Single Server Markovian Queue Research Articles

Algorithms for Determination of Sample Sizes for Bayesian Estimations in Single-Server Markovian Queues

Although the single-server Markovian queues are one of the simplest models in Queue Theory, they have important practical applications. One of the initial steps for its application includes the determination of the necessary sample sizes for an interval estimation of its parameters. This includes the traffic intensity, which is defined as the ratio between the arrival rate and the service rate. In this article, we develop Bayesian algorithms to determine the size of samples that must be collected to guarantee a pre-specified mean amplitude or mean coverage for the traffic intensity. These samples are composed of the number of arrivals during service times, a practical way to collect data. Monte Carlo simulations attest to the efficiency and effectiveness of the algorithms proposed.

Equilibrium analysis of a partially observable priority queue

This study considers a single-server Markovian queue with preemptive priority discipline. In front of a server, two queues are formed: a normal queue for ordinary customers and a priority queue for priority customers. When a customer arrives, they can observe the length of the normal queue, but not the priority queue. By considering the length of the normal queue and the expected waiting time in the priority queue, an arriving customer will choose either the priority queue by paying a priority fee or the normal queue for free. We investigate the customers’ equilibrium strategies. Specifically, we find the necessary and sufficient conditions for a threshold-type strategy to be an equilibrium. We also investigate the stability of equilibria. In addition, we examine the effect of information disclosure regarding the length of the priority queue on equilibrium behavior.

Equilibrium and social optimality in queues with service rate and customers’ joining decisions

ABSTRACT This paper treats a single server Markovian queueing system with changeable service rates and customer choice. When a customer arrives at an empty system, he is served by the server at a low service rate. This low service rate continues until the queue length reaches a threshold. Then the server switches to a high service rate to serve customers who join the system. The high service rate continues until the system becomes empty and a low service rate starts again. Under a service reward and waiting cost structure, customers decide to join or to balk based on their net utility of joining the system. The customers’ equilibrium and socially optimal joining strategies are studied under two information scenarios which are observable and unobservable queues, respectively. Such a model fits a stochastic service system where the server can make his service rate decision and customers can make their joining or baking decisions and thus has wide applications in service industries. Our focus is on examining the equilibrium performance of the queue under the interaction between the server’s service rate and customers’ joining decisions. We develop the steady-state distribution and performance measures of the system by formulating two-dimensional continuous-time Markov chains.

Classical and Bayesian estimations of performance measures in a single server Markovian queueing system based on arrivals during service times

The present study considers a single-server Markovian queueing system by observing the number of customer arrivals during the service time of a customer. We estimate the traffic intensity in this queueing system along with the average queue length and the expected number of customers in the system. We propose classical and Bayesian frameworks to estimate the parameters of interest. In the Bayesian setup, three forms of prior distributions for ρ and two loss functions are considered. Furthermore, the predictive distribution of the number of customer arrivals during the service time of a customer and the equal-tailed credible region of ρ are obtained. The aforementioned approaches are illustrated with numerical examples based on simulation studies.

Join, balk, or jettison? The effect of flexibility and ranking knowledge in systems with batch arrivals

Families that visit theme parks like Disneyland are debating on two aspects when they try to determine whether they prefer to join an activity of interest or would rather balk: (1) Is it better to join or balk as a group or allow the flexibility to get separated and jettison some members? and (2) Will it make any difference if they set a ranking among themselves beforehand as to who will be served first, second, etc.? We tackle the effect of flexibility and ranking knowledge and answer the above questions considering a single server Markovian queue with a generic batch size distribution. We consider two levels of flexibility: an inflexible setting, under which a family makes a common decision, and a flexible setting, under which each member makes her own decision. We pair each level with two sublevels with respect to the ranking knowledge: the case where the members set their ranking beforehand, and the case where they do not and assume they will be served according to a random order. We provide a full analytical characterization of the equilibrium and socially optimal strategies, and a comprehensive analysis of the intricate interplay among flexibility, ranking knowledge, and batch size variability, notions that do not exist in single‐ins arrival systems. We offer insights as to under which circumstances entity jettison is preferable. We investigate the corresponding implications of the above on system throughput and social welfare and determine which setting is preferable for the customers and which for the society, depending on the objective and the system dynamics. Further, we highlight key differences between single versus batch‐arrival models and provide high‐level guidelines for managers and policymakers as to how they can influence customer decisions so that they move toward the preferable setting (e.g., by revealing/concealing the ranking, encouraging flexibility, pricing, etc.).

Bayesian analysis for single-server Markovian queues based on the No-U-Turn sampler

Traffic intensity is one of the most critical parameters of single-server Markovian queues. This paper deals with the Bayesian inference for the M/M/1 queue by sampling from the posterior distribution. The No-U-Turn Sampler (NUTS) is a recently developed Markov Chain Monte Carlo (MCMC) algorithm, which is proposed to compute the traffic intensity by observing the number of customers in the system at the departure epoch. Numerical results show that the NUTS outperforms the other algorithms in the literature.

Analysis and optimisation of a M/M/1/WV queue with Bernoulli schedule vacation interruption and customer’s impatience

Abstract In this investigation, we establish a steady-state solution of an infinite-space single-server Markovian queueing system with working vacation (WV), Bernoulli schedule vacation interruption, and impatient customers. Once the system becomes empty, the server leaves the system and takes a vacation with probability p or a working vacation with probability 1 − p, where 0 ≤ p ≤ 1. The working vacation period is interrupted if the system is non empty at a service completion epoch and the server resumes its regular service period with probability 1 − q or carries on with the working vacation with probability q. During vacation and working vacation periods, the customers may be impatient and leave the system. We use a probability generating function technique to obtain the expected number of customers and other system characteristics. Stochastic decomposition of the queueing model is given. Then, a cost function is constructed by considering different cost elements of the system states, in order to determine the optimal values of the service rate during regular busy period, simultaneously, to minimize the total expected cost per unit time by using a quadratic fit search method (QFSM). Further, by taking illustration, numerical experiment is performed to validate the analytical results and to examine the impact of different parameters on the system characteristics.

Transient analysis and ANFIS computing of unreliable single server queueing model with multiple stage service and functioning vacation

A single server Markovian queueing system with multiple stage service and functioning vacation is examined where server is prone to breakdown in both busy period and vacation period. The service is provided in ‘M’ stages and all ‘M’ stages are essential for every job to finish the entire service process. The service provider can go for vacation only after the completion of the Mth stage of service of an individual customer. Runge–Kutta technique is employed for working out the transient probabilities of various states and performance indices of Markov model. A numerical experiment is carried out to investigate the sensitivity and associated cost of the system. The comparison of neuro-fuzzy results obtained by implementing a technique called ‘Adaptive neuro-fuzzy interface system (ANFIS)’ and the results obtained using Runge–Kutta method is carried out for validation purposes.

Equilibrium analysis of cloud user request based on the Markov queue with variable vacation and vacation interruption

This paper considers the equilibrium balking behavior of customers in a single-server Markovian queue with variable vacation and vacation interruption, where the server can switch across four states: vacation, working vacation, idle period, and busy period. Once the queue becomes empty, the server commences a working vacation and slows down its service rate. However, this period may be interrupted anytime by the vacation interruption. Upon the completion of a working vacation, the server takes a vacation in a probability-based manner and stops service if the system is empty. The system stays idle after a vacation until a new customer arrives. The comparisons between the equilibrium balking strategy of customers and the optimal expected social benefit per time unit for each type of queue are elucidated and the inconsistency between the individual optimization and the social optimization is revealed. Moreover, the sensitivity of the expected social benefit and the equilibrium threshold with respect to the several parameters as well as diverse precision levels is illustrated through numerical examples in a competitive cloud environment.

Impact of Bernoulli and non-Bernoulli working vacation schedule on single server Markovian queue with disaster

We discussed SSMQM (Single Server Markovian Queueing Model) with disaster. In Bernoulli trial, vacation period for working, after work the server either wait for the next service with probability p (or) go for a vacation with probability q. By letting the disaster to happen in busy state, time dependent probabilities of the model under Bernoulli and Non-Bernoulli are investigated under Laplace transform techniques and recurrence relation for generating function at power series.

Equilibrium Joining Strategies of Positive Customers in a Markovian Queue with Negative Arrivals and Working Vacations

We study the economic analysis of a single server Markovian queueing system with positive and negative customers and multiple working vacations. Both positive and negative customers arrive in the system according to a Poisson process. Upon arrival, positive customers acquire some system information and decide whether to join or to balk the system based on the acquired information and a linear cost-reward structure. Negative customers on arrival break the server and kill the positive customer in service. The server is immediately sent for repair, and no customers are allowed during a repair. The server takes multiple working vacations after serving all the positive waiting customers. We obtain the equilibrium strategies and social benefit of positive customers under four different information situations. Numerical experiments are presented to show the effects of model parameters and information levels on the equilibrium joining behavior of positive customers.

M/M/1 Queueing Model with Working Vacation and Two Type of Server Breakdown

In this paper the steady state probability distribution of the number of customers in single server Markovian queue is obtained by using matrix geometric approach with working vacation where server may breakdown in working vacation state as well as in busy state. The arrival of the customer is depends upon the server’s state and arrival follows FCFS discipline. The server is made available for rendering alternate service to customers. The inter-arrival time of the customers, service time, vacation duration, and lifetime and repair time of the server follows an exponential distribution. Numerical illustrations are made to examine the validity of analytical results. Sensitivity analysis is also made of in order to discover the outcome of various parameters on the system performance physical appearance.

ON MARKOVIAN QUEUES WITH SINGLE WORKING VACATION AND BERNOULLI INTERRUPTIONS

This paper considers the customers’ equilibrium and socially optimal joining–balking behavior in a single-server Markovian queue with a single working vacation and Bernoulli interruptions. The model is motivated by practical service systems where the service rate can be adjusted according to whether or not the system is empty. Specifically, we focus on a single-server queue in which the server's service rate is reduced from a regular to a lower one when the system becomes empty. This lower rate period is called a working vacation for the server which may represent that part of the service facility is under a maintenance process or works on other non-queueing job, or simply for saving the energy (for a machine server case). In this paper, we assume that the working vacation period is terminated after a random period or with probability p after serving a customer in a non-empty system. Such a system is called a queue with single working vacation and Bernoulli interruptions. Customers are strategic and can make choice of joining or balking based on different levels of system information. We consider four scenarios: fully observable, almost observable, almost unobservable, and fully unobservable queue cases. Under a reward-cost structure, we analyze the customer's equilibrium and social-optimal strategies. In addition, the effects of system parameters on optimal strategies are illustrated by numerical examples.

Impact of Bernoulli and non-Bernoulli working vacation schedule on single server Markovian queue with disaster

Impact of Bernoulli and non-Bernoulli working vacation schedule on single server Markovian queue with disaster

Taylor series solution of a single server queueing model with feedback

In multi-access systems, scheduling mechanism often require a proper feedback policy. In this article, we compute transient state probabilities of the classical single server Markovian queueing system with feedback. The method demonstrated in present study involves direct and practical steps for deriving explicit expression for queue size distribution. The present method avoids involvement of any special function (e.g., Bessel function), transformation (e.g., Laplace transform) or/and complex analysis. The resulting Taylor series for queue size distribution are proved to converge for all time under arbitrary initial condition. This approach is easy to understand and to extend for a more advanced queueing system otherwise another solution technique is very complicated when compared to this technique. For example, if we use Laplace technique then Rouch theorem will be used to find the roots and it is very difficult to obtain the roots by using any other method.

Optimal strategies and pricing analysis in M/M/1 queues with a single working vacation and multiple vacations

This paper considers the customers’ equilibrium and socially optimal joining-balking behavior in single-server Markovian queues with a single working vacation and multiple vacations. Arriving customers decide whether to join the system or balk based on the system states and a linear reward-cost structure, which incorporates the desire of customers for service and their dislike to wait. We consider that the system states are almost unobservable and fully unobservable, respectively. For these two cases, we first analyze the stationary behavior of the system, and get the equilibrium strategies of the customers and compare them to socially optimal balking strategies using numerical examples. We also study the pricing problem that maximizes the server’s profit and derive the optimal pricing strategy. Finally, the social benefits of the almost and fully unobservable queues are compared by numerical examples.

Analysis of cost model with queuing system under uncertainty

ABSTRACT In this study, we analyze the proficiency of a single-server Markovian queuing system M/M/1/N with unpleasant services and encouraged arrivals to compare the policies for service delivery, which is a context of social planning to improve the social welfare. Then, we examine the economic analysis of the cost in an uncertain environment. In this regard, the entry rate, the service rate, and the processing time are presumed as vague numbers and developed a concept that is an essential condition of vague set that is applied. In addition, taking into account to the expected costs, server maintenance costs, customer dissuasion cost, and financing costs in this investigation can enhance the managers capability in order to adopt the decision in organizations.

Dynamic Aspect of Two Dimensional Single Server Markovian Queueing Model With Multiple Vacations and Reneging

This work deals with “the dynamic analysis of Two-dimensional M/M/1 queuing model with reneging and multiple vacations”. Customers renege according to negative exponential distribution. Dynamic aspect is more appropriate in understanding the behaviour of the system. Two dimensions represent respectively the number of arrivals at, and departure from, the system at a given time. The system starts with “i units at the time “t=0”.Allowing server to take vacationmakes queuing model more feasible in studying the waiting time system appropriately. For example in ticket booking counter, messages to be delivered, patients form queue to have appointments before the clinic open or arrival of doctor. The solution for this model is obtained recursively with the help of Laplace transformation and results are achieved_without involving complex functions.

Equilibrium threshold strategies and social benefits in the fully observable Markovian queues with partial breakdowns and interruptible setup/closedown policy

ABSTRACT This paper studies the equilibrium balking strategies in the single-server Markovian queues with partial breakdowns and interruptible/setup closedown policy. That means the server may fail at any point in time, no matter the server is idle or busy and the service continues at a slower rate instead of stopping service totally. When the system becomes empty, the server ceases to operate by a closedown time. If a customer arrives during the closedown period, the system starts the service immediately without a setup time. After the closedown period, the server needs a setup time before providing the service upon arrival. We presume that the customers decide whether to join or balk immediately after their arrival based on the reward-cost structure of the system. The fully observable case is discussed and the equilibrium threshold strategies and the equilibrium social benefits for all customers are derived. A numerical example and a case study for a bank system are investigated in this regard. Eventually, the effects of several parameters on the equilibrium thresholds, social benefits, balking probability, probability of empty system and stationary distribution are expressed.

Transient Behavior of a Single-Server Markovian Queue with Balking and Working Vacation Interruptions

This paper studies the time-dependent analysis of an M/M/1 queueing model with single, multiple working vacation, balking and vacation interruptions. Whenever the system becomes empty, the server commences working vacation. During the working vacation period, if the queue length reaches a positive threshold value ‘k’, the working vacation of the server is interrupted and it immediately starts the service in an exhaustive manner. During working vacations, the customers become discouraged due to the slow service and possess balking behavior. The transient system size probabilities of the proposed model are derived explicitly using the method of generating function and continued fraction. The performance indices such as average and variance of system size are also obtained. Further, numerical simulations are presented to analyze the impact of system parameters.