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Node Cardinality Estimation in a Heterogeneous Wireless Network Deployed Over a Large Region Using a Mobile Base Station

Sachin Kadam, Kaustubh S. Bhargao, Gaurav S. Kasbekar

TL;DR

The paper addresses rapid per-type node cardinality estimation in a heterogeneous wireless network distributed over a large region using a mobile base station. It extends the SRC_M protocol to two heterogeneous schemes, HSRC-M1 and HSRC-M2, each consisting of two phases per stop, and proves that the final per-type estimates equal those obtained by running the homogeneous protocol $T$ times, while reducing the required time slots. It provides closed-form expressions for expected slot counts and node energy under HSRC-M1, proves the Optimal MBS Tour (OMT) problem is NP-complete, offers a greedy solution, and validates substantial slot-saving gains via simulations compared to naive $T$-repetitions. The work enables energy-efficient, wide-area node cardinality estimation with mobile infrastructure, combining accuracy guarantees with practical routing considerations for large HWNs.

Abstract

We consider the problem of estimation of the node cardinality of each node type in a heterogeneous wireless network with $T$ types of nodes deployed over a large region, where $T \ge 2$ is an integer. A mobile base station (MBS), such as that mounted on an unmanned aerial vehicle, is used in such cases since a single static base station is not sufficient to cover such a large region. The MBS moves around in the region and makes multiple stops, and at the last stop, it is able to estimate the node cardinalities for the entire region. In this paper, two schemes, viz., HSRC-M1 and HSRC-M2, are proposed to rapidly estimate the number of nodes of each type. Both schemes have two phases, and they are performed at each stop. We prove that the node cardinality estimates computed using our proposed schemes are equal to, and hence as accurate as, the estimates that would have been obtained if a well-known estimation protocol designed for homogeneous networks in prior work were separately executed $T$ times. Closed-form expressions for the expected number of slots required by HSRC-M1 to execute and the expected energy consumption of a node under HSRC-M1 are computed. The problem of finding the optimal tour of the MBS around the region, which covers all the nodes and minimizes the travel cost of the MBS, is formulated and shown to be NP-complete, and a greedy algorithm is provided to solve it. Using simulations, it is shown that the numbers of slots required by the proposed schemes, HSRC-M1 and HSRC-M2, for computing node cardinality estimates are significantly less than the number of slots required for $T$ separate executions of the above estimation protocol for homogeneous networks.

Node Cardinality Estimation in a Heterogeneous Wireless Network Deployed Over a Large Region Using a Mobile Base Station

TL;DR

The paper addresses rapid per-type node cardinality estimation in a heterogeneous wireless network distributed over a large region using a mobile base station. It extends the SRC_M protocol to two heterogeneous schemes, HSRC-M1 and HSRC-M2, each consisting of two phases per stop, and proves that the final per-type estimates equal those obtained by running the homogeneous protocol times, while reducing the required time slots. It provides closed-form expressions for expected slot counts and node energy under HSRC-M1, proves the Optimal MBS Tour (OMT) problem is NP-complete, offers a greedy solution, and validates substantial slot-saving gains via simulations compared to naive -repetitions. The work enables energy-efficient, wide-area node cardinality estimation with mobile infrastructure, combining accuracy guarantees with practical routing considerations for large HWNs.

Abstract

We consider the problem of estimation of the node cardinality of each node type in a heterogeneous wireless network with types of nodes deployed over a large region, where is an integer. A mobile base station (MBS), such as that mounted on an unmanned aerial vehicle, is used in such cases since a single static base station is not sufficient to cover such a large region. The MBS moves around in the region and makes multiple stops, and at the last stop, it is able to estimate the node cardinalities for the entire region. In this paper, two schemes, viz., HSRC-M1 and HSRC-M2, are proposed to rapidly estimate the number of nodes of each type. Both schemes have two phases, and they are performed at each stop. We prove that the node cardinality estimates computed using our proposed schemes are equal to, and hence as accurate as, the estimates that would have been obtained if a well-known estimation protocol designed for homogeneous networks in prior work were separately executed times. Closed-form expressions for the expected number of slots required by HSRC-M1 to execute and the expected energy consumption of a node under HSRC-M1 are computed. The problem of finding the optimal tour of the MBS around the region, which covers all the nodes and minimizes the travel cost of the MBS, is formulated and shown to be NP-complete, and a greedy algorithm is provided to solve it. Using simulations, it is shown that the numbers of slots required by the proposed schemes, HSRC-M1 and HSRC-M2, for computing node cardinality estimates are significantly less than the number of slots required for separate executions of the above estimation protocol for homogeneous networks.
Paper Structure (32 sections, 2 theorems, 31 equations, 14 figures, 4 tables, 1 algorithm)

This paper contains 32 sections, 2 theorems, 31 equations, 14 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Network Model, Problem Formulation, and Background
  4. Network Model and Problem Formulation
  5. Review of the Multiple-set Simple RFID Counting () Protocol
  6. Review of Phase 1 of the Protocol
  7. Review of Phase 2 of the Protocol
  8. Proposed Node Cardinality Estimation Schemes
  9. Phase 2 of the Heterogeneous (HSRC-M1) Scheme
  10. Phase 2 of the Heterogeneous (HSRC-M2) Scheme
  11. Computation of Node Cardinality Estimates
  12. Performance Analysis
  13. Expected Number of Slots Required in HSRC-M1
  14. Expected Energy Consumption of a Node under HSRC-M1
  15. Expected Energy Consumption of a Node in Phase 1 at a Given Stop
  16. ...and 17 more sections

Key Result

Theorem 1

The final node cardinality estimate, $\hat{n}_{b}$, of each type $b \in \{1, \ldots, T\}$, obtained using any one of the proposed schemes, viz., HSRC-M1 and HSRC-M2, equals, and hence is as accurate as, the estimate that would have been obtained if phases 1 and 2 of the SRC$_M$ protocol were separat

Figures (14)

  • Figure 1: The figure shows $M = 4$ locations (stops) of a mobile base station (MBS) and $T = 5$ types of nodes in a region. The coverage range of the MBS at a stop is the area inside the circle with that stop as the centre.
  • Figure 2: The figure shows the frame structure used in the SRC$_M$ protocol when the MBS is at stop $m$.
  • Figure 3: The figure shows the frame structure used in the 3-Step Protocol (3-SP) for the case $T=4$.
  • Figure 4: The figure shows the symbol combination used by each node type in HSRC-M1. The symbol $0$ indicates "no transmission".
  • Figure 5: The figure shows the frame structure used in the 2-Step Protocol (2-SP) for the case $T=4$.
  • ...and 9 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof