Queueing Theory and Stochastic Models

Queueing theory and stochastic models provide a mathematical framework for analysing systems in which tasks, customers or data packets arrive randomly and await service. Originating with simple single‐server models such as M/M/1 and M/G/1 queues, the field has expanded to encompass networks of queues, retrial systems, priority schemes, customer impatience behaviours and server vacations or breakdowns. Stochastic processes—particularly continuous‐time Markov chains—underpin exact analyses, while diffusion approximations and heavy‐traffic limits yield tractable descriptions in complex or non‐stationary settings. Contemporary research integrates inventory control, energy‐aware server operation and correlated arrival patterns, reflecting the demands of telecommunications, cloud computing, manufacturing logistics and healthcare. Key performance measures include queue length distributions, waiting‐time statistics, throughput, loss probabilities and reliability indices. Analytical techniques such as probability‐generating functions, matrix‐analytic methods and perturbation bounds remain central, even as simulation and numerical schemes support systems with time‐varying rates or non‐exponential service processes. This blend of rigour and applicability ensures that queueing theory continues to inform the design and optimisation of diverse service systems worldwide.

Research from Nature Portfolio

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Research from all publishers

Recent studies have enriched models of service systems with inventory and retrial features under non‐standard server regimes. One line of work examines batch arrivals and batch service in a single‐server queueing–inventory system wherein the server produces stock during vacation periods. Using phase‐type distributions and Markov chains, researchers derived steady‐state probabilities and explored how correlation in successive arrivals and service phases alters performance measures and optimal control policies. Another investigation considers a queue‐dependent service rate model with a finite waiting room and an infinite retrial orbit. Adopting a classic (s,Q) inventory replenishment policy, this analysis employs a matrix‐geometric approach to obtain stability conditions, steady‐state distributions and waiting‐time laws for new and returning customers, illustrating trade‐offs between service speed and holding costs. A third contribution focuses on an M/G/1 retrial queue with delayed repairs and feedback under a working‐vacation policy, allowing customers to balk or renege. By constructing supplementary‐variable equations for orbit and system size, the study reveals how repair delays, vacation schedules and impatience parameters jointly determine queue dynamics and reliability indices. Together, these developments highlight the versatility of matrix‐analytic methods and the importance of integrating operational constraints into stochastic service models.

Queueing Theory and Stochastic Models publication trend

The graph below shows the total number of articles in queueing theory and stochastic models across all publications each year (not limited to Nature Index journals).

Technical terms

Continuous‐time Markov chain: A stochastic process that undergoes transitions at random times between a countable set of states, with the future evolution depending only on the present state.

Phase‐type distribution: A probabilistic model representing a mixture of exponential stages, used to approximate general holding‐time distributions in queues.

Retrial queue: A system in which customers who find the server busy join an orbit and retry for service after random intervals.

Working vacation: A server state in which service continues at a reduced rate, often used to model maintenance or energy‐saving modes.

Matrix‐geometric method: An analytical technique for obtaining steady‐state probabilities of quasi‐birth–death processes through rate matrices and geometric series.

Balking: The phenomenon whereby arriving customers decide not to enter the queue upon observing its length.

Reneging: The abandonment of a waiting customer before receiving service due to impatience or external constraints.

References

1. Analysis of a Batch Arrival, Batch Service Queuing-Inventory System with Processing of Inventory While on Vacation. Mathematics (2021).
2. Stochastic modeling on M/M/1/N inventory system with queue-dependent service rate and retrial facility. AIMS Mathematics (2021).
3. Analysis of an M/G/1 Retrial Queue with Delayed Repair and Feedback under Working Vacation policy with Impatient Customers. Symmetry (2022).