Markov chains governed by complicated renewal processes

1970 ◽  
Vol 2 (02) ◽  
pp. 287-322
Author(s):  
T. Gergely ◽  
I. N. Tsukanow ◽  
I. I. Yezhow

In this work Markov chains governed by complicated processes are introduced and investigated (Section 1). In Section 2 an ergodic theorem for these processes is formulated, while in Section 3 the sojourn time of the process in a fixed region is studied; in Section 4 some examples are considered. The processes studied are of practical importance in the description of mass service systems and the theory of reliability for which the time intervals between successive demands cannot be assumed to be mutually independent random variables. It is shown that the dependence parameter r of these processes, if it is sufficiently large, allows us to formulate a relationship between the time intervals in question.

1970 ◽  
Vol 2 (2) ◽  
pp. 287-322 ◽  
Author(s):  
T. Gergely ◽  
I. N. Tsukanow ◽  
I. I. Yezhow

In this work Markov chains governed by complicated processes are introduced and investigated (Section 1). In Section 2 an ergodic theorem for these processes is formulated, while in Section 3 the sojourn time of the process in a fixed region is studied; in Section 4 some examples are considered. The processes studied are of practical importance in the description of mass service systems and the theory of reliability for which the time intervals between successive demands cannot be assumed to be mutually independent random variables. It is shown that the dependence parameter r of these processes, if it is sufficiently large, allows us to formulate a relationship between the time intervals in question.


1972 ◽  
Vol 9 (3) ◽  
pp. 681-683
Author(s):  
Leon Podkaminer

The probabilities of the occurrence of n events in a certain time period are calculated under the assumptions that the time intervals between the neighbouring events are mutually independent random variables, satisfying some analytic conditions.


1967 ◽  
Vol 4 (2) ◽  
pp. 402-405 ◽  
Author(s):  
H. D. Miller

Let X(t) be the position at time t of a particle undergoing a simple symmetrical random walk in continuous time, i.e. the particle starts at the origin at time t = 0 and at times T1, T1 + T2, … it undergoes jumps ξ1, ξ2, …, where the time intervals T1, T2, … between successive jumps are mutually independent random variables each following the exponential density e–t while the jumps, which are independent of the τi, are mutually independent random variables with the distribution . The process X(t) is clearly a Markov process whose state space is the set of all integers.


Author(s):  
Ю.В. Горгуца

При проектировании рейдовых причалов, строительство которых получило широкое развитие в настоящее время, невозможно воспользоваться методами, предлагаемыми ныне действующими Нормами технологического проектирования, так как они были выполнены для традиционных защищённых акваторий и опираются на статистический материал, полученный по существующим портам. Для разработки методов определения простоев судов при обработке судов на рейдовых причалах с учётом потока помех от метеофакторов (штормов) как потока случайных событий в данной статье описывается исследование новых моделей систем массового обслуживания. Используется метод суперпозиций – находятся решения для простых моделей, которые затем используются для получения решений по более сложным моделям. Первоначально рассматривается простейшая модель, состоящая из потоков вызовов (штормов) и прибора (порта). Поток вызовов - пуассоновский. Время обслуживания – произвольное с преобразованием Лапласа-Стилтьеса Полученные результаты используются для исследования модели с потоками помех от ветров двух различных направлений. Далее исследуется однолинейная модель с «ненадёжным» прибором. Входящий поток – пуассоновский поток подходящих к порту судов. Время обслуживания - длительность интервалов времени между освобождением места у причала для судна, ожидающих на рейде. Выход из строя прибора, как в свободном, так и в занятом обслуживанием состоянии определяется наступлением шторма – событием пуассоновского потока с интервалами между событиями – интервалами между наступлением штормов. Длительность восстановления работоспособности прибора – определяемая в первой модели длительность простоя причала из-за воздействия метеофакторов. Суда, оказавшиеся в порту при наступлении шторма «дообслуживаются» после его окончания Итоговая модель – многоканальная с параллельно работающими приборами (причалам) и экспоненциальным временем обслуживания судов. Полученные результаты сравнивались со статистическими и показали их высокую сходимость, что доказывает их достоверность. While designing offshore terminals, which are being built quite widely in recent time, it is impossible to use methods, proposed by current technological design norms, because they were created for traditional protected waters and are based on statistical data, acquired by existing ports. This article describes the research of new models of mass service systems to develop methods of defining demurrage while processing vehicles on offshore terminals, taking into account disturbance flow from weather factors (storms) as flow of random events. Method of superpositions is used - to find solutions for simple models, which are used afterwards for getting solutions for more complicated models. Initially the basic model is reviewed, consisting of flow of challenges (storms) and device (port). Challenges flow is Poisson. Service time - arbitrary with transformation of Laplace-Stiltjes. Results acquired are used for researching the model with disturbance flows from windows of various directions. Next the unilineal model with “unreliable” device is researched. Incoming flow is Poisson flow of incoming vehicles. Service time - length of time intervals between berths exemption for vehicles awaing on raid. Device failure, both in free and in maintenance mode was defined by storm incoming - the event of Poisson flow with intervals between events - intervals between storms. Duration of device efficiency recovery - is the defined in the first model duration of terminal demurrage due to weather influence. Vessels, caught up in the port during storm will be maintained after its end. Final model is multi-channel one with working devices (terminals) and exponential time of vessel service. Acquired results have been compared with statistical data, which showed they high convergence, proving their reliability.


1972 ◽  
Vol 9 (03) ◽  
pp. 681-683
Author(s):  
Leon Podkaminer

The probabilities of the occurrence of n events in a certain time period are calculated under the assumptions that the time intervals between the neighbouring events are mutually independent random variables, satisfying some analytic conditions.


1967 ◽  
Vol 4 (02) ◽  
pp. 402-405 ◽  
Author(s):  
H. D. Miller

Let X(t) be the position at time t of a particle undergoing a simple symmetrical random walk in continuous time, i.e. the particle starts at the origin at time t = 0 and at times T 1, T 1 + T 2, … it undergoes jumps ξ 1, ξ 2, …, where the time intervals T 1, T 2, … between successive jumps are mutually independent random variables each following the exponential density e–t while the jumps, which are independent of the τi , are mutually independent random variables with the distribution . The process X(t) is clearly a Markov process whose state space is the set of all integers.


1965 ◽  
Vol 2 (02) ◽  
pp. 352-376 ◽  
Author(s):  
Samuel Karlin ◽  
James McGregor

In the Ehrenfest model with continuous time one considers two urns and N balls distributed in the urns. The system is said to be in stateiif there areiballs in urn I, N −iballs in urn II. Events occur at random times and the time intervals T between successive events are independent random variables all with the same negative exponential distributionWhen an event occurs a ball is chosen at random (each of theNballs has probability 1/Nto be chosen), removed from its urn, and then placed in urn I with probabilityp, in urn II with probabilityq= 1 −p, (0 <p< 1).


1980 ◽  
Vol 12 (04) ◽  
pp. 1000-1018 ◽  
Author(s):  
J. Walrand ◽  
P. Varaiya

Consider an open multiclass Jacksonian network in equilibrium and a path such that a customer travelling along it cannot be overtaken directly by a subsequent arrival or by the effects of subsequent arrivals. Then the sojourn times of this customer in the nodes constituting the path are all mutually independent and so the total sojourn time is easily calculated. Two examples are given to suggest that the non-overtaking condition may be necessary to ensure independence when there is a single customer class.


2021 ◽  
Author(s):  
Cristian Suteanu

<p>Characterizing properties of wind speed variability and their dependence on the temporal scale is important: from sub-second intervals (for the design and monitoring of wind turbines) to longer time scales – months, years (for the evaluation of the wind power potential). Wind speed data are usually reported as averages over time intervals of various length (minutes, days, months, etc). The research project presented in this paper addressed the following questions: What aspects of the wind pattern are changed, in what ways and to what extent, in the process of producing time-averaged values? What precautions should be considered when time-averaged values are used in the assessment of wind variability? What are the conditions to be fulfilled for a meaningful comparison of wind pattern characteristics obtained in distinct studies? Our research started from wind speed records sampled at 0.14 second intervals, which were averaged over increasingly longer time intervals. Variability evaluation was based on statistical moments, L-moments, and detrended fluctuation analysis. We present the change suffered by characteristics of temporal variability as a function of sampling rate and the averaging time interval. In particular, the height dependence of wind speed variability, which is of theoretical and practical importance, is shown to be progressively erased when averaging intervals are increased. The paper makes recommendations regarding the interpretation of wind pattern characteristics obtained at different sites as a function of sampling rate and time-averaging intervals.</p>


1981 ◽  
Vol 18 (3) ◽  
pp. 747-751
Author(s):  
Stig I. Rosenlund

For a time-homogeneous continuous-parameter Markov chain we show that as t → 0 the transition probability pn,j (t) is at least of order where r(n, j) is the minimum number of jumps needed for the chain to pass from n to j. If the intensities of passage are bounded over the set of states which can be reached from n via fewer than r(n, j) jumps, this is the exact order.


Sign in / Sign up

Export Citation Format

Share Document