Single Vacation Research Articles

Analysis of MAP/G/1/N queue with two phases of service under single (multiple) vacation(s)

This paper deals with a finite buffer MAP/G/1 queue with an additional second phase of optional service with single and multiple vacation policies. Arrival of customers follows a Markovian arrival process (MAP). After completion of the first phase of essential service, the customer may or may not opt for second phase of service. If the server finds an empty system at a service completion epoch, he goes for single or multiple vacations. The queue length distributions at various epochs, such as pre-arrival, arbitrary and departure have been obtained using embedded Markov chain technique and supplementary variable technique. Waiting time of a customer in the queue is also obtained. Numerical illustrations are presented.

On optional deterministic server vacations in a batch arrival queueing system with a single server providing first essential service followed by one of the two types of additional optional service

We analyze a batch arrival queue with a single server providing first essential service (FES) followed by one of the two types of additional optional service (AOS). After completion of the FES, a customer has the option to leave the system or to choose one of the two types of AOS and as soon as a customer leaves (either after the FES or after completing one of the chosen AOS, the server may take a vacation or may continue staying in the system. The vacation times are assumed to be deterministic and the server vacations are based on Bernoulli schedules under a single vacation policy. We obtain explicit queue size distribution at a random epoch under the steady state. In addition, some important performance measures such as the steady state expected queue size and the expected waiting time of a customer at a random epoch are obtained. Further, some interesting particular cases are also discussed.

Queueing Analysis of Machine Repair Problem with Controlled Rates and Working Vacation Under F-Policy

This paper deals with the queueing characterization of a machine repair problem with single working vacation and controlled arrival of failed machines under F-policy. The server allows the failed machines to enter the system till full capacity K and thereafter the arrival of the failed machines is prohibited which continues till the queue length is reduced to the threshold value F. The server takes some time to start the repair of the failed machines. The life time of the machines, startup time and repair time of the server are exponentially distributed. When the system is empty, the server goes for a single vacation but it keeps on serving the failed units even during vacation at a different pace, and is said to be on working vacation. The matrix method has been used to obtain the steady state probabilities of the system states. Various performance characteristics have been derived. The numerical results have been obtained and shown in tables and graphs.

Analysis of a batch arrival queue with vacation policy and exceptional service

ABSTRACTWe consider the distributions of operating characteristics of an M[x]/G/1 queue under vacation policies, where the first customer of each busy period receives an exceptional service. When all the customers are served in the system exhaustively, the server deactivates and operates one of two vacation policies: (1) multiple vacation policy and (2) single vacation policy. We develop the performance measures for both systems. Finally, some numerical illustrations are also given. These two vacation models have potential applications in day-to-day life, such as post offices, banks, hospitals, etc.

A Novel Approach for Analyzing Single Buffer Queueing Systems with State-Dependent Vacation and Correlated Input Process under Four Different Service Disciplines

The author presents performance analysis of a single buffer multiple-queue system. Four different types of service disciplines (i.e., non-preemptive, pre-emptive repeat different, state dependent random polling and globally gated) are analyzed. His model includes correlated input process and three different types of non-productive time (i.e., switchover, vacation and idle time). Special cases of the model includes server with mixed multiple and single vacations, stopping server with delayed vacation and stopping server with alternating vacation and idle time. For each of the four service disciplines the key performance measures such as average customer waiting time, loss probability, and throughput are computed. The results permit a detailed discussion of how these performance measures depends on the customer arrival rate, the customer service time, the switchover time, the vacation time, and the idle time. Moreover, extensive numerical results are presented and the four service disciplines are compared with respect to the performance measure. Previous studies of the single buffer multiple-queue systems tend to provide separate analysis for the two cases of zero and nonzero switchover time. The author is able to provide a unified analysis for the two cases. His results generalize and improve a number of known results on single buffer multiple-queue systems. Furthermore, this method does not require differentiation while it is needed if one uses the probability generating function approach. Lastly, the author's approach works for all single buffer multiple-queue systems in which the next queue to be served is determines solely on the basis of the occupancy states at the end of the cycle time.

Analysis of an M/M/1 queue with vacations and impatience timers which depend on the server's states

We consider an M/M/1 queueing system with vacations and impatient customers. Whenever a customer arrives at the system, it activates an random ``impatience timer. If the customer's service has not been completed before the customer's impatience timer expires, the customer abandons the queue, and never returns. It is assumed that the impatience timer depends on the server's states. We analyze both multiple and single vacation scenarios and derive the probability generating functions of the number of customers in the system when the server is in vacation period and busy period. Then, we obtain explicit expressions for various performance measures such as the mean system sizes when the server is either on vacation or busy, the proportion of customers served, and the average rate of abandonments due to impatience. We present some numerical results for multiple vacation scenario to show the effects of the parameters of impatience timers on some performance measures. Finally, we show some inequalities on some performances under the single vacation policy and under multiple vacation policy.

A batch arrival multi phase queueing system with random feedback in service and single vacation policy

We study a batch arrival Poisson input, k phases heterogeneous services with randomly feedback in services. The first phase of service is essential for all customers, but with probability γ1 a tagged customer chose second phase, with probability α1 feedback to tail of original queue or with probability β1 = 1 − γ1 − α1 leave the system. Also, after completion of the second phase, with probability γ2 the customer leads to the third service, or with probability α2 feedback to tail of original queue, or leaves the system with probability β2 = 1 − γ2 − α2. Like that, in kth phase feedback with probability αk or leaves the system with probability 1 − αk. After completion of each phases, the server either goes for a vacation with probability 𝜃(0 ≤ 𝜃 ≤ 1), or may continue to serve the next unit with probability 1 − 𝜃, if any. Otherwise, it remains in the system until a customer arrives, which is single vacation policy. We assume restricted admissibility of arriving batches in which not all batches are allowed to join the system at all times. In this paper we derive the steady- state equations, PGF’s of the system, and measures of sysytem.

Analysis of an M/G/1 queue with N-policy, single vacation, unreliable service station and replaceable repair facility

This paper considers an M/G/1 repairable queueing system with N-policy and single vacation, in which the service station is subject to random breakdowns. Once the service station breaks down, it is repaired by a repair facility. Moreover, the repair facility may fail during the repair period which results in repair interruptions. Failed repair facility resumes repair after a random period of time. Applying the renewal process theory and the probability decomposition technique, the probability that the service station is broken, the rate of occurrence of breakdowns of the service station, the probability that the repair facility is being replaced, the rate of occurrence of failures of the repair facility and the stochastic decomposition property of the reliability measures are obtained. Following the construction of the long-run expected cost function per unit time, we numerically determined the optimal threshold N ∗ for minimizing the cost function.

Transient workload distribution in the $$M/G/1$$ M / G / 1 finite-buffer queue with single and multiple vacations

A finite-buffer \(M/G/1\)-type queueing system with exhaustive service and server vacations is considered. The single and multiple vacation policies are investigated separately. In the former case a single vacation is being initialized every time when the system becomes empty. After the vacation the server waits on standby for packets to start the service process. In the multiple vacation policy the server begins successive single vacation periods until at least one packet is present in the system. For both vacation policies systems of integral equations for the transient virtual waiting time \(v(t)\) distributions, conditioned by the numbers of packets present in the system at the start epoch, are found. The representations for the twofold Laplace transforms of the conditional distributions of \(v(t)\) are obtained and written in compact forms, convenient in numerical treatment. From these formulae stationary distributions of \(v(t)\) and their means can be computed via usual operations on transforms. Some numerical illustrations of theoretical results are attached as well.

Equilibrium Mixed Strategies in a Discrete-Time Markovian Queue Under Multiple and Single Vacation Policies

We study the Nash equilibrium behavior of customers in a discrete-time single-server queue under multiple vacation policy or single vacation policy. Every arriving customer either joins the queue or balks based on his or her service utility. Using a Quasi-Birth-Death Processes (QBD) model, we obtain the stationary distribution of the queue length via the Matrix-Geometric Solution method. We analyze the equilibrium mixed threshold strategies under two different vacation policies. By presenting numerical examples, we examine the impacts of system parameters on the equilibrium customer behavior and compare the single and multiple vacation policies in terms of the social welfare.

M/G/1 Vacation Queueing Systems with Server Timeout

We consider a single-server vacation queueing system that operates in the following manner. When the server returns from a vacation, it observes the following rule. If there is at least one customer in the system, the server commences service and serves exhaustively before taking another vacation. If the server finds the system empty, it waits a fixed time c. At the expiration of this time, the server commences another vacation if no customer has arrived; otherwise, it serves exhaustively before commencing another vacation. Analytical results are derived for the mean waiting time in the system. The timeout scheme is shown to be a generalized scheme of which both the single vacation and multiple vacations schemes are special cases, with c=∞ and c=0, respectively. The model is extended to the N-policy vacation queueing system.

A < V, p>-Policy Queueing System with Startup/Closedown Times

This study considers a <V, p>-policy Geo/G/1 queue with startup and closedown times. The < V, p>-policy operates as follows. On completion of the vacation, if there are customers in the queue, the server is either to activate with probability p or to stay dormant in the system with probability 1-p for waiting a new arrival; and if no customer presents in the queue, the server obeys classical single vacation policy to stay dormant in the system until one arrival. We give analytic expressions for the stationary system size distributions of the various states of the server, the length distributions of various state periods and the queue waiting time distribution of an arbitrary customer. Furthermore, we also demonstrate the stochastic decomposition property of the system size and waiting time in the queue. Some numerical examples of the mean system sizes and the mean waiting times in the queue with respect to startup/closedown times, startup times, closedown times, and both no startup and no closedown times, are presented. Finally, with the vacation of fixed length time (say T), the long run average cost function per unit time is analytically developed to determine the joint optimal values of T and p at a minimum cost.

Retrial queueing system with balking, optional service and vacation

In this study, we propose a single server retrial queueing system with balking, second optional service and single vacation. At the arrival epoch, if the server is busy, the arriving job join the orbit or balks the system whereas if the server is free, then the arriving job starts its service immediately. For each job, the server provides two phases of service. All the jobs demand the first essential service, whereas only some of the jobs demand for the second optional service. If the system is empty, then the server becomes inactive and begins a single vacation. If server comes back from the vacation, it does not go for another vacation even if the system is still empty at that time. The steady state distributions of the server state and the number of jobs in the orbit are obtained along with other performance measures. The effects of various parameters on the system performance are analyzed numerically. A general decomposition law for this retrial queueing system is established.

A finite-buffer queue with a single vacation policy: An analytical study with evolutionary positioning

Abstract In this paper, application of an evolutionary strategy to positioning a GI/M/1/N-type finite-buffer queueing system with exhaustive service and a single vacation policy is presented. The examined object is modeled by a conditional joint transform of the first busy period, the first idle time and the number of packets completely served during the first busy period. A mathematical model is defined recursively by means of input distributions. In the paper, an analytical study and numerical experiments are presented. A cost optimization problem is solved using an evolutionary strategy for a class of queueing systems described by exponential and Erlang distributions.

Reliability analysis of a k-out-of-n:G repairable system with single vacation

This paper analyzes a k-out-of-n:G repairable system with one repairman who takes a single vacation, the duration of which follows a general distribution. The working time of each component is an exponentially distributed random variable and the repair time of each failed component is governed by an arbitrary distribution. Moreover, we assume that every component is “as good as new” after being repaired. Under these assumptions, several important reliability measures such as the availability, the rate of occurrence of failures, and the mean time to first failure of the system are derived by employing the supplementary variable technique and the Laplace transform. Meanwhile, their recursive expressions are obtained. Furthermore, through numerical examples, we study the influence of various parameters on the system reliability measures. Finally, the Monte Carlo simulation and two special cases of the system which are (n-1)-out-of-n:G repairable system and 1-out-of-n:G repairable system are presented to illustrate the correctness of the analytical results.

Cost optimization of a repairable M/G/1 queue with a randomized policy and single vacation

This paper deals with the (p,N)-policy M/G/1 queue with an unreliable server and single vacation. Immediately after all of the customers in the system are served, the server takes single vacation. As soon as N customers are accumulated in the queue, the server is activated for services with probability p or deactivated with probability (1−p). When the server returns from vacation and the system size exceeds N, the server begins serving the waiting customers. If the number of customers waiting in the queue is less than N when the server returns from vacation, he waits in the system until the system size reaches or exceeds N. It is assumed that the server is subject to break down according to a Poisson process and the repair time obeys a general distribution. This paper derived the system size distribution for the system described above at a stationary point of time. Various system characteristics were also developed. We then constructed a total expected cost function per unit time and applied the Tabu search method to find the minimum cost. Some numerical results are also given for illustrative purposes.

On transient queue-size distribution in the batch-arrivals system with a single vacation policy

A queueing system with batch Poisson arrivals and single vacations with the exhaustive service discipline is investigated. As the main result the representation for the Laplace transform of the transient queue-size distribution in the system which is empty before the opening is obtained. The approach consists of few stages. Firstly, some results for a ``usual'' system without vacations corresponding to the original one are derived. Next, applying the formula of total probability, the analysis of the original system on a single vacation cycle is brought to the study of the ``usual'' system. Finally, the renewal theory is used to derive the general result. Moreover, a numerical approach to analytical results is discussed and some illustrative numerical examples are given.

Equilibrium Customer Strategies in the Geo/Geo/1 Queue with Single Working Vacation

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.

Multi-server machine repair problems under a (V, R) synchronous single vacation policy

This paper considers a machine repair problem with M operating machines and S standbys, in which R repairmen are responsible for supervising these machines and operate a (V,R) vacation policy. With such policy, if the number of the failed machines is reduced to R−V (R>V) (there exists V idle repairmen) at a service completion, these V idle servers will together take a synchronous vacation (or leave for other secondary job). Upon returning from the vacation, they do not take a vacation again and remain idle until the first arriving failed machine arrives. The steady-state probabilities are solved in terms of matrix forms and the system performance measures are obtained. Algorithmic procedures are provided to deal with the optimization problem of discrete/continuous decision variables while maintaining a minimum specified level of system availability.

Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy

In this paper, we consider an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy, where we examine the case that customer impatience is due to the servers' vacation. Whenever a system becomes empty, the server takes a vacation. However, the server is allowed to take a maximum number $K$ of vacations if the system remains empty after the end of a vacation. This vacation policy includes both a single vacation and multiple vacations as special cases. We derive the probability generating functions of the steady-state probabilities and obtain the closed-form expressions of the system sizes when the server is in different states. We further make comparisons between the mean system sizes under the variant vacation policy and the mean system sizes under the single vacation policy or the multiple vacation policy. In addition, we obtain the closed-form expressions for other important performance measures and discuss their monotonicity with respect $K$. Finally, we present some numerical results to show the effects of some parameters on some performance measures.