Exhaustive Service Discipline Research Articles

Achieving optimal resource utilization in industrial environments: Combining polling methods and an exhaustive service discipline with an unreliable service

Achieving optimal resource utilization in industrial environments: Combining polling methods and an exhaustive service discipline with an unreliable service

A MAP/PH(1), PH(2)/2 Production Inventory System with Multiple Servers and Varying Service Rates

The paper comprises an (s,S) production inventory facility in which multiple vacations and varying service rates are considered for the multiple servers. The arrival of customers constitutes a Markovian Arrival Process (MAP) and service completion times follow the phase type distributions with representations ( α, S ) and ( β, T ) respectively. Manufacturing needs to begin at the moment when the stock position falls to s. Service is offered at a lower rate if the stock level ranges from 0 to s and the service time distribution has the representations (α, η 1 S) and (β, η 2 T ), 0 < η 1, η 2 < 1 respectively. If the stock level reaches the maximum S, production is ceased to operate. 1-limited service policy, Bernoulli service schedule, and Exhaustive service disciplines are considered for the servers. A suitable cost function is outlined as per performance assessment. The impact of negative correlated inter arrival times on cost function is illustrated. Also, a relative study of expected cost function on different service modes is presented.

Equilibrium Strategies of a Polling Network with Non-zero Switch-over Times

In this paper, we consider a cyclic two-queue polling system with non-zero switchover times under exhaustive service discipline. Arriving customers decide whether to join queue 1 ( Q1 ) or queue 2 ( Q2 ) under two different information levels: fully unobservable case and fully observable case. With the help of the waiting costs, we study the equilibrium strategies for arriving customers in both cases. Furthermore, we analyze the expected waiting cost of an arriving customer in fully unobservable case by numerical examples.

The analysis of batch sojourn-times in polling systems

We consider a cyclic polling system with general service times, general switch-over times, and simultaneous batch arrivals. This means that at an arrival epoch, a batch of customers may arrive simultaneously at the different queues of the system. For the exhaustive service discipline, we study the batch sojourn-time, which is defined as the time from an arrival epoch until service completion of the last customer in the batch. We obtain exact expressions for the Laplace–Stieltjes transform of the steady-state batch sojourn-time distribution, which can be used to determine the moments of the batch sojourn-time and, in particular, its mean. However, we also provide an alternative, more efficient way to determine the mean batch sojourn-time, using mean value analysis. We briefly show how our framework can be applied to other service disciplines: locally gated and globally gated. Finally, we compare the batch sojourn-times for different service disciplines in several numerical examples. Our results show that the best performing service discipline, in terms of minimizing the batch sojourn-time, depends on system characteristics.

Stationary analysis of a BMAP/R/1 queue with R-type multiple working vacations

ABSTRACTWe consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple working vacation policy. The service time during a working vacation is generally distributed random variable which is independent of the service times during a normal busy period as well as the arrival process. Duration of service times during a normal busy period and duration of working vacation times follow the class of distributions whose Laplace-Stieltjes transforms are rational functions (R-type distributions). The service time during a normal busy period, working vacation time, and the service time during a working vacation are independent of each other as well as of the arrival process. If a working vacation terminates while service is going on for a customer at head of the queue in vacation mode then, the server switches to normal mode and the customer at head of the queue is entitled to receive a full service time in the normal busy period irrespective of the amount of service received by the customer at head of the queue during the previous working vacation period. We obtain system-length distributions at various epoch, such as post-departure, pre-arrival, arbitrary, and pre-service. The proposed analysis is based on the use of matrix-analytic procedure to obtain system-length distribution at post-departure epoch. Later, we use supplementary variable technique and simple algebraic manipulations to obtain system-length distribution at arbitrary epoch using the system-length distribution at post-departure epoch. Some important performance measures, such as mean system lengths and mean waiting time have been obtained. Finally, some numerical results have been presented in the form of tables and graphs to show the applicability of the results obtained in this article. The model has potential application in areas of computer and communication networks, such as ethernet passive optical network (EPON).

G-RAND: A phase-type approximation for the nonstationary [formula omitted] queue

We present a Markov model to analyze the queueing behavior of the nonstationary G(t)/G(t)/s(t)+G(t) queue. We assume an exhaustive service discipline (where servers complete their current service before leaving) and use acyclic phase-type distributions to approximate the general interarrival, service, and abandonment time distributions. The time-varying performance measures of interest are: (1) the expected number of customers in queue, (2) the variance of the number of customers in queue, (3) the expected number of abandonments, and (4) the virtual waiting time distribution of a customer arriving at an arbitrary moment in time. We refer to our model as G-RAND since it analyzes a general queue using the randomization method. A computational experiment shows that our model allows the accurate analysis of small- to medium-sized problem instances.

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.

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.

Asymptotic behavior of the loss probability for an M/G/1/N queue with vacations

In this paper, asymptotic properties of the loss probability are considered for an M/G/1/N queue with server vacations and exhaustive service discipline, denoted by an M/G/1/N-(V,E)-queue. Exact asymptotic rates of the loss probability are obtained for the cases in which the traffic intensity is smaller than, equal to and greater than one, respectively. When the vacation time is zero, the model considered degenerates to the standard M/G/1/N queue. For this standard queueing model, our analysis provides new or extended asymptotic results for the loss probability. In terms of the duality relationship between the M/G/1/N and GI/M/1/N queues, we also provide asymptotic properties for the standard GI/M/1/N model.

On the stability of a polling system with an adaptive service mechanism

We consider a single-server cyclic polling system with three queues where the server follows an adaptive rule: if it finds one of queues empty in a given cycle, it decides not to visit that queue in the next cycle. In the case of limited service policies, we prove stability and instability results under some conditions which are sufficient but not necessary, in general. Then we discuss open problems with identifying the exact stability region for models with limited service disciplines: we conjecture that a necessary and sufficient condition for the stability may depend on the whole distributions of the primitive sequences, and illustrate that by examples. We conclude the paper with a section on the stability analysis of a polling system with either gated or exhaustive service disciplines.

Analysis of single working vacation in GI/M/1/N and GI/M/1/&amp;infin; queueing systems

We consider a finite-buffer GI/M/1 queue with exhaustive service discipline and single working vacation. Service time in a vacation, in a service period and vacation time all are exponentially distributed random variables independent of each other. Queue length distributions at pre-arrival and arbitrary epoch with some important performance measures such as, probability of blocking, mean waiting time in the system, etc. have been obtained using the method of embedded Markov chain and supplementary variable. The corresponding infinite-buffer GI/M/1 queue with exhaustive service discipline and single working vacation has also been analysed. For this model, we also obtain pre-arrival and arbitrary epoch probability along with some important performance measures. These queueing models have potential application in the area of computer and communication network where a single channel is allotted for more than one type of job.

Unified analysis of BMAP/G/1 cyclic polling models

In this paper we present a unified analysis of the BMAP/G/1 cyclic polling model and its application to the gated and exhaustive service disciplines as examples. The applied methodology is based on the separation of the analysis into service discipline independent and dependent parts. New expressions are derived for the vector-generating function of the stationary number of customers and for its mean in terms of vector quantities depending on the service discipline. They are valid for a broad class of service disciplines and both for zero- and nonzero-switchover-times polling models. We present the service discipline specific solution for the nonzero-switchover-times model with gated and exhaustive service disciplines. We set up the governing equations of the system by using Kronecker product notation. They can be numerically solved by means of a system of linear equations. The resulting vectors are used to compute the service discipline specific vector quantities.

A Survey and Experimental Comparison of Service-Level-Approximation Methods for NonstationaryM(t)/M/s(t)Queueing Systems with Exhaustive Discipline

We compare the performance of seven methods in computing or approximating service levels for nonstationary M(t)/M/s(t) queueing systems: an exact method (a Runge-Kutta ordinary-differential-equation solver), the randomization method, a closure (or surrogate-distribution) approximation, a direct infinite-server approximation, a modified-offered-load infinite-server approximation, an effective-arrival-rate approximation, and a lagged stationary approximation. We assume an exhaustive service discipline, where service in progress when a server is scheduled to leave is completed before the server leaves. We used all of the methods to solve the same set of 640 test problems. The randomization method was almost as accurate as the exact method and used about half the computational time. The closure approximation was less accurate, and usually slower, than the randomization method. The two infinite-server-based approximations, the effective-arrival-rate approximation, and the lagged stationary approximation were less accurate but had computation times that were far shorter and less problem-dependent than the other three methods.

On the GI/M/1/N queue with multiple working vacations—analytic analysis and computation

We consider finite buffer single server GI / M /1 queue with exhaustive service discipline and multiple working vacations. Service times during a service period, service times during a vacation period and vacation times are exponentially distributed random variables. System size distributions at pre-arrival and arbitrary epoch with some important performance measures such as, probability of blocking, mean waiting time in the system etc. have been obtained. The model has potential application in the area of communication network, computer systems etc. where a single channel is allotted for more than one source.

M/G/1 queue with multiple working vacations

We study an M/G/1 queue with multiple vacations and exhaustive service discipline such that the server works with different service times rather than completely stopping service during a vacation. Both service times in a vacation and in a service period are generally distributed random variables. It is assumed that the Laplace–Stieltjes transform (LST) for the distribution of the vacation length is a rational function. We derive the distributions for the queue size and the system time for an arbitrary customer in the steady state. Several special cases, namely, exponentially distributed vacation lengths and/or exponentially distributed service times in a vacation, are considered. Finally some numerical examples are presented.

On the Queue Length Distribution for the GI/G/1/K/VM Queue

We present a transform-free distribution of the steady-state queue length for the GI/G/1/K queueing system with multiple vacations under exhaustive FIFO service discipline. The method we use is a modified supplementary variable technique and the result we obtain is expressed in terms of conditional expectations of the remaining service time, the remaining interarrival time, and the remaining vacation, conditional on the queue length at the embedded points. The case K → ∞ is also considered.

Analysis of random polling system with an infinite number of coupled servers and correlated input process

We introduce a simple approach for modeling and analyzing a random polling system with infinite servers. We assume that the infinite number of servers is coupled together and they visit the queues as one processing unit. It is assumed that the customer arrival processes at all queues are correlated. Two classes of service disciplines, exhaustive and gated, are considered. We will derive several performance measures of the system. These performance measures include the mean cycle time and the expected delay observed by a customer. For the special case of M/ D/∞ vacation queue, we also provide a new proof of a known result. The numerical results indicate that when the expected number of busy servers is high gated service produces mean waiting times less than those given by exhaustive service discipline. This result differs significantly from the known result for single server polling system and it is due to the assumption of coupled servers. Scope and purpose The Information Community's insatiable appetite for timely information has led to a number of new services: voice, data, teleconferencing, entertainment video, local area networks bridging, and distributed data processing. For both economic reasons and sophistication of services provided, a multicast feature must be provided. An essential component of multicasting is communication link sharing, an example of which is a self-routing network. Furthermore, the dramatic increase in transmission speed due to advancing technology has significantly altered many of the operating assumptions of communication systems. In particular, the quality of services provided by these communication systems can be improved by redesigning the access control mechanisms. Thus, the design and analysis of high-speed communication networks is drawing increasing attention. As a result, a number of approaches has been proposed for controlling the high-speed communication systems typified by parallel communication in fixed time synchronous mode. The parallel communication aspect is a result of advancing technology. For example, widespread use of fiber optics in parallel buses, parallel rings and star topology networks using frequency or wavelength division multiplexing have made high-speed parallel communication a reality. Furthermore, it has been demonstrated that by using parallel high-speed communication systems the total system capacity can be increased beyond that given by the single channel high-speed communication system. The use of fixed synchronous time division control in high-speed parallel communication systems is dictated by the speed limitations of electronics. However, the implementation of fixed synchronous time division control in high-speed parallel communication systems at fiber optics communication rates has been shown to be feasible. Although a number of performance studies on parallel high-speed communication systems has appeared in the literature; the performance of parallel high-speed communication system is not well understood. Therefore, there is a need to develop models of parallel high-speed communication systems that enable one to analyze their performance. Furthermore, from the perspectives of design and control of the communication system, there is a need for medium access control protocols that will allow designers to prioritize the different stations and thus improve the overall system performance. Motivated by these observations, the objective of this paper is to introduce a new approach to modeling and analyzing parallel high-speed communication systems.

Performance of Bluetooth Slave/Slave Bridge

The performance of a Bluetooth scatternet with a Slave/Slave bridge is analyzed using the tools of queueing theory. We analyze two possible scheduling policies: limited service and exhaustive service, and derive analytical results for the probability distribution of access delay (i.e., the time that a packet has to wait before being serviced) and end-to-end delay for both intra- and inter-piconet bursty traffic. The exhaustive service discipline was found to provide lower values for all delay variables, over a wide range of parameter values. All analytical results have been confirmed through simulations.

Polling systems in the critical regime

We study polling systems with Poisson arrival streams and exhaustive service discipline in the critical load case ρ=1. We show that the system may exhibit null-recurrence or transient behavior and give the corresponding conditions in terms of the first two moments of the distribution of service and switching times. For models with two nodes we establish existence of moments of small orders for the return time to the origin of the corresponding embedded Markov chain in the case of null-recurrence.

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.