scholarly journals Hitting Times and the Running Maximum of Markovian Growth-Collapse Processes

2011 ◽  
Vol 48 (02) ◽  
pp. 295-312 ◽  
Author(s):  
Andreas Löpker ◽  
Wolfgang Stadje
Keyword(s):  
Power Series Expansion ◽  
Inverse Function ◽  
Hitting Times ◽  
Asymptotic Results ◽  
Rapid Variation ◽  
State Dependent ◽  
Special Cases ◽  
The Laplace Transform ◽  
Running Maximum ◽  
Explicit Formulae

We consider the level hitting times τy= inf{t≥ 0 |Xt=y} and the running maximum processMt= sup{Xs| 0 ≤s≤t} of a growth-collapse process (Xt)t≥0, defined as a [0, ∞)-valued Markov process that grows linearly between random ‘collapse’ times at which downward jumps with state-dependent distributions occur. We show how the moments and the Laplace transform of τycan be determined in terms of the extended generator ofXtand give a power series expansion of the reciprocal of Ee−sτy. We prove asymptotic results for τyandMt: for example, ifm(y) = Eτyis of rapid variation thenMt/m-1(t) →w1 ast→ ∞, wherem-1is the inverse function ofm, while ifm(y) is of regular variation with indexa∈ (0, ∞) andXtis ergodic, thenMt/m-1(t) converges weakly to a Fréchet distribution with exponenta. In several special cases we provide explicit formulae.

scholarly journals Hitting Times and the Running Maximum of Markovian Growth-Collapse Processes

2011 ◽  
Vol 48 (2) ◽  
pp. 295-312 ◽  
Author(s):  
Andreas Löpker ◽  
Wolfgang Stadje
Keyword(s):  
Power Series Expansion ◽  
Inverse Function ◽  
Hitting Times ◽  
Asymptotic Results ◽  
Rapid Variation ◽  
State Dependent ◽  
Special Cases ◽  
The Laplace Transform ◽  
Running Maximum ◽  
Explicit Formulae

We consider the level hitting times τy = inf{t ≥ 0 | Xt = y} and the running maximum process Mt = sup{Xs | 0 ≤ s ≤ t} of a growth-collapse process (Xt)t≥0, defined as a [0, ∞)-valued Markov process that grows linearly between random ‘collapse’ times at which downward jumps with state-dependent distributions occur. We show how the moments and the Laplace transform of τy can be determined in terms of the extended generator of Xt and give a power series expansion of the reciprocal of Ee−sτy. We prove asymptotic results for τy and Mt: for example, if m(y) = Eτy is of rapid variation then Mt / m-1(t) →w 1 as t → ∞, where m-1 is the inverse function of m, while if m(y) is of regular variation with index a ∈ (0, ∞) and Xt is ergodic, then Mt / m-1(t) converges weakly to a Fréchet distribution with exponent a. In several special cases we provide explicit formulae.

scholarly journals A Markovian growth-collapse model

2006 ◽  
Vol 38 (1) ◽  
pp. 221-243 ◽  
Author(s):  
Onno Boxma ◽  
David Perry ◽  
Wolfgang Stadje ◽  
Shelemyahu Zacks
Keyword(s):  
General Theory ◽  
Rate Function ◽  
Current Level ◽  
Hitting Times ◽  
Dependent Rate ◽  
State Dependent ◽  
Long Run ◽  
Special Cases ◽  
Collapse Model ◽  
Markov Modulated

We consider growth-collapse processes (GCPs) that grow linearly between random partial collapse times, at which they jump down according to some distribution depending on their current level. The jump occurrences are governed by a state-dependent rate function r(x). We deal with the stationary distribution of such a GCP, (Xt)t≥0, and the distributions of the hitting times Ta = inf{t ≥ 0 : Xt = a}, a > 0. After presenting the general theory of these GCPs, several important special cases are studied. We also take a brief look at the Markov-modulated case. In particular, we present a method of computing the distribution of min[Ta, σ] in this case (where σ is the time of the first jump), and apply it to determine the long-run average cost of running a certain Markov-modulated disaster-ridden system.

scholarly journals A Markovian growth-collapse model

2006 ◽  
Vol 38 (01) ◽  
pp. 221-243 ◽  
Author(s):  
Onno Boxma ◽  
David Perry ◽  
Wolfgang Stadje ◽  
Shelemyahu Zacks
Keyword(s):  
General Theory ◽  
Rate Function ◽  
Current Level ◽  
Hitting Times ◽  
Dependent Rate ◽  
State Dependent ◽  
Long Run ◽  
Special Cases ◽  
Collapse Model ◽  
Markov Modulated

We consider growth-collapse processes (GCPs) that grow linearly between random partial collapse times, at which they jump down according to some distribution depending on their current level. The jump occurrences are governed by a state-dependent rate functionr(x). We deal with the stationary distribution of such a GCP, (Xt)t≥0, and the distributions of the hitting timesTa= inf{t≥ 0 :Xt=a},a> 0. After presenting the general theory of these GCPs, several important special cases are studied. We also take a brief look at the Markov-modulated case. In particular, we present a method of computing the distribution of min[Ta, σ] in this case (where σ is the time of the first jump), and apply it to determine the long-run average cost of running a certain Markov-modulated disaster-ridden system.

State-dependent signalling in queueing networks

1994 ◽  
Vol 26 (02) ◽  
pp. 436-455 ◽  
Author(s):  
W. Henderson ◽  
B. S. Northcote ◽  
P. G. Taylor
Keyword(s):  
Queueing Networks ◽  
State Dependence ◽  
Product Form ◽  
Batch Size ◽  
Single Server ◽  
Negative Customers ◽  
Affect Control ◽  
State Dependent ◽  
Special Cases ◽  
Equilibrium Distributions

It has recently been shown that networks of queues with state-dependent movement of negative customers, and with state-independent triggering of customer movement have product-form equilibrium distributions. Triggers and negative customers are entities which, when arriving to a queue, force a single customer to be routed through the network or leave the network respectively. They are ‘signals' which affect/control network behaviour. The provision of state-dependent intensities introduces queues other than single-server queues into the network. This paper considers networks with state-dependent intensities in which signals can be either a trigger or a batch of negative customers (the batch size being determined by an arbitrary probability distribution). It is shown that such networks still have a product-form equilibrium distribution. Natural methods for state space truncation and for the inclusion of multiple customer types in the network can be viewed as special cases of this state dependence. A further generalisation allows for the possibility of signals building up at nodes.

On the Theory of Thermally Weighted Electron-Phonon Transition Probabilities

1980 ◽  
Vol 35 (9) ◽  
pp. 902-914
Author(s):  
J. Schupfner
Keyword(s):  
Transition Probabilities ◽  
Algebraic Method ◽  
Calculation Technique ◽  
Radiative Transitions ◽  
Special Cases ◽  
Explicit Formulae ◽  
The Common ◽  
Common Transition ◽  
Franck Condon ◽  
Refined Calculation

Abstract We present a refined calculation method for the phonon part (Franck-Condon Overlaps) of the transition probabilities of electron-phonon radiative and non-radiative transitions in crystals. The evaluation of the thermal averaged Franck-Condon integrals is a purely algebraic method and the transition probabilities we use are derived from first principles and completely atomistic. For the electronic transitions we take into account the frequency shift of the lattice and the change of the phonon normal coordinates. Explicit formulae of the phonon parts are derived and it is shown that the common transition probabilities used in literature are special cases of our functional calculation technique.

EXPLICIT FUNCTIONS OF FUZZY CONTROL SYSTEMS

1996 ◽  
Vol 04 (06) ◽  
pp. 515-535 ◽  
Author(s):  
LÁSZLÓ T. KÓCZY ◽  
MICHIO SUGENO
Keyword(s):  
Computational Complexity ◽  
Fuzzy Control ◽  
Control Systems ◽  
Mathematical Description ◽  
Membership Functions ◽  
Mathematical Functions ◽  
Low Computational Complexity ◽  
Special Cases ◽  
Single Input ◽  
Explicit Formulae

Fuzzy control systems have proved their applicability in many areas. Their user-friend-liness and transparency certainly belong to their main advantages, and these two enable developing and tuning such controllers easily, without knowing their exact mathematical description. Nevertheless, it is of interest to know, what mathematical functions hide behind a set of fuzzy rules and an inference machine. For practical purposes it is necessary to consider real, implementable fuzzy control systems with reasonably low computational complexity. This paper discusses the problem of what types of functions are generated by realistic fuzzy control systems. In the paper the explicit formulae of the transference functions for practically important special cases are determined, controllers having rules with triangular and trapezoidal membership functions, and crisp consequents. Here we restrict our investigations to rules with a single input.

A Novel Approach for Analyzing Single Buffer Queueing Systems with State-Dependent Vacation and Correlated Input Process under Four Different Service Disciplines

2015 ◽  
Vol 6 (3) ◽  
pp. 19-59 ◽  
Author(s):  
Thomas Yew Sing Lee
Keyword(s):  
Performance Measures ◽  
Probability Generating Function ◽  
Performance Measure ◽  
Idle Time ◽  
Separate Analysis ◽  
Input Process ◽  
State Dependent ◽  
Special Cases ◽  
Service Disciplines ◽  
Different Types

The author presents performance analysis of a single buffer multiple-queue system. Four different types of service disciplines (i.e., non-preemptive, pre-emptive repeat different, state dependent random polling and globally gated) are analyzed. His model includes correlated input process and three different types of non-productive time (i.e., switchover, vacation and idle time). Special cases of the model includes server with mixed multiple and single vacations, stopping server with delayed vacation and stopping server with alternating vacation and idle time. For each of the four service disciplines the key performance measures such as average customer waiting time, loss probability, and throughput are computed. The results permit a detailed discussion of how these performance measures depends on the customer arrival rate, the customer service time, the switchover time, the vacation time, and the idle time. Moreover, extensive numerical results are presented and the four service disciplines are compared with respect to the performance measure. Previous studies of the single buffer multiple-queue systems tend to provide separate analysis for the two cases of zero and nonzero switchover time. The author is able to provide a unified analysis for the two cases. His results generalize and improve a number of known results on single buffer multiple-queue systems. Furthermore, this method does not require differentiation while it is needed if one uses the probability generating function approach. Lastly, the author's approach works for all single buffer multiple-queue systems in which the next queue to be served is determines solely on the basis of the occupancy states at the end of the cycle time.

Sequential urn schemes and birth processes

1985 ◽  
Vol 17 (02) ◽  
pp. 257-279
Author(s):  
Lars Holst ◽  
Jürg Hüsler
Keyword(s):  
Waiting Times ◽  
Asymptotic Results ◽  
Special Cases ◽  
Urn Scheme ◽  
Birth Processes

An urn contains balls of different colours, which are randomly drawn one at a time. After each draw the number of balls in the urn with the same colour as the ball last drawn is changed. Special cases are sampling with and without replacement, and Pólya sampling. The drawing stops when a given number of colours has been drawn a given number of times. The number of times the different colours have been drawn is studied in this paper by imbedding the urn scheme in birth processes. Both exact and asymptotic results are obtained. In particular, waiting times for sampling with and without replacement and for Pólya sampling are considered.

A Non‐Bayesian Theory of State‐Dependent Utility

Econometrica ◽  
2019 ◽  
Vol 87 (4) ◽  
pp. 1341-1366 ◽  
Author(s):  
Brian Hill
Keyword(s):  
Expected Utility ◽  
State Dependence ◽  
Comparative Statics ◽  
Bayesian Theory ◽  
Subjective Expected Utility ◽  
State Dependent ◽  
Special Cases

Many decision situations involve two or more of the following divergences from subjective expected utility: imprecision of beliefs (or ambiguity), imprecision of tastes (or multi‐utility), and state dependence of utility. This paper proposes and characterizes a model of uncertainty averse preferences that can simultaneously incorporate all three phenomena. The representation supports a principled separation of (imprecise) beliefs and (potentially state‐dependent, imprecise) tastes. Moreover, the representation permits comparative statics separating the roles of beliefs and tastes, and is modular: it easily delivers special cases involving various combinations of the phenomena, as well as state‐dependent multi‐utility generalizations covering popular ambiguity models.

scholarly journals Stochastic Game Theoretic Formulation for a Multi-Period DC Pension Plan with State-Dependent Risk Aversion

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 108 ◽  
Author(s):  
Liyuan Wang ◽  
Zhiping Chen
Keyword(s):  
Risk Aversion ◽  
Stochastic Game ◽  
Pension Plan ◽  
Time Inconsistency ◽  
State Variables ◽  
Economic Point ◽  
State Dependent ◽  
Equilibrium Control ◽  
Special Cases ◽  
Game Theoretic

When facing a multi-period defined contribution (DC) pension plan investment problem during the accumulation phase, the risk aversion attitude of a mean-variance investor may depend on state variables. In this paper, we propose a state-dependent risk aversion model which is a linear function of the current wealth level after contribution. This risk aversion model is reasonable from both the dimensional analysis and the economic point of view. Moreover, we incorporate the wage income factor into our model. In the field of dynamic investment analysis, most studies have irrational situations in their models because of the lack of the positiveness for the wealth process. In view of it, we further improve the work of Wang and Chen by completely eliminating the irrationality of the model. Due to the time-inconsistency of the resulting stochastic control problem, we derive the explicit expressions of the equilibrium control and the corresponding equilibrium value function by adopting the game theoretic framework developed in Björk and Murgoci. Further, two special cases are discussed. Finally, using a more realistic risk aversion coefficient, we provide a series of empirical tests based on the real data from the American market and compare our results with the relevant results in the literature.

Sign in / Sign up

Export Citation Format

Share Document