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  • 1
    Online Resource
    Online Resource
    Multiple Decision Procedures Based on Ranks for Certain Problems in Analysis of Variance
    Institute of Mathematical Statistics ; 1969
    In:  The Annals of Mathematical Statistics Vol. 40, No. 2 ( 1969-04), p. 619-632
    Details
    In: The Annals of Mathematical Statistics, Institute of Mathematical Statistics, Vol. 40, No. 2 ( 1969-04), p. 619-632
    Type of Medium: Online Resource
    ISSN: 0003-4851
    URL: Article
    DOI: 10.1214/aoms/1177697730
    Language: English
    Publisher: Institute of Mathematical Statistics
    Publication Date: 1969
    detail.hit.zdb_id: 184679-6
    detail.hit.zdb_id: 2010060-7
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  • 2
    Online Resource
    Online Resource
    On the limit behavior of certain quantities in a subcritical storage model
    Cambridge University Press (CUP) ; 1985
    In:  Advances in Applied Probability Vol. 17, No. 2 ( 1985-06), p. 443-459
    Details
    In: Advances in Applied Probability, Cambridge University Press (CUP), Vol. 17, No. 2 ( 1985-06), p. 443-459
    Abstract: The limit behavior of the content of a subcriticai storage model defined on a semi-Markov process is examined. This is achieved by creating a renewal equation using a regeneration point ( i 0 ,0) of the process. By showing that the expected return time to ( i 0 , 0) is finite, the conditions needed for the basic renewal theorem are established. The joint asymptotic distribution of the content of the storage at time t and the accumulated amount of the unmet (lost) demands during (0, t ) is then established by showing the asymptotic independence of these two.
    Type of Medium: Online Resource
    ISSN: 0001-8678 , 1475-6064
    URL: Article
    DOI: 10.2307/1427150
    RVK:
    SA 1600
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1985
    detail.hit.zdb_id: 1474602-5
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  • 3
    Online Resource
    Online Resource
    On the limit behavior of certain quantities in a subcritical storage model
    Cambridge University Press (CUP) ; 1985
    In:  Advances in Applied Probability Vol. 17, No. 02 ( 1985-06), p. 443-459
    Details
    In: Advances in Applied Probability, Cambridge University Press (CUP), Vol. 17, No. 02 ( 1985-06), p. 443-459
    Abstract: The limit behavior of the content of a subcriticai storage model defined on a semi-Markov process is examined. This is achieved by creating a renewal equation using a regeneration point ( i 0 ,0) of the process. By showing that the expected return time to ( i 0 , 0) is finite, the conditions needed for the basic renewal theorem are established. The joint asymptotic distribution of the content of the storage at time t and the accumulated amount of the unmet (lost) demands during (0, t ) is then established by showing the asymptotic independence of these two.
    Type of Medium: Online Resource
    ISSN: 0001-8678 , 1475-6064
    URL: Article
    DOI: 10.1017/S0001867800015068
    RVK:
    SA 1600
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1985
    detail.hit.zdb_id: 1474602-5
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  • 4
    Online Resource
    Online Resource
    A method for studying the integral functionals of stochastic processes with applications: I. Markov chain case
    Cambridge University Press (CUP) ; 1971
    In:  Journal of Applied Probability Vol. 8, No. 02 ( 1971-06), p. 331-343
    Details
    In: Journal of Applied Probability, Cambridge University Press (CUP), Vol. 8, No. 02 ( 1971-06), p. 331-343
    Abstract: The subject of this paper is the study of the distribution of integrals of the type where { X ( t ); t ≧ 0} is some appropriately defined continuous-time parameter stochastic process, and f is a suitable non-negative function of its arguments. This subject has also sometimes been labelled as “the occupation time or the sojourn time problem” in literature. These integrals arise in several domains of applications such as in the theory of inventories and storage (see Moran [14], Naddor [15] ), in the study of the cost of the flow-stopping incident involved in the automobile traffic jams (see Gaver [8], Daley [3] , Daley and Jacobs [4]). The author encountered such integrals while studying certain stochastic models suitable for the study of response time distributions arising in various live situations. In fact in [19] , it was shown that such a distribution is equivalent to the study of an integral of the type (1). Again, in the study of response of host to injection of virulent bacteria, Y ( t ) with f ( X ( t ), t ) = bX ( t ), with b & gt; 0, could be regarded as a measure of the total amount of toxins produced by the bacteria during (0, t ), assuming a constant toxin-excretion rate per bacterium. Here X ( t ) denotes the number of live bacteria at time t , the growth of which is governed by a birth and death process (see Puri [16], [17] and [18]).
    Type of Medium: Online Resource
    ISSN: 0021-9002 , 1475-6072
    URL: Article
    DOI: 10.1017/S0021900200035348
    RVK:
    SA 6100
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1971
    detail.hit.zdb_id: 1474599-9
    detail.hit.zdb_id: 219147-7
    SSG: 3,2
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  • 5
    Online Resource
    Online Resource
    On a generalized storage model with moment assumptions
    Cambridge University Press (CUP) ; 1981
    In:  Journal of Applied Probability Vol. 18, No. 02 ( 1981-06), p. 473-481
    Details
    In: Journal of Applied Probability, Cambridge University Press (CUP), Vol. 18, No. 02 ( 1981-06), p. 473-481
    Abstract: This paper considers a semi-infinite storage model, of the type studied by Senturia and Puri [13] and Balagopal [2] , defined on a Markov renewal process, {( X n , T n ), n = 0, 1, ·· ·}, with 0 = T 0 & lt; T 1 & lt; · ··, almost surely, where X n takes values in the set {1, 2, ·· ·}. If at T n , X n = j , then there is a random ‘input' V n ( j ) (a negative input implying a demand) of ‘type' j , having distribution function F j (·). We assume that { V n ( j )} is an i.i.d. sequence of random variables, taken to be independent of {( X n , T n )} and of { V n ( k )}, for k ≠ j , and that V n ( j ) has first and second moments. Here the random variables V n ( j ) represent instantaneous ‘inputs' (a negative value implying a demand) of type j for our storage model. Under these assumptions, we establish certain limit distributions for the joint process ( Z ( t ), L ( t )), where Z ( t ) (defined in (2)) is the level of storage at time t and L ( t ) (defined in (3)) is the demand lost due to shortage of supply during [0, t ]. Different limit distributions are obtained for the cases when the ‘average stationary input' ρ , as defined in (5), is positive, zero or negative.
    Type of Medium: Online Resource
    ISSN: 0021-9002 , 1475-6072
    URL: Article
    DOI: 10.1017/S0021900200098120
    RVK:
    SA 6100
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1981
    detail.hit.zdb_id: 1474599-9
    detail.hit.zdb_id: 219147-7
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  • 6
    Online Resource
    Online Resource
    Poisson mixtures and quasi-infinite divisibility of distributions
    Cambridge University Press (CUP) ; 1979
    In:  Journal of Applied Probability Vol. 16, No. 01 ( 1979-03), p. 138-153
    Details
    In: Journal of Applied Probability, Cambridge University Press (CUP), Vol. 16, No. 01 ( 1979-03), p. 138-153
    Abstract: Any probability distribution on [0,∞) can function as the mixing distribution for a Poisson mixture , i.e. a mixture of Poisson distributions. The mixing distribution is called quasi-infinitely divisible ( q.i.d. ) if it renders the Poisson mixture infinitely divisible, or λ -q.i.d. if it does so after scaling by a factor λ & gt; 0, or ∗-q.i.d. if it is λ-q.i.d. for some λ. These classes of distributions include the infinitely divisible distributions, and each exhibits many of the properties of the latter class but in weakened form. The paper presents the main properties of the classes and the class of Poisson mixtures, including characterisations of membership, relation with cumulants, and closure properties. Examples are given that establish among other things strict inclusions between the classes of mixing distributions.
    Type of Medium: Online Resource
    ISSN: 0021-9002 , 1475-6072
    URL: Article
    DOI: 10.1017/S0021900200046271
    RVK:
    SA 6100
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1979
    detail.hit.zdb_id: 1474599-9
    detail.hit.zdb_id: 219147-7
    SSG: 3,2
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  • 7
    Online Resource
    Online Resource
    Interconnected birth and death processes
    Cambridge University Press (CUP) ; 1968
    In:  Journal of Applied Probability Vol. 5, No. 2 ( 1968-08), p. 334-349
    Details
    In: Journal of Applied Probability, Cambridge University Press (CUP), Vol. 5, No. 2 ( 1968-08), p. 334-349
    Abstract: Two cases of multiple linearly interconnected linear birth and death processes are considered. It is found that in general the solution of the Kolmogorov differential equations for the probability generating function (p.g.f) g of the random variables involved is not obtainable by standard methods, although one can obtain moments of the random variables from these equations. A method is considered for obtaining an approximate solution for g. This is based on the introduction of a sequence of stochastic processes such that the sequence { f ( n ) } of their p.g.f.'s tends to g as n → ∞ in an appropriate manner. The method is applied to the simple case of two birth and death processes with birth and death rates λ i and μ i , i = 1,2, interconnected linearly with transition rates v and δ (see Figure 2). For this case some limit theorems are established and the probability of ultimate extinction of both the processes is considered. In addition, for the special cases (i) λ 1 = δ = 0, with the remaining rates time dependent and (ii) λ 2 = δ = 0, with the remaining rates constant, explicit solutions for g are obtained and studied.
    Type of Medium: Online Resource
    ISSN: 0021-9002 , 1475-6072
    URL: Article
    DOI: 10.2307/3212256
    RVK:
    SA 6100
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1968
    detail.hit.zdb_id: 1474599-9
    detail.hit.zdb_id: 219147-7
    SSG: 3,2
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  • 8
    Online Resource
    Online Resource
    Some new results in the mathematical theory of phage-reproduction
    Cambridge University Press (CUP) ; 1969
    In:  Journal of Applied Probability Vol. 6, No. 03 ( 1969-12), p. 493-504
    Details
    In: Journal of Applied Probability, Cambridge University Press (CUP), Vol. 6, No. 03 ( 1969-12), p. 493-504
    Abstract: In the theory of phage reproduction, the mathematical models considered thus far (see Gani [5]) assume that the bacterial burst occurs a fixed time after infection, after a fixed number of generations of phage multiplication, or when the number of mature bacteriophages has reached a fixed threshold. In the present paper, a more realistic assumption is considered: given that until time t the bacterial burst has not taken place, its occurence between t and t + Δ t is a random event with probability f (· | t )Δ t + o (Δ t ), where f is a non-negative and non-decreasing function of the number X ( t ) of vegetative phages and of Z ( t ), the number of mature bacteriophages at time t. More specifically it is assumed that f = b ( t ) X ( t ) + c ( t ) Z ( t ) with b ( t ), c ( t ) ≦ 0. Here X ( t ) denotes the survivors in a linear birth and death process and Z ( t ) the number of deaths until time t. The joint distribution of X T and Z T , the respective numbers of vegetative and mature bacteriophages at the burst time is considered. The distribution of Z T is then fitted to some observed data of Delbrück [2].
    Type of Medium: Online Resource
    ISSN: 0021-9002 , 1475-6072
    URL: Article
    DOI: 10.1017/S0021900200026565
    RVK:
    SA 6100
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1969
    detail.hit.zdb_id: 1474599-9
    detail.hit.zdb_id: 219147-7
    SSG: 3,2
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  • 9
    Online Resource
    Online Resource
    Some further results on the birth-and-death process and its integral
    Cambridge University Press (CUP) ; 1968
    In:  Mathematical Proceedings of the Cambridge Philosophical Society Vol. 64, No. 1 ( 1968-01), p. 141-154
    Details
    In: Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press (CUP), Vol. 64, No. 1 ( 1968-01), p. 141-154
    Abstract: In a simple homogeneous birth-and-death process with λ and μ as the constant birth and death rates respectively, let X ( t ) denote the population size at time t , Z ( t ) the number of deaths and N ( t ) the number of events (births and deaths combined) occurring during (0, t ). Also let . The results obtained include the following: ( a ) An explicit formula for the characteristic quasi-probability generating function of the joint distribution of X ( t ), Y ( t ) and Z ( t ). ( b ) Let X (0) = 1. It is shown that, if t → ∞ while λ ≤ μ, N ( t ) ↑ N a.s., where N takes only positive odd integral values. If λ 〉 μ, then P [ N ( t ) ↑ ∞] = 1 − μ/λ. Given that N ( t ) ∞, the limiting distribution of N ( t ) is similar to that of N . It was reported earlier (Puri (11)), that the limiting distribution of Y ( t ) is a weighted average of certain chi-square distributions. It is now found that these weights are nothing but the probabilities P [ N = 2 k + 1] ( k = 0, 1,…). ( c ) Let λ = μ, and M X ω), M Y ω and M Z ω be defined as in (36), then as where the c.f. of ( X * ; Y * ; Z * ) is given by (38). ( d ) Exact expressions for the p.d.f. of Y ( t ) are derived for the cases (i) λ = 0, μ 〉 0, (ii) λ 〉 0, μ = 0. For the case (iii) λ gt; 0, μ 〉 0, since the complete expression is complicated, only the procedure of derivation is indicated. ( e ) Finally, it is shown that the regressions of Y ( t ) and of Z ( t ) on X ( t ) are linear for X ( t ) ≥ 1.
    Type of Medium: Online Resource
    ISSN: 0305-0041 , 1469-8064
    URL: Article
    DOI: 10.1017/S0305004100042651
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1968
    detail.hit.zdb_id: 1483586-1
    detail.hit.zdb_id: 209201-3
    SSG: 17,1
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  • 10
    Online Resource
    Online Resource
    A semi-Markov storage model
    Cambridge University Press (CUP) ; 1973
    In:  Advances in Applied Probability Vol. 5, No. 2 ( 1973-08), p. 362-378
    Details
    In: Advances in Applied Probability, Cambridge University Press (CUP), Vol. 5, No. 2 ( 1973-08), p. 362-378
    Abstract: In this paper a storage model is described in which fluctuations in the content are governed by a sequence of independent identically distributed (i.i.d.) random inputs and i.i.d. random releases. This sequence proceeds according to an underlying semi-Markov process. Laplace transforms of the exact distribution of the content are given for the case of negative exponential distributions for both inputs and releases. Exact expressions for limiting (in time) content distributions are found. In the general case, the asymptotic behavior of the content is described for critical and supercritical limiting conditions.
    Type of Medium: Online Resource
    ISSN: 0001-8678 , 1475-6064
    URL: Article
    DOI: 10.2307/1426041
    RVK:
    SA 1600
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1973
    detail.hit.zdb_id: 1474602-5
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