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On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

The risk processes considered in this paper are generated by an underlying Markov process with a regenerative structure and an independent sequence of independent and identically distributed claims. Between the arrivals of claims the process increases at a rate which is a nonnegative function of the present value of the Markov process. The intensity for a claim to occur is another nonnegative function of the value of the Markov process. The claim arrival times are the regeneration times for the Markov process. Two-sided claims are allowed, but the distribution of the positive claims is assumed to have a Laplace transform that is a rational function. The main results describe the joint Laplace transform of the time at ruin and the deficit at ruin. The method used consists in finding partial eigenfunctions for the generator of the joint process consisting of the Markov process and the accumulated claims process, a joint process which is also Markov. These partial eigenfunctions are then used to find a martingale that directly leads to an expression for the desired Laplace transform. In the final section, three examples are given involving different types of the underlying Markov process.

In this paper we consider two classes of reflected Ornstein–Uhlenbeck (OU) processes: the reflected OU process with jumps and the Markov-modulated reflected OU process. We prove that their stationary distributions exist. Furthermore, for the jump reflected OU process, the Laplace transform (LT) of the stationary distribution is given. As for the Markov-modulated reflected OU process, we derive an equation satisfied by the LT of the stationary distribution.

Chiu and Yin (2005) found the Laplace transform of the last time a spectrally negative Lévy process, which drifts to ∞, is below some level. The main motivation for the study of this random time stems from risk theory: what is the last time the risk process, modeled by a spectrally negative Lévy process drifting to ∞, is 0? In this paper we extend the result of Chiu and Yin, and we derive the Laplace transform of the last time, before an independent, exponentially distributed time, that a spectrally negative Lévy process (without any further conditions) exceeds (upwards or downwards) or hits a certain level. As an application, we extend a result found in Doney (1991).

The present paper generalises some results for spectrally negative Lévy processes to the setting of Markov additive processes (MAPs). A prominent role is assumed by the first passage times, which will be determined in terms of their Laplace transforms. These have the form of a phase-type distribution, with a rate matrix that can be regarded as an inverse function of the cumulant matrix. A numerically stable iteration to compute this matrix is given. The theory is first developed for MAPs without positive jumps and then extended to include positive jumps having phase-type distributions. Numerical and analytical examples show agreement with existing results in special cases.

In this paper, we present an iterative procedure to calculate explicitly the Laplace transform of the distribution of the maximum for a Lévy process with positive jumps of phase type. We derive error estimates showing that this iteration converges geometrically fast. Subsequently, we determine the Laplace transform of the law of the upcrossing ladder process and give an explicit pathwise construction of this process.

We analyse in this paper the fluid weighted fair queueing system with two classes of customers, who arrive according to Poisson processes and require arbitrarily distributed service times. In a first step, we express the Laplace transform of the joint distribution of the workloads in the two virtual queues of the system by means of unknown Laplace transforms. Such an unknown Laplace transform is related to the distribution of the workload in one queue provided that the other queue is empty. We explicitly compute the unknown Laplace transforms by means of a Wiener—Hopf technique. The determination of the unknown Laplace transforms can be used to compute some performance measures characterizing the system (e.g. the mean waiting time for each class) which we compute in the exponential service case.

We show in this paper how the Laplace transform θ* of the duration θ of an excursion by the occupation process {Λt} of an M/M/∞ system above a given threshold can be obtained by means of continued fraction analysis. The representation of θ* by a continued fraction is established and the [m−1/m] Padé approximants are computed by means of well known orthogonal polynomials, namely associated Charlier polynomials. It turns out that the continued fraction considered is an S fraction and as a consequence the Stieltjes transform of some spectral measure. Then, using classic asymptotic expansion properties of hypergeometric functions, the representation of the Laplace transform θ* by means of Kummer's function is obtained. This allows us to recover an earlier result obtained via complex analysis and the use of the strong Markov property satisfied by the occupation process {Λt}. The continued fraction representation enables us to further characterize the distribution of the random variable θ.

In this paper we consider phase-type distributions, their Laplace transforms which are rational functions and their representations which are finite-state Markov chains with an absorbing state. We first prove that, in any representation, the minimal number of states which are visited before absorption is equal to the difference of degree between denominator and numerator in the Laplace transform of the distribution. As an application, we prove that when the Laplace transform has a denominator with n real poles and a numerator of degree less than or equal to one the distribution has order n. We show that, in general, this result can be extended neither to the case where the numerator has degree two nor to the case of non-real poles.

This paper is concerned with a model for the effect of erosion on crop production. Crop yield in the year n is given by X(n) = YnLn, where is a sequence of strictly positive i.i.d. random variables such that E{Y 1} is a Markov chain with stationary transition probabilities, independent of . When suitably normalized, leads to a martingale which converges to 0 almost everywhere (a.e.) as n → ∞. In addition, for large n, the distribution of Ln is approximately lognormal. The conditional expectations and probabilities of , given the past history of the process, are determined. Finally, the asymptotic behaviour of the total crop yield is discussed. It is established that under certain regularity conditions Sn converges a.e. to a finite-valued random variable S whose Laplace transform can be obtained as the solution of a Volterra-type linear integral equation.

This paper establishes the existence of the limit as t → ∞ of the distribution of the residual lifetime at time t of a two-unit warm-standby reliability system supported by a single repair facility, conditioned on the event that the system has not been down at any time in (0, t). The conditioned limit distribution is proved to be exponential irrespective of the distributions of the lifetimes and repair times of the individual units, provided that their Laplace transforms are rational functions of their arguments.

SummaryA weir of capacity K is considered in which the water inflow is a process with stationary independent increments. Unless the weir is empty, there is a continuous release of water at unit rate; if K is finite the weir may become full in which case the excess water overflows instantaneously. A weir for which K is infinite will be referred to as infinite dam. For the latter the transient behaviour is well known if the input possesses a second moment (cf. e.g., Prabhu [7]) and serves as the starting point for the present paper. This result is first extended to yield the Laplace transform (L.T.) of the trivariate Laplace-Stieltjes transform (L.S.T.) of the content v(t) at time t, the input X(t) in (0, t) and the total time d(t) in the interval (0, t) during which the dam is dry. (Incidentally, the last two quantities, for relevant time intervals, will be carried throughout.) Then we use a relation between the latter and the L.S.T. of the expected number of downward level y crossings of the v(t) process established in Roes [9]. Since the dam processes considered are Markov processes, we have therewith the L.S.T. of the renewal function of the renewal process imbedded at level y. From this, one finds the L.S.T.'s of first entrance and taboo first entrance times (for their definition see introduction). Next we calculate the first skip times for the infinite dam from the first entrance times and the L.T. of the L.S.T. of v(t). It is then a routine matter to determine the taboo first skip times. From the (taboo) first entrance and skip times we derive the first entrance times for the finite dam, which in turn lead to the renewal functions of the renewal processes imbedded in the finite dam content process v*(t) and hence to the transient behaviour of the finite dam.The advantage of the present approach over the one given in Roes [8] is that it is entirely probabilistic and avoids involved analytic arguments. As a result, the question of uniqueness of the solution does not arise, while more insight is obtained in the structure. The L.S.T. of several first entrance times and first skip times have been derived by Cohen [2] for compound Poisson input.