Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$X_1 ,X_2 ,\ldots ,X_n $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> </mrow> </math> </EquationSource> </InlineEquation> be a sequence of Markov Bernoulli trials (MBT) and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\underline{X}_n =( {X_{n,k_1 } ,X_{n,k_2 } ,\ldots ,X_{n,k_r } })$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <munder> <mi>X</mi> <mo>̲</mo> </munder> <mi>n</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mrow> <msub> <mi>X</mi> <mrow> <mi>n</mi> <mo>,</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow> <mi>n</mi> <mo>,</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow> <mi>n</mi> <mo>,</mo> <msub> <mi>k</mi> <mi>r</mi> </msub> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> be a random vector where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$X_{n,k_i } $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>X</mi> <mrow> <mi>n</mi> <mo>,</mo> <msub> <mi>k</mi> <mi>i</mi> </msub> </mrow> </msub> </math> </EquationSource> </InlineEquation> represents the number of occurrences of success runs of length <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$k_i \,( {i=1,2,\ldots ,r})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>k</mi> <mi>i</mi> </msub> <mspace width="0.166667em"/> <mrow> <mo stretchy="false">(</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation>. In this paper the joint distribution of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\underline{X}_n $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <munder> <mi>X</mi> <mo>̲</mo> </munder> <mi>n</mi> </msub> </math> </EquationSource> </InlineEquation> in the sequence of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>n</mi> </math> </EquationSource> </InlineEquation> MBT is studied using method of conditional probability generating functions. Five different counting schemes of runs namely non-overlapping runs, runs of length at least <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$k$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>k</mi> </math> </EquationSource> </InlineEquation>, overlapping runs, runs of exact length <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$k$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>k</mi> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$\ell $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>ℓ</mi> </math> </EquationSource> </InlineEquation>-overlapping runs (i.e. <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$\ell $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>ℓ</mi> </math> </EquationSource> </InlineEquation>-overlapping counting scheme), <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$$0\le \ell >k$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>0</mn> <mo>≤</mo> <mi>ℓ</mi> <mo>></mo> <mi>k</mi> </mrow> </math> </EquationSource> </InlineEquation> are considered. The pgf of joint distribution of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$$\underline{X}_n $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <munder> <mi>X</mi> <mo>̲</mo> </munder> <mi>n</mi> </msub> </math> </EquationSource> </InlineEquation> is obtained in terms of matrix polynomial and an algorithm is developed to get exact probability distribution. Numerical results are included to demonstrate the computational flexibility of the developed results. Various applications of the joint distribution of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$$\underline{X}_n $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <munder> <mi>X</mi> <mo>̲</mo> </munder> <mi>n</mi> </msub> </math> </EquationSource> </InlineEquation> such as in evaluation of the reliability of <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$$( {n,f,k})\!\!:\!\!G$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <mrow> <mi>n</mi> <mo>,</mo> <mi>f</mi> <mo>,</mo> <mi>k</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="-0.166667em"/> <mspace width="-0.166667em"/> <mo>:</mo> <mspace width="-0.166667em"/> <mspace width="-0.166667em"/> <mi>G</mi> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$$>n,f,k>\!:\!\!G$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo>></mo> <mi>n</mi> <mo>,</mo> <mi>f</mi> <mo>,</mo> <mi>k</mi> <mo>></mo> <mspace width="-0.166667em"/> <mo>:</mo> <mspace width="-0.166667em"/> <mspace width="-0.166667em"/> <mi>G</mi> </mrow> </math> </EquationSource> </InlineEquation> system, in evaluation of quantities related to start-up demonstration tests, acceptance sampling plans are also discussed. Copyright Springer-Verlag Berlin Heidelberg 2014
