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We study a queueing system of an unreliable server providing services to customers with two different service rates. When the system is empty of customers, the server takes a vacation and provides ...

In this study, two servers namely server1 and server2 with working vacations are considered where one server is exclusively used for high priority customers and another for low priority customers. The modified working vacation is considered for server1 and delayed working vacation for server2 which is a main feature of this model. A high priority customer demands both item and service whereas a low priority customer demands only service. Items are replenished under ( s , Q ) ordering policy. In this system, the arrival of both types of customers is of independent Poisson processes and the service times of both types of customers are independent exponential distributions in which the service rates of both servers differ in regular service and working vacation time as well. The joint probability distribution of the inventory level, the statuses of both servers, the number of high priority customers in queue 1 and low priority customers in queue 2 are found to be in a steady state. Also, the distributions of waiting time of high priority and low priority customers are individually analyzed by Laplace–Stieltjes transform. The various measures of system performance in the steady state are obtained. The consequences are exemplified with numerical evidences. Mainly, some evidences portray the advantages of the feature of modified working vacation of the model compared to the features, like, simply vacations and non-delayed working vacations.

This article analyzes an infinite buffer discrete-time single server queueing system with variant working vacations in which customers arrive according to a geometric process. As soon as the system becomes empty, the server takes working vacations. The server will take a maximum number K of working vacations until either he finds at least on customer in the queue or the server has exhaustively taken all the vacations. The service times during regular busy period, working vacation period and vacation times are assumed to be geometrically distributed. The probability generating function of the steady-state probabilities and the closed form expressions of the system size when the server is in different states have been derived. In addition, some other performance measures, their monotonicity with respect to K and a cost model are presented to determine the optimal service rate during working vacation.

This paper concerns the cost optimisation analysis of a discrete-time finite-capacity multiserver queueing system with Bernoulli feedback, synchronous multiple and single working vacations, balking, and reneging during both busy and working vacation periods. A reneged customer can be retained in the system by employing certain persuasive mechanism for completion of service. Using recursive method, the explicit expressions for the stationary state probabilities are obtained. Various system performance measures are presented. Further, a cost model is formulated. Then, the optimization of the model is carried out using quadratic fit search method (QFSM). Finally, the impact of various system parameters on the performance measures of the queueing system is shown numerically.

This paper analyses an M/M/c queueing system with variant working vacations. The service times during regular busy periods, working vacation periods and vacation times are exponentially distributed...

This paper compares the performance of exhaustive and nonexhaustive M/M/1/N queues with working vacation and threshold policy. In an exhaustive queue, the server slows down its service rate only when no customers exist in the system, and turns to normal service until the number of customers achieves a threshold. However, in a nonexhaustive queue, the server switches service rate between a low and a high value depending on system congestion. To get equilibrium arrival rate of customers and social welfare for the two types of queues, we first derive queue length distributions and expected busy circle. Then, by making sensitivity analysis of busy circle, system cost, arrival rate and optimal social welfare, we find that customers tend to join exhaustive queues instead of nonexhaustive queues, and the optimal threshold in an exhaustive queue is probably inconsistent with the one in a nonexhaustive queue. Moreover, in general, whether to consider system cost or not in social welfare will obviously affect the tendencies of optimal arrival rate and optimal social welfare with the threshold and system capacity for the two types of queues, especially for the nonexhaustive queues, and then affect the final decisions of social planner or system manager.

In this paper, we consider the impatient customers in M/M/1 queueing model under variant working vacation policy. The customer’s impatience is due to its arrival during a working vacation period, where the service rate of the customer is lower than a normal busy period. If the system is non-empty when the server returns from the working vacation, the server resumes the normal service period. Otherwise, the server will take successive working vacations till it reaches the maximum number of K working vacations and then the server remains idle until the next arrival. Closed-form probabilities are obtained by using the identities involving beta functions and degenerate hypergeometric functions, and the performance measures of the system are derived using generating functions. The stochastic decomposition structures of the mean queue length and mean waiting time are verified. The effects of the system parameters on some performance measures had been numerically illustrated.

An \(M/M/1\) queueing model with multiple exponential working vacation and threshold policy is considered. Here, the server will go for a working vacation after completing the service for all waiting customers in the system. During the working vacation, the service rate in which customers are served is slow compared to normal service rate. The server will move to a busy state from working vacation state when he finds \(N\) or more number of customers in the system. In this paper, we derived explicit analytical expressions of time-dependent probabilities of the model explained above.

A single server queue with Poisson arrival and exponential service times subject to a policy of single working vacation with customer impatience is considered. The service times are different for an active phase and a working vacation phase. The customer is allowed to leave the system during the working vacation phase. Catastrophes, when they occur, wipe out the system which results in the system being inactive for a random period of time. Explicit expressions for the transient probabilities of the close-down period, maintenance state, active state, working vacation state and system size for active phase and working vacation phase have been obtained. The corresponding steady-state analysis and performance measures are also obtained. The effects of various parameters on the system performance measures are studied using numerical examples.

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.

This paper studies an M/M/1 queue with single, multiple working vacations and customers’ variant impatient behavior. During working vacations, the arriving customers are served with slower service rate than the service rate of non-vacation period. An arriving customer, during working vacation, finds the system empty and gets his service immediately, does not become impatient. The only customers who are waiting for service, during working vacation, become impatient. The transient system size probabilities of this model are derived explicitly in the closed form using continued fraction. The time-dependent mean and variance are also computed. Numerical examples are provided to visualize the analytical results.

This paper analyzes an MX/M/1 variant working vacation queue with server breakdowns and repair. The server takes working vacations as soon as the system becomes empty. The service times dur...

This paper deals with the new type of retrial queueing system with working vacations and working breakdowns. The system may become defective by disasters at any point of time when the regular busy server is in operation. The occurrence of disasters forces all customers to leave the system and causes the main server to fail. At a failure instant, the main server is sent to the repair and the repair period immediately begins. As soon as the orbit becomes empty at regular service completion instant or disaster occurs in the regular busy server, the server goes for a working vacation and working breakdown (called lower speed service period). During this period, the server works at a lower service rate to arriving customers. Using the supplementary variable technique, we analyze the steady state probability generating function of system size. Some important system performance measures are obtained. Finally, some numerical examples and cost optimization analysis are presented.

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.

This investigation is concerned with the performance modeling of machining system operating under the admission control F-policy and server working vacation policy. The repair of failed machines is provided by an unreliable server, who also renders the service with the slower rate rather than completely terminating the service during the vacation period. The failed machines are allowed to enter the system till the system capacity (K) is full; then after failed machines are not allowed to join the system until the system size again decreases to the prespecified threshold level ‘F’. At that instant, the server takes start-up time in order to start allowing the failed machines to enter into the system for the repair job. Numerical method based on successive over-relaxation is applied to obtain the steady-state probabilities and various performance indices including the cost function. The numerical simulation is performed to explore the sensitivity of the system indices with respect to various parameters. Quasi-Newton method and direct search method are used to determine the optimal service rate and threshold parameter.

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,...

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 current study deals with machine repair problem with service interruptions and working vacation. The server being working vacation of random length when there are no machines to be repaired. The service process is not completely stopped during the vacation period. Instead, different repair rates are used during normal busy period and vacation state. The repair time, vacation time, failure times are assumed to be exponentially distributed. Various performance measures are calculated using Runge-Kutta method with MATLAB software.

This paper deals with the analysis of an M/G/1 retrial queue with starting failure and working vacation which arise in many industries. The single server provides service in both a normal busy period and in a vacation period, which turn as working vacations. We assume that, during the normal busy period, the server faces unreliability due to starting failure. In working vacation period the server will provide service lower rate rather than completely stopping the service. After the vacation completion instant, the server will come back to the normal working and starts serving the customers. The necessary and suffcient condition for the system to be stable have been derived. Using supplementary variable method, we obtain the stationary probability distribution and some performance measures. To validate the proposed model some numerical examples are illustrated. Further, we carry out some special cases for the proposed model.

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.

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.

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 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 models wherein having a backup server during the absences (caused by vacations and breakdowns) of the main server have found many applications in practice. Such services offered by a backup server can be viewed as the (main) server working during a vacation or during a breakdown period. A backup server offering services can be thought of as the main server working (at a reduced rate) on vacations/repairs. Using Neuts’ versatile point process for the arrivals and modeling the service times with phase type distributions, we propose a model that generalizes some of the previously published ones on working-vacation-breakdown-repair queues. We carry out the steady-state analysis and report interesting illustrative numerical examples. We also prove decomposition results for the rate matrix and the mean number in the system under some special cases.

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.

In this paper, we model a discrete-time renewal input queue with change over times and Bernoulli schedule vacation interruption under batch service (a, c, b) policy. The service, working vacation and change over times are geometrically distributed. At the instants of a service completion, the vacation is interrupted and the server is resumed to a regular busy period with probability 1 − q if there are c or more customers in the system, or continues the vacation with probability q. Employing the supplementary variable and recursive techniques, we have derived the steady state queue length distributions at various epochs. Some performance measures and a cost model have been presented and an optimum service rate has been obtained using geneticalgorithm. Numerical results showing the effect of the parameters of the modelon the key performance measures are presented.

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

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.

In this paper, we analyze an M/M/1 queueing system with working vacations and impatient customers. We examine the case that the customers' impatience is due to a working vacation. During a working vacation, customers are served at a slower than usual service rate and are likely to become impatient. Whenever a customer arrives in the system and realizes that the server is on vacation, the customer activates an ``impatience timer" which is exponentially distributed. If a customer's service has not been completed before the customer's timer expires, the customer leaves the queue, never to return. By analyzing this model, we derive the probability generating functions of the number of customers in the system when the server is in a service period and a working vacation, respectively. We further obtain the closed-form expressions for various performance measures, including the mean system size, the mean sojourn time of a customer served, the proportion of customers served and the rate of abandonment due to impatience. Finally, we present some numerical results to demonstrate effects of some parameters on these performance measures of the system.

Consider a Geo/Geo/1 retrial queue with working vacations and vacation interruption, and assume requests in the orbit try to get service from the server with a constant retrial rate. During the working vacation period, customers can be served at a lower rate. If there are customers in the system after a service completion instant, the vacation will be interrupted and the server comes back to the normal working level. 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.

We derive an explicit closed form of the stationary distribution of an M/M/1 queue with unreliable service and a working vacation. We also show that the work in (Patterson & Korzeniowski, 2018) can be obtained as a special case of this model. Future work remains to be done; specifically, it may be possible to use the explicit stationary distribution given here to decompose the queue length into the sum of independent random variables. Consequently, it may then be possible to utilize Little’s Law (Little, 1961) to decompose the customer waiting time as well.

This paper deals with steady state analysis of single server priority retrial queue with Bernoulli working vacation, where the regular busy server can be subjected to breakdown and repair. There are two types of customers are considered, which are priority customers and ordinary customers. As soon as orbit becomes empty, the server goes for a working vacation (WV). The server works at a lower service rate during working vacation period. If there are customers in the system at the end of each vacation, the server becomes idle and ready for serving new arrivals with probability p (single WV) or it remains on vacation with probability q (multiple WVs). Using the supplementary variable technique, we obtained the steady state probability generating functions for the system and its orbit. Important system performance measures, the mean busy period and the mean busy cycle are discussed. Finally, some numerical examples are presented.

Consider a single server retrial queueing system with preemptive priority service and single working vacation in which two types of customers arrive in a Poisson process with arrival rates λ1 for low and λ2 for high priority customers. We assume that regular service times follow an exponential distribution with parameters μ1 and μ2 correspondingly. The retrial is introduced for low priority customers only. During working vacation the server serve’s the arriving customers with lesser service rates µ3 and µ4 respectively. These service rates µ3 and µ4 follow an exponential distribution. However at any time the server may return from the working vacation with a working vacation rate α which follows the exponential distribution. The access from orbit to the service facility follows the classical retrial policy and the high priority customers will be governed by the pre-emptive priority service. This model is solved by using Matrix geometric Technique. Numerical study have been done in elaborate manner for finding the Mean number of customers in the orbit, Probabilities that server is idle, busy during working vacation and normal period.

We consider an M/M/R queue with vacations, in which the server works with different service rates rather than completely terminates service during his vacation period. Service times during vacation period, service times during service period and vacation times are all exponentially distributed. Neuts’ matrix–geometric approach is utilized to develop the computable explicit formula for the probability distributions of queue length and other system characteristics. A cost model is derived to determine the optimal values of the number of servers and the working vacation rate simultaneously, in order to minimize the total expected cost per unit time. Under the optimal operating conditions, numerical results are provided in which several system characteristics are calculated based on assumed numerical values given to the system parameters.

In this paper, we study the M/M/1 queue with working vacations and vacation interruptions. The working vacation is introduced recently, during which the server can still provide service on the original ongoing work at a lower rate. Meanwhile, we introduce a new policy: the server can come back from the vacation to the normal working level once some indices of the system, such as the number of customers, achieve a certain value in the vacation period. The server may come back from the vacation without completing the vacation. Such policy is called vacation interruption. We connect the above mentioned two policies and assume that if there are customers in the system after a service completion during the vacation period, the server will come back to the normal working level. In terms of the quasi birth and death process and matrix-geometric solution method, we obtain the distributions and the stochastic decomposition structures for the number of customers and the waiting time and provide some indices of systems.

In this paper, we consider a GI/Geo/1 queue with working vacations and vacation interruption. The server takes the original work at the lower rate rather than completely stopping during the vacation period. Meanwhile, we introduce vacation interruption policy: the server can come back to the normal working level once there are customers after a service completion during the vacation period, thus the server may not accomplish a complete vacation. Using matrix-geometric solution method, we obtain the steady-state distributions for the number of customers in the system at arrival epochs, and waiting time for an arbitrary customer. Meanwhile, we explain the stochastic decomposition properties of queue length and waiting time.

This paper deals with a single server retrial queueing system with working vacations and vacation interruption. There are three different types of customers are considered, which are priority customers, ordinary customers and negative customers. The priority customers do not form any queue and have an exclusive preemptive priority to receive their services over ordinary customers. The negative customer is arriving during the service time of any positive customer, will remove the positive customer from the service. If the interrupted customer is an ordinary customer, he will leave the system. As soon as the orbit becomes empty at the time of service completion, the server goes for a working vacation. The server works at a lower speed during a working vacation period. Using the supplementary variable technique, the steady-state probability generating a function of the system and its orbit are found. Some numerical examples are presented.

In this paper, we consider a cost optimisation analysis of M/G/1 retrial queue with k optional phases of service under multiple working vacations and vacation interruption, where each phase consists of an optional re-service. An arriving customer may balk the system at some particular times. When the orbit becomes empty at regular service completion instant, the server goes for a working vacation. During a working vacation period, the server works in lower service rate. The normal busy server may get to breakdown and the service channel will fail for a short interval of time. We analyse the steady state probability generating function for the system size by using the supplementary variable method. Some important system performance measures are obtained. Finally, some numerical examples and cost optimisation analysis are presented.

This paper investigates queue with working vacation and vacation interruptions with k-phases essential service. All k-phases are essential for each customer to complete the whole service process. According to working vacation, the sever works at a lower rate rather than completely stopping the service during the vacation period. Various performance measures are determined using matrix geometric method (MGM). Moreover, numerical results have also been provided. Finally, conclusion has been given.

In this work, we consider a single server retrial queueing system with multiple working vacations and vacation interruption, where the regular busy server is subjected to breakdown due to the arrival of negative customers. After the completion of regular service the positive customers may demand re-service of the previous service without joining the orbit or may leave the system. When the orbit becomes empty at service completion instant for a positive customer; the server goes for multiple working vacations. The steady state probability generating function for the system size and orbit size are obtained by using the method of supplementary variable technique. Some important system performance measures and the mean busy period are obtained. The conditional stochastic decomposition law is shown for good for this model. Finally, the effects of various parameters on the system performance are analysed numerically.