Discrete-time single-server finite-buffer queues under discrete Markovian arrival process with vacations

This paper treats a discrete-time single-server finite-buffer exhaustive (single- and multiple-) vacation queueing system with discrete-time Markovian arrival process (D-MAP). The service and vacation times are generally distributed random variables and their durations are integral multiples of a slot duration. We obtain the queue-length distributions at departure, service completion, vacation termination, arbitrary and prearrival epochs. Several performance measures such as probability of blocking, average queue-length and the fraction of time the server is busy have been discussed. Finally, the analysis of actual waiting time under the first-come-first-served discipline is also carried out.     

Stationary distributions and optimal control of queues with batch Markovian arrival process under multiple adaptive vacations

We consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple adaptive vacation policy. That is, the server serves until system emptied and after that server takes a random maximum number H different vacations until either he finds at least one customer in queue or the server have exhaustively taken all the vacations. The maximum number H of vacations taken by the server is a discrete random variable. We obtain queue-length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, post-departure and pre-service. The proposed analysis is based on the use of matrix-analytic procedure to obtain queue-length distribution at a post-departure epoch. Later we use supplementary variable method and simple algebraic manipulations to obtain the queue-length distribution at other epochs using queue-length distribution at post-departure epoch. Some important performance measures, like mean queue lengths and mean waiting times have been obtained. Several other vacation queueing models can be obtained as a special case of our model, e.g., single-, multiple-vacation model and queues with exceptional first vacation time. Finally, the total expected cost function per unit time is considered to determine a locally optimal multiple adaptive vacation policy at a minimum cost.     

Performance analysis of an M/G/1 queueing system under Bernoulli vacation schedules with server setup and close down periods

This paper is concerned with the analysis of a single server queueing system subject to Bernoulli vacation schedules with server setup and close down periods. An explicit expression for the probability generating function of the number of customers present in the system is obtained by using imbedded Markov chain technique. The steady state probabilities of no customer in the system at the end of vacation termination epoch and a service completion epoch are derived. The mean number of customers served during a service period and the mean number of customers in the system at an arbitrary epoch are investigated under steady state. Further, the Laplace-Stieltjes transform of the waiting time distribution and its corresponding mean are studied. Numerical results are provided to illustrate the effect of system parameters on the performance measures.     

Markovian bulk service queue with delayed vacations

An M/M(a, b)/1 queueing system with multiple vacations is studied, in which if the number of customers in the queue is a - 1 either at a service completion epoch or at a vacation completion point, the server will wait for an exponential time in the system which is called the changeover time. During this changeover time if there is an arrival the server will start service immediately, otherwise at the end of the changeover time the server will go for a vacation. The duration of vacation is also exponential. This paper is concerned with the determination of the stationary distribution of the number of customers in the queue and the waiting time distribution of an arriving customer. The expected queue length is also obtained. Sample numerical illustrations are given.     

An MX/G/1 queue with randomized working vacations and at most J vacations

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.     

A batch arrival queue under randomised multi-vacation policy with unreliable server and repair

This article examines an M^[x]/G/1 queueing system with an unreliable server and a repair, in which the server operates a randomised vacation policy with multiple available vacations. Upon the system being found to be empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1???p. When one or more customers arrive when the server is idle, the server immediately starts providing service for the arrivals. It is possible that an unpredictable breakdown may occur in the server, in which case a repair time is requested. For such a system, we derive the distributions of several important system characteristics, such as the system size distribution at a random epoch and at a departure epoch, the system size distribution at the busy period initiation epoch, and the distribution of the idle and busy periods. We perform a numerical analysis for changes in the system characteristics, along with changes in specific values of the system parameters. A cost effectiveness maximisation model is constructed to show the benefits of such a queueing system.     

Models of Perishable Queueing-Inventory Systems with Server Vacations

The model of perishable queueing-inventory system with server vacations is studied. Upon service completion, server takes vacation if there are no customers in the queue and it starts service at the end of the vacation if the number of customers in the system exceeds some threshold; otherwise, it takes new vacation. Exact and approximate methods are proposed to calculate the characteristics of the system.     

Cost-minimization analysis of a working vacation queue with N-policy and server breakdowns

This paper investigates the N-policy M/M/1 queueing system with working vacation and server breakdowns. As soon as the system becomes empty, the server begins a working vacation. The server works at a lower service rate rather than completely stopping service during a vacation period. The server may break down with different breakdown rates during the idle, working vacation, and normal busy periods. It is assumed that service times, vacation times, and repair times are all exponentially distributed. We analyze this queueing model as a quasi-birth–death process. Furthermore, the equilibrium condition of the system is derived for the steady state. Using the matrix-geometric method, we find the matrix-form expressions for the stationary probability distribution of the number of customers in the system and system performance measures. The expected cost function per unit time is constructed to determine the optimal values of the system decision variables, including the threshold N and mean service rates. We employ the particle swarm optimization algorithm to solve the optimization problem. Finally, numerical results are provided, and an application example is given to demonstrate the applicability of the queueing model.     

The optimal control of an M/G/1 queueing system with server vacations, startup and breakdowns

This paper studies the control policy of the N policy M/G/1 queue with server vacations, startup and breakdowns, where arrivals form a Poisson process and service times are generally distributed. The server is turned off and takes a vacation whenever the system is empty. If the number of customers waiting in the system at the instant of a vacation completion is less than N, the server will take another vacation. If the server returns from a vacation and finds at least N customers in the system, he requires a startup time before providing service until the system is again empty. It is assumed that the server breaks down according to a Poisson process and his repair time has a general distribution. The system characteristics of such a model are analyzed and the total expected cost function per unit time is developed to determine the optimal threshold of N at a minimum cost.     

An algorithmic analysis of multi-server vacation model with service interruptions

Motivated by the trade-off between reliability and utilization level of a stochastic service system, we considers a Markovian multi-server vacation queueing system with c unreliable servers. In such a system, some servers may not be available due to either planned stoppage (vacations) or unplanned service interruptions (server failures). The vacations are controlled by a threshold policy. With this policy, at a service completion instant, if d (?c) servers become idle, they take a vacation together and will keep taking vacations until they find at least c − d + 1 customers are in the system at a vacation completion instant, and then they return to serve the queue. In addition, all on-duty servers are subject to failures and can be repaired within a random period of time. We formulate a quasi-birth–death (QBD) process, establish the stability condition, and develop a computational algorithm to obtain the stationary performance measures of the system. Numerical examples are presented to show the performance evaluation and optimization of such a system. The insights gained from this model help practitioners make capacity and operating decisions for this type of waiting line systems.     

A retrial queue for modeling fault-tolerant systems with checkpointing and rollback recovery

In this paper we analyze a retrial queue that can be used to model fault-tolerant systems with checkpointing and rollback recovery. We assume that the service time of each job is decomposed into N modules, at the end of each of which a checkpoint is established. Checkpointing and rollback recovery consists, basically, of saving periodically the state of the system on a secure device so that, upon recovery from a system failure, the system can resume the computation from the most recent checkpoint, rather than from the beginning. Upon a successful service completion of a job, the server activates a timer and remains awake. If the timer expires without a request, the server departs for a vacation. Upon returning from the vacation, the server activates the timer again. Furthermore, both idle and vacation periods can be interrupted by the server in order to perform secondary jobs. Applications of this model can be found in power saving of mobile devices in a half-duplex communication system operating in wireless environment, and in long-running software applications. We investigate stability condition and steady state analysis. We also apply a mean value analysis to obtain useful performance measures, and prove that the model satisfies the stochastic decomposition property. Useful energy metrics are determined and constrained optimization problems are formulated and used to obtain extensive numerical results.     

The combined gated-exhaustive vacation system in discrete time

We consider a discrete-time gated vacation system. The available buffer space is divided into two subsequent queues separated by a gate and new customers arrive either before or after this gate. Whenever all customers after the gate are served, the server takes a vacation. After each vacation, the gate opens which causes all waiting customers to move to the buffer space after the gate. The model under investigation allows to capture performance of a.o. the exhaustive and the gated queueing systems with multiple or single vacations. Using a probability generating functions approach, we obtain expressions for performance measures such as moments of system contents at various epochs in equilibrium and of customer delay. We conclude with a numerical example.     

A discrete-time single-server queue with a modified N -policy

We consider a discrete-time single-server queue where the idle server waits for reaching a level N in the queue size to start a batch service of N messages, although the following arrivals during the busy period receive single services. We find the stationary distributions of the queue and system lengths as well as some performance measures. The vacation and busy periods of the system and the number of messages served during a busy period are also analyzed. The stationary distributions of the time spent waiting in the queue and in the system are studied too. Finally, a total expected cost function is developed to determine the optimal operating N-policy at minimum cost.     

Analytic and numerical aspects of batch service queues with single vacation

This paper deals with an M/G/1 batch service queue where customers are served in batches of maximum size b with a minimum threshold value a. The server takes a single vacation when he finds less than a customers after the service completion. The vacation time of the server is arbitrarily distributed. Using the supplementary variable method we obtain the probability generating functions of the queue length distributions at various epochs. We also obtain relations among queue length distributions at arbitrary, service (vacation) termination epochs. Further their evaluation is also discussed. Finally, some numerical results and graphs are presented.     

On a batch arrival Poisson queue with a random setup time and vacation period

The single server queue with vacation has been extended to include several types of extensions and generalisations, to which attention has been paid by several researchers (e.g. see Doshi, B. T., Single server queues with vacations — a servey. Queueing Systems, 1986, 1, 29–66; Takagi, H., Queueing Analysis: A Foundation of Performance evaluation, Vol. 1, Vacation and Priority systems, Part. 1. North Holland, Amsterdam, 1991; Medhi, J., Extensions and generalizations of the classical single server queueing system with Poisson input. J. Ass. Sci. Soc., 1994, 36, 35–41, etc.). The interest in such types of queues have been further enhanced in resent years because of their theoretical structures as well as their application in many real life situations such as computer, telecommunication, airline scheduling as well as production/inventory systems. This paper concerns the model building of such a production/inventory system, where machine undergoes extra operation (such as machine repair, preventive maintenance, gearing up machinery, etc.) before the processing of raw material is to be started. To be realistic, we also assume that raw materials arrive in batch. This production system can be formulated as an M^x/M/1 queues with a setup time. Further, from the utility point of view of idle time this model can also be formulated as a case of multiple vacation model, where vacation begins at the end of each busy period. Besides, the production/inventory systems, such a model is generally fitted to airline scheduling problems also. In this paper an attempt has been made to study the steady state behavior of such an M^x/M/1 queueing system with a view to provide some system performance measures, which lead to remarkable simplification when solving other similar types of queueing models.This paper deals with the steady state behaviour of a single server batch arrival Poisson queue with a random setup time and a vacation period. The service of the first customer in each busy period is preceded by a random setup period, on completion of which service starts. As soon as the system becomes empty the server goes on vacation for a random length of time. On return from vacation, if he finds customer(s) waiting, the server starts servicing the first customer in the queue. Otherwise it takes another vacation and so on. We study the steady state behaviour of the queue size distribution at random (stationary) point of time as well as at departure point of time and try to show that departure point queue size distribution can be decomposed into three independent random variables, one of which is the queue size of the standard M^x/M/1 queue. The interpretation of the other two random variables will also be provided. Further, we derive analytically explicit expressions for the system state (number of customers in the system) probabilities and provide their appropriate interpretations. Also, we derive some system performance measures. Finally, we develop a procedure to find mean waiting time of an arbitrary customer.     

Impatient customers in an M/M/1 queue with single and multiple working vacations

We consider an M/M/1 queue with impatient customers and two different types of working vacations. The working vacation policy is the one in which the server serves at a lower rate during a vacation period rather than completely stop serving. The customer’s impatience is due to its arrival during a working vacation period, in which the customer service rate is lower than the normal busy period. We analyze the queue for two different working vacation termination policies, a multiple working vacation policy and a single working vacation policy. Closed-form solutions and various performance measures like, the mean queue lengths and the mean waiting times are derived. The stochastic decomposition properties are verified for both multiple and single working vacation cases. A comparison of both the models is carried out to capture their performances with the change in system parameters.     

A repairable queueing model with two-phase service,start-up times and retrial customers

A repairable queueing model with a two-phase service in succession, provided by a single server, is investigated. Customers arrive in a single ordinary queue and after the completion of the first phase service, either proceed to the second phase or join a retrial box from where they retry, after a random amount of time and independently of the other customers in orbit, to find a position for service in the second phase. Moreover, the server is subject to breakdowns and repairs in both phases, while a start-up time is needed in order to start serving a retrial customer. When the server, upon a service or a repair completion finds no customers waiting to be served, he departs for a single vacation of an arbitrarily distributed length. The arrival process is assumed to be Poisson and all service and repair times are arbitrarily distributed. For such a system the stability conditions and steady state analysis are investigated. Numerical results are finally obtained and used to investigate system performance.     

Single server retrial queue with group admission of customers

We consider a retrial queueing system with a single server and novel customer׳s admission discipline. The input flow is described by a Markov Arrival Process. If an arriving customer meets the server providing the service, it goes to the orbit and repeats attempts to get service in random time intervals whose duration has exponential distribution with parameter dependent on the customers number in orbit. Server operates as follows. After a service completion epoch, customers admission interval starts. Duration of this interval has phase type distribution. During this interval, primary customers and customers from the orbit are accepted to the pool of customers which will get service after the admission interval. Capacity of this pool is limited and after the moment when the pool becomes full before completion of admission interval all arriving customers move to the orbit. After completion of an admission interval, all customers in the pool are served simultaneously by the server during the time having phase type distribution depending on the customers number in the pool. Using results known for Asymptotically Quasi-Toeplitz Markov Chains, we derive stability condition of the system, compute the stationary distribution of the system states, derive formulas for the main performance measures and numerically show advantages of the considered customer׳s admission discipline (higher throughput, smaller average number of customers in the system, higher probability to get a service without visiting the orbit) in case of proper choice of the capacity of the pool and the admission period duration.     

An advanced queueing model to analyze appointment-driven service systems

Many service systems are appointment-driven. In such systems, customers make an appointment and join an external queue (also referred to as the “waiting list”). At the appointed date, the customer arrives at the service facility and receives service. Important measures of interest include the size of the waiting list as well as the time spent in the waiting list. We develop a model to assess these performance measures. The model may be used to support strategic decisions concerning server capacity (e.g. how often should a server be online, how many customers should be served during each service session, etc.). The model is a vacation model that uses efficient algorithms and matrix analytical techniques to obtain waiting list performance measures.