Batch sojourn time in polling systems on a circle

TL;DR

This work analyzes a continuous-circle polling system with Poisson batch arrivals and uniformly located batch members, focusing on the mean number of waiting customers and the mean batch sojourn time. The authors develop a mean-value-analysis framework combined with a branching-process construction and an integral-equation approach to derive a closed-form expression for the mean batch sojourn time $\mathbb{E}[S^B]$ and a compact formula for the steady-state mean number of waiting customers $\mathbb{E}[L]$, under the stability condition $\rho=\lambda\mathbb{E}[K]\mathbb{E}[B]<1$. They obtain a linear solution for the key integral equation and prove uniqueness, yielding explicit expressions that accommodate special cases such as unit batch size and light traffic, as well as various batch-size distributions. Numerical results compare the continuous model to symmetric discrete polling, showing rapid convergence and highlighting how batch-size variance and workload affect $\mathbb{E}[S^B]$, providing insights for optimization and design of continuous-polling systems. The approach offers a tractable, explicit framework for analyzing similar continuum polling models and supports further extensions to more general arrival-location distributions.

Abstract

In this paper, we analyze a polling system on a circle. Random batches of customers arrive at a circle, where each customer, independently, obtains a uniform location. A single server cyclically travels over the circle to serve all customers. Using mean value analysis, we derive the expected number of waiting customers within a given distance of the server and a closed form expression for the mean batch sojourn time.

Figures (4)

• Figure 1: Illustration of the polling model and corresponding range within distance $x$ of the server. In this example $L(x) = 2$.
• Figure 2: Illustration of the waiting time of a tagged customer (green) that is generated by a service (of the orange customer) and the corresponding branching process. During the service of the orange customer, blue customers arrive, of which only the first two are considered. During the service of the first blue customer, the red customers arrive, of which only one will be served before the tagged customer.
• Figure 3: The expected batch sojourn time for deterministic batch sizes and exponential service requirements with unit mean, comparing the discrete and continuous polling model.
• Figure 4: Comparison of the expected batch sojourn time for different batch size distributions: geometric, Poisson, negative binomial with 5 successes, binomial with 15 trials and deterministic. Services take $1/5$ time units and the average batch size is taken $5$.

Theorems & Definitions (25)

• Lemma 1
• proof
• Remark 1
• Remark 2
• Definition 1
• Definition 2
• Lemma 2
• proof
• Remark 3
• Lemma 3: Little's law