Spatial Queues with Nearest Neighbour Shifts

TL;DR

This paper characterizes the fraction of servers that bear a larger load as compared to when the users do not follow any strategy, i.e., they join the queue they arrive at, called overloaded servers.

Abstract

This work studies queues in a Euclidean space. Consider $N$ servers that are distributed uniformly in $[0,1]^d$. Customers arrive at the servers according to independent stationary processes. Upon arrival, they probabilistically decide whether to join the queue they arrived at, or shift to one of the nearest neighbours. Such shifting strategies affect the load on the servers, and may cause some of the servers to become overloaded. We derive a law of large numbers and a central limit theorem for the fraction of overloaded servers in the system as the total number of servers $N \to \infty$. Additionally, in the one-dimensional case ($d=1$), we evaluate the expected fraction of overloaded servers for any finite $N$. Numerical experiments are provided to support our theoretical results. Typical applications of the results include electric vehicles queueing at charging stations, and queues in airports or supermarkets.

Figures (6)

• Figure 1: The star graph $K_i$ formed by a node with in-degree $i$.
• Figure 2: Region of integration.
• Figure 3: Illustration of $Q^N_{1,1,0}=Q^N_{1,1,2}$. An arrow $i \to j$ indicates $j$ is the nearest neighbour of $i$.
• Figure 4: Histogram of the number of overloaded servers and the servers with no change in load for different probability values of the $(1,p)$-NNS strategy in (a) one dimension and (b) two dimension.
• Figure 5: Spatial distribution of $100$ charging stations (top blue), underloaded stations (middle orange) and overloaded stations (bottom green) in $[0,1]$ for different probability values.

Theorems & Definitions (20)

• Definition 2.1
• Definition 2.2
• Theorem 2.1
• Corollary 2.1
• Theorem 2.2
• Theorem 2.3
• Definition 3.1
• Theorem 3.1: bahadirNumberWeaklyConnected2021
• Proposition 3.2
• proof