Stationary Queue Length Research Articles

Matched queues with matching batch pair [formula omitted]

In this paper, we develop the matrix-analytic method to discuss an interesting but challenging bilateral stochastic matching problem: A matched queue with matching batch pair (m,n) and two types of impatient customers, where the two types of customers arrive according to two independent Poisson processes. Once m A-customers and n B-customers are matched as a group, the m+n customers immediately leave the system. We show that this matched queue can be expressed as a novel bidirectional level-dependent quasi-birth-and-death (QBD) process whose analysis has its own interests, and specifically, computing the maximal non-positive inverse matrices of bidirectional infinite sizes by using the RG-factorizations. Based on this, we can provide an effective matrix-analytic method to deal with this matched queue, including the system stability, the average stationary queue lengths, the average sojourn times, and the departure process. We believe that the methodology and results developed in this paper can be applicable to studying more general matched queueing systems, which are widely encountered in many practical areas.

Performance analysis and ANFIS computing of an unreliable Markovian feedback queueing model under a hybrid vacation policy

This study covers the stationary analysis of an unreliable Markovian queueing system with feedback based on a hybrid vacation policy. Customers arrive at the station at random using a Poisson process. Service time is distributed exponentially. The necessary and sufficient requirements for system stability have been demonstrated. Many performance metrics of this queueing system, such as the stationary queue length distribution, have been obtained from the steady-state probability distribution of the queueing model and the steady-state queue length using the matrix geometric method. To establish the model’s performance indicators, a few formulas have also been developed. The Adaptive Neuro-Fuzzy Inference System (ANFIS) was used to develop rules from the input datasets and generate different performance indices. Finally, the impact of system parameters and cost analysis was studied using numerical examples.

Cost Optimization using Genetic Algorithm in Customers Intolerance Markovian Model with Working Vacation and Multiple Working Breakdowns

In this paper we consider a single server queueing model with working vacation and multiple working breakdowns. When on vacation, the server works at a different pace. Disturbances occur in the server due to multiple breakdowns. In working breakdowns server works at a different rate. During the time of interruption caused by working vacation and working breakdowns, the main server can find many implementations in operation. Both the server's lifespan and the time it takes to repair it are considered to be exponentially dispersed. Also, we have considered balking and reneging behaviours of customers. The stationary queue length distribution is computed using a matrix-analytic technique. Using Genetic Algorithm (GA) we optimize the cost function. The predicted length of a busy period, the expected length of a working vacation period, the expected length of a working breakdown period, the mean waiting time, and the average delay are all established. We compute numerical results to verify the analytical point of view. The effect of individual parameters is investigated using sensitivity analysis.

Asymptotic optimality of speed-aware JSQ for heterogeneous service systems

The Join-the-Shortest-Queue (JSQ) load-balancing scheme is known to minimise the average delay of jobs in homogeneous systems consisting of identical servers. However, it performs poorly in heterogeneous systems where servers have different processing rates. Finding a delay optimal scheme remains an open problem for heterogeneous systems. In this paper, we consider a speed-aware version of the JSQ scheme for heterogeneous systems and show that it achieves delay optimality in the fluid limit. One of the key issues in establishing this optimality result for heterogeneous systems is to show that the sequence of steady-state distributions indexed by the system size is tight in an appropriately defined space. The usual technique for showing tightness by coupling with a suitably defined dominant system does not work for heterogeneous systems. To prove tightness, we devise a new technique that uses the drift of exponential Lyapunov functions. Using the non-negativity of the drift, we show that the stationary queue length distribution has an exponentially decaying tail — a fact we use to prove tightness. Another technical difficulty arises due to the complexity of the underlying state-space and the separation of two time-scales in the fluid limit. Due to these factors, the fluid-limit turns out to be a function of the invariant distribution of a multi-dimensional Markov chain which is hard to characterise. By using some properties of this invariant distribution and using the monotonicity of the system, we show that the fluid limit is has a unique and globally attractive fixed point.

Double-ended queues with non-Poisson inputs and their effective algorithms

It is interesting and challenging to study double-ended queues with First-Come-First-Match discipline under customers’ impatient behavior and non-Poisson inputs. The system stability can be guaranteed by the customers’ impatient behavior, while the existence of impatient customers makes analysis of such double-ended queues more difficult or even impossible to find an explicitly analytic solution, thus it becomes more and more important to develop effective numerical methods in a variety of practical matching problems. This paper studies a block-structured double-ended queue, whose block structure comes from two independent Markovian arrival processes (MAPs), which are non-Poisson inputs. We show that such a queue can be expressed as a new bilateral quasi birth-and-death (QBD) process which has its own interest. Based on this, we provide a detailed analysis for both the bilateral QBD process and the double-ended queue, including the system stability, the queue size distributions, the average stationary queue lengths, and the sojourn time of any arriving customers. Furthermore, we develop three effective algorithms for computing the performance measures (i.e., the probabilities of stationary queue lengths, the average stationary queue lengths, and the average sojourn times) of the double-ended queue with non-Poisson inputs. Finally, we use some numerical examples in tabular and graphical to illustrate how the performance measures are influenced by some key system parameters. We believe that the methodology and results described in this paper can be applicable to deal with more general double-ended queues in practice, and develop some effective algorithms for the purpose of many actual uses.

ANFIS and Cost Optimization for Markovian Queue with Operational Vacation

In this paper, we investigate the M/M/1/N single server finite capacity Markovian queueing model with operational vacation and impatient behavior of the customers. To recover the server broken down during a busy period, M-threshold recovery policy along with set-up is used. Using the inflow and outflow transition rates, the state probabilities equations for different system states are constructed. For computing the stationary queue length, matrix-geometric analytic is performed. The sensitivity analysis is carried for the validation of the system performance measures. To examine the scope of the adaptive neuro-fuzzy inference system (ANFIS), computational results are presented using matric-geometric and ANFIS approaches.

NUMERICAL IMPLEMENTATION OF THE AUGMENTED TRUNCATION APPROXIMATION TO SINGLE-SERVER QUEUES WITH LEVEL-DEPENDENT ARRIVALS AND DISASTERS

This paper considers the computation of the stationary queue length distribution in single-server queues with level-dependent arrivals and disasters. We assume that service times follow a general distribution and therefore, we consider the stationary queue length distribution via an imbedded Markov chain. Because this imbedded Markov chain has infinitely many states, level dependence, and bidirectional jumps of levels, it is hard to compute the solution of the global balance equation exactly. We thus consider the augmented truncation approximation. In particular, we focus on the computation of the truncated state transition probability matrix of the imbedded Markov chain, assuming that the underlying continuous-time absorbing Markov chain during a service time is not uniformizable. Under some stability conditions, we develop a computational procedure for the truncated transition probability matrix, where the upper bound of errors owing to truncation can be set in advance. We also provide some numerical examples and demonstrate that our procedure works well.

An alternative approach in finding the stationary queue length distribution of a queueing system with negative customers

A single-server queueing system with negative customers is considered in this paper. One positive customer will be removed from the head of the queue if any negative customer is present. The distribution of the interarrival time for the positive customer is assumed to have a rate that tends to a constant as time t tends to infinity. An alternative approach will be proposed to derive a set of equations to find the stationary probabilities. The stationary probabilities will then be used to find the stationary queue length distribution. Numerical examples will be presented and compared to the results found using the analytical method and simulation procedure. The advantage of using the proposed alternative approach will be discussed in this paper.

Alternative approach based on roots for computing the stationary queue-length distributions in GIX/M(1,b)/1 single working vacation queue

The purpose of this paper is to present an alternative algorithm for computing the stationary queue-length and system-length distributions of a single working vacation queue with renewal input batch arrival and exponential holding times. Here we assume that a group of customers arrives into the system, and they are served in batches not exceeding a specific number b. Because of batch arrival, the transition probability matrix of the corresponding embedded Markov chain for the working vacation queue has no skip-free-to-the-right property. Without considering whether the transition probability matrix has a special block structure, through the calculation of roots of the associated characteristic equation of the generating function of queue-length distribution immediately before batch arrival, we suggest a procedure to obtain the steady-state distributions of the number of customers in the queue at different epochs. Furthermore, we present the analytic results for the sojourn time of an arbitrary customer in a batch by utilizing the queue-length distribution at the pre-arrival epoch. Finally, various examples are provided to show the applicability of the numerical algorithm.

Analysis of G-Queue with Pseudo-Fault and Multiple Working Vacations

This paper presents a new model of discrete time Geo/Geo/1 repairable queueing system with pseudo-fault, negative customers and multiple working vacations. The authors assume that system service may be interrupted by breakdown or pseudo-fault, this system may become disabled only when it is in a regular busy period, and negative customers adopt two types of typical killing strategies. In this paper, the authors know that the evolution of the system can be described by a two-dimensional Markov chain, and the two-dimensional Markov chain satisfies the condition of quasi birth and death chains. Based on the method of matrix-geometric solution, the authors obtain distributions for the stationary queue length in RCH and RCE strategy, respectively. Moreover, the reliability of the system is analyzed and the number of customers and waiting time of a customer in the system in steady state are obtained. The authors analyze the impact of two killing strategies on the system comparatively. This paper studies the individually and socially optimal behaviors of positive customers, and presents a pricing policy for positive customers, therefore, the authors obtain the socially optimal arrival rate. Various numerical results are provided to show the change of performance measures.

Decomposition of M/M/1 With Unreliable Service and a Working Vacation

We use the stationary distribution for the M/M/1 with Unreliable Service and aWorking Vacation (M/M/1/US/WV) given explicitly in (Patterson & Korzeniowski, 2019) to find a decomposition of the stationary queue length N. By applying the distributional form of Little's Law the Laplace-tieltjes Transform of the stationary customer waiting time W is derived. The closed form of the expected value and variance for both N and W is found and the relationship of the expected stationary waiting time as a function of the service failure rate is determined.

Infinite-server systems with Coxian arrivals

We consider a network of infinite-server queues where the input process is a Cox process of the following form: The arrival rate is a vector-valued linear transform of a multivariate generalized (i.e., being driven by a subordinator rather than a compound Poisson process) shot-noise process. We first derive some distributional properties of the multivariate generalized shot-noise process. Then these are exploited to obtain the joint transform of the numbers of customers, at various time epochs, in a single infinite-server queue fed by the above-mentioned Cox process. We also obtain transforms pertaining to the joint stationary arrival rate and queue length processes (thus facilitating the analysis of the corresponding departure process), as well as their means and covariance structure. Finally, we extend to the setting of a network of infinite-server queues.

Double-sided matching queues: Priority and impatient customers

We analyze a double-sided queue with priority that serves patient customers and customers with zero patience (i.e., impatient customers). In a two-sided market, high and low priority customers arrive to one side and match with queued customers on the opposite side. Impatient customers match with queued patient customers; when there is no queue, they leave the system unmatched. All arrivals follow independent Poisson processes. We derive exact formulae for the stationary queue length distribution and several steady-state performance measures.

A central limit theorem for a Markov-modulated infinite-server queue with batch Poisson arrivals and binomial catastrophes

This paper considers the stationary queue length distribution of a Markov-modulated MX/M/∞queue with binomial catastrophes. When a binomial catastrophe occurs, each customer is either removed with a probability or is retained with the complementary probability. We focus on our model under a heavy traffic regime because its exact analysis is difficult if not impossible. We establish a central limit theorem for the stationary queue length of our model in a heavy traffic regime. The central limit theorem can be used to approximate the queue length distribution of our model with large arrival rates.

Stationary queue length distribution of a continuous-time queueing system with negative arrival

This paper studies a continuous-time single-server infinite capacity queueing system with two types of customer: positive and negative customers. Positive customers are ordinary customers that receives service in the server. A negative customer that arrives to the system according to a Poisson process with rate γ will remove one positive customer at the head upon its arrival. By assuming that the interarrival time and service time distributions tend to a constant when time t goes to infinity, a set of equations will be derived by using an alternative approach to find the stationary queue length distribution. Numerical results obtained by the alternative approach will be compared to those obtained by the existing method and verified by the simulation procedure.

A central limit theorem for a Markov-modulated infinite-server queue with batch Poisson arrivals and binomial catastrophes

This paper considers the stationary queue length distribution of a Markov-modulated MX/M/∞ queue with binomial catastrophes. Binomial catastrophes occur according to a Poisson process, and each customer is removed with a probability and retained with the complementary probability upon the arrival of a binomial catastrophe. We focus on our model under a heavy traffic regime because its exact analysis is difficult if not impossible. We establish a central limit theorem for the stationary queue length of our model in the heavy traffic regime. The central limit theorem can be used to approximate the queue length distribution of our model with large arrival rates.

Analysis of a Discrete-Time Geo/G/1 Queue in a Multi-Phase Service Environment with Disasters

Abstract This paper considers a discrete-time Geo/G/1 queue in a multi-phase service environment, where the system is subject to disastrous breakdowns, causing all present customers to leave the system simultaneously. At a failure epoch, the server abandons the service and the system undergoes a repair period. After the system is repaired, it jumps to operative phase i with probability qi, i = 1, 2 ⋯, n. Using the supplementary variable technique, we obtain the distribution for the stationary queue length at the arbitrary epoch, which are then used for the computation of other performance measures. In addition, we derive the expected length of a cycle time, the generating function of the sojourn time of an arbitrary customer, and the generating function of the server’s working time in a cycle. We also give the relationship between the discrete-time queueing system to its continuous-time counterpart. Finally, some examples and numerical results are presented.

Analysis of an N–Policy GI/M/1 Queue in a Multi–Phase Service Environmentwith Disasters

Abstract This paper investigates an N-policy GI/M/1 queue in a multi-phase service environment with disasters, where the system tends to suffer from disastrous failures while it is in operative service environments, making all present customers leave the system simultaneously and the server stop working completely. As soon as the number of customers in the queue reaches a threshold value, the server resumes its service and moves to the appropriate operative service environment immediately with some probability. We derive the stationary queue length distribution, which is then used for the computation of the Laplace-Stieltjes transform of the sojourn time of an arbitrary customer and the server’s working time in a cycle. In addition, some numerical examples are provided to illustrate the impact of several model parameters on the performance measures.

Repairable Queue with Non-exponential Interarrival Time and Variable Breakdown Rates

This paper considers a single server queue in which the service time is exponentially distributed and the service station may breakdown according to a Poisson process with the rates γ and γ' in busy period and idle period respectively. Repair will be performed immediately following a breakdown. The repair time is assumed to have an exponential distribution. Let g(t) and G(t) be the probability density function and the cumulative distribution function of the interarrival time respectively. When t tends to infinity, the rate of g(t)/[1 – G(t)] will tend to a constant. A set of equations will be derived for the probabilities of the queue length and the states of the arrival, repair and service processes when the queue is in a stationary state. By solving these equations, numerical results for the stationary queue length distribution can be obtained.

A Highly Efficient Big Data Mining Algorithm Based on Stock Market

This article proposes a new algorithm which includes two stages. First, the Pearson correlation coefficient is used to calculate the similarity data, and the activity of stock money flow was calculated by combined the probability generating function (P.G.F.) of stationary waiting time and stationary queue length. Second, the discrete time Geo/G/1 queue with a Bernoulli gated service is proposed in calculating money flow by data mining of stock. The new algorithm could calculate data in real time, and each investor could see the real-time data mining graphics. Investors could establish their quantitative trading strategies based on the new money flow model. The proposed algorithm exploits the nature behind stock data. The experimental results show that the authors' approach can be automatically implemented by the investment strategy and know the future trend of the stock market, as well as the economic development of the region, according to the results of the stock data mining in a certain region.