Exhaustive Service Research Articles

On finite capacity queueing systems with a general vacation policy

We consider a Poisson arrival queueing system with finite capacity and a general vacation policy as described in Loris‐Teghem [Queueing Systems 3 (1988), 41‐52]. From our previous results regarding the stationary queue length distributions immediately after a departure and at an arbitrary epoch, we derive a relation between both distributions which extends a result given in Frey and Takahashi [Operations Research Letters 21 (1997), 95‐100] for the particular case of an exhaustive service multiple vacation policy.

Delay in polling systems with large switch-over times

We study the delay in cyclic polling systems with mixtures of gated and exhaustive service, and with deterministic switch-over times. We show that, under proper scalings, the waiting-time distribution at each of the queues converges to a uniform distribution over a known interval when the switch-over times tend to infinity.

Delay in polling systems with large switch-over times

We study the delay in cyclic polling systems with mixtures of gated and exhaustive service, and with deterministic switch-over times. We show that, under proper scalings, the waiting-time distribution at each of the queues converges to a uniform distribution over a known interval when the switch-over times tend to infinity.

Delay in polling systems with large switch-over times

We study the delay in cyclic polling systems with mixtures of gated and exhaustive service, and with deterministic switch-over times. We show that, under proper scalings, the waiting-time distribution at each of the queues converges to a uniform distribution over a known interval when the switch-over times tend to infinity.

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

Analysis of a time-limited polling system

In this paper, we consider a cyclic polling system in which arrivals are governed by the Markovian arrival process. Each queue is visited according to the exhaustive time-limited service discipline to coincide with IEEE 802.5 and 802.4 standards. Using the decomposition approach, each queue is analyzed as a single server queue with vacation. By exploiting the properties of the discrete time phase distribution we construct the vacation period from the visit period of the other queues in the polling system. Using an iterative procedure we were able to compute the queue length distribution and the average waiting time for polling systems with finite capacity (or infinite capacity) in all queues. Comparison of the mean waiting time with simulation results shows that the proposed models give reasonable results.

An MX/GI/1/N queue with close‐down and vacation times

An MX/GI/1/N finite capacity queue with close‐down time, vacation time and exhaustive service discipline is considered under the partial batch acceptance strategy as well as under the whole batch acceptance strategy. Applying the supplementary variable technique the queue length distribution at an arbitrary instant and at a departure epoch is obtained under both strategies, where no assumption on the batch size distribution is made. The loss probabilities and the Laplace‐Stieltjes transforms of the waiting time distribution of the first customer and of an arbitrary customer of a batch are also given. Numerical examples give some insight into the behavior of the system.

AN EXPLICIT SOLUTION FOR AN M/GI/1/N QUEUE WITH VACATION TIME AND EXHAUSTIVE SERVICE DISCIPLINE

We consider an M/GI/l/N queue with vacation time and exhaustive service discipline. We show that the embedded Markov chain formed by the customer departure epochs makes the analysis simple and it enables us to find an explicit solution for the server-vacation queue. We also find an explicit solution for the ordinary (non-vacation) queue.

Arbitrary open queueing networks with server vacation periods and blocking

A cost-effective decomposition algorithm is proposed for the approximate analysis of arbitrary first-come-first-served (FCFS) open queueing network models (QNMs) with Generalised Exponential (GE) external interarrival time and service time distributions, multiple vacation periods, exhaustive service scheme and repetitive service blocking of random destination (RS-RD). The algorithm is based on a new product-form approximation suggested by the information theoretic principle of Minimum Relative Entropy (MRE), subject to marginal mean value constraints and a prior joint distribution estimate of the corresponding open QNM with infinite capacity. The censored GEyGEy1yN queue with exponential server vacations is solved via relative entropy minimisation and in conjunction with associated effective flow streams (departure, splitting, merging), plays the role of an efficient building block in the solution process. An exhaustive validation study against simulation is included to illustrate the credibility of the MRE results.

A note on an M/GI/1/N queue with vacation time and exhaustive service discipline

We consider an M/GI/1/N queue with vacation time and exhaustive service discipline, the same queue as in Lee [Oper. Res. 32 (1984) 774–784]. Focusing only on the service completion epochs (as opposed to both service completion epochs and vacation completion epochs by Lee), we present a simple analysis for the queue length distribution at an arbitrary time as well as for the waiting time distribution.

Analysis of an M/G/1 queue with constant repeated attempts and server vacations

We consider an M/G/1 queue with repeated attempts in which the server operates under a general exhaustive service vacation policy. We develop a comprehensive analysis of the system including ergodicity, limiting behaviour, stochastic decomposition and optimal control. We would like to point out that the system size distribution decomposes into three random variables which are respectively associated with the vacation time, the retrial policy and the ordinary M/G/1 queue without vacations and repeated attempts.

Machine interference problem with warm spares, server vacations and exhaustive service

We consider the machine interference problem with warm spares in which the server takes a vacation of random duration every time the repair facility becomes empty. We address the cases of multiple vacations, single vacation and hybrid multiple/single vacation schemes with exhaustive service. Vacation models that have been studied in the past generally assume infinite population. In this paper, we provide new, transform free, closed form expressions for the probability distribution of the number of machines in the repair facility and the performance measures for machine interference problem with warm spares and server vacations. By adjusting the parameters, closed form expressions for machine interference problem with cold spares, machine interference problem without spares, finite capacity and infinite capacity vacation models can also be derived. An extremely powerful and efficient algorithm is presented to determine the steady state probability distribution of the number of customers in the system as well as the performance measures for all these models. An example is considered and some insight is also provided.

M( n)/ G/1/ N queues with generalized vacations

We consider M/ G/1/ N queues with generalized vacations and exhaustive service, where arrival rates depend on the number of customers in the system. An efficient recursive algorithm with overall computational complexity O ( N 2) for computing the exact stationary queue length distribution is developed. Based on the stationary queue length distribution, some other performance characteristics such as the Laplace transform of busy period and moments of virtual waiting time at arbitrary time can be obtained. Our numerical investigations demonstrate the strength of the algorithm.

On the optimality of exhaustive service policies in multiclass queueing systems with modulated arrivals and switchovers

Consider a single-server multiclass queueing system withK classes where the individual queues are fed byK-correlated interrupted Poisson streams generated in the states of aK-state stationary modulating Markov chain. The service times for all the classes are drawn independently from the same distribution. There is a setup time (and/or a setup cost) incurred whenever the server switches from one queue to another. It is required to minimize the sum of discounted inventory and setup costs over an infinite horizon. We provide sufficient conditions under which exhaustive service policies are optimal. We then present some simulation results for a two-class queueing system to show that exhaustive, threshold policies outperform non-exhaustive policies.

Message delay for a priority-based automatic meter reading network

In this paper, we model a data acquisition system as cyclic polling systems with two priority levels. Two types of polling system are studied, exhaustive and time limited service, with application to automatic meter reading. In the case of the exhaustive service strategy, stations are served exhaustively for both priorities, with high priority messages served first on a non-preemptive basis. For the case of the time limited discipline, each station is served for no more than τ max but not less than τ min. Approximate results are obtained for the high and low priority mean waiting time for polling systems with constant switchover times, N stations, general service times distributions, and independent arrival processes for each priority level. Comparisons of the approximations with simulation results for different service distributions, traffic intensities and load combinations show a high degree of accuracy.

A POLLING MODEL WITH RETRIAL CUSTOMERS

A polling model with n stations and switchover times is considered. The customers are of n different types, arrive to the system according to the Poisson distribution in batches of random size, and if they find the server unavailable, they start to make retrials until succeed to find a position for service. Each batch may contain customers of different types while the numbers of customers belonging to each type in a batch are distributed according to a multivariate general distribution. The server, upon polling a station, stays there for an exponential period of time and if a customer asks for service before this time expires, the customer is served and a new "stay period" begins. Finally the service times and the switchover times are both arbitrarily distributed with different distributions for the different stations. For such a model we obtain formulas for the expected number of retrial customers in each station in a steady state. Our results can be easily adapted to hold for zero switchover times and also in the case of the ordinary exhaustive service polling model with (without) switchover times and correlated batch arrivals. In all cases mentioned above (retrial model, exhaustive model, switchover times, zero switchover times) to find the expected queue lengths we need finally to solve a set of only n linear equations (O(n^3) arithmetic operations to compute the coefficients). Tables of numerical values are finally obtained and used to observe the system performance when we vary the values of the parameters.

The Impact of Setup Times on the Performance of Multiclass Service and Production Systems

We characterize the impact of the means and distributions of setup times in polling systems. Starting with cyclic systems with exhaustive service, we show that the vector of average setup times affects the complete distribution of waiting times and queue sizes only via its sum. In addition, all moments of all waiting times and queue sizes are reduced if the higher order moments of one or more of the setup times is reduced while maintaining their means. We also obtain a fundamental decomposition result of the queue sizes at polling instants which is reminiscent of those obtained in vacation models. We show that the above results continue to apply under more general and more efficient protocols, e.g., general polling tables, systems with gated service, and systems with mixed service.

Polling systems with station breakdowns

None of the works in the literature on Polling Systems deals with the important phenomenon of failing nodes. This paper addresses this issue. We assume that there are two modes of station breakdowns: node-dependent and general. Each node is subject to a random failure process and a failure of a node during a visit of the server causes the latter to move to the next station. We carry performance analyses of both the Gated and the Exhaustive service regimes and derive probability generating functions, Laplace transforms and means of variables such as queue sizes at polling and at arbitrary instants, durations of busy periods, number of jobs left behind at server's station-departure moments, and cycle times. The analysis of the Exhaustive case is achieved via an explicit solution of an infinite set of linear equations, where the unknowns are the state-dependent joint transforms of two key characteristics: the number of arrivals to a station during a busy period starting with a given number of jobs, and the number of jobs successfully served during that period. Stability conditions are indicated for each regime.

Polling systems with station breakdowns

Abstract None of the works in the literature on Polling Systems deals with the important phenomenon of failing nodes. This paper addresses this issue. We assume that there are two modes of station breakdowns: node-dependent and general. Each node is subject to a random failure process and a failure of a node during a visit of the server causes the latter to move to the next station. We carry performance analyses of both the Gated and the Exhaustive service regimes and derive probability generating functions, Laplace transforms and means of variables such as queue sizes at polling and at arbitrary instants, durations of busy periods, number of jobs left behind at server's station-departure moments, and cycle times. The analysis of the Exhaustive case is achieved via an explicit solution of an infinite set of linear equations, where the unknowns are the state-dependent joint transforms of two key characteristics: the number of arrivals to a station during a busy period starting with a given number of jobs, and the number of jobs successfully served during that period. Stability conditions are indicated for each regime.

Analysis of M/G/1 queues with delayed vacations and exhaustive service discipline

We consider M/G/1 queues with exhaustive service and delayed vacations, where at the end of every busy period the server stays idle in the system for a period of time called changeover time and then follows a mixed vacation policy from a given vacation policy set if there is no arrival during the changeover time. The successive vacations are assumed to be independent but not necessarily identical. This vacation policy includes many existing vacation policies as special cases. We derive the Laplace transform of the joint distribution of the queue length and the remaining service (or vacation) time at arbitrary time as well as that of the virtual waiting time distribution. It results in the stochastic decomposition property immediately. We also provide an explicit formula for the Laplace transform of the additional delay for our model. It enables us to analyze and unify many new and existing vacation policies easily.