Bounds on the mean and squared coefficient of variation of phase-type distributions

2021 ◽  
Vol 58 (4) ◽  
pp. 880-889
Author(s):  
Qi-Ming He
Keyword(s):  
Stochastic Modeling ◽  
Open Problem ◽  
Coefficient Of Variation ◽  
Poisson Processes ◽  
Counting Processes ◽  
Event Time ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Mean ◽  
Markov Modulated

AbstractWe consider a class of phase-type distributions (PH-distributions), to be called the MMPP class of PH-distributions, and find bounds of their mean and squared coefficient of variation (SCV). As an application, we have shown that the SCV of the event-stationary inter-event time for Markov modulated Poisson processes (MMPPs) is greater than or equal to unity, which answers an open problem for MMPPs. The results are useful for selecting proper PH-distributions and counting processes in stochastic modeling.

On identifiability and order of continuous-time aggregated Markov chains, Markov-modulated Poisson processes, and phase-type distributions

1996 ◽  
Vol 33 (3) ◽  
pp. 640-653 ◽  
Author(s):  
Tobias Rydén
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Continuous Time ◽  
Poisson Processes ◽  
Hitting Times ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Minimum Number ◽  
Markov Modulated ◽  
Two Parameters

An aggregated Markov chain is a Markov chain for which some states cannot be distinguished from each other by the observer. In this paper we consider the identifiability problem for such processes in continuous time, i.e. the problem of determining whether two parameters induce identical laws for the observable process or not. We also study the order of a continuous-time aggregated Markov chain, which is the minimum number of states needed to represent it. In particular, we give a lower bound on the order. As a by-product, we obtain results of this kind also for Markov-modulated Poisson processes, i.e. doubly stochastic Poisson processes whose intensities are directed by continuous-time Markov chains, and phase-type distributions, which are hitting times in finite-state Markov chains.

On identifiability and order of continuous-time aggregated Markov chains, Markov-modulated Poisson processes, and phase-type distributions

1996 ◽  
Vol 33 (03) ◽  
pp. 640-653 ◽  
Author(s):  
Tobias Rydén
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Continuous Time ◽  
Poisson Processes ◽  
Hitting Times ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Minimum Number ◽  
Markov Modulated ◽  
Two Parameters

An aggregated Markov chain is a Markov chain for which some states cannot be distinguished from each other by the observer. In this paper we consider the identifiability problem for such processes in continuous time, i.e. the problem of determining whether two parameters induce identical laws for the observable process or not. We also study the order of a continuous-time aggregated Markov chain, which is the minimum number of states needed to represent it. In particular, we give a lower bound on the order. As a by-product, we obtain results of this kind also for Markov-modulated Poisson processes, i.e. doubly stochastic Poisson processes whose intensities are directed by continuous-time Markov chains, and phase-type distributions, which are hitting times in finite-state Markov chains.

scholarly journals Preservation of phase-type distributions under Poisson shock models

1992 ◽  
Vol 24 (01) ◽  
pp. 223-225 ◽  
Author(s):  
M. Manoharan ◽  
Harshinder Singh ◽  
Neeraj Misra
Keyword(s):  
Continuous Phase ◽  
Finite Mixture ◽  
Life Distribution ◽  
Mean Values ◽  
Shock Models ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Discrete Phase ◽  
The Mean ◽  
The Relationship

In this paper, we consider the life distribution H(t) of a device subject to shocks governed by a finite mixture of homogeneous Poisson processes. It will be shown that if (pk ), the probabilities that the device fails on the kth shock, has a discrete phase-type (DPH) distribution, then H(t) is continuous phase-type (CPH). The relationship between the mean values of (pk ) and H(t) is established.

A versatile Markovian point process

1979 ◽  
Vol 16 (4) ◽  
pp. 764-779 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Point Processes ◽  
Probability Distributions ◽  
Poisson Processes ◽  
Renewal Processes ◽  
Probability Models ◽  
Phase Type ◽  
Finite State ◽  
Markov Point Processes ◽  
Arrival Processes ◽  
Markov Modulated

We introduce a versatile class of point processes on the real line, which are closely related to finite-state Markov processes. Many relevant probability distributions, moment and correlation formulas are given in forms which are computationally tractable. Several point processes, such as renewal processes of phase type, Markov-modulated Poisson processes and certain semi-Markov point processes appear as particular cases. The treatment of a substantial number of existing probability models can be generalized in a systematic manner to arrival processes of the type discussed in this paper.Several qualitative features of point processes, such as certain types of fluctuations, grouping, interruptions and the inhibition of arrivals by bunch inputs can be modelled in a way which remains computationally tractable.

scholarly journals A Review on Phase-type Distributions and their Use in Risk Theory

2005 ◽  
Vol 35 (01) ◽  
pp. 145-161 ◽  
Author(s):  
Mogens Bladt
Keyword(s):  
Stochastic Modeling ◽  
Ruin Probability ◽  
Risk Theory ◽  
Jump Processes ◽  
Positive Real ◽  
Markov Jump ◽  
Premium Rate ◽  
Markov Jump Processes ◽  
Phase Type ◽  
Phase Type Distributions

Phase-type distributions, defined as the distributions of absorption times of certain Markov jump processes, constitute a class of distributions on the positive real axis which seems to strike a balance between generality and tractability. Indeed, any positive distribution may be approximated arbitrarily closely by phase-type distributions whereas exact solutions to many complex problems in stochastic modeling can be obtained either explicitly or numerically. In this paper we introduce phase-type distributions and retrieve some of their basic properties through appealing probabilistic arguments which, indeed, constitute their main feature of being mathematically tractable. This is illustrated in an example where we calculate the ruin probability for a rather general class of surplus processes where the premium rate is allowed to depend on the current reserve and where claims sizes are assumed to be of phase-type. Finally we discuss issues concerning statistical inference for phase-type distributions and related functionals such as e.g. a ruin probability.

scholarly journals A Review on Phase-type Distributions and their Use in Risk Theory

2005 ◽  
Vol 35 (1) ◽  
pp. 145-161 ◽  
Author(s):  
Mogens Bladt
Keyword(s):  
Stochastic Modeling ◽  
Ruin Probability ◽  
Risk Theory ◽  
Jump Processes ◽  
Positive Real ◽  
Markov Jump ◽  
Premium Rate ◽  
Markov Jump Processes ◽  
Phase Type ◽  
Phase Type Distributions

Phase-type distributions, defined as the distributions of absorption times of certain Markov jump processes, constitute a class of distributions on the positive real axis which seems to strike a balance between generality and tractability. Indeed, any positive distribution may be approximated arbitrarily closely by phase-type distributions whereas exact solutions to many complex problems in stochastic modeling can be obtained either explicitly or numerically. In this paper we introduce phase-type distributions and retrieve some of their basic properties through appealing probabilistic arguments which, indeed, constitute their main feature of being mathematically tractable. This is illustrated in an example where we calculate the ruin probability for a rather general class of surplus processes where the premium rate is allowed to depend on the current reserve and where claims sizes are assumed to be of phase-type. Finally we discuss issues concerning statistical inference for phase-type distributions and related functionals such as e.g. a ruin probability.

scholarly journals Preservation of phase-type distributions under Poisson shock models

1992 ◽  
Vol 24 (1) ◽  
pp. 223-225 ◽  
Author(s):  
M. Manoharan ◽  
Harshinder Singh ◽  
Neeraj Misra
Keyword(s):  
Continuous Phase ◽  
Finite Mixture ◽  
Life Distribution ◽  
Mean Values ◽  
Shock Models ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Discrete Phase ◽  
The Mean ◽  
The Relationship

In this paper, we consider the life distribution H(t) of a device subject to shocks governed by a finite mixture of homogeneous Poisson processes. It will be shown that if (pk), the probabilities that the device fails on the kth shock, has a discrete phase-type (DPH) distribution, then H(t) is continuous phase-type (CPH). The relationship between the mean values of (pk) and H(t) is established.

A versatile Markovian point process

1979 ◽  
Vol 16 (04) ◽  
pp. 764-779 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Point Processes ◽  
Probability Distributions ◽  
Poisson Processes ◽  
Renewal Processes ◽  
Probability Models ◽  
Phase Type ◽  
Finite State ◽  
Markov Point Processes ◽  
Arrival Processes ◽  
Markov Modulated

We introduce a versatile class of point processes on the real line, which are closely related to finite-state Markov processes. Many relevant probability distributions, moment and correlation formulas are given in forms which are computationally tractable. Several point processes, such as renewal processes of phase type, Markov-modulated Poisson processes and certain semi-Markov point processes appear as particular cases. The treatment of a substantial number of existing probability models can be generalized in a systematic manner to arrival processes of the type discussed in this paper. Several qualitative features of point processes, such as certain types of fluctuations, grouping, interruptions and the inhibition of arrivals by bunch inputs can be modelled in a way which remains computationally tractable.

Design and development of manually operated gladiolus planter

2018 ◽  
Vol 5 (01) ◽  
Author(s):  
TAPAN K. KHURA ◽  
H. L. KUSHWAHA ◽  
SATISH D LANDE ◽  
PKSAHOO . ◽  
INDRA L . KUSHWAHA
Keyword(s):  
Coefficient Of Variation ◽  
Field Capacity ◽  
Power Source ◽  
Forward Speed ◽  
Cereal Crop ◽  
Maximum Coefficient ◽  
Coefficient Of Uniformity ◽  
The Mean ◽  
The Cost ◽  
Immense Potential

Floriculture is an age-old farming activity in India having immense potential for generating selfemployment and income to farmers. However, the cost of cultivation of flower is high as compared to cereal crop. Level of mechanization for different field operations is one but foremost reason for the higher cost of cultivation. As most of the Indian farmers are marginal and small, a need for manually operated gladiolus planter was felt. The geometric properties of gladiolus corm were determined for designing the seed metering system and seed hopper of the planter. The planter was evaluated in the field when pulled by two persons as a power source and guided by a person. The coefficient of variation and highest deviation from the mean spacing was observed as 12.93% and 2.65cm respectively. The maximum coefficient of uniformity of 90.59% was observed for a nominal corm spacing of 15cm at 0.56 kmh-1 forward speed. An average MISS percentage was observed as 2.65 and 2.25 for nominal corm spacing of 15 and 20 cm. The multiple index was zero for two levels corm spacing and forward speed of operation. The QFI was found in the range of 97.2 and 97.9 percent. The average field capacity of the planter was observed as 0.02 hah-1.The average draft requirement of the planter was found as 821 ± 50.3 N.

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