[Paper Review] On the first-visit-time problem for birth and death processes with catastrophes

Antonio Di Crescenzo, Virginia Giorno+2 more|ArXiv.org|Jul 15, 2003

Advanced Queuing Theory Analysis16 references3 citations

TL;DR

This paper investigates the first-visit time distribution for birth-death processes subject to catastrophes, deriving the Laplace transform of the first-visit time density to any state $k \in \{r, r+1, \ldots\}$, and computes its mean and variance. By relating the process with catastrophes to an equivalent process without catastrophes, the authors obtain exact expressions for the first-visit time distribution and the first catastrophe occurrence time, including explicit formulas for the expected values and steady-state distributions under various parameter regimes.

ABSTRACT

For a birth-death process subject to catastrophes, defined on the state-space $S=\{r,r+1,r+2,...\}$, with $r$ a positive integer or zero, the first-visit time to a state $k\in S$ is considered and the Laplace transform of its probability density function is determined, use of which is then made to obtain mean and variance. The Laplace transform of the probability density function of the first effective catastrophe occurrence time and its expected value are also obtained. Some extensions to time-non-homogeneous processes are then provided. Finally, certain additional results concerning the determination of the steady-state distribution and the representation of the transition probabilities are worked out, while some applications to particular birth-death processes are shown in the Appendix.

Motivation & Objective

• To determine the first-visit time distribution to any preassigned non-zero state in a birth-death process subject to catastrophes.
• To derive the Laplace transform of the probability density function for the first-visit time to state $k$, and compute its mean and variance.
• To analyze the first occurrence time of a catastrophe, obtaining its Laplace transform and expected value.
• To extend results to time-non-homogeneous processes and derive the steady-state distribution of the process.
• To provide explicit representations of transition probabilities and illustrate the framework through specific birth-death processes in the appendix.

Proposed method

• The authors establish functional relations between the birth-death process with catastrophes and a corresponding process without catastrophes, using the latter as a reference for analytical tractability.
• They derive the Laplace transform of the first-visit time density by exploiting the relationship between the two processes and solving the resulting integral equations.
• For the first catastrophe occurrence time, the Laplace transform is obtained via a similar reduction technique, leveraging the memoryless property of Poisson-distributed catastrophes.
• The steady-state distribution is derived using confluent hypergeometric functions and beta functions, depending on the relative values of birth, death, and catastrophe rates.
• Explicit expressions for the expected value of $N(t)$ are obtained by substituting parameters in the mean formula of the reference process.
• The method is extended to time-non-homogeneous processes by adapting the generating function approach used in prior work.

Experimental results

Research questions

• RQ1 What is the distribution of the first time a birth-death process with catastrophes reaches a given non-absorbing state $k \geq r$?
• RQ2 How does the presence of catastrophes affect the expected time to first visit a target state compared to the process without catastrophes?
• RQ3 What is the Laplace transform of the first-occurrence time of a catastrophe, and what is its expected value?
• RQ4 Under what conditions does a steady-state distribution exist for the process with catastrophes, and what is its explicit form?
• RQ5 How do the transition probabilities and first-visit time distributions change when the process is time-inhomogeneous?

Key findings

• The Laplace transform of the first-visit time density to state $k$ is derived in closed form, enabling exact computation of the mean and variance of the first-visit time.
• For the case $\alpha \neq \beta + \xi$, the expected value of $N(t)$ given $N(0) = j$ is $j e^{(\alpha - \beta - \xi)t} + \frac{\nu (e^{(\alpha - \beta - \xi)t} - 1)}{\alpha - \beta - \xi}$, which reduces to the non-catastrophe case when $\xi = 0$.
• The steady-state distribution exists when $\alpha < \beta + \xi$, and is given by $q_n = \xi \widehat{\pi}_{0,n}(\xi)$, with $\widehat{\pi}_{0,n}(\lambda)$ expressed via hypergeometric and beta functions depending on the parameter regime.
• When $\alpha = \beta$, the steady-state distribution involves the confluent hypergeometric function of the second kind, with $q_0 = \left(\frac{\xi}{\alpha}\right)^{\nu/\alpha} e^{\xi/\alpha} \Gamma\left(1 - \frac{\nu}{\alpha}, \frac{\xi}{\alpha}\right)$.
• For $\alpha > \beta$, the steady-state distribution is expressed using the hypergeometric function $F$ and the beta function, with parameters adjusted to reflect the dominance of birth over death.
• The expected value of the first catastrophe occurrence time is derived as the inverse of the catastrophe rate $\xi$, consistent with the memoryless property of Poisson processes.