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This paper considers an infinite buffer renewal input queue with multiple working vacation policy wherein customers are served by a single server according to general bulk service (a, b)-rule (1 ≤ a ≤ b). If the number of waiting customers in the system at a service completion epoch (during a normal busy period) is lower than ‘a’, then the server starts a vacation. During a vacation if the number of waiting customers reaches the minimum threshold size ‘a’, then the server starts serving this batch with a lower rate than that of the normal busy period. After completion of a batch service during working vacation, if the server finds less than ‘a’ customers accumulated in the system, then the server takes another vacation, otherwise the server continues to serve the available batch with that lower rate. The maximum allowed size of a batch in service is ‘b’. The authors derive both queue-length and system-length distributions at pre-arrival epoch using both embedded Markov chain approach and the roots method. The arbitrary epoch probabilities are obtained using the classical argument based on renewal theory. Several performance measures like average queue and system-length, mean waiting-time, cost and profit optimization are studied and numerically computed.

In this paper we consider an M/G/1 queueing model with single working vacation and vacation interruption. By studying the spectral properties of the corresponding operator we obtain that the time-dependent solution of the model strongly converges to its steady-state solution.

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.

Optimal scheduling of working vacation (WV) in queueing system is complex because of the stochasticity of arrival-service process. In this study, we suggest a neural network based scheduling (NNS) model because the neural network has the capability of handling the complexity of the system. We focus on the MX/M/1/N system where X is a random variable and the vacation time is dependent on the operation time. The neural network is used to estimate the performance of each action, and the scheduling rule is made from this estimation. Since the target of neural network to train is not obtainable, we contrived a mathematical model to feed the neural network with target. The experimental results show that estimated value from neural network follows the trend of true value, and the performance of NNS was shown to outperform a common exhaustive vacation (EV) scheduling baseline in most cases. We also identified the settings in which we can expect our NNS to achieve high performance.

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.

This paper analyses an M / M / 1 queueing system with a variant of working vacations wherein an arriving customer may balk. The service times during regular busy periods, working vacation periods and vacation times are exponentially distributed and are mutually independent. In working vacation periods, the customer may renege due to impatience and the impatient timer follows an exponential distribution. We derive the probability generating function of the steady-state probabilities and obtain the closed form expressions of the system size. In addition, some performance measures and the sojourn time of a tagged customer are derived.

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,...

This paper deals with an M/M/1 queueing system with a single working vacation and reneging of impatient customers and fuzzy parameters. For this fuzzy queueing model, researches fetch performance measure of interest such as the mean number of customer in the system during working vacation, the mean number of customers in the system during regular busy period and the mean of reneging rate of the system. Finally numerical results are presented to show the effects of system parameter.

In this paper, a variant working vacation queueing system with second optional service, unreliable server, and retention of reneged customers is studied. This model has potential applications in tele-services, banks, manufacturing processes, etc. The expressions of the steady-state probabilities and the system sizes when the server is in different states are obtained using probability generating functions. This paper also discusses an optimisation problem under a given cost structure. Direct search method and quadratic fit search method (QFSM) are applied to obtain the optimum number of working vacations and optimum service rates during regular busy period and working vacation period, respectively.

The loss of a node in a computing network, either for unplanned reasons, such as hardware failure, or planned outages, such as upgrades, can result in degraded computing network performance or loss of redundancy. To reduce this possibility, a spare node is kept powered on and visible on the network. In this paper, we study the Markovian warm-spare nodes provisioning computing network where spare nodes may also be predisposed to failure in switching from standby state to the operating state. In addition to switching failure and common cause failure, the realistic and economical maintenance server’s modified multiple working vacation policy and failed nodes’ synchronized reneging are also considered. Reliability characteristics of the computing network for I/O operations have been derived using transient-state probabilities which have been computed using the theory of the Quasi-Birth-and-Death process, Laplace transforms, Eigenvalue and Eigenvector. The critical analysis of reliability characteristics has also been done and the paper has been enriched with numerical results in the form of tables and graphs to provide a glance at the investigation. The concluding remarks and future scope have also been included.

An M/M/1 queue with working vacation state has been considered. If the system is in busy state, it functions as a single server Markovian queue. When it is on vacation, again it functions as a single server Markovian queue but with different arrival and service rates. In addition, during busy period, the arrival and service are generated by K distinct randomly varying environments. The server takes vacation whenever there are no one in the system. But, the vacationing server serves at a lower rate as an arrival occur. The vacation policy is multiple vacation policy and the vacation period follows negative exponential. For this model using Matrix-Geometric method the probability distribution of number of customers in the queue in steady state has been obtained. Some illustrative examples are also provided.

In this paper, we consider a bulk arrival retrial queue with non - persistent customers and exponentially distributed multiple working vacation. If the primary customer finds the server busy then the customer becomes impatient and may leave the system forever without service. After service completion epoch there is no customers in the orbit then the server takes vacation. During the vacation period, customers can be served at a lower rate. We obtain the probability generating function for the number of customers and the average number of customers in the orbit. Also, we discuss some particular cases of the model.

We consider a bulk arrival retrial queue with orbital search and exponentially distributed multiple working vacation. After each service completion, the server searches for customers in the orbit. During the vacation period, customers can be served at a lower rate. Using supplementary variable method, we obtain the probability generating function for the number of customers in the orbit. Some particular cases are discussed.

An M/M/1 retrial queue with working vacation interruption is considered. Upon the arrival of a customer, if the server is busy, it would join the orbit of infinite size. The customers in the orbit will try for service one by one when the server is idle under the classical retrial policy with retrial ratenα, wherenis the size of the orbit. During a working vacation period, if there are customers in the system at a service completion instant, the vacation will be interrupted. Under the stable condition, the probability generating functions of the number of customers in the orbit are obtained. Various system performance measures are also developed. Finally, some numerical examples and cost optimization analysis are presented.

We study the paper of Gao and Liu (2013). In the paper, the authors pay little attention to the customer’s sojourn time in their model although both the queue length and the busy cycle are seriously analyzed. The objective of this note is to derive the sojourn time distribution of Gao and Liu’s model without considering Bernoulli schedule. A simple numerical example is also given to demonstrate our result.

This paper investigates equilibrium threshold balking strategies of customers in a renewal input batch arrival queue with multiple and single working vacation of the server. The vacation period, service period during normal service and vacation period are considered to be independent and exponentially distributed. Upon arriving, the customers decide whether to join or balk the queue based on observation of the system-length and status of the server. The waiting time in the system is associated with a linear cost-reward structure for estimating the net benefit if a customer wishes to participate in the system. Equilibrium customer strategy is studied under four cases: fully observable, almost observable, almost unobservable and fully unobservable. Using embedded Markov chain approach and system cost analysis, we obtain the equilibrium threshold. The analysis of unobservable cases is based on the roots of the characteristics equations formed using the probability generating function of embedded pre-arrival epoch probabilities. Equilibrium balking strategy may be useful in quality of service for EPON (ethernet passive optical network) as a multiple working vacation model and accounting through gatekeeper based H.323 protocols as a single working vacation model.

This paper analyzes a batch arrival infinite-buffer single server queueing system with variant working vacations in which customers arrive according to a Poisson process. As soon as the system becomes empty, the server takes working vacation. The service rate during regular busy period, working vacation period and vacation times are assumed to be exponentially distributed. We derive the probability generating function of the steady-state probabilities and obtain the closed form expressions of the system size when the server is in different states. In addition, we obtain some other performance measures and discuss their monotonicity and a cost model is formulated to determine the optimal service rate during working vacation.

This paper studies a cold standby repairable system with working vacations and vacation interruptions. The repairman׳s multiple vacation, the working vacation policy and the vacation interruption policy are considered simultaneously. The lifetime of component follows a Phase-type (PH) distribution. The repair time in the regular repair period and the working vacation period are other two Phase-type distributions with different representations, and the successive vacation times are governed by a Markovian arrival process (MAP). For this system, a vector-valued Markov process is constructed. We obtain several important performance measures for the system in transient and stationary regimes applying the matrix-analytic method. Finally, a numerical example is given to illustrate the results obtained in the paper.

This paper studies a Markovian queue with multiple working vacations and controlled vacation interruption. If there are at least N customers waiting upon completion of a service at a lower rate, the vacation is interrupted and the server returns to the system to resume the normal working level. Otherwise, the server continues the vacation until the system is non-empty after a vacation ends or there are at least N customers after a service ends. Moreover, the variable arrival rate of the customers is taken into account. Under such assumptions, by using the quasi-birth-and-death process, the matrix- geometric method and the difference equation theory, the steady-state queue length distribution along with various performance measures are derived. Additionally, under a certain cost structure, the optimal threshold N* that minimizes the long-run expected cost function per unit time is numerically determined.

In this investigation, a priority retrial queue with working vacations and negative customers is addressed. The priority clients don’t shape any line and have an elite preemptive priority to get their services over normal customers. When the orbit is noticeably empty at the season of service consummation, the server takes for a working vacation. In working vacation period, the server works at a lower rate of service. Utilizing the supplementary variable technique (SVT), the probability generating function (PGF) of the system capacity is found. Some important special cases are discussed.

This paper concentrates on single server bulk arrival queue system with batch size dependent service and working vacation. The server provides service in two service modes depending upon the queue length. The server provides single service if the queue length is at least ‘a’. On the other hand the server provides fixed batch service if the queue length is at least ‘k’ (k > a). Batch service is provided with some fixed batch size ‘k’. After completion of service if the queue length is less than ‘a’ then the server leaves for working vacation. During working vacation customers are served with lower service rate than the regular service rate. Service during working vacation also contains two service modes. For the proposed model probability generating function of the queue length at an arbitrary time will be obtained by using supplementary variable technique. Some performance measures will also be presented with suitable numerical illustrations.This paper concentrates on single server bulk arrival queue system with batch size dependent service and working vacation. The server provides service in two service modes depending upon the queue length. The server provides single service if the queue length is at least ‘a’. On the other hand the server provides fixed batch service if the queue length is at least ‘k’ (k > a). Batch service is provided with some fixed batch size ‘k’. After completion of service if the queue length is less than ‘a’ then the server leaves for working vacation. During working vacation customers are served with lower service rate than the regular service rate. Service during working vacation also contains two service modes. For the proposed model probability generating function of the queue length at an arbitrary time will be obtained by using supplementary variable technique. Some performance measures will also be presented with suitable numerical illustrations.

This paper presents the analysis of a renewal input multiple working vacations queue with balking, reneging and heterogeneous servers. Whenever the system becomes empty the second server leaves for a working vacation whereas the first server remains idle in the system. During a working vacation the second server provides service at a slower rate rather than completely stopping service. The steady-state probabilities of the model are obtained using supplementary variable and recursive techniques. Various performance measures of the model such as expected system length, expected balking rate, etc., have been discussed. Finally, some numerical results have been presented to show the effect of model parameters on the system performance measures.

Queueing phenomenon is prevalent in all areas of society and a reasonable queueing design can improve service efficiency and customer satisfaction. In order to adapt to the new requirements of web service system, this paper studies a Geo/Geo/1 retrial queue with working vacation interruption and nonpersistent customers. Firstly, a series of assumptions about the queueing system are put forward and the corresponding transition probability matrix is obtained. Then the stationary condition of the queueing system is derived. After that, the stationary distribution and the performance measures are obtained by using the matrix-analytic method. Finally, numerical analysis is conducted to discuss the effect of parameters on performance measures, furthermore, the performance of the queueing model is optimized to obtain the best parameters and the minimum cost.

We study a finite buffer N-policy GI/M(n)/1 queue with Bernoulli-schedule vacation interruption. The server works with a slower rate during vacation period. At a service completion epoch during working vacation, if there are at least N customers present in the queue, the server interrupts vacation and otherwise continues the vacation. Using the supplementary variable technique and recursive method, we obtain the steady state system length distributions at prearrival and arbitrary epochs. Some special cases of the model, various performance measures, and cost analysis are discussed. Finally, parameter effect on the performance measures of the model is presented through numerical computations.

This paper analyzes a discrete-time finite buffer GI/Geo/1 queue with multiple working vacations wherein the customers may balk due to impatience. The service times during working vacation and vacation times are assumed to be geometrically distributed. Embedded Markov chain and supplementary variable techniques have been adopted to obtain the steady-state system length distributions at pre-arrival and arbitrary epochs, respectively. Various performance measures of the model and waiting-time distribution are also presented. A variety of numerical results showing the effect of model parameters on key performance measures are demonstrated.

This paper investigates a discrete-time single server finite buffer queue with single working vacation wherein the customers decide either to join the queue or balk. The inter-arrival times, service times and vacation times are assumed to be independent and geometrically distributed. Using matrix method, the steady-state probabilities are obtained. Various performance measures of the model and average sojourn time are presented. An optimization problem is considered using particle swarm optimization (PSO) through a cost function. Further, maximum likelihood estimate (MLE) for the balking parameter is obtained. Numerical results are presented in the form of tables and graphs to show the effect of parameters on the system performance.

This paper analyses an M/G/1 retrial queue with working vacation and constant retrial policy. As soon as the system becomes empty, the server begins a working vacation. The server works with different service rates rather than completely stopping service during a vacation. We construct the mathematical model and derive the steady-state queue distribution of number of customer in the retrial group. The effects of various performance measures are derived.

The objective of this paper is to analyze a finite buffer M/M/2 working vacations queue with balking and reneging wherein the servers operate under a triadic (0,Q,N,M) policy. As soon as the system becomes empty, both the servers leave for working vacations with only one of the two servers being active during the vacation. The service times during the working vacation and the vacation times are assumed to be exponentially distributed. Various performance measures are discussed and cost analysis is carried out using the quadratic fit search method. Finally, using some numerical results we present the parameter effect on the performance measures of the model.

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.

This paper is concerned with the equilibrium balking strategies of customers in a Geo/Geo/1 queue with single working vacation. Instead of completely stopping service, the server works with a small probability during the working vacation period. As soon as no customers exist in the system, the server takes a single vacation. The customers decide for themselves whether to enter the system or balk based on a natural reward-cost structure, the information available about the status of the server, and the queue length on hand upon arrival. We obtain the equilibrium balking strategies in two cases: fully observable and fully unobservable cases, which depend on whether the customers know both the queue length and the state of the server or none of them. Finally, we present several numerical experiments that demonstrate the effect of some parameters on the equilibrium behavior.

This paper analyzes renewal input batch arrival queues with multiple and single working vacation policy. We obtain the steady state system-length distributions at pre-arrival and arbitrary epochs. The Laplace-Stieltjes transforms (LST) of the sojourn time distribution for the first customer and an arbitrary customer in an accepted batch have been derived. Also the probability density function of sojourn time distribution for the first customer in an accepted batch has been obtained. The proposed analysis is based on the roots of the characteristics equations formed using the probability generating functions (p.g.f.) of embedded pre-arrival epoch probabilities. Several performance measures such as the mean system-length and mean sojourn times of first and an arbitrary customer in an accepted batch are presented. The suggested queues have potential application in ethernet passive optical network (EPON).

This paper treats a bulk arrival queue with randomized working vacation policy. Whenever the system becomes empty, the server takes a vacation. During the vacation period, customers are to be served at a lower rate. Once the vacation ends, the server will return to the normal working state and begin to serve the customers in the system if any. Otherwise, the server either remains idle with probability p or leaves for another vacation with probability 1−p. This pattern continues until the number of vacations taken reaches J. If the system is empty at the end of the Jth vacation, the server will wait idly for a new arrival. By using supplementary variable technique, we derive the system size distribution at arbitrary epoch, at departure epoch and at busy period initial epoch, as well as some important system characteristics. Numerical examples are provided to illustrate the influence of system parameters on several performance measures.

Crowdsourcing has been used in different domains such as healthcare, computer science, environmental sciences, business and marketing. However, only recently, queueing models useful in the context of crowdsourcing have been studied. These studies involve queueing models of the type M/M/c, MAP/PH/1, and MAP/PH/c. In this paper we introduce vacation and working vacation in the context of MAP/PH/1 with crowdsourcing and highlight the qualitative aspects of the model through illustrative examples.

In this paper, we consider a single server feedback retrial queueing system with multiple working vacations and vacation interruption. An arriving customer may balk the system at some particular times. As soon as orbit becomes empty at regular service completion instant, the server goes for a working vacation. The server works at a lower service rate during working vacation (WV) period. After completion of regular service, the unsatisfied customer may rejoin into the orbit to get another service as feedback customer. The normal busy server may get to breakdown and the service channel will fail for a short interval of time. The steady state probability generating function for the system size is obtained by using the supplementary variable method. Some important system performance measures are obtained. Finally, some numerical examples and cost optimization analysis are presented. Keywords: Retrial queues, Feedback, Working vacation, Balking, Server breakdown

We two had year-long research leaves in Japan, working together fulltime with several Japanese plus Tony De Groot back in Livermore and Harald Posch in Vienna. We summarize a few of the high spots from that very productive year ( 1989-1990 ), followed by an additional fifteen years' work in Livermore, with extensive travel. Next came our retirement in Nevada in 2005, which has turned out to be a long-term working vacation. Carol narrates this part of our history together.

This paper treats a discrete-time batch arrival queue with geometrically working vacations and vacation interruption. The main purpose of this paper is to present a performance analysis of this system. For this purpose, we first derive the probability generating function (PGF) of the stationary system length at the departure epochs based on the embedded Markov chain technique. Next, we present the stationary distributions of the system lengths as well as some performance measures at random epochs by using the supplementary variable method. Thirdly, still based on the supplementary variable method we give a distribution for the number of the customers at the beginning of a busy period and give a stochastic decomposition formula for the PGF of the stationary system length at the departure epochs. Additionally, we investigate the relation between our discrete-time system and its continuous counterpart. Finally, some numerical examples show the influence of the parameters on some crucial performance c...

This paper analyzes a finite buffer multiple working vacations queue with balking, reneging, and vacation interruption underN-policy. In the working vacation, a customer is served at a lower rate and at the instants of a service completion; if there are at leastNcustomers in the queue, the vacation is interrupted and the server switches to regular busy period otherwise continues the vacation. Using Markov process and recursive technique, we derive the stationary system length distributions at arbitrary epoch. Various performance measures and some special models of the system are presented. Cost analysis is carried out using particle swarm optimization and quadratic fit search method. Finally, some numerical results showing the effect of model parameters on key performance measures of the system are presented.

Consider an M/M/1 retrial queue with collisions and working vacation interruption under N-policy. We use a quasi birth and death process to describe the considered system and derive a condition for the stability of the model. Using the matrix-analytic method, we obtain the stationary probability distribution and some performance measures. Furthermore, we prove the conditional stochastic decomposition for the queue length in the orbit. Finally, some numerical examples are presented.

An M/M/1 queueing system with second optional service and unreliable server is studied. We consider that the server works at different rate rather than being idle during the vacation period. The customers arrive to the system according to Poisson process with state dependent rates depending upon the server’s status. All customers demand the first essential service whereas only some of them demand the second optional service. A customer either may leave the system after the first essential service with probability (1-r) or at the completion of the first essential service go for second optional service with probability r (0≤r≤1). The server may breaks down according to Poisson process during the busy and working vacation duration. Both service times in vacation and in service period are exponentially distributed. The matrix geometric technique is used for the analysis of the concerned queueing system. The sensitive analysis is also performed to examine the variation of the system performance characteristics with various input parameters.