Stationary Queue Length Distribution Research Articles

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.

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 stationary queue-length distributions of the BMAP/R(a,b)/1 queue

This paper investigates an infinite waiting space single-server queueing system in which customers arrive according to a batch Markovian arrival process. The server serves the customers in batches of maximum size ‘b’ with a minimum threshold value ‘a’. The service time of each batch follows R-type distribution, where R-type distribution represents a class of distributions whose Laplace-Stieltjes transform is rational or approximated rational function, which is independent of service batch size and the arrival process. We determine the queue-length distribution at departure epoch in terms of roots of the associated characteristic equation of the vector probability generating function. To show the strength and advantage of roots method, we also provide a comprehensive analysis of the queue-length distribution at departure epoch using the matrix-analytic method (MAM). To determine the queue-length distribution at random epoch, we obtain a relation between the queue-length distributions at departure and random epochs using the supplementary variable technique with remaining service time of a batch in service as the supplementary variable. We also derive the queue-length distributions at pre-arrival epoch of an arrived batch and post-arrival epoch of a random customer of an arrived batch. Some numerical results are demonstrated for different service-time distributions to show the key performance measures of the system.

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.

Continuous-type [formula omitted]-inventory model with an attached M/M/1 queue and lost sales

In this study, we consider an M/M/1 queuing model with an attached continuous-type inventory. Customers arrive in the system according to a Poisson process and are served individually on a first-come, first-served basis. The service times of customers are assumed to be independent and identically distributed exponential random variables. Along with the queue, there is an internal finite storage for the inventory and each service requires an exponentially distributed random amount H of inventory from the storage. Therefore, a customer leaves the system with H amount of item at his/her service completion time. The inventory is replenished by an outside supplier with a random lead time under an (s,Q) inventory control policy. We assume that the customers who arrive during stock-out periods are lost (lost sales). For this queuing-inventory system, we derive the stationary joint probability distribution of queue length and inventory level in explicit product form. Numerical examples followed by a cost model are also 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.

An analysis of the M $$^X$$ X /M/1 queue with multiple working vacations by GI/M/1 type Markov process

In this paper, we analyze an M/M/1 queue with batch arrival and multiple working vacations. We describe the queueing model by a special GI/M/1 type Markov process with infinite phases, and by the matrix analytic method, we not only give the stationary queue length distribution of the model, but also obtain the exact number of vacations completed by the server.

Stationary queue length of a single-server queue with negative arrivals and nonexponential service time distributions

In this paper, a single-server queue with negative customers is considered. The arrival of a negative customer will remove one positive customer that is being served, if any is present. An alternative approach will be introduced to derive a set of equations which will be solved to obtain the stationary queue length distribution. We assume that the service time distribution tends to a constant asymptotic rate when time t goes to infinity. This assumption will allow for finding the stationary queue length of queueing systems with non-exponential service time distributions. Numerical examples for gamma distributed service time with fractional value of shape parameter will be presented in which the steady-state distribution of queue length with such service time distributions may not be easily computed by most of the existing analytical methods.

Conveyor Merges in Zone Picking Systems: A Tractable and Accurate Approximate Model

Sequential zone picking systems are popular conveyor-based picker-to-parts order picking systems that divide the order picking area in work zones. When designing a zone picking system, it is important to know whether the throughput capability of the system can meet customer demand. However, the performance and maximum throughput capability of a zone picking system is largely determined by congestion and blocking that occur at the various conveyor merges in the system. In this paper we develop an analytical model to study the impact of conveyor merges in sequential zone picking systems. Because of finite buffers, blocking, recirculation, and merging, the resulting queueing model does not have a product-form stationary queue-length distribution which makes exact analysis practically infeasible. Therefore, we develop an approximate solution by using an aggregation technique and matrix-geometric methods to study the throughput capability of the system. The model is suitable to support rapid design of complex zone picking systems, in terms of number and length of zones, input and output buffer capacities, and storage allocation of products to zones to meet prespecified performance targets. Comparison of the approximation results to simulation show that for a wide range of parameters the mean relative error in the system throughput is typically less than 5%. The model accurately predicts the loss in throughput due to congestion and blocking at the merges, and can be used to allocate input and output buffer spaces to maximize the throughput capability of the system.

Computation of the moments of queue length in the [formula omitted] queue

The BMAP∕SM∕1 queue is the most general single-server queueing model which can be analysed analytically. Problem of computation of stationary distributions of queue length is solved in the literature. However, the problem of computation of the moments of these distributions is not enough addressed. This problem is more complicated than its particular case when the service times are independent identically distributed random variables due to reducibility of some involved matrices. In this communication, we solve this problem.

A single-server queue with batch arrivals and semi-Markov services

We investigate the transient and stationary queue length distributions of a class of service systems with correlated service times. The classical M^X/G/1 queue with semi-Markov service times is the most prominent example in this class and serves as a vehicle to display our results. The sequence of service times is governed by a modulating process J(t). The state of J(cdot ) at a service initiation time determines the joint distribution of the subsequent service duration and the state of J(cdot ) at the next service initiation. Several earlier works have imposed technical conditions, on the zeros of a matrix determinant arising in the analysis, that are required in the computation of the stationary queue length probabilities. The imposed conditions in several of these articles are difficult or impossible to verify. Without such assumptions, we determine both the transient and the steady-state joint distribution of the number of customers immediately after a departure and the state of the process J(t) at the start of the next service. We numerically investigate how the mean queue length is affected by variability in the number of customers that arrive during a single service time. Our main observations here are that increasing variability may reduce the mean queue length, and that the Markovian dependence of service times can lead to large queue lengths, even if the system is not in heavy traffic.

Unreliable M/M/1/1 retrial queues with set-up time

We consider a single-server retrial queue with set-up time and server breakdown. There is no waiting space in front of the server, and customers who cannot occupy the server upon arrival will go into a retrial orbit and they retry independently of each other after an exponentially distributed time. To save power, the server is turned off immediately after a service if there is no customer in the orbit. If the orbit is not empty, the server will stay idle to wait a customer coming from the external or from the orbit. A newly coming customer can reactivate the off server, and the server needs some set-up time to work. The server may fail to restart. Once failed, it will be repaired immediately and the repair time is exponentially distributed. Two models are considered according to whether the server can be perfectly repaired or not. The first model is concerned with imperfect repair and the server may still fail to activate after repair. In the second model, the server is perfectly repaired. In both models, we get the explicit expressions of stationary distribution of queue length in the system. Cost minimization is studied numerically and several numerical examples are presented.