quasi-birth-death modeling

This work proposes a parameter estimation method based on the Expectation–Maximization (EM) algorithm for Markovian Arrival Process (MAP)-driven Quasi-Birth–Death (QBD) queueing systems, tailored to realistic scenarios where only coarse-grained data such as system utilization are available. Within a maximum likelihood framework, the approach infers sufficient statistics—including sojourn times, phase transitions, and service dynamics—underlying the hidden states directly from utilization time series. To the best of our knowledge, this is the first method capable of fully estimating MAP-QBD model parameters using solely utilization data. The study further introduces an innovative use of the Akaike Information Criterion (AIC) to automatically select the number of MAP phases, thereby mitigating overfitting. Experimental results demonstrate that the method accurately recovers both arrival and service parameters, offering a practical performance modeling tool for real-world systems lacking fine-grained event logs.

This study addresses the complex interplay of dynamically evolving customer classes, abandonment behavior, and dynamic prioritization in finite-capacity, multi-server queueing systems. To tackle this challenge, the authors propose a scalable continuous-time Markov chain (CTMC) modeling framework that integrates quasi-birth–death processes, matrix-analytic methods, and Krylov subspace approximations to efficiently compute both conditional and steady-state waiting time distributions for two customer classes. Notably, this work is the first to incorporate dynamic customer-type evolution and reneging into waiting time analysis for such systems. The model’s validity is demonstrated using real-world data from a tertiary referral hospital in Australia, where it successfully quantifies the disparity in waiting times between complex and routine patients, thereby offering actionable, quantitative insights for healthcare operational decision-making.

This paper addresses the ambulance fleet sizing problem in emergency medical services, where time-varying demand and mixed urgent/non-urgent call arrivals complicate resource allocation. To tackle this, we propose an analytical queueing modeling framework. Methodologically, we innovatively incorporate first-passage time theory of one-dimensional random walks to model nonstationary call arrival processes and develop a category-conditioned probabilistic framework to separately characterize service performance for urgent and non-urgent calls. The model accommodates both stationary and nonstationary operational regimes and enables KPI-driven quantitative analysis. Our key contribution is a closed-form, analytically tractable formula for the required number of ambulances, which significantly improves the accuracy of resource provisioning and response timeliness. The resulting model provides both theoretical foundations and a practical tool for real-time, dynamic ambulance dispatch and fleet management.

Existing characterizations of product-form steady-state distributions in Markov chains lack structural, graph-theoretic foundations. Method: We introduce the novel concept of *graph-based product-form*, wherein the steady-state probability decomposes as a product of functions depending only on the ancestral sets induced by a directed graph structure. Leveraging graph cuts and joint ancestor independence, we establish multiple equivalent characterizations and design an $O(|V|^2|E|)$ graph-traversal algorithm to automatically verify the condition for any state pair. We further generalize the theory to higher-order product-forms and unify the framework under a graph-structural decision procedure. Results: The approach is validated on classical models—including Jackson networks and BCMP queues—demonstrating both effectiveness and broad applicability. This work provides the first systematic, computationally tractable, graph-theoretic methodology for determining the existence of product-form steady-state distributions.

This paper addresses the degradation of early-learning performance in queueing systems caused by parameter uncertainty. To quantify this issue, it introduces the *Queue Learning Cost* (CLQ)—a novel metric capturing the maximum transient growth in time-averaged queue length under unknown system parameters. Methodologically, the work establishes a unified analytical framework integrating Lyapunov stability theory with multi-armed bandit analysis, leveraging asymptotic and non-asymptotic probabilistic tools, stochastic bandwidth modeling, and queueing network techniques. The CLQ is fully characterized for single-queue, multi-server systems and extended to multi-queue, multi-server configurations and general queueing networks. The resulting theory provides the first unified, verifiable early-time performance guarantees for a broad class of learning-based scheduling algorithms—thereby filling a fundamental theoretical gap in the transient analysis of learning-controlled queueing systems.

This study addresses the absence of a unified closed-form solution and systematic numerical validation for the M/PH₂/1 queueing system by deriving, for the first time, explicit analytical expressions for both the queue-length and sojourn-time distributions. It fully specifies the coefficient formulas for the PH₂-type service-time distribution and establishes PH₂ as the boundary of algebraic solvability. The approach integrates algebraic derivation, matrix-analytic methods, and symbolic computation, complemented by large-scale numerical experiments implemented via BuTools. Results demonstrate that queue-length computations achieve machine precision even under extreme conditions—such as high traffic intensity (ρ ≈ 0.999) and very high service-time variability (Cₛ² > 10⁴)—while sojourn-time calculations retain at least 10 significant digits of accuracy, thereby enabling reliable sensitivity analysis, threshold optimization, and tail-probability guarantees.

This paper addresses the quantification of behavioral similarity between states in continuous-time Markov processes—particularly diffusion processes. To this end, it introduces, for the first time, a second class of behavioral pseudometrics based on trajectory-level semantics, complementing the existing first class grounded in time-indexed Markov kernels. The two classes are rigorously constructed via functional iteration and real-valued logical distance, respectively, and unified within a fixed-point theoretical framework. Theoretical analysis establishes that both pseudometrics are functionally equivalent and logically expressible, and that each converges to behavioral equivalence. This work not only extends the theoretical foundations of behavioral metrics for continuous-time stochastic systems but also provides the first trajectory-driven characterization of behavioral similarity for diffusion processes, thereby bridging the long-standing theoretical gap between kernel-based and path-based approaches.

This study addresses the challenge of performance evaluation and stability characterization in multi-server queueing systems with general arrival and service times. By leveraging stochastic recursive equations and ergodic theory, the authors analyze monotonicity and decomposability under First-Come-First-Served (FCFS) scheduling, and for the first time extend Loynes’ theorem to multi-server job models. Building on this theoretical foundation, they propose a near-perfect sampling algorithm amenable to large-scale GPU parallelization and further generalize it to complex systems with typed resources. The resulting framework substantially improves the efficiency of workload sampling and the accuracy of stability condition estimation in cloud environments, offering a scalable computational approach for performance analysis of large-scale queueing systems.

Traditional Markov-modulated fluid queues struggle to capture complex queueing systems that require event memory or suffer from state-space explosion. This work proposes a colored Markov-modulated fluid queue model augmented with a fluid jump mechanism, where “colors” serve as an additional memory dimension to record the fluid state at specific event occurrences. By incorporating this memory-aware structure, the model significantly enhances its capacity to represent history-dependent behaviors. The approach mitigates the curse of dimensionality while establishing a novel theoretical framework capable of analytically tractable performance evaluation for previously intractable queueing systems. Consequently, it broadens the applicability of fluid queueing models in performance analysis of computer and communication systems.

This work addresses the challenge of causal inference with continuous-time marked point process data, for which existing methods lack a suitable identification framework. Building on martingale theory, the authors extend the core assumptions of discrete-time causal inference—consistency, exchangeability, and positivity—to the continuous-time setting. They formulate a dynamic treatment strategy and a potential outcomes model tailored to marked point processes and establish corresponding causal identification conditions. Leveraging this foundation, they derive a novel marginal g-formula that enables nonparametric identification of causal effects. The proposed framework subsumes existing results for discrete-time and counting process settings as special cases, demonstrating both theoretical compatibility and extensibility, thereby unifying survival analysis and causal inference within a coherent paradigm.