arXiv subjects

Qihui Bu

Publications and source records attributed to Qihui Bu.

5 recordsLinked to original sources

arXiv2026

Queues with Correlated Service Times -- the $M/M_D/c$ Model

This paper studies multi-server queueing systems with correlated service times, modeled as the $M/M_D/c$ queue, which is a natural extension of the recent work by Thapa and Zhao \cite{Thapa-Zhao:2026}. In this model, arrivals follow a Poisson process, while service times across servers exhibit dependence captured by the Marshall--Olkin multivariate exponential distribution (MO-MVED). We first develop a rigorous sample-path construction of the system and establish that the resulting queueing process is a continuous-time Markov chain. We then analyze the stationary behavior of the $M/M_D/c$ model. In the homogeneous case, we derive a complete solution via geometric tail structure and explicit boundary equations, recovering a tractable one-dimensional representation. In the heterogeneous case, we establish a general framework combining a geometric tail with a finite boundary system, and prove existence, uniqueness, and nonnegativity of the stationary distribution. The above results provide a unified analytic framework extending classical $M/M/c$ theory to correlated-service settings, and reveal how dependence among service times fundamentally affects system performance and structure. Beyond the $M/M_D/c$ model, We next study the interplay between Marshall--Olkin service dependence and queue-state Markovianity. On the one hand, Marshall--Olkin dependent service completions are shown to preserve Markovianity for a broad class of queueing systems. On the other hand, if a queueing process admits a Markovian state description without tracking service ages, residual service times, or service phases, then its service mechanism must satisfy a weak multivariate lack-of-memory property and consequently belongs to the Marshall--Olkin family. These results provide a probabilistic foundation for the use of Marshall--Olkin multivariate exponential service times in Markovian queueing models.

math.PR↗

arXiv2026

GTH Algorithm, Censored Markov Chains, and $RG$-Factorization in Block-Form

In 1985, Grassmann, Taksar, and Heyman published their celebrated paper, in which they introduced a numerically stable algorithm for computing the stationary probabilities of a finite-state Markov chain, one of the key performance quantities in both theory and applications. This algorithm later became the well-known GTH algorithm (or the state-reduction method) in the literature, becoming one of the standard algorithms in applied probability. Later, this algorithm was extended to deal with the stationary distributions of block-structured Markov chains with repeating rows. In this paper, we focus on the block-form GTH algorithm and organize it into two parts. In the first part, we connect the block-form GTH algorithm to censored Markov chains and the block-form $RG$-factorization. We show that the forward block-elimination and back block-form substitution of the block-form GTH algorithm are equivalent to solving a system formulated using the $RG$-factorization in two steps. We also show that this connection remains valid when the block-form GTH algorithm is extended to infinite-state Markov chains. It is well known that censoring an infinite-state Markov chain to a finite state space yields a stationary distribution that provides a best approximation to the stationary distribution of the original infinite-state Markov chain. In the second part, we first derive an explicit expression for the censored Markov chain from the infinite state space to a finite space for Markov chains of $M/G/1$ type. Based on this expression, we propose a renormalized approximated censored transition matrix (RA-CM). The resulting stationary distribution is shown to be asymptotically optimal in terms of approximation error. We compare the approximation error of the RA-CM with the error arising from the last-block-column augmentation.

math.NA↗

arXiv2020

Approximations for a Queueing Game Model with Join-the-Shortest-Queue Strategy

This paper investigates a partially observable queueing system with $N$ nodes in which each node has a dedicated arrival stream. There is an extra arrival stream to balance the load of the system by routing its customers to the shortest queue. In addition, a reward-cost structure is considered to analyze customers' strategic behaviours. The equilibrium and socially optimal strategies are derived for the partially observable mean field limit model. Then, we show that the strategies obtained from the mean field model are good approximations to the model with finite $N$ nodes. Finally, numerical experiments are provided to compare the equilibrium and socially optimal behaviours, including joining probabilities and social benefits for different system parameters.

math.PR↗

arXiv2020

Join-the-Shortest-Queue Model as a Mean Field Approximation to a Local Balancing Network

In this paper, we consider a queueing network with $N$ nodes, each of which has a fixed number $k$ of neighboring nodes, referred to as the $N$ node network with local balancing. We assume that to each of the $N$ nodes, an incoming job (or task) chooses the shortest queue from this node and its neighboring nodes. We construct an appropriate Markov process for this network and find a mean field approximation to this network as $N\rightarrow\infty$, which turns out to be the standard join-the-shortest-queue model.

math.PR↗

arXiv2018

Mean Field Approximations to a Queueing System with Threshold-Based Workload Control Scheme

In this paper, motivated by considerations of server utilization and energy consumptions in cloud computing, we investigate a homogeneous queueing system with a threshold-based workload control scheme. In this system, a virtual machine will be turned off when there are no tasks in its buffer upon the completion of a service by the machine, and turned on when the number of tasks in its buffer reaches a pre-set threshold value. Due to complexity of this system, we propose approximations to system performance measures by mean field limits. An iterative algorithm is suggested for the solution to the mean field limit equations. In addition, numerical and simulation results are presented to justify the proposed approximation method and to provide a numerical analysis on the impact of the system performances by system parameters.

math.PR↗