Applications of the age distribution in age dependent branching processes

This paper considers asymptotic properties of increasing population age dependent branching processes which have a limiting age distribution. In Section I, the Bellman-Harris model [2] is altered in accord with a suggestion by Kendall [3]. The effect of this modification on the applicable results of [3] is indicated. An extension to the case where each cell passes through a sequence of states to mitosis or proceeds from state to state in accord with a semi-Markov process to mitosis is considered. Section 1.1 considers asymptotic moments for the total number of cells. Section 1.2 treats moments in a sequence-of-states model [4]. Section 1.3 extends the results of 1.2 to a semi-Markov model. Section 1.4 treats various asymptotic lifetime distributions and fraction of cells in a given state for both the sequence-of-states and semi-Markov models.

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On age dependent branching processes

Journal of Applied Probability ◽

10.2307/3212127 ◽

1966 ◽

Vol 3 (2) ◽

pp. 383-402 ◽

Cited By ~ 11

Author(s):

Howard Weiner

Keyword(s):

Branching Processes ◽

Asymptotic Properties ◽

Age Dependent

This paper deals with asymptotic properties of various models of age dependent branching processes, relying heavily on Harris [3].

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Loss tandem networks with blocking - a semi-Markov approach

Bulletin of the Polish Academy of Sciences Technical Sciences ◽

10.2478/v10175-010-0072-9 ◽

2010 ◽

Vol 58 (4) ◽

pp. 673-681

Author(s):

W. Oniszczuk

Keyword(s):

Steady State ◽

Markov Process ◽

Markov Model ◽

Markov Models ◽

State Graph ◽

State Equations ◽

Semi Markov Process ◽

Tandem Networks ◽

Measures Of Effectiveness

Loss tandem networks with blocking - a semi-Markov approachBased on the semi-Markov process theory, this paper describes an analytical study of a loss multiple-server two-station network model with blocking. Tasks arrive to the tandem in a Poisson fashion at a rate λ, and the service times at the first and second stations are non-exponentially distributed with means sAand sB, respectively. Between these two stations there is a buffer with finite capacity. In this type of network, if the buffer is full, the accumulation of new tasks (jobs) by the second station is temporarily suspended (blocking factor) and tasks must wait on the first station until the transmission process is resumed. Any new task that finds all service lines at the first station occupied is turned away and is lost (loss factor). Initially, in this document, a Markov model of the loss tandem with blocking is investigated. Here, a two-dimensional state graph is constructed and a set of steady-state equations is created. These equations allow the calculation of state probabilities for each graph state. A special algorithm for transforming the Markov model into a semi-Markov process is presented. This approach allows calculating steady-state probabilities in the semi-Markov model. In the next part of the paper, the algorithms for calculation of the main measures of effectiveness in the semi-Markov model are presented. Finally, the numerical part of this paper contains an investigation of some special semi-Markov models, where the results are presented of the calculation of the quality of service (QoS) parameters and the main measures of effectiveness.

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Mean-square and almost-sure convergence of supercritical age-dependent branching processes

Journal of Applied Probability ◽

10.1017/s002190020011811x ◽

1974 ◽

Vol 11 (04) ◽

pp. 678-686

Author(s):

Edgar Z. Ganuza ◽

S. D. Durham

Keyword(s):

Branching Process ◽

Branching Processes ◽

Age Distribution ◽

Almost Sure Convergence ◽

Mean Square ◽

Malthusian Parameter ◽

Age Dependent ◽

Mean Square Convergence ◽

Watson Process ◽

Successive Generations

Letting Z(t) be the number of objects alive at time t in a general supercritical age-dependent branching process generated by a single ancestor born at time 0, one achieves (Theorem 1) mean-square convergence of Z(t)/E[Z(t)] provided and , where N(t) is the number of offspring of the initial ancestor born by time t and α is the (positive) Malthusian parameter defined by . If the stronger conditions that (Theorem 2) and hold also, then the convergence is almost-sure. It is of interest that the embedded Galton-Watson process of successive generations need not have a finite mean for the conditions of the above theorems to hold. Similar results are obtained for the age-distribution as well.

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Semi-Markov-Based Approach for the Analysis of Open Tandem Networks with Blocking and Truncation

International Journal of Applied Mathematics and Computer Science ◽

10.2478/v10006-009-0014-6 ◽

2009 ◽

Vol 19 (1) ◽

pp. 151-164 ◽

Cited By ~ 4

Author(s):

Walenty Oniszczuk

Keyword(s):

Steady State ◽

Markov Process ◽

Markov Model ◽

Markov Models ◽

Network Models ◽

State Graph ◽

State Equations ◽

Semi Markov Process ◽

Tandem Networks ◽

Measures Of Effectiveness

Semi-Markov-Based Approach for the Analysis of Open Tandem Networks with Blocking and TruncationThis paper describes an analytical study of open two-node (tandem) network models with blocking and truncation. The study is based on semi-Markov process theory, and network models assume that multiple servers serve each queue. Tasks arrive at the tandem in a Poisson fashion at the rate λ, and the service times at the first and the second node are non-exponentially distributed with meanssAandsB, respectively. Both nodes have buffers with finite capacities. In this type of network, if the second buffer is full, the accumulation of new tasks by the second node is temporarily suspended (a blocking factor) and tasks must wait on the first node until the transmission process is resumed. All new tasks that find the first buffer full are turned away and are lost (a truncation factor). First, a Markov model of the tandem is investigated. Here, a two-dimensional state graph is constructed and a set of steady-state equations is created. These equations allow calculating state probabilities for each graph state. A special algorithm for transforming the Markov model into a semi-Markov process is presented. This approach allows calculating steady-state probabilities in the semi-Markov model. Next, the algorithms for calculating the main measures of effectiveness in the semi-Markov model are presented. In the numerical part of this paper, the author investigates examples of several semi-Markov models. Finally, the results of calculating both the main measures of effectiveness and quality of service (QoS) parameters are presented.

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Semi-Markov Models of Safety of the Renewal Systems Operation

Journal of Konbin ◽

10.2478/v10040-008-0064-0 ◽

2008 ◽

Vol 6 (3) ◽

pp. 153-176

Author(s):

Grabski Franciszek ◽

Jaźwiński Jerzy

Keyword(s):

Markov Process ◽

Markov Models ◽

Renewal Function ◽

System Operation ◽

Reliability Characteristics ◽

Operation Process ◽

Alternating Renewal Process ◽

Semi Markov Process ◽

Semi Markov Processes ◽

Special Case

Semi-Markov Models of Safety of the Renewal Systems Operation Usually, the renewal systems are mathematically described by the alternating renewal process, which is special case of the semi-Markov process. It allows us to find the basic reliability characteristics like the renewal function, the operational availability of the system and many other. Many papers are devoted to that problem. Very often the operation of the system is perturbed by the danger events. The operation process of the system is broken or stopped. In that case we can say that the unsafety event occurred. Semi-Markov model of the safety of operation is constructed in this paper. The safety characteristics of the system operation are calculated by using the properties of the semi-Markov processes.

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Mean-square and almost-sure convergence of supercritical age-dependent branching processes

Journal of Applied Probability ◽

10.2307/3212551 ◽

1974 ◽

Vol 11 (4) ◽

pp. 678-686 ◽

Cited By ~ 2

Author(s):

Edgar Z. Ganuza ◽

S. D. Durham

Keyword(s):

Branching Process ◽

Branching Processes ◽

Age Distribution ◽

Almost Sure Convergence ◽

Mean Square ◽

Malthusian Parameter ◽

Age Dependent ◽

Mean Square Convergence ◽

Watson Process ◽

Successive Generations

Letting Z(t) be the number of objects alive at time t in a general supercritical age-dependent branching process generated by a single ancestor born at time 0, one achieves (Theorem 1) mean-square convergence of Z(t)/E[Z(t)] provided and , where N(t) is the number of offspring of the initial ancestor born by time t and α is the (positive) Malthusian parameter defined by . If the stronger conditions that (Theorem 2) and hold also, then the convergence is almost-sure. It is of interest that the embedded Galton-Watson process of successive generations need not have a finite mean for the conditions of the above theorems to hold. Similar results are obtained for the age-distribution as well.

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