Online reservation queueing–inventory system with two distinct services and client types

N. Suresh Kumar; N. Anbazhagan; S. Amutha; Gyanendra Prasad Joshi; Woong Cho; N. Suresh Kumar; N. Anbazhagan; S. Amutha; Gyanendra Prasad Joshi; Woong Cho

doi:10.3934/math.2025869

AIMS Mathematics

2025, Volume 10, Issue 8: 19460-19494. doi: 10.3934/math.2025869

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Research article

Online reservation queueing–inventory system with two distinct services and client types

1.
Department of Mathematics, Alagappa University, karaikudi-630003, India
2.
Ramanujan Centre for Higher Mathematics, Alagappa University, karaikudi-630003, India
3.
Department of Electronic and AI System Engineering, Kangwon National University, Samcheok 25913, Republic of Korea

Received: 23 December 2024 Revised: 20 July 2025 Accepted: 14 August 2025 Published: 26 August 2025
MSC : 60K25, 90B05, 91B70

This study examines a queueing–inventory system offering premium and non-premium services for a single commodity, where both services have individual waiting areas with finite capacities. To enhance premium service, the system includes an online reservation facility with a limited level. An online client initially pre-books a spot in the premium waiting area (PWA) and, after a random duration, either joins the PWA or cancels the reservation. An offline client directly visits the system and chooses any one of the services based on their needs. The arrivals of both client types follow independent Markovian arrival processes (MAPs). Further, the waiting times of a client in the premium and non-premium waiting areas are derived using the Laplace–Stieltjes transform. The steady-state probabilities are computed, and the system's essential performance metrics are calculated. Subsequently, the optimal total expected cost is determined through numerical analysis and visually represented in a graph.
- online client,
- offline client,
- online reservation,
- Markovian arrival process,
- premium service,
- non-premium service
Citation: N. Suresh Kumar, N. Anbazhagan, S. Amutha, Gyanendra Prasad Joshi, Woong Cho. Online reservation queueing–inventory system with two distinct services and client types[J]. AIMS Mathematics, 2025, 10(8): 19460-19494. doi: 10.3934/math.2025869

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Abstract

This study examines a queueing–inventory system offering premium and non-premium services for a single commodity, where both services have individual waiting areas with finite capacities. To enhance premium service, the system includes an online reservation facility with a limited level. An online client initially pre-books a spot in the premium waiting area (PWA) and, after a random duration, either joins the PWA or cancels the reservation. An offline client directly visits the system and chooses any one of the services based on their needs. The arrivals of both client types follow independent Markovian arrival processes (MAPs). Further, the waiting times of a client in the premium and non-premium waiting areas are derived using the Laplace–Stieltjes transform. The steady-state probabilities are computed, and the system's essential performance metrics are calculated. Subsequently, the optimal total expected cost is determined through numerical analysis and visually represented in a graph.

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