Second Order Properties of Thinned Counts in Finite Birth--Death Processes

TL;DR

The paper analyzes the second-order properties of thinned counts in a finite-state birth–death process by deriving a closed-form expression for the asymptotic index of dispersion $\mathcal{D}$ of the thinned counting process $N_q$. Using regenerative arguments and an explicit inverse of a certain matrix $\mathbf W$, the authors express $\mathcal{D}$ as $\mathcal{D}=1+2\sum_{k=0}^{J-1} R_k$ with $R_k=(P_k-\Lambda_k)(\frac{\overline{\lambda}(P_k-\Lambda_k)}{\pi_k\lambda_k}+q_k^+-q_{k+1}^-)$, where $P_k$ and $\Lambda_k$ are cumulative distributions tied to the stationary distribution and thinning, and $\overline{\lambda}$ is the thinned rate. The results connect to BRAVO-type variance reduction in queueing systems with state-dependent thinning and extend known pure-death results; the paper also conjectures and explores a countably infinite state-space analogue with numerical illustrations. The techniques provide a tractable way to quantify variability in thinned counting processes and may have broader applicability via MAP/renewal–reward frameworks. The explicit matrix-inverse construction is highlighted as a potential independent tool for related birth–death analyses.

Abstract

The paper studies the counting process arising as a subset of births and deaths in a birth--death process on a finite state space. Whenever a birth or death occurs, the process is incremented or not depending on the outcome of an independent Bernoulli experiment whose probability is a state-dependent function of the birth and death and also depends on whether it is a birth or death that has occurred. We establish a formula for the asymptotic variance rate of this process, also presented as the ratio of the asymptotic variance and the asymptotic mean. Several examples including queueing models illustrate the scope of applicability of the results. An analogous formula for the countably infinite state space is conjectured and tested.

Figures (2)

• Figure 1: The asymptotic index of dispersion for M/M/$s$/K$+$M systems with $K=20$, $s=10$, and $\mu = 1$. When the offered load $\rho \approx 1$, the asymptotic variability of the output process is reduced in comparison to the Poisson process ($\mathcal{D} = 1$) case.
• Figure 2: The asymptotic index of dispersion for a counting process of interactions upon arrival to a finite population infinite-service system.

Theorems & Definitions (10)

• Theorem 1
• Corollary 1
• proof
• Lemma 1
• proof
• Example 1
• Example 2
• Example 3
• Example 4
• Example 5