On the optimal control of parallel systems and queues

Applied Mathematical Modelling, 2009

This paper deals with the steady-state behaviour of an M/G/1 queue with an additional second phase of optional service subject to breakdowns occurring randomly at any instant while serving the customers and delayed repair. This model generalizes both the classical M/G/1 queue subject to random breakdown and delayed repair as well as M/G/1 queue with second optional service and server breakdowns. For this model, we first derive the joint distributions of state of the server and queue size, which is one of chief objectives of the paper. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch as a classical generalization ofPollaczek-Khinchin formula. Next, we derive Laplace Stieltjes transform of busy period distribution and waiting time distribution. Finally, we obtain some important performance measures and reliability indices of this model. Choudhury and Paul [5] investigated such a model under Bernoulli feedback mechanism. In this context Krishnakumar and Arivudainambi in [6] obtained the explicit expression for transient probabilities for this type of finite capacity model M/G/1/1 Bernoulli feedback queue and M/G/1/1 queue with unreliable server . Recently, Wang [8] investigated such a model with the assumption that the server is subject to breakdowns and repairs, and some critical reliability indices are obtained. More recently, Ke [9] extended the result for a multi-optional service system where concept of setup time is also introduced.

Journal of physics, 2018

In this paper, we examine a two-stage queueing system where the arrivals are Poisson with rate depends on the condition of the server to be specific: vacation, pre-service, operational or breakdown state. The service station is liable to breakdowns and deferral in repair because of non-accessibility of the repair facility. The service is in two basic stages, the first being bulk service to every one of the customers holding up on the line and the second stage is individual to each of them. The server works under N-policy. The server needs preliminary time (startup time) to begin batch service after a vacation period. Startup times, uninterrupted service times, the length of each vacation period, delay times and service times follows an exponential distribution. The closed form of expressions for the mean system size at different conditions of the server is determined. Numerical investigations are directed to concentrate the impact of the system parameters on the ideal limit N and the minimum base expected unit cost.

2019 Winter Simulation Conference (WSC)

While high levels of automation in modern manufacturing systems increase the reliability of production, tool failure and preventive maintenance (PM) events remain a significant source of production variability. It is well known for production systems, such as the M/G/1 queue, that optimal PM policies possess a threshold structure. Much less is known for networks of queues. Here we consider the prototypical tandem queue consisting of two exponential servers in series subject to health deterioration leading to failure and repair. We model the PM decision problem as a Markov decision process (MDP) with a discounted infinitehorizon cost. We conduct numerical studies to assess the structure of optimal policies. Simulation is used to assess the value of the optimal PM policy relative to the use of a PM policy derived by considering each queue in isolation. Our simulation studies demonstrate that the mean cycle time and discounted operating costs are 10% superior.

OPSEARCH, 2012

In this paper, we consider a single server Markovian queueing system with a finite buffer. In addition to a Poisson stream of positive arrivals we assume that there is a also a Poisson stream of negative arrivals into the system. These negative arrivals which may be called as catastrophes may occur at any instant of time, whether the server is idle or busy. The time dependent performance measures and the busy period of the system are discussed. The corresponding steady state results are derived. We present a few numerical examples to illustrate the behavior of the time dependent probabilities, the time dependent expected system size and the time dependent variance of the system size distribution.

Journal of Applied Probability, 1999

Performance Evaluation, 1990

The queueing system to be analysed is a model of a multi-terminal system subject to random breakdowns. All random variables involved here are independent and exponentially distributed. Although the stochastic process describing the system's behaviour is a Markov chain, the number of states becomes very large. The main contribution of this paper is a recursive computational approach to solve the steady-state equations concerning the problem. In equilibrium, the main performance measures of the system, such as the mean number of jobs residing at the CPU, the mean number of functional terminals, the average number of busy servers, the expected response times of jobs, and utilizations are obtained. Finally, some numerical results illustrate the problem in question.

IEEE Transactions on Information Theory

Service reliability in a distributed queuing system can be assessed by means of the probability of successfully serving queued tasks in a heterogeneous network of nodes. Moreover, randomness in delays imposed by the communication medium adds to the uncertainties in the shared information among these faulty nodes, which complicates the dynamics of the queuing system. A novel approach based on stochastic regeneration is used to obtain renewal equations that characterize the service reliability. The theory is applied to develop optimal load-balancing policies for a two-node distributed-computing system, which is then systematically extended to a multi-node system yielding a computationally efficient suboptimal solution.

In this paper we study a class of open queueing network where servers suffer breakdowns and are subsequently repaired. The network topology is a pipeline with feedback from the final node to the first. Each node consists of a number of queues each with an unreliable server. There are no losses from the queues in this system, however jobs are routed according to the distribution of operational servers at each node in the pipeline. This model is in general intractable, however an iterative technique is presented which combines a number of earlier results to generate an approximation to steady state measures found by simulation.

2014

Here we will study bulk service to customer under optimal operation of a single removable and non-reliable server in Markovian queueing system under steady-state conditions. The decision maker can turn a server on at customer’s arrival or off at service completion. Here it is assumed that the server may breakdown only if working and requires repair at repair facility. Inter-arrival and service time distributions of the customers are assumed to be exponentially distributed. Breakdown and repair time distributions of the server are assumed to be exponentially distributed. The following cost structure is incurred to be system; a holding cost for each customer in the system per unit time, cost per unit time when a server fails, and fixed costs for turning the server on or off. The expected cost function per unit time is developed to obtain the optimal operating policy at minimum cost.

Microelectronics Reliability, 1997

We analyze the long-run availability of a two-unit parallel system sustained by a cold standby unit and attended by two identical repairmen. The system satisfies the usual conditions (i.i.d. random variables, perfect repair, instantaneous and perfect switch, queueing). Each operative unit has a constant failure rate and a general repair time distribution. We use Hokstad's supplementary variable method to construct a system of simultaneous partial differential equations. We present a first-order numerical scheme and an iterative algorithm for the solution of the equations. A particular but important numerical example (Weibull repair), illustrated by computer-plotted graphs, motivates the proposed approximation.

Quality Technology & Quantitative Management, 2017

In this paper, we study the steady state behaviour of an M/G/1 queue with two types of general heterogeneous service and optional repeated service subject to server's breakdowns occurring randomly at any instant while serving the customers and delayed repair. We assume that customers arrive to the system according to a Poisson process with rate 'λ' and the server provides two types of general heterogeneous service. At the beginning of a service, a customer has the option to choose any one type of service. After completion of either type of service, the customer has the further option to repeat the same type of service. For this model, we first derive the joint distribution of state of the server and queue size by considering both elapsed and remaining time, which is one of the objectives of this paper. Secondly, we derive the probability generating function of the stationary queue size distribution at departure epoch. Next, we derive Laplace-Stieltjes transform of busy period distribution and waiting time distribution. Finally, we obtain some important performance measure and reliability indices of this model.

Fourth International Conference on Advances in Information Processing and Communication Technology - IPCT 2016, 2016

We consider an M X /G/1 queuing system with breakdown and repairs, where batches of customers are assumed to arrive in the system according to a compound poisson process. While the server is being repaired, the customer in service either remains the service position or enters a service orbit and keeps returning, after repair the server must wait for the customer to return. The server is not allowed to accepte new customers until the customer in service leaves the system. We find a stability condition for this system. In the steady state the joint distribution of the server state and queue length is obtained, and some performance mesures of the system, such as the mean number of customers in the retrial queue and waiting time, and some numerical results are presented to illustrate the effect of the system parameters on the developed performance measures. Keywords-batch arrival, break down, repair. I. Introduction Retrial queuing systems have been widely used to model many practical problems arising in telephone switching systems, telecommunication networks, and computer systems. The main characteristic of these queues is that a customer who find the sever busy upon arrival joins the retrial group called orbit to repeat his request for service after some random time. For a systematic account of the fundamental methods and results on this topic the reader can refer to the survey papers of (

International Journal of Mathematics in Operational Research, 2012

In this paper, we study the modelling and analysis of unreliable M/M/2/K queuing system under N-and F-policy with multi-optional phase repair and start-up. According to the (N, F) policy, the first server is turned on only when there are 'N' or more customers are available in the system. The second server is turned on when the queue length of the customers reaches its system capacity K. Furthermore, no more customers are allowed in the system till the number of the customers again drops up to a certain threshold level 'F'. The servers may break down during busy period. There is a provision of 'l' phase repairs to restore the servers. The first-phase repair is necessary but remaining (l-1) phases are optional. The lifetime, start-up time and repair time of the servers are assumed to be exponentially distributed. Using matrix method, transient probabilities of the system states are determined. To examine the effect of different parameters on various performance indices, the numerical results are provided by taking an illustration.

Mathematics

In this paper, we discuss a non-Markovian batch arrival general bulk service single-server queueing system with server breakdown and repair, a stand-by server, multiple vacation and re-service. The main server's regular service time, re-service time, vacation time and stand-by server's service time are followed by general distributions and breakdown and repair times of the main server with exponential distributions. There is a stand-by server which is employed during the period in which the regular server remains under repair. The probability generating function of the queue size at an arbitrary time and some performance measures of the system are derived. Extensive numerical results are also illustrated.

Probability in the Engineering and …, 1990