Single Server Markovian Queue Research Articles

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.

STRATEGIC CUSTOMERS IN MARKOVIAN QUEUES WITH VACATIONS AND SYNCHRONIZED ABANDONMENT

We study impatient customers’ joining strategies in a single-server Markovian queue with synchronized abandonment and multiple vacations. Customers receive the system information upon arrival, and decide whether to join or balk, based on a linear reward-cost structure under the acquired information. Waiting customers are served in a first-come-first-serve discipline, and no service is rendered during vacation. Server’s vacation becomes the cause of impatience for the waiting customers, which leads to synchronous abandonment at the end of vacation. That is, customers consider simultaneously but independent of others, whether to renege the system or to remain. We are interested to study the effect of both information and reneging choice on the balking strategies of impatient customers. We examine the customers’ equilibrium and socially optimal balking strategies under four cases of information: fully/almost observable and fully/almost unobservable cases, assuming the linear reward-cost structure. We compare the social benefits under all the information policies.

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.

Impatient customers in Markovian queue with Bernoulli feedback and waiting server under variant working vacation policy

This paper deals with customers’ impatience behaviour for single server Markovian queueing system under K-variant working vacation policy, waiting server, Bernoulli feedback, balking, reneging, and retention of reneged customers. Using probability generating function (PGF) technique, we obtain the steady-state solution of the system. In addition, we prove the stochastic decomposition properties. Useful performance measures of the considered queueing system are derived. A cost model is developed. Then, the parameter optimisation is carried out numerically, using quadratic fit search method (QFSM). Finally, numerical examples are provided in order to visualize the analytical results.

Analysis of a Markovian queueing system with single working vacation and impatience of customers

In this paper, we study an infinite capacity single server Markovian queue with a single working vacation and reneging of impatient customers in the queue during working vacation period. Customers arrive to the system following a Poisson distribution. The server goes to vacation when the system is empty and stay in vacation for a random period of time that is exponentially distributed. During the working vacation period, the server still continue providing service with a slow service rate. After the completion of the vacation, the server returns back to the regular service period and continue providing service with the regular busy period service rate if there are one or more customers in the system or it will stay idle until a new customer arrives to the system. During working vacation, customers in the queue get impatient and renege from the system and the reneging time is assumed to follow an exponential distribution. The system is modeled as a quasi-birth-death process and the stationary probabilities of the model are obtained using probability generating function approach. Some numerical analysis is also carried out to show the effect of some of the parameters on some selected performance measures of the system.

Social Optimization and Pricing Strategies in Unobservable Queues with Delayed Multiple Vacations

The paper considers the single-server Markovian queues with delayed multiple vacations. Once the system becomes empty, the server will experience a changeover time to begin the vacation. By using the matrix-geometric solution method, we obtain the stationary probability distribution and the mean queue length under almost unobservable and fully unobservable cases. Based on the system status information and a linear reward-cost structure reflecting the desire of customers for service and their unwillingness for waiting, the social welfare function is determined. We also investigate optimal pricing strategies of the server under the ex-postpayment scheme and the ex-antepayment scheme, and we get the customer’s equilibrium strategy under optimal pricing strategy. Finally, we illustrate the effect of several parameters and information levels on socially optimal strategies and optimal social benefit by numerical examples.

Serve the shortest queue and Walsh Brownian motion

We study a single-server Markovian queueing model with $N$ customer classes in which priority is given to the shortest queue. Under a critical load condition, we establish the diffusion limit of the nominal workload and queue length processes in the form of a Walsh Brownian motion (WBM) living in the union of the $N$ nonnegative coordinate axes in $\mathbb{R}^{N}$ and a linear transformation thereof. This reveals the following asymptotic behavior. Each time that queues begin to build starting from an empty system, one of them becomes dominant in the sense that it contains nearly all the workload in the system, and it remains so until the system becomes (nearly) empty again. The radial part of the WBM, given as a reflected Brownian motion (RBM) on the half-line, captures the total workload asymptotics, whereas its angular distribution expresses how likely it is for each class to become dominant on excursions. As a heavy traffic result, it is nonstandard in three ways: (i) In the terminology of Harrison (In Stochastic Networks (1995) 1–20 Springer), it is unconventional, in that the limit is not a RBM. (ii) It does not constitute an invariance principle, in that the limit law (specifically, the angular distribution) is not determined solely by the first two moments of the data, and is sensitive even to tie breaking rules. (iii) The proof method does not fully characterize the limit law (specifically, it gives no information on the angular distribution).

Equilibrium customers strategies in the Markovian working vacation queue with setup times

In this paper, we research the customers' equilibrium behaviour in the single server Markovian queue with setup times and working vacation. In such an M/M/1 queueing system, the arriving customers' decision is whether to enter the system or balk based on the reward-cost structure, which includes their desire for service and their unwillingness to wait. We separately discuss the fully observable and fully unobservable cases. For each of case, we acquire the related equilibrium balking strategies of customers and the expected social benefits per time unit. Finally, we obtain some numerical examples to illustrate the effect of several parameters on the equilibrium and optimal strategy.

Analysis of a Single Server Markovian Queueing System with Working Vacations and Vacation Interruptions with Setup Times

Analysis of a Single Server Markovian Queueing System with Working Vacations and Vacation Interruptions with Setup Times

Equilibrium balking strategies in Markovian queues with a single working vacation and vacation interruption

In this work, we study customer equilibrium balking strategies in single-server Markovian queueing systems with a single working vacation and vacation interruption. That is, the server undertakes workload with a lower service rate rather than completely stops working during the vacation period. At the instant of completing the service during a vacation period, if there are any customers in the system, the server ends the vacation compulsively and returns to the normal working level, i.e. vacation interruption occurs. Upon arrival, the customers must make a decision whether to join or balk the system based on the information about the number of customers in the system and the server status, along with the reward–cost structure of the system. Accordingly, four cases with respect to different levels of information are introduced and the corresponding Nash equilibria are presented. In addition, the impact of the information levels, as well as of several system parameters, on the equilibrium thresholds and equilibrium joining probabilities is illustrated by numerical examples.

Strategic joining in an M/M/1 queue with risk-sensitive customers

To analyze a stochastic service system with customers choosing to join or balk upon arrival, we model the system as a single server Markovian queue with a quadratic utility function for customers. In contrast to classical models with risk-neutral customers, we focus on the queueing model with risk-sensitive ones and study customer strategies under individual interest equilibrium, server’s profit optimization, and social welfare optimization. The quadratic utility function allows us to take the risk and return tradeoff into account in analyzing customer joining strategies. We show that while some of the well-known results for the risk-neutral customer situation apply, others may fail to hold in some realistic risk-sensitive customer situations. Furthermore, we examine the queue length information effect on different performance measures from server’s profit and social welfare perspectives. A practical implication of this study is that managers of service systems should be very cautious about relying on classical stationary queueing analysis when customers are risk-sensitive.

A note on Bayesian estimation of traffic intensity in single-server Markovian queues

ABSTRACTIn queuing theory, a major interest of researchers is studying the behavior and formation process and analyzing the performance characteristics of queues, particularly the traffic intensity, which is defined as the ratio between the arrival rate and the service rate. How these parameters can be estimated using some statistical inferential method is the mathematical problem treated here. This article aims to obtain better Bayesian estimates for the traffic intensity of M/M/1 queues, which, in Kendall notation, stand for Markovian single-server infinity queues. The Jeffreys prior is proposed to obtain the posterior and predictive distributions of some parameters of interest. Samples are obtained through simulation and some performance characteristics are analyzed. It is observed from the Bayes factor that Jeffreys prior is competitive, among informative and non-informative prior distributions, and presents the best performance in many of the cases tested.

Equilibrium behaviour and social optimization in Markovian queues with impatient customers and variant of working vacations

We study customers’ equilibrium behaviour and social optimization in a single-server Markovian queue with impatient customers and variant of multiple working vacations, where the impatience is due to slow service rate. Under the variant of multiple working vacations, the server takes a working vacation as soon as the system gets empty. When an arriving customer joins the vacation system, it activates an impatience timer. If its patience timer expires before it gets service, the customer abandons the system, and never returns. The server is allowed to take at most J successive working vacations, if at the end of a working vacation the system remains empty. An arriving customer takes a decision on the basis of available information whether to join or to balk, which unifies wish for the service as well as reluctance to wait. We discuss equilibrium threshold strategies on the basis of linear reward-cost structure in the fully observable and fully unobservable cases. We present numerical results that establish the impact of the information level as well as various parameters on the equilibrium balking strategies and social benefits. The research outputs may be useful for decision makers to convey information to customers in net benefit maximization and for examining the corresponding social optimization problems.

Strategic customer behavior in a queueing system with delayed observations

We consider the single-server Markovian queue with infinite waiting space and assume that there exists a certain reward-cost structure that reflects the customers' desire for service and their disl...

Optimal revenue management in two class pre-emptive delay dependent Markovian queues

In this paper, we present a comparative study on the total revenue generated with pre-emptive and non pre-emptive priority scheduler for a fairly generic problem of pricing the server’s surplus capacity in a single server Markovian queue. The specific problem is to optimally price the server’s surplus capacity by introducing a new class of customers (secondary class) without affecting the pre-specified service level of its current customers (primary class) when pre-emption is allowed. Pre-emptive scheduling is used in various applications. First, a finite step algorithm is proposed to obtain global optimal operating and pricing parameters for this problem. These optimal operating and pricing parameters constitute a unique Nash equilibrium in a certain two player non cooperative game. We then describe the range of service level where pre-emptive scheduling gives feasible solution and generates some revenue while non pre-emptive scheduling has infeasible solution. Further, some complementary conditions are identified to compare revenue analytically for certain range of service level where strict priority to secondary class is optimal. Our computational examples show that the complementary conditions adjust in such a way that pre-emptive scheduling always generates more revenue. Theoretical analysis is found to be intractable for the range of service level when pure dynamic policy is optimal. Hence, extensive numerical examples are presented to describe different instances. It is noted in numerical examples that pre-emptive scheduling generates at least as much revenue as non pre-emptive scheduling. A certain range of service level is identified where improvement in revenue is quite significant.

Equilibrium and Socially Optimal Balking Strategies in Markovian Queues with Vacations and Sequential Abandonment

In this paper, we consider customers’ equilibrium and socially optimal behavior in a single-server Markovian queue with multiple vacations and sequential abandonments. Upon arrival customers decide for themselves whether to join or balk, based on the level of information available to them. During the server’s vacation, present customers become impatient and decide sequentially whether they will abandon the system or not upon the availability of a secondary transport facility. Assuming the linear reward-cost structure, we analyze the equilibrium balking strategies of customers under four cases: fully and almost observable as well as fully and almost unobservable. In all the above cases, the individual and social optimal strategies are derived. Finally, the dependence of performance measures on system parameters are demonstrated via numerical experiments.

Equilibrium and optimal balking strategies of customers in unobservable queues with double adaptive working vacations

This paper studies the customers’ equilibrium and socially optimal balking behaviour in some single-server Markovian queues with double adaptive working vacations. Once the system becomes empty, the server takes a working vacation with a low service rate. If there are customer arrivals during this period, a regular busy period begins as soon as he finishes the vacation. Otherwise, he takes another working vacation with a much lower service rate than that in the first working vacation. After completing the second vacation, the server either stays idle or begins a busy period. We discuss two types of unobservable queues: the almost unobservable queues and the fully unobservable queues, respectively, and arriving customers can’t observe system occupancy in both cases. For each type of queues, we get both the customers’ equilibrium and socially optimal balking strategies and make numerical comparisons between them. We observe that their positive and stable equilibrium strategy and optimal strategy are unique, and especially, the customers’ actual arrival rates in equilibrium in vacation states are not necessarily smaller than that in busy state in the almost unobservable queues. Moreover, we also find that whether the system information should be revealed to customers depends on the potential demand arrivals.