Single Server Vacation Research Articles

On a Single Server Vacation Queue with Two Types of Service and Two Types of Vacation

We study a single server queueing system that receives singly arriving customers according to a Poisson process. The server offers one of the two types of heterogeneous services. Before the beginning of a service, , the customer can choose an exponential service with probability p1 or a deterministic service with probability p2, where p1+p2=1 Immediately after a service is completed, the server has a choice of taking a vacation with probability δ, or, with probability 1-δ, the server may continue staying in the system. We further assume that if the server opts to take a vacation, then with probability α1, he may take a vacation of an exponential duration with mean vacation time 1/u (u>0) or with probability he may want to take a deterministic vacation with constant duration d>0, where α1+α2=1. After a vacation is complete, the server instantly starts providing service if there is at least one customer in the system or the server remains idle in the system till a new customer arrives for service. We find a steady state solution in terms of the generating function of the queue length as well as the steady state probabilities for all different states of the system.

Game-theoretic analysis of the single vacation queue with negative customers

ABSTRACT We present a game-theoretic analysis of an M/M/1 queueing system with negative customers and single server vacation. Both positive and negative customers arrive according to a Poisson process and the server stars a vacation when the system is empty. Whenever a negative customer arrives, the positive customer being served (if any) is forced to abandon the system and the server suffers a breakdown, immediately after, a repair is required. During the repair process, positive customers are not allowed to join the system. Besides, they decide whether to join or to balk the system based on a reward-cost structure under four cases of different levels of information. We derive the equilibrium joining strategies of positive customers in each case. Specifically, we obtain the equilibrium threshold in the observable queue and mixed joining probability in the unobservable queue. Finally, the effects of different information levels and several parameters on the equilibrium threshold and mixed joining probabilities are illustrated by numerical examples.

A Study on IFM/IFG/1 Vacation Queueing System with Breakdown, Repair and Server Timeout in (Triangular, Trapezoidal & Pentagon) Intuitionistic Fuzzy Set using α-Cuts

In this, we studied and investigated IFM/IFG/1 vacation queueing system/waiting line with server breakdowns, repair and server timeout, “by using (Triangular Trapezoidal and Pentagonal) IF (Intuitionistic Fuzzy) numbers with the application of IFS (Intuitionistic fuzzy set). Here we operate single server vacation queueing system in the following manner; when the system finds empty, the server waits for fixed time ’c’ known as server timeout. At the expiration of this time, if no one arrives into the system, then server takes the vacation. If anyone arrived in the system during the timeout period as well as in vacation the server commences the service otherwise, he will go for another vacation. If the system had occurred with a breakdown, just after a break down the server undergoes for repair. After the repaired process is completed the server restarts the service to the arrived customer. By the approach of IFS properties, we develop the membership function of the system performance are of fuzzy nature. Based on IFS α-cut approach the Intuitionistic fuzzy queues are reduced to a family of ICS (Intuitionistic Crisp Set). The numerical results are illustrated to the model.

MX/G/1 Vacation Queueing System with Two Types of Repair facilities and Server Timeout

We consider a single server vacation queue with two types of repair facilities and server timeout. Here customers are in compound Poisson arrivals with general service time and the lifetime of the server follows an exponential distribution. The server find if the system is empty, then he will wait until the time ‘c’. At this time if no one customer arrives into the system, then the server takes vacation otherwise the server commence the service to the arrived customers exhaustively. If the system had broken down immediately, it is sent for repair. Here server failure can be rectified in two case types of repair facilities, case1, as failure happens during customer being served willstays in service facility with a probability of 1-q to complete the remaining service and in case2 it opts for new service also who joins in the head of the queue with probability q. Obtained an expression for the expected system length for different batch size distribution and also numerical results are shown.

The single server vacation queue with geometric abandonments and Bernoulli’s feedbacks

In this article the behaviour of a single server vacation queue with geometric abandonments and Bernoulli’s feedbacks is carried out and various important performance measures are derived. Some numerical experiments are presented to study how the parameters of the model influence the state of the system.

System capacity optimization design and optimal threshold $N^{*}$ for a $GEO/G/1$ discrete-time queue with single server vacation and under the control of Min($N, V$)-policy

In this paper, we deal with a discrete-time $Geo/G/1$ queueing system under the control of Min($N, V$)-policy in which the server takes single vacation whenever the system becomes empty. The Min($N, V$)-policy means that the server commences its service once the number of waiting customers reaches threshold $N$ or when its vacation time ends with at least one but less than $N$ customers waiting for processing, whichever occurs first. Otherwise, if no customer is presenting at the end of the server vacation, the server remains idle until the first arrival occurs. Under these assumptions, the $z$-transform expressions for the transient queue size distribution at time epoch $n^+$ are obtained by employing the renewal process theory and the total probability decomposition technique. Based on the transient analysis, the explicit recursive formulas of the steady-state queue length distribution at time epochs $n^+$, $n$, $n^-$ and outside observer's time epoch are derived, respectively. Additionally, the stochastic decomposition structure is presented and some other performance measures are also discussed. Furthermore, some computational experiments are implemented to demonstrate the significant application value of the recursive formulas for the steady-state queue size in designing system capacity. Finally, the optimal threshold of $N$ for economizing the system cost is numerically determined.

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.

Maximum Entropy Approach for Un-Reliable Server Vacation Queueing Model with Optional Bulk Service

In this study, we consider a single server vacation queueing model with optional bulk service and an un-reliable server. A single server provides first essential service (FES) to all arriving customers one by one; apart from essential service, he can also facilitate the additional phase of optional service (OS) in batches of fixed size b( ≥ 1), in case when the customers request for it. The server may take a single vacation whenever he finds no customers waiting in the queue to be served. Moreover, the server is subjected to unpredictable breakdown while providing the first essential service. The vacation time, service time and repair time of the server are exponentially distributed. The steady state results are obtained in terms of probability generating function for queue size distributions. By using the maximum entropy analysis (MEA), we derive various system performance measures. A comparative study is performed between the exact and approximate waiting time of the system. By taking the numerical illustrations, the sensitivity analysis is done to explore the effect of different descriptors on various performance measures.

Recursive solution of queue length distribution for [formula omitted] queue with single server vacation and variable input rate

In this paper we consider discrete time G e o / G / 1 queue with single server vacation and variable input rate. Using renewal process, probability decomposition technique and u -transform, we derive the recursive expressions of the queue length distributions at epochs n − , n + , and n . The results obtained in this paper indicate that the equilibrium queue length distribution no longer follows the stochastic decomposition discipline. Furthermore we derive the important relations between equilibrium queue length distributions at different epochs ( n − , n + , n ).

The single server vacation queueing model with geometric abandonments

Several recent papers have investigated the customer reneging behavior in queueing systems with vacations, where the customers become impatient during the absence of the server(s). These studies have treated the cases of independent and synchronized abandonments. In the case of independent abandonments, the customers have their own independent patience times and abandon the system when such times expire. In the case of synchronized abandonments, the abandonment opportunities occur according to a certain point process and then all present customers decide simultaneously but independently whether they will abandon the system or not. In the present paper, we complement these studies by considering the case of geometric abandonments. This case arises when the abandonment opportunities occur according to a certain point process and the customers decide sequentially whether they will leave the system or not. We derive explicit expressions and computational schemes for various performance descriptors, including the number of customers in system, the sojourn time of a customer, the duration and the maximum number of customers in a busy period.

The recursive solution for Geom/G/1(E, SV) queue with feedback and single server vacation

Using recursive method, this paper studies the queue size properties at any epoch n + in Geom/G/1(E, SV) queueing model with feedback under LASDA (late arrival system with delayed access) setup. Some new results about the recursive expressions of queue size distribution at different epoch (n +, n, n −) are obtained. Furthermore the important relations between stationary queue size distribution at different epochs are discovered. The results are different from the relations given in M/G/1 queueing system. The model discussed in this paper can be widely applied in many kinds of communications and computer network.

On The Transient Departure Process of M x /G/1 Queueing System with Single Server Vacation

This paper studies the transient departure process of Mx/G/1 queueing system with single server vacation. We present a simple probability decomposition method to derive the expected number of departures occurring in finite time interval from any initial state and the asymptotic expansion of the expected number. Especially, we derive some more practical results for some special cases.

Performance analysis of GI/M/1 queue with working vacations and vacation interruption

In this paper, we analyze a single-server vacation queue with a general arrival process. Two policies, working vacation and vacation interruption, are connected to model some practical problems. The GI/M/1 queue with such two policies is described and by the matrix analysis method, we obtain various performance measures such as mean queue length and waiting time. Finally, using some numerical examples, we present the parameter effect on the performance measures and establish the cost and profit functions to analyze the optimal service rate η during the vacation period.

Entropy maximization and the busy period of some single-server vacation models

In this paper, information theoretic methodology for system modeling is applied to investigate the probability density function of the busy period in M/G/1 vacation models operating under the N-, T- and D-policies. The information about the density function is limited to a few mean value constraints (usually the first moments). By using the maximum entropy methodology one obtains the least biased probability density function satisfying the system's constraints. The analysis of the three controllable M/G/1 queueing models provides a parallel numerical study of the solution obtained via the maximum entropy approach versus “classical” solutions. The maximum entropy analysis of a continuous system descriptor (like the busy period) enriches the current body of literature which, in most cases, reduces to discrete queueing measures (such as the number of customers in the system).

Vacation Models in Discrete Time

A class of single server vacation queues which have single arrivals and non-batch service is considered in discrete time. It is shown that provided the interarrival, service, vacation, and server operational times can be cast with Markov-based representation then this class of vacation model can be studied as a matrix–geometric or a matrix-product problem – both in the matrix–analytic family – thereby allowing us to use well established results from Neuts (1981). Most importantly it is shown that using discrete time approach to study some vacation models is more appropriate and makes the models much more algorithmically tractable. An example is a vacation model in which the server visits the queue for a limited duration. The paper focuses mainly on single arrival and single unit service systems which result in quasi-birth-and-death processes. The results presented in this paper are applicable to all this class of vacation queues provided the interarrival, service, vacation, and operational times can be represented by a finite state Markov chain.

A vacation queue with setup and close-down times and batch Markovian arrival processes

We consider a finite-capacity single-server vacation queue with close-down/setup times and batch Markovian arrival process (BMAP), where both the service time, the vacation time, the setup time, and the close-down time are generally distributed. The queueing model has potential applications in SVC (switched virtual connection)-based IP-over-ATM networks and multiple protocol label switched (MPLS) networks. By applying the supplementary variable technique, we develop a unified solution to both the single-vacation and multiple-vacation models and for either the PBAS ( partial batch acceptance strategy) or the WBAS ( whole batch acceptance strategy) service disciplines. For both models, we obtain the queue length distribution at batch arrival epochs and that at an arbitrary time instant, the loss probability of a whole batch or an arbitrary customer in a batch, server setup rate, server utilization ratio, and the LST of the waiting time distribution. Through the numerical examples, we find that: (1) there is a trade-off between the user’s quality-of-service (e.g., loss probabilities, wanting times) and the system performance (e.g., server setup rate, server utilization ratio); (2) the system performance is closely related not only to the first and second order moments of the arrival process but also the pattern (distribution) of the customer arrivals; (3) mean batch size is a much more critical factor to influence the queueing system’s performance than the type of batch size distribution. These conclusions are of instructive meanings in the design of IP-over-ATM or more generally MPLS-based networks.

M/M/1 queues with working vacations (M/M/1/WV)

The classical single server vacation model is generalized to consider a server which works at a different rate rather than completely stops during the vacation period. Simple explicit formulae for the mean, variance, and distribution of the number and time in the system are presented. The distributional results generalize the classical vacation decomposition with no service during a vacation. This model approximates a multi-queue system whose service rate is one of the two speeds for which the fast speed mode cyclically moves from queue to queue with an exhaustive schedule. This work is motivated and illustrated by the analysis of a WDM optical access network using multiple wavelengths which can be reconfigured.

THE QUEUE-LENGTH DISTRIBUTION FOR Mx/G/1 QUEUE WITH SINGLE SERVER VACATION

This paper studies the bulk-arrival Mx/G/1 queue with single server vacation. By introducing the server busy period and using the Laplace transform, the recursion expression of the Laplace transform of the transient queue-length distribution is derived. Furthermore, the distribution and stochastic decomposition result of the queue length at a random point in equilibrium are obtained. Especially some results for the single-arrival M/G/1 queue with single server vacation and bulk-arrival Mx/G/1 queue but with no server vacation can be derived directly by the results obtained in this paper.

A finite-capacity queue with exhaustive vacation/close-down/setup times and Markovian arrival processes

We consider a finite-capacity single-server vacation model with close-down/setup times and Markovian arrival processes (MAP). The queueing model has potential applications in classical IP over ATM or IP switching systems, where the close-down time corresponds to an inactive timer and the setup time to the time delay to set up a switched virtual connection (SVC) by the signaling protocol. The vacation time may be considered as the time period required to release an SVC or as the time during which the server goes to set up other SVCs. By using the supplementary variable technique, we obtain the queue length distribution at an arbitrary instant, the loss probability, the setup rate, as well as the Laplace–Stieltjes transforms of both the virtual and actual waiting time distributions.

The BMAP/G/1 vacation queue with queue-length dependent vacation schedule

AbstractWe treat a single-server vacation queue with queue-length dependent vacation schedules. This subsumes the single-server vacation queue with exhaustive service discipline and the vacation queue with Bernoulli schedule as special cases. The lengths of vacation times depend on the number of customers in the system at the beginning of a vacation. The arrival process is a batch-Markovian arrival process (BMAP). We derive the queue-length distribution at departure epochs. By using a semi-Markov process technique, we obtain the Laplace-Stieltjes transform of the transient queue-length distribution at an arbitrary time point and its limiting distribution