arXiv subjects

Quan-Lin Li

Publications and source records attributed to Quan-Lin Li.

At least 37 records · Page 2Linked to original sources

arXiv2019

Markov Processes in Blockchain Systems

In this paper, we develop a more general framework of block-structured Markov processes in the queueing study of blockchain systems, which can provide analysis both for the stationary performance measures and for the sojourn times of any transaction and block. Note that an original aim of this paper is to generalize the two-stage batch-service queueing model studied in Li et al. \cite{Li:2018} both ``from exponential to phase-type" service times and ``from Poisson to MAP" transaction arrivals. In general, the MAP transaction arrivals and the two stages of PH service times make our blockchain queue more suitable to various practical conditions of blockchain systems with crucial random factors, for example, the mining processes, the block-generations, the blockchain-building and so forth. For such a more general blockchain queueing model, we focus on two basic research aspects: (1) By using the matrix-geometric solution, we first obtain a sufficient stable condition of the blockchain system. Then we provide simple expressions for the average number of transactions in the queueing waiting room, and the average number of transactions in the block. (2) However, comparing with Li et al. \cite{Li:2018}, analysis of the transaction-confirmation time becomes very difficult and challenging due to the complicated blockchain structure. To overcome the difficulties, we develop a computational technique of the first passage times by means of both the PH distributions of infinite sizes and the $RG$-factorizations. Finally, we hope that the methodology and results given in this paper will open a new avenue to queueing analysis of more general blockchain systems in practice, and can motivate a series of promising future research on development of lockchain technologies.

cs.CR↗

arXiv2018

Blockchain Queueing Theory

Blockchain has many benefits including decentralization, availability, persistency, consistency, anonymity, auditability and accountability, and it also covers a wide spectrum of applications ranging from cryptocurrency, financial services, reputation system, Internet of Things, sharing economy to public and social services. Not only may blockchain be regarded as a by-product of Bitcoin cryptocurrency systems, but also it is a type of distributed ledger technology through using a trustworthy, decentralized log of totally ordered transactions. By summarizing the literature of blockchain, it is found that more papers focus on engineering implementation and realization, while little work has been done on basic theory, for example, mathematical models (Markov processes, queueing theory and game models), performance analysis and optimization of blockchain systems. In this paper, we develop queueing theory of blockchain systems and provide system performance evaluation. To do this, we design a Markovian batch-service queueing system with two different service stages, while the two stages are suitable to well express the mining process in the miners pool and the building of a new blockchain. By using the matrix-geometric solution, we obtain a system stable condition and express three key performance measures: (a) The number of transactions in the queue, (b) the number of transactions in a block, and (c) the transaction-confirmation time. Finally, We use numerical examples to verify computability of our theoretical results. Although our queueing model is simple under exponential or Poisson assumptions, our analytic method will open a series of potentially promising research in queueing theory of blockchain systems.

cs.CE↗

arXiv2018

A $c/μ$-Rule for Service Resource Allocation in Group-Server Queues

In this paper, we study a dynamic on/off server scheduling problem in a queueing system with multi-class servers, where servers are heterogeneous and can be classified into $K$ groups. Servers in the same group are homogeneous. A scheduling policy determines the number of working servers (servers that are turned on) in each group at every state $n$ (number of customers in the system). Our goal is to find the optimal scheduling policy to minimize the long-run average cost, which consists of an increasing convex holding cost and a linear operating cost. We use the sensitivity-based optimization theory to characterize the optimal policy. A necessary and sufficient condition of the optimal policy is derived. We also prove that the optimal policy has monotone structures and a quasi bang-bang control is optimal. We find that the optimal policy is indexed by the value of $c - μG(n)$, where $c$ is the operating cost rate, $μ$ is the service rate for a server, and $G(n)$ is a computable quantity called perturbation realization factor. Specifically, the group with smaller negative $c - μG(n)$ is more preferred to be turned on, while the group with positive $c - μG(n)$ should be turned off. However, the preference ranking of each group is affected by $G(n)$ and the preference order may change with the state $n$, the arrival rate, and the cost function. Under a reasonable condition of scale economies, we further prove that the optimal policy obeys a so-called $c$/$μ$-rule. That is, the servers with smaller $c$/$μ$ should be turned on with higher priority and the preference order of groups remains unchanged. This rule can be viewed as a sister version of the famous $cμ$-rule for polling queues. With the monotone property of $G(n)$, we further prove that the optimal policy has a multi-threshold structure when the $c$/$μ$-rule is applied.

math.OC↗

arXiv2018

Optimal Energy-Efficient Policies for Data Centers through Sensitivity-Based Optimization

In this paper, we propose a novel dynamic decision method by applying the sensitivity-based optimization theory to find the optimal energy-efficient policy of a data center with two groups of heterogeneous servers. Servers in Group 1 always work at high energy consumption, while servers in Group 2 may either work at high energy consumption or sleep at low energy consumption. An energy-efficient control policy determines the switch between work and sleep states of servers in Group 2 in a dynamic way. Since servers in Group 1 are always working with high priority to jobs, a transfer rule is proposed to migrate the jobs in Group 2 to idle servers in Group 1. To find the optimal energy-efficient policy, we set up a policy-based Poisson equation, and provide explicit expressions for its unique solution of performance potentials by means of the RG-factorization. Based on this, we characterize monotonicity and optimality of the long-run average profit with respect to the policies under different service prices. We prove that the bang-bang control is always optimal for this optimization problem, i.e., we should either keep all servers sleep or turn on the servers such that the number of working servers equals that of waiting jobs in Group 2. As an easy adoption of policy forms, we further study the threshold-type policy and obtain a necessary condition of the optimal threshold policy. We hope the methodology and results derived in this paper can shed light to the study of more general energy-efficient data centers.

math.OC↗

arXiv2018

A Stochastic Model for File Lifetime and Security in Data Center Networks

Data center networks are an important infrastructure in various applications of modern information technologies. Note that each data center always has a finite lifetime, thus once a data center fails, then it will lose all its storage files and useful information. For this, it is necessary to replicate and copy each important file into other data centers such that this file can increase its lifetime of staying in a data center network. In this paper, we describe a large-scale data center network with a file d-threshold policy, which is to replicate each important file into at most d-1 other data centers such that this file can maintain in the data center network under a given level of data security in the long-term. To this end, we develop three relevant Markov processes to propose two effective methods for assessing the file lifetime and data security. By using the RG-factorizations, we show that the two methods are used to be able to more effectively evaluate the file lifetime of large-scale data center networks. We hope the methodology and results given in this paper are applicable in the file lifetime study of more general data center networks with replication mechanism.

cs.PF↗

arXiv2018

A Mean-Field Matrix-Analytic Method for Bike Sharing Systems under Markovian Environment

To reduce automobile exhaust pollution, traffic congestion and parking difficulties, bike-sharing systems are rapidly developed in many countries and more than 500 major cities in the world over the past decade. In this paper, we discuss a large-scale bike-sharing system under Markovian environment, and propose a mean-field matrix-analytic method in the study of bike-sharing systems through combining the mean-field theory with the time-inhomogeneous queues as well as the nonlinear QBD processes. Firstly, we establish an empirical measure process to express the states of this bike-sharing system. Secondly, we apply the mean-field theory to establishing a time-inhomogeneous MAP(t)/MAP(t)/1/K+2L+1 queue, and then to setting up a system of mean-field equations. Thirdly, we use the martingale limit theory to show the asymptotic independence of this bike-sharing system, and further analyze the limiting interchangeability as N goes to infinity and t goes to infinity. Based on this, we discuss and compute the fixed point in terms of a nonlinear QBD process. Finally, we analyze performance measures of this bike-sharing system, such as, the mean of stationary bike number at any station and the stationary probability of problematic stations. Furthermore, we use numerical examples to show how the performance measures depend on the key parameters of this bike-sharing system. We hope the methodology and results of this paper are applicable in the study of more general large-scale bike-sharing systems.

math.PR↗

arXiv2017

Group-Server Queues

By analyzing energy-efficient management of data centers, this paper proposes and develops a class of interesting {\it Group-Server Queues}, and establishes two representative group-server queues through loss networks and impatient customers, respectively. Furthermore, such two group-server queues are given model descriptions and necessary interpretation. Also, simple mathematical discussion is provided, and simulations are made to study the expected queue lengths, the expected sojourn times and the expected virtual service times. In addition, this paper also shows that this class of group-server queues are often encountered in many other practical areas including communication networks, manufacturing systems, transportation networks, financial networks and healthcare systems. Note that the group-server queues are always used to design effectively dynamic control mechanisms through regrouping and recombining such many servers in a large-scale service system by means of, for example, bilateral threshold control, and customers transfer to the buffer or server groups. This leads to the large-scale service system that is divided into several adaptive and self-organizing subsystems through scheduling of batch customers and regrouping of service resources, which make the middle layer of this service system more effectively managed and strengthened under a dynamic, real-time and even reward optimal framework. Based on this, performance of such a large-scale service system may be improved greatly in terms of introducing and analyzing such group-server queues. Therefore, not only analysis of group-server queues is regarded as a new interesting research direction, but there also exists many theoretical challenges, basic difficulties and open problems in the area of queueing networks.

cs.PF↗

arXiv2017

Fluid and Diffusion Limits for Bike Sharing Systems

Bike sharing systems have rapidly developed around the world, and they are served as a promising strategy to improve urban traffic congestion and to decrease polluting gas emissions. So far performance analysis of bike sharing systems always exists many difficulties and challenges under some more general factors. In this paper, a more general large-scale bike sharing system is discussed by means of heavy traffic approximation of multiclass closed queueing networks with non-exponential factors. Based on this, the fluid scaled equations and the diffusion scaled equations are established by means of the numbers of bikes both at the stations and on the roads, respectively. Furthermore, the scaling processes for the numbers of bikes both at the stations and on the roads are proved to converge in distribution to a semimartingale reflecting Brownian motion (SRBM) in a $N^{2}$-dimensional box, and also the fluid and diffusion limit theorems are obtained. Furthermore, performance analysis of the bike sharing system is provided. Thus the results and methodology of this paper provide new highlight in the study of more general large-scale bike sharing systems.

math.PR↗

arXiv2017

A Nonlinear Solution to Closed Queueing Networks for Bike Sharing Systems with Markovian Arrival Processes and under an Irreducible Path Graph

As a favorite urban public transport mode, the bike sharing system is a large-scale and complicated system, and there exists a key requirement that a user and a bike should be matched sufficiently in time. Such matched behavior makes analysis of the bike sharing systems more difficult and challenging. This paper considers a more general large-scale bike sharing system from two important views: (a) Bikes move in an irreducible path graph, which is related to geographical structure of the bike sharing system; and (b) Markovian arrival processes (MAPs) are applied to describe the non-Poisson and burst behavior of bike-user (abbreviated as user) arrivals, while the burstiness demonstrates that the user arrivals are time-inhomogeneous and space-heterogeneous in practice. For such a complicated bike sharing system, this paper establishes a multiclass closed queueing network by means of some virtual ideas, for example, bikes are abstracted as virtual customers; stations and roads are regarded as virtual nodes. Thus user arrivals are related to service times at station nodes; and users riding bikes on roads are viewed as service times at road nodes. Further, to deal with this multiclass closed queueing network, we provide a detailed observation practically on physical behavior of the bike sharing system in order to establish the routing matrix, which gives a nonlinear solution to compute the relative arrival rates in terms of the product-form solution to the steady-state probabilities of joint queue lengths at the virtual nodes. Based on this, we can compute the steady-state probability of problematic stations, and also deal with other interesting performance measures of the bike sharing system. We hope that the methodology and results of this paper can be applicable in the study of more general bike sharing systems through multiclass closed queueing networks.

math.PR↗

arXiv2017

Groups of Repairmen and Repair-based Load Balancing in Supermarket Models with Repairable Servers

Supermarket models are a class of interesting parallel queueing networks with dynamic randomized load balancing and real-time resource management. When the parallel servers are subject to breakdowns and repairs, analysis of such a supermarket model becomes more difficult and challenging. In this paper, we apply the mean-field theory to studying four interrelated supermarket models with repairable servers, and numerically indicate impact of the different repairman groups on performance of the systems. First, we set up the systems of mean-field equations for the supermarket models with repairable servers. Then we prove the asymptotic independence of the supermarket models through the operator semi-group and the mean-field limit. Furthermore, we show that the fixed points of the supermarket models satisfy the systems of nonlinear equations. Finally, we use the fixed points to give numerical computation for performer analysis, and provide valuable observations on model improvement. Therefore, this paper provides a new and effective method in the study of complex supermarket models.

cs.PF↗

arXiv2016

Queueing Analysis of a Large-Scale Bike Sharing System through Mean-Field Theory

The bike sharing systems are fast increasing as a public transport mode in urban short trips, and have been developed in many major cities around the world. A major challenge in the study of bike sharing systems is that large-scale and complex queueing networks have to be applied through multi-dimensional Markov processes, while their discussion always suffers a common difficulty: State space explosion. For this reason, this paper provides a mean-field computational method to study such a large-scale bike sharing steps: Firstly, a multi-dimensional Markov process is set up for expressing the states of the bike sharing system, and the empirical process of the multi-dimensional Markov process is given to partly overcome the difficulty of state space explosion. Based on this, the mean-field equations are derived by means of a virtual time-inhomogeneous M(t)/M(t)/1/K queue whose arrival and service rates are determined by the mean-field computation. Secondly, the martingale limit is employed to investigate the limiting behavior of the empirical process, the fixed point is proved to be unique so that it can be computed by means of a nonlinear birth-death process, the asymptotic independence of this system is discussed simply, and specifically, these lead to numerical computation of the steady-state probability of the problematic (empty or full) stations. Finally, some numerical examples are given for valuable observation on how the steady-state probability of the problematic stations depends on some crucial parameters of the bike sharing system.

math.PR↗

arXiv2016

A Unified Framework for Analyzing Closed Queueing Networks in Bike Sharing Systems

During the last decade bike sharing systems have emerged as a public transport mode in urban short trips in more than 500 major cities around the world. For the mobility service mode, many challenges from its operations are not well addressed yet, for example, how to develop the bike sharing systems to be able to effectively satisfy the fluctuating demands both for bikes and for vacant lockers. To this end, it is a key to give performance analysis of the bike sharing systems. This paper first describes a large-scale bike sharing system. Then the bike sharing system is abstracted as a closed queueing network with multi-class customers, where the virtual customers and the virtual nodes are set up, and the service rates as well as the relative arrival rates are established. Finally, this paper gives a product-form solution to the steady state joint probabilities of queue lengths, and gives performance analysis of the bike sharing system. Therefore, this paper provides a unified framework for analyzing closed queueing networks in the study of bike sharing systems. We hope the methodology and results of this paper can be applicable in the study of more general bike sharing systems.

cs.PF↗

arXiv2016

Nonlinear Markov Processes in Big Networks

Big networks express various large-scale networks in many practical areas such as computer networks, internet of things, cloud computation, manufacturing systems, transportation networks, and healthcare systems. This paper analyzes such big networks, and applies the mean-field theory and the nonlinear Markov processes to set up a broad class of nonlinear continuous-time block-structured Markov processes, which can be applied to deal with many practical stochastic systems. Firstly, a nonlinear Markov process is derived from a large number of interacting big networks with symmetric interactions, each of which is described as a continuous-time block-structured Markov process. Secondly, some effective algorithms are given for computing the fixed points of the nonlinear Markov process by means of the UL-type RG-factorization. Finally, the Birkhoff center, the Lyapunov functions and the relative entropy are used to analyze stability or metastability of the big network, and several interesting open problems are proposed with detailed interpretation. We believe that the results given in this paper can be useful and effective in the study of big networks.

eess.SY↗

arXiv2016

Reward Processes and Performance Simulation in Supermarket Models with Different Servers

Supermarket models with different servers become a key in modeling resource management of stochastic networks, such as, computer networks, manufacturing systems and transportation networks. While these different servers always make analysis of such a supermarket model more interesting, difficult and challenging. This paper provides a new novel method for analyzing the supermarket model with different servers through a multi-dimensional continuous-time Markov reward processes. Firstly, the utility functions are constructed for expressing a routine selection mechanism that depends on queue lengths, on service rates, and on some probabilities of individual preference. Then applying the continuous-time Markov reward processes, some segmented stochastic integrals of the random reward function are established by means of an event-driven technique. Based on this, the mean of the random reward function in a finite time period is effectively computed by means of the state jump points of the Markov reward process, and also the mean of the discounted random reward function in an infinite time period can be calculated through the same event-driven technique. Finally, some simulation experiments are given to indicate how the expected queue length of each server depends on the main parameters of this supermarket model.

cs.PF↗

arXiv2014

Block-Structured Supermarket Models

Supermarket models are a class of parallel queueing networks with an adaptive control scheme that play a key role in the study of resource management of, such as, computer networks, manufacturing systems and transportation networks. When the arrival processes are non-Poisson and the service times are non-exponential, analysis of such a supermarket model is always limited, interesting, and challenging. This paper describes a supermarket model with non-Poisson inputs: Markovian Arrival Processes (MAPs) and with non-exponential service times: Phase-type (PH) distributions, and provides a generalized matrix-analytic method which is first combined with the operator semigroup and the mean-field limit. When discussing such a more general supermarket model, this paper makes some new results and advances as follows: (1) Providing a detailed probability analysis for setting up an infinite-dimensional system of differential vector equations satisfied by the expected fraction vector, where "the invariance of environment factors" is given as an important result. (2) Introducing the phase-type structure to the operator semigroup and to the mean-field limit, and a Lipschitz condition can be obtained by means of a unified matrix-differential algorithm. (3) The matrix-analytic method is used to compute the fixed point which leads to performance computation of this system. Finally, we use some numerical examples to illustrate how the performance measures of this supermarket model depend on the non-Poisson inputs and on the non-exponential service times. Thus the results of this paper give new highlight on understanding influence of non-Poisson inputs and of non-exponential service times on performance measures of more general supermarket models.

cs.PF↗

arXiv2014

The Chaos of Propagation in a Retrial Supermarket Model

When decomposing the total orbit into $N$ sub-orbits (or simply orbits) related to each of $N$ servers and through comparing the numbers of customers in these orbits, we introduce a retrial supermarket model of $N$ identical servers, where two probing-server choice numbers are respectively designed for dynamically allocating each primary arrival and each retrial arrival into these orbits when the chosen servers are all busy. Note that the designed purpose of the two choice numbers can effectively improve performance measures of this retrial supermarket model. This paper analyzes a simple and basic retrial supermarket model of N identical servers, that is, Poisson arrivals, exponential service and retrial times. To this end, we first provide a detailed probability computation to set up an infinite-dimensional system of differential equations (or mean-field equations) satisfied by the expected fraction vector. Then, as N goes to infinity, we apply the operator semigroup to obtaining the mean-field limit (or chaos of propagation) for the sequence of Markov processes which express the state of this retrial supermarket model. Specifically, some simple and basic conditions for the mean-field limit as well as for the Lipschitz condition are established through the first two moments of the queue length in any orbit. Finally, we show that the fixed point satisfies a system of nonlinear equations which is an interesting networking generalization of the tail equations given in the M/M/1 retrial queue, and also use the fixed point to give performance analysis of this retrial supermarket model through numerical computation.

cs.NI↗

arXiv2013

A Computational Framework for the Mixing Times in the QBD Processes with Infinitely-Many Levels

In this paper, we develop some matrix Poisson's equations satisfied by the mean and variance of the mixing time in an irreducible positive-recurrent discrete-time Markov chain with infinitely-many levels, and provide a computational framework for the solution to the matrix Poisson's equations by means of the UL-type of $RG$-factorization as well as the generalized inverses. In an important special case: the level-dependent QBD processes, we provide a detailed computation for the mean and variance of the mixing time. Based on this, we give new highlight on computation of the mixing time in the block-structured Markov chains with infinitely-many levels through the matrix-analytic method.

math.PR↗

arXiv2013

Tail Probabilities in Queueing Processes

In the study of large scale stochastic networks with resource management, differential equations and mean-field limits are two key techniques. Recent research shows that the expected fraction vector (that is, the tailed probability vector) plays a key role in setting up mean-field differential equations. To further apply the technique of tailed probability vector to deal with resource management of large scale stochastic networks, this paper discusses tailed probabilities in some basic queueing processes including QBD processes, Markov chains of GI/M/1 type and of M/G/1 type, and also provides some effective and efficient algorithms for computing the tailed probabilities by means of the matrix-geometric solution, the matrix-iterative solution, the matrix-product solution and the two types of RG-factorizations. Furthermore, we consider four queueing examples: The M/M/1 retrial queue, the M(n)/M(n)/1 queue, the M/M/1 queue with server multiple vacations and the M/M/1 queue with repairable server, where the M/M/1 retrial queue is given a detailed discussion, while the other three examples are analyzed simply. Note that the results given in this paper will be very useful in the study of large scale stochastic networks with resource management, including the supermarket models and the work stealing models.

math.PR↗