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1.
We solve main two-boundary problems for a random walk. The generating function of the joint distribution of the first exit
time of a random walk from an interval and the value of the overshoot of the random walk over the boundary at exit time is
determined. We also determine the generating function of the joint distribution of the first entrance time of a random walk
to an interval and the value of the random walk at this time. The distributions of the supremum, infimum, and value of a random
walk and the number of upward and downward crossings of an interval by a random walk are determined on a geometrically distributed
time interval. We give examples of application of obtained results to a random walk with one-sided exponentially distributed
jumps.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1485–1509, November, 2007. 相似文献
2.
For a homogeneous process with independent increments, we determine the integral transforms of the joint distribution of the
first-exit time from an interval and the value of a jump of a process over the boundary at exit time and the joint distribution
of the supremum, infimum, and value of the process.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1359–1384, October, 2005. 相似文献
3.
《Stochastic Processes and their Applications》2020,130(7):3967-3989
For a spectrally positive strictly stable process with index in (1, 2), we obtain (i) the sub-probability density of its first exit time from an interval by hitting the interval’s lower end before jumping over its upper end, and (ii) the joint distribution of the time, undershoot, and jump of the process when it makes the first exit the other way around. The density of the exit time is expressed in terms of the roots of a Mittag-Leffler function. Some theoretical applications of the results are given. 相似文献
4.
Several two-boundary problems are solved for a special Lévy process: the Poisson process with an exponential component. The
jumps of this process are controlled by a homogeneous Poisson process, the positive jump size distribution is arbitrary, while
the distribution of the negative jumps is exponential. Closed form expressions are obtained for the integral transforms of
the joint distribution of the first exit time from an interval and the value of the overshoot through boundaries at the first
exit time. Also the joint distribution of the first entry time into the interval and the value of the process at this time
instant are determined in terms of integral transforms. 相似文献
5.
In this paper we solve a two-sided exit problem for a difference of a compound Poisson process and a compound renewal process. More specifically, we determine the Laplace transforms of the joint distribution of the first exit time, the value of the overshoot and the value of a linear component at this time instant. The results obtained are applied to solve the two-sided exit problem for a particular class of stochastic processes, i.e. the difference of the compound Poisson process and the renewal process whose jumps are exponentially distributed. The advantage is that these results are in a closed form, in terms of resolvent sequences of the process. We determine the Laplace transforms of the busy period of the systems M ? |G δ |1|B, G δ |M ? |1|B in case when δ~exp?(λ). Additionally, we prove the weak convergence of the two-boundary characteristics of the process to the corresponding functionals of the standard Wiener process. 相似文献
6.
We consider a random walk generated by a sequence of independent identically distributed random variables. We assume that the distribution function of a jump of the random walk equals an exponential polynomial on the negative half-axis. For double transforms of the joint distribution of the first exit time from an interval and overshoot, we obtain explicit expressions depending on finitely many parameters that, in turn, we can derive from the system of linear equations. The principal difference of the present article from similar results in this direction is the rejection of using factorization components and projection operators connected with them. 相似文献
7.
A bulk-arrival single server queueing system with second multi-optional service and unreliable server is studied in this paper. Customers arrive in batches according to a homogeneous Poisson process, all customers demand the first "essential" service, whereas only some of them demand the second "multi-optional" service. The first service time and the second service all have general distribution and they are independent. We assume that the server has a service-phase dependent, exponentially distributed life time as well as a servicephase dependent, generally distributed repair time. Using a supplementary variable method, we obtain the transient and the steady-state solutions for both queueing and reliability measures of interest. 相似文献
8.
We consider the following Type of problems. Calls arrive at a queue of capacity K (which is called the primary queue), and attempt to get served by a single server. If upon arrival, the queue is full and
the server is busy, the new arriving call moves into an infinite capacity orbit, from which it makes new attempts to reach
the primary queue, until it finds it non-full (or it finds the server idle). If the queue is not full upon arrival, then the
call (customer) waits in line, and will be served according to the FIFO order. If λ is the arrival rate (average number per
time unit) of calls and μ is one over the expected service time in the facility, it is well known that μ > λ is not always
sufficient for stability. The aim of this paper is to provide general conditions under which it is a sufficient condition.
In particular, (i) we derive conditions for Harris ergodicity and obtain bounds for the rate of convergence to the steady
state and large deviations results, in the case that the inter-arrival times, retrial times and service times are independent
i.i.d. sequences and the retrial times are exponentially distributed; (ii) we establish conditions for strong coupling convergence
to a stationary regime when either service times are general stationary ergodic (no independence assumption), and inter-arrival
and retrial times are i.i.d. exponentially distributed; or when inter-arrival times are general stationary ergodic, and service
and retrial times are i.i.d. exponentially distributed; (iii) we obtain conditions for the existence of uniform exponential
bounds of the queue length process under some rather broad conditions on the retrial process. We finally present conditions
for boundedness in distribution for the case of nonpatient (or non persistent) customers.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
9.
David C.M. Dickson 《Insurance: Mathematics and Economics》2012,50(3):334-337
We use probabilistic arguments to derive an expression for the joint density of the time to ruin and the number of claims until ruin in the classical risk model. From this we obtain a general expression for the probability function of the number of claims until ruin. We also consider the moments of the number of claims until ruin and illustrate our results in the case of exponentially distributed individual claims. Finally, we briefly discuss joint distributions involving the surplus prior to ruin and deficit at ruin. 相似文献
10.
In this paper we examine the joint distributions of several actuarial diagnostics which are important to insurers’ running in the classical risk model. They include the time of the surplus process leaving zero ultimately (simply, the ultimately leaving-time), the number of zero, the surplus immediately prior to ruin, the deficit at ruin, the supreme and minimum profits before ruin, the supreme profits and deficit until it leaves zero ultimately and so on. We obtain explicit expressions for their joint distributions mainly by strong Markov property of the surplus process—a technique used by Wu et al. (2002) [J. Appl. Math., in press], which is completely different from former contributions on this topic. Further, we give the exact calculating results for them when the individual claim amounts are exponentially distributed. 相似文献
11.
On the basis of a given sequence of independent identically distributed pairs of random variables, we construct the step process
of semi-Markov random walk that is later delayed by a screen at zero. For this process, we obtain the Laplace transform of
the distribution of the time of the first hit of the level zero.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 912–919, July, 2007. 相似文献
12.
该文主要讨论带干扰古典风险模型的破产瞬间余额和破产赤字的边际及联合分布.借助于修正阶梯高度的结果,得到了它们的表达式.当索赔服从指数分布时,给出它们的精确表达. 相似文献
13.
Murtuza Ali Abidini Onno Boxma Bara Kim Jeongsim Kim Jacques Resing 《Queueing Systems》2017,87(3-4):293-324
We consider gated polling systems with two special features: (i) retrials and (ii) glue or reservation periods. When a type-i customer arrives, or retries, during a glue period of the station i, it will be served in the following service period of that station. Customers arriving at station i in any other period join the orbit of that station and will retry after an exponentially distributed time. Such polling systems can be used to study the performance of certain switches in optical communication systems. When the glue periods are exponentially distributed, we obtain equations for the joint generating functions of the number of customers in each station. We also present an algorithm to obtain the moments of the number of customers in each station. When the glue periods are generally distributed, we consider the distribution of the total workload in the system, using it to derive a pseudo-conservation law which in turn is used to obtain accurate approximations of the individual mean waiting times. We also investigate the problem of choosing the lengths of the glue periods, under a constraint on the total glue period per cycle, so as to minimize a weighted sum of the mean waiting times. 相似文献
14.
Consider a tandem queue model with a single server who can switch instantaneously from one queue to another. Customers arrive according to a Poisson process with rate λ . The amount of service required by each customer at the ith queue is an exponentially distributed random variable with rate μi. Whenever two or more customers are in the system, the decision as to which customer should be served first depends on the optimzation criterion. In this system all server allocation policies in the finite set of work conserving deterministic policies have the same expected first passage times (makespan) to empty the system of customers from any initial state. However, a unique policy maximizes the first passage probability of empty-ing the system before the number of customers exceeds K, for any value of K, and it stochastically minimizes (he number of customers in the system at any time t > 0 . This policy always assigns the server to the non empty queue closest to the exit 相似文献
15.
Sheldon M Ross 《Stochastic Processes and their Applications》1976,4(2):167-173
Consider an n-component reliability system having the property that at any time each of its components is either up (i.e., working) or down (i.e., being repaired). Each component acts independently and we suppose that each time the ith component goes up it remains up for an exponentially distributed time having mean μi, and each time it goes down it remains down for an exponentially distributed time having mean υi. We further suppose that whether or not the system itself is up at any time depends only on which components are up at that time. We are interested in the distribution of the time of first system failure when all components are initially up at time zero. In section 2 we show that this distribution has the NBU (i.e., new better than used) property, and in Section 3 we make use of this and other results to obtain a lower bound to the mean time until first system failure. 相似文献
16.
We study a BMAP/>SM/1 queue with batch Markov arrival process input and semi‐Markov service. Service times may depend on arrival
phase states, that is, there are many types of arrivals which have different service time distributions. The service process
is a heterogeneous Markov renewal process, and so our model necessarily includes known models. At first, we consider the first
passage time from level {κ+1} (the set of the states that the number of customers in the system is κ+1) to level {κ} when a batch arrival occurs at time 0 and then a customer service included in that batch simultaneously starts. The service
descipline is considered as a LIFO (Last‐In First‐Out) with preemption. This discipline has the fundamental role for the analysis
of the first passage time. Using this first passage time distribution, the busy period length distribution can be obtained.
The busy period remains unaltered in any service disciplines if they are work‐conserving. Next, we analyze the stationary
workload distribution (the stationary virtual waiting time distribution). The workload as well as the busy period remain unaltered
in any service disciplines if they are work‐conserving. Based on this fact, we derive the Laplace–Stieltjes transform for
the stationary distribution of the actual waiting time under a FIFO discipline. In addition, we refer to the Laplace–Stieltjes
transforms for the distributions of the actual waiting times of the individual types of customers. Using the relationship
between the stationary waiting time distribution and the stationary distribution of the number of customers in the system
at departure epochs, we derive the generating function for the stationary joint distribution of the numbers of different types
of customers at departures.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
17.
In this paper, we study absolute ruin questions for the perturbed compound Poisson risk process with investment and debit
interests by the expected discounted penalty function at absolute ruin, which provides a unified means of studying the joint
distribution of the absolute ruin time, the surplus immediately prior to absolute ruin time and the deficit at absolute ruin
time. We first consider the stochastic Dirichlet problem and from which we derive a system of integro-differential equations
and the boundary conditions satisfied by the function. Second, we derive the integral equations and a defective renewal equation
under some special cases, then based on the defective renewal equation we give two asymptotic results for the expected discounted
penalty function when the initial surplus tends to infinity for the light-tailed claims and heavy-tailed claims, respectively.
Finally, we investigate some explicit solutions and numerical results when claim sizes are exponentially distributed. 相似文献
18.
In this paper, we consider a renewal risk model with stochastic premiums income. We assume that the premium number process and the claim number process are a Poisson process and a generalized Erlang (n) processes, respectively. When the individual stochastic premium sizes are exponentially distributed, the Laplace transform and a defective renewal equation for the Gerber-Shiu discounted penalty function are obtained. Furthermore, the discounted joint distribution of the surplus just before ruin and the deficit at ruin is given. When the claim size distributions belong to the rational family, the explicit expression of the Gerber-Shiu discounted penalty function is derived. Finally, a specific example is provided. 相似文献
19.
This paper investigates the first exit time and the ruin time of a risk reserve process with reserve-dependent income under the assumption that the claims arrive as a Poisson process. We show that the Laplace transform of the distribution of the first exit time from an interval satisfies an integro-differential equation. The exact solution for the classical model and for the Embrechts–Schmidli model are derived. 相似文献
20.
D.F. Holman M.L. Chaudhry B.R.K. Kashyap 《European Journal of Operational Research》1983,13(2):142-145
In the present paper we consider the service system MX/G/∞ characterized by an infinite number of servers anda general service time distribution. The customers arrive at the system in groups of size X, which is a random variable, the time between group arrivals being exponentially distributed. Using simple probability arguments, we obtain probability generating functions (p.g.f.'s) of the number of busy servers at time t and the number that depart by time t. Several other properties of these random variables are also discussed. 相似文献