Polling on a circle with non-uniform batch arrivals

TL;DR

This work analyzes a continuous circle polling model with batch arrivals and non-uniform locations, addressing two service policies: globally gated and exhaustive. It provides exact Laplace-Stieltjes transform results for batch sojourn time and time to delivery under Globally Gated, and develops a mean-value analysis with a fixed-point integral equation plus an iterative solver for Exhaustive. The authors derive explicit light- and heavy-traffic limits for both policies, and demonstrate via numerical results that arrival-location distributions have limited impact on average performance, with practical implications for milkrun warehouse systems. The combination of exact transform results, a constructive solving procedure, and limiting analyses offers actionable insights for design and performance evaluation of continuous polling systems in logistics and related domains.

Abstract

In this paper, we analyze a polling system on a circle with. Random batches of customers arrive at a circle, where each customer, independently, obtains a location according to a general distribution. A single server cyclically travels over the circle to serve all customers. We analyze the experienced delay of batches for two service policies: globally gated and exhaustive. The Laplace-Stieltjes transform of the experienced delay is found under the former policy. For the latter policy, we propose a mean-value analysis, resulting in an algorithmic approach for the evaluation of the mean experienced delay. Light- and heavy-traffic limits are derived exactly for the system performance.

Figures (12)

• Figure 1: Illustration of the extra 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 2: The expected batch sojourn time of the continuous polling model, under $B\equiv 0.01$, for varying arrival location distributions and loads. The results under exhaustive service are presented as a solid line, the dashed line refers to the globally gated policy.
• Figure 3: The expected time to delivery of the continuous polling model, under $B\equiv 0.01$, for varying arrival location distributions and loads. The results under exhaustive service are presented as a solid line, the dashed line refers to the globally gated policy.
• Figure 4: The expected batch sojourn time of the continuous polling model, under $B\equiv 1$, for varying arrival location distributions and loads. The results under exhaustive service are presented as a solid line, the dashed line refers to the globally gated policy.
• Figure 5: The expected time to delivery of the continuous polling model, under $B\equiv 1$, for varying arrival location distributions and loads. The results under exhaustive service are presented as a solid line, the dashed line refers to the globally gated policy.

Theorems & Definitions (70)

• Theorem 1
• proof
• Corollary 1
• proof
• Remark 1
• Theorem 2
• proof
• Corollary 2
• proof
• Remark 2