑓)] can be obtained by applying (6).

The GERT network for 𝑚=2 (say) nonoverlapping occurrence of the pattern Λ^𝑘1,𝑘2_𝑓 is represented by Figure 3. Each node represents a specific state as described earlier in Figure 2.

Figure 3 

GERT network representing the 𝑚th (𝑚=2, here) nonoverlapping occurrence of the pattern Λ𝑘1,𝑘2𝑓=𝑆𝐹𝐹⋯𝐹|{z}𝑘1≤𝑘𝑓≤𝑘2𝑆.

Theorem 3. 𝑊^𝑜_𝑆𝐹𝐹⋯𝐹|{z}_{𝑘1𝑓2≤𝑘≤𝑘}^𝑆(0,𝑧)_𝑚, the probability generating function for the waiting time distribution of the mth overlapping occurrence of the pattern Λ^𝑘1,𝑘2_𝑓 involving Homogenous Markov Dependence is given by 𝑊^𝑜_𝑆𝐹𝐹⋯𝐹|{z}_{𝑘1𝑓2≤𝑘≤𝑘}^𝑆(0,𝑧)_𝑚=𝑞₀𝑝₂𝑧²+𝑝₀𝑧1−𝑞₂𝑧×𝑞₁𝑝₂𝑞^𝑘1₂⁻¹𝑧^𝑘1+11−𝑞₂𝑧^{𝑘2−𝑘1+1𝑚}×1−𝑞₂𝑧1−𝑞₂𝑧−𝑝₁𝑧−𝑞₁𝑝₂𝑧²+𝑞₂𝑝₁𝑧²−𝑞₁𝑝₂𝑞^𝑘2₂𝑧^𝑘2+2+𝑞₁𝑝₂𝑞₂^𝑘1−1𝑧^{𝑘1+1−𝑚}. (9) Then, 𝐸𝑊^𝑜Λ^𝑘1,𝑘2_𝑓=𝐸𝑚𝑖𝑛𝑖𝑚𝑢𝑚𝑛𝑢𝑚𝑏𝑒𝑟𝑜𝑓𝑡𝑟𝑖𝑎𝑙𝑠𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑𝑡𝑜𝑜𝑏𝑡𝑎𝑖𝑛𝑡ℎ𝑒𝑚𝑡ℎ𝑜𝑣𝑒𝑟𝑙𝑎𝑝𝑝𝑖𝑛𝑔𝑜𝑐𝑐𝑢𝑟𝑟𝑒𝑛𝑐𝑒𝑜𝑓𝑝𝑎𝑡𝑡𝑒𝑟𝑛Λ^𝑘1,𝑘2_𝑓=𝑞₁𝑞^𝑘1₂1−𝑞^{𝑘2−𝑘1}₂⁺¹𝑞₀+𝑝₂+𝑚𝑞₂𝑞₁+𝑝₂𝑞₁𝑝₂𝑞^𝑘1₂1−𝑞^{𝑘2−𝑘1}₂⁺¹. (10) and Var[𝑊^𝑜(Λ^𝑘1,𝑘2_𝑓)] can be obtained by applying  (6).

The GERT network for 𝑚th (𝑚=2, say), overlapping occurrence of the pattern Λ^𝑘1,𝑘2_𝑓 is represented by Figure 4. Each node represents a specific state as described earlier in Figure 2.

Figure 4 

GERT network representing the 𝑚th (𝑚=2, here) overlapping occurrence of the pattern Λ𝑘1,𝑘2𝑓=𝑆𝐹𝐹⋯𝐹|{z}𝑘1≤𝑘𝑓≤𝑘2𝑆.

Particular Cases. (i) For 𝑘₁=𝑘_𝑓=𝑘₂=𝑘−2, 𝑝₀=𝑝₁=𝑝₂=𝑝, and 𝑞₀=𝑞₁=𝑞₂=𝑞, in the pattern Λ^𝑘1,𝑘2_𝑓, that is, for a run of at most 𝑘−2 failures bounded by successes, the probability generating function becomes 𝑊_𝑆𝐹𝐹⋯𝐹|{z}_𝑘−2^𝑆()=0,𝑧𝑝𝑧²1−𝑞𝑧^𝑘−11−𝑞𝑧1−𝑧+𝑝𝑧1−𝑞𝑧^𝑘−1, (11) verifying the results of Koutras [3, Theorem 3.2].

(ii) If 𝑘₁=1, 𝑘_𝑓=𝑘₂=𝑘, that is, pair of successes are separated by a run of failures of length at least one and at most 𝑘, that is, 𝑆𝐹𝐹⋯𝐹|{z}_1≤𝑘𝑆 then (4), (7), and (9), respectively, become 𝑊_𝑆𝐹𝐹⋯𝐹|{z}_1≤𝑘^𝑆()=𝑞0,𝑧₀𝑝₂𝑧²+𝑝₀𝑧1−𝑞₂𝑧𝑝₂𝑞₁𝑧²1−𝑞₂𝑧^𝑘1−𝑞₂𝑧1−𝑞₂𝑧−𝑝₁𝑧+𝑞₂𝑝₁𝑧²−𝑞₁𝑝₂𝑞₂^𝑘𝑧^𝑘+2.𝑊𝐹_𝑆^𝑛𝑜𝐹⋯𝐹|{z}_1≤𝑘^𝑆=𝑞₀𝑝₂𝑧²+𝑝₀𝑧1−𝑞₂𝑧𝑞₁𝑝₂𝑧²+𝑝₁𝑧1−𝑞₂𝑧^𝑚−1×𝑞₁𝑝₂𝑧²1−𝑞₂𝑧^𝑘𝑚1−𝑞₂𝑧^−𝑚×1−𝑝₁𝑧−𝑞₂𝑧+𝑝₁𝑞₂𝑧²−𝑝₂𝑞₁𝑞^𝑘₂𝑧^{𝑘+2−𝑚}.𝑊^𝑜_𝑆𝐹𝐹⋯𝐹|{z}_1≤𝑘^𝑆()=𝑞0,𝑧₀𝑝₂𝑧²+𝑝₀𝑧1−𝑞₂𝑧𝑝₂𝑞₁𝑧²1−𝑞₂𝑧^𝑘𝑚1−𝑞₂𝑧1−𝑞₂𝑧−𝑝₁𝑧+𝑞₂𝑝₁𝑧²−𝑝₂𝑞₁𝑞₂^𝑘𝑧^𝑘+2𝑚. (12)

(iii) If 𝑘₁=𝑘 and 𝑘₂→∞, then it reduces to the pattern Λ^≥𝑘=𝑆𝐹𝐹⋯𝐹|{z}_≥𝑘𝑆, that is, 𝑊_𝑆𝐹𝐹⋯𝐹|{z}_≥𝑘^𝑆()=𝑞0,𝑧₀𝑝₂𝑧²+𝑝₀𝑧1−𝑞₂𝑧𝑝₂𝑞₁𝑞₂^𝑘−1𝑧^𝑘+1×1−𝑞₂𝑧1−𝑞₂𝑧−𝑝₁𝑧+𝑞₂𝑝₁𝑧²−𝑞₁𝑝₂𝑧²1−𝑞₂𝑧^𝑘−1−1. (13)

(iv) If 𝑘₁=𝑘_𝑓=𝑘₂=𝑘,

then (4), (7), (9) reduce to the results of Mohan et al. [19], that is, 𝑊_𝑆𝐹𝐹⋯𝐹|{z}_𝑘^𝑆()=𝑞0,𝑧₀𝑝₂𝑧²+𝑝₀𝑧1−𝑞₂𝑧𝑞₁𝑝₂𝑞₂^𝑘−1𝑧^𝑘+1×1−𝑞₂𝑧−𝑝₁𝑧−𝑝₂𝑞₁𝑧²+𝑝₁𝑞₂𝑧²−𝑞₁𝑝₂𝑞^𝑘₂𝑧^𝑘+2+𝑞₁𝑝₂𝑞₂^𝑘−1𝑧^𝑘+1−1,𝑊𝐹_𝑆^𝑛𝑜𝐹⋯𝐹|{z}_𝑘^𝑆(0,𝑧)_𝑚=𝑞₀𝑝₂𝑧²+𝑝₀𝑧1−𝑞₂𝑧𝑝₂𝑞₁𝑞₂^𝑘−1𝑧^𝑘+1𝑚×𝑝₁𝑧1−𝑞₂𝑧+𝑝₂𝑞₁𝑧^2𝑚−1×1−𝑞₂𝑧−𝑝₁𝑧−𝑞₁𝑝₂𝑧²+𝑞₂𝑝₁𝑧²−𝑞₁𝑝₂𝑞^𝑘₂𝑧^𝑘+2+𝑞₁𝑝₂𝑞₂^𝑘−1𝑧^{𝑘+1−𝑚},𝑊^𝑜_𝑆𝐹𝐹⋯𝐹|{z}_𝑘^𝑆(0,𝑧)_𝑚=𝑞₀𝑝₂𝑧²+𝑝₀𝑧1−𝑞₂𝑧𝑝₂𝑞₁𝑞₂^𝑘−1𝑧^𝑘+1𝑚1−𝑞₂𝑧^𝑚−1×1−𝑞₂𝑧−𝑝₁𝑧−𝑞₁𝑝₂𝑧²+𝑞₂𝑝₁𝑧²−𝑞₁𝑝₂𝑞^𝑘₂𝑧^𝑘+2+𝑞₁𝑝₂𝑞₂^𝑘−1𝑧^{𝑘+1−𝑚}. (14)

5. Conclusion

In this paper we proposed Graphical Evaluation and Review Technique (GERT) to study probability generating functions of the waiting time distributions of 1st, and 𝑚th nonoverlapping and overlapping occurrences of the pattern Λ^𝑘1,𝑘2_𝑓=𝑆𝐹𝐹⋯𝐹|{z}_{𝑘1≤𝑘𝑓≤𝑘2}𝑆(𝑘₁>0), 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 2(𝑘₂−𝑘₁+1) paths to reach state 0¹ from the starting node S, see Table 1.

Table 1 

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 

However, paths at Nos. 𝑘₂−𝑘₁+2, 𝑘₂−𝑘₁+3, ⋯, 2(𝑘₂−𝑘₁+1) also contain first order nontouching loop (1^𝐴 to 1^𝐴), whose value is given by 𝑞₂𝑧. Further, first order loop No. 𝑘₁+3 forms first order nontouching loop to both paths, whose value is given by 𝑞₂𝑧.

Also second order loops corresponding to first order loops from No. 2 to No. 𝑘₁+3 are given (since first order loop no. 1, that is, 1^𝐴 to 1^𝐴 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 

Thus, by applying Mason’s rule we obtain the following generating function 𝑊_𝑆𝐹𝐹⋯𝐹|{z}_{𝑘1𝑓2≤𝑘≤𝑘}^𝑆(0,𝑧) of the waiting time for the occurrence of the pattern Λ^𝑘1,𝑘2_𝑓 as follows: 𝑊_𝑆𝐹𝐹⋯𝐹|{z}_{𝑘1𝑓2≤𝑘≤𝑘}^𝑆()=𝑞0,𝑧₀𝑝₂𝑧²+𝑝₀𝑧1−𝑞₂𝑧1−𝑞₂𝑧×𝑞₁𝑝₂𝑞^𝑘1₂⁻¹z^𝑘1+11+𝑞₂𝑧+𝑞₂𝑧²+⋯+𝑞₂𝑧^{𝑘2−𝑘1}×1−𝑞₂𝑧1−𝑞₂𝑧−𝑝₁𝑧+𝑞₂𝑝₁𝑧²−𝑞₁𝑝₂𝑧²+𝑞₁𝑝₂𝑞^𝑘1₂⁻¹𝑧^𝑘1+1−𝑞₁𝑝₂𝑞^𝑘2₂𝑧^𝑘2+2−1, (A.1) which yields (4).

Similarly, for the 𝑚th 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 0₂ is directly reached with conditional probability 𝑝₁ otherwise state 1^𝐴 with conditional probability 𝑞₁ and continues to move in that state with conditional probability 𝑞₂ until a success occurs at which it moves to state 0₂ with conditional probability 𝑝₂. Thus, there are two paths to reach node 0₂ from node 0¹. Now, on reaching node 0₂ 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 4⋅(𝑘₂−𝑘₁+1)² paths (in general for any 𝑚 there are 2⋅2^𝑚−1(𝑘₂−𝑘₁+1)^𝑚 paths) for reaching state 0² (0^𝑚) from the starting node 𝑆. Thus, proceeding as in Theorem 1 and applying Mason’s rule we obtain the following generating function 𝑊𝐹_𝑆^𝑛𝑜𝐹⋯𝐹|{z}_{𝑘1𝑓2≤𝑘≤𝑘}^𝑆(0,𝑧)₂ of the waiting time for the 2nd nonoverlapping occurrence of the required pattern Λ^𝑘1,𝑘2_𝑓 as follows: 𝑊𝐹_𝑆^𝑛𝑜𝐹⋯𝐹|{z}_{𝑘1𝑓2≤𝑘≤𝑘}^𝑆(0,𝑧)₂=𝑞₀𝑝₂𝑧²+𝑝₀𝑧1−𝑞₂𝑧𝑝₁𝑧1−𝑞₂𝑧+𝑞₁𝑝₂𝑧²×𝑝₂𝑞₁𝑞^𝑘1₂⁻¹𝑧^𝑘1+11−𝑞₂𝑧^{𝑘2−𝑘1+12}×1−𝑞₂𝑧⁻²1−𝑞₂𝑧−𝑝₁𝑧+𝑞₂𝑝₁𝑧²−𝑞₁𝑝₂𝑧²+𝑞₁𝑝₂𝑞^𝑘1₂⁻¹𝑧^𝑘1+1−𝑞₁𝑝₂𝑞^𝑘2₂𝑧^𝑘2+2−2. (A.2) Proceeding similarly as above, we can obtain generating function 𝑊𝐹_𝑆^𝑛𝑜𝐹⋯𝐹|{z}_{𝑘1𝑓2≤𝑘≤𝑘}^𝑆(0,𝑧)_𝑚 as given by (7).

For 𝑚th 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 𝑞₁ otherwise it continues to move in state 0¹ with conditional probability 𝑝₁ until a failed trial occurs resulting in occurrence of state 1 with conditional probability 𝑞₁. Now, on reaching node 1 if the next contiguous trial results in failure then state 2 occurs with conditional probability 𝑞₂ otherwise state 0¹ with conditional probability 𝑝₂. 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 2⋅(𝑘₂−𝑘₁+1)² (in general for any 𝑚, there are 2⋅(𝑘₂−𝑘₁+1)^𝑚) paths for reaching state 0²(0^𝑚) from the starting node 𝑆.

Now pairs (0₁,0¹), (0¹,0²) are identical. Thus, proceeding as in the Theorem 1 and by applying Mason’s rule the generating function 𝑊^𝑜_𝑆𝐹𝐹⋯𝐹|{z}_{𝑘1𝑓2≤𝑘≤𝑘}^𝑆(0,𝑧)₂ of the waiting time for the 2nd overlapping occurrence of the required pattern is given by𝑊^𝑜_𝑆𝐹𝐹⋯𝐹|{z}_{𝑘1𝑓2≤𝑘≤𝑘}^𝑆(0,𝑧)₂=𝑞₀𝑝₂𝑧²+𝑝₀𝑧1−𝑞₂𝑧𝑞₁𝑝₂𝑞^𝑘1₂⁻¹𝑧^𝑘1+11−𝑞₂𝑧^{𝑘2−𝑘1+12}1−𝑞₂𝑧1−𝑞₂𝑧−𝑝₁𝑧+𝑞₂𝑝₁𝑧²−𝑞₁𝑝₂𝑧²−𝑞₁𝑝₂𝑞₂^𝑘2𝑧^𝑘2+2+𝑞₁𝑝₂𝑞₂^𝑘1−1𝑧^𝑘1+12. (A.3)Similarly, proceeding as above we can obtain generating function 𝑊^𝑜_𝑆𝐹𝐹⋯𝐹|{z}_{𝑘1𝑓2≤𝑘≤𝑘}^𝑆(0,𝑧)_𝑚 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.