Jobs Of Different Classes Research Articles

Stochastic Monotonicity of Markovian Multiclass Queueing Networks

Multiclass queueing networks (McQNs) extend the classical concept of the Jackson network by allowing jobs of different classes to visit the same server. Although such a generalization seems rather natural, from a structural perspective, there is a significant gap between the two concepts. Nice analytical features of Jackson networks, such as stability conditions, product–form equilibrium distributions, and stochastic monotonicity, do not immediately carry over to the multiclass framework. The aim of this paper is to shed some light on this structural gap, focusing on monotonicity properties. To this end, we introduce and study a class of Markov processes, which we call Q-processes, modeling the time evolution of the network configuration of any open, work-conservative McQN having exponential service times and Poisson input. We define a new monotonicity notion tailored for this class of processes. Our main result is that we show monotonicity for a large class of McQN models, covering virtually all instances of practical interest. This leads to interesting properties that are commonly encountered for “traditional” queueing processes, such as (i) monotonicity with respect to external arrival rates and (ii) star-convexity of the stability region (with respect to the external arrival rates); such properties are well known for Jackson networks but had not been established at this level of generality. This research was partly motivated by the recent development of a simulation-based method that allows one to numerically determine the stability region of a McQN parameterized in terms of the arrival-rates vector.

Optimal control strategies for single-machine family scheduling with sequence-dependent batch setup and controllable processing times

The problem of scheduling jobs on an unreliable single machine is considered in this paper. The scheduling problem is characterized by the following features: jobs are grouped into classes of equivalent jobs; the generalized due date model is adopted for each class of jobs; it is possible to reduce the processing time of a job, at the price of the payment of an extra cost; a costly setup is required when switching between jobs of different classes. The scheduling problem is solved from a perspective which is different from the traditional determination of an optimal sequence of jobs; in fact, the objective of the paper is to determine optimal control strategies (functions of the system state) which allow generating the optimal decisions during the evolution of the system, taking into account the actual system state. In this way, optimal decisions can be promptly taken also in the presence of perturbations which affect the single machine (such as breakdowns and slowdowns). To this aim, a specific optimal control problem is stated and solved in the paper.

Modified Algorithm for Scheduling Problem

The problem of scheduling n jobs on a single machine is considered, where the jobs are divided into two classes and a machine set up is necessary between jobs of different classes. Jobs i (i= 1,…, n) becomes available for processing at time zero, requires a positive processing time . Disjoint subsets N1 and N2 define the partition of jobs into two classes. If two jobs in the same class are sequenced in adjacent positions, then no set up time between these jobs in necessary. We address the bicriterion (multi objective) scheduling problem, the two criteria are the minimization of flow time ( ) and the minimization maximum Tardiness ( ). We characterized the set of all efficient points and the optimal solution. A modified algorithm presented to find efficient solutions for the problem with set up times. A relation found between number of efficient solutions and range of ‘tardiness of shortest processing time ( ), tardiness of early due date ( )’. This algorithm treats with a case that the set up time in rule is in increasing order. A counter example presented to show that the algorithm will fail if the set up time in rule is in decreasing order. Our task is to present the decision makers with all possible solutions and let them make the final selection. The decision maker has two objectives in mind ( ) , ( ) and some solutions (efficient), we will choose the best one from the efficient solutions depending on his experiences.

On the Gittins index in the M/G/1 queue

For an M/G/1 queue with the objective of minimizing the mean number of jobs in the system, the Gittins index rule is known to be optimal among the set of non-anticipating policies. We develop properties of the Gittins index. For a single-class queue it is known that when the service time distribution is of type Decreasing Hazard Rate (New Better than Used in Expectation), the Foreground---Background (First-Come-First-Served) discipline is optimal. By utilizing the Gittins index approach, we show that in fact, Foreground---Background and First-Come-First-Served are optimal if and only if the service time distribution is of type Decreasing Hazard Rate and New Better than Used in Expectation, respectively. For the multi-class case, where jobs of different classes have different service distributions, we obtain new results that characterize the optimal policy under various assumptions on the service time distributions. We also investigate distributions whose hazard rate and mean residual lifetime are not monotonic.

A heuristic approach for tow-machine no-wait flowshop scheduling with due dates and class setups

This paper studies the two-machine flowshop scheduling problem with class setups in a no-wait processing environment to minimize the maximum lateness. The jobs are divided into job classes and a setup is required at the initial processing of a job or between the processing of jobs of different classes. In a no-wait environment, a job must be processed from start to finish without interruptions on a machine or between the machines. A batch is a maximal subset of consecutively processed jobs of the same class. Several properties concerning the beneficial merging of batches and some dominance rules that improve the objective function are derived. Since the problem is NP-hard, a heuristic is proposed and evaluated computationally. The numerical results demonstrate that the heuristic can produce near-optimal solutions quickly for realistic-sized problems.

A heuristic approach for single-machine scheduling with due dates and class setups

The single-machine sequence-independent class setup scheduling problem is examined in this paper. It is assumed that jobs are classified into classes and a setup is required between jobs of different classes, but not of the same class. Furthermore, this setup time is fixed and depends only on the current job. Since the problem is NP-hard, a heuristic algorithm is proposed to find an approximate schedule that minimizes the maximum lateness on a set of jobs. The algorithm can easily be modified to solve the maximum tardiness problems as well. The accuracy of the heuristic algorithm in generating near optimal solutions is empirically evaluated. Scope and purposeFor batch manufacturing, it maybe desirable to produce many items of the same type, or class, at the same run in order to save the setup cost. However, committing facilities to long production runs for one product may inevitably make others tardy. Small batch size may conform urgent jobs to their delivery date, but one of the consequences would be the loss of productive efficiency due to numerous setups. Therefore, scheduling is basically a trade-off between the inherently conflicting efficiency measure and due-date compliance. This paper considers a single-machine scheduling problem in which jobs are classified into classes and a setup is required between jobs of different classes. The setup time is fixed and depends only on the current job. This problem is called a sequence-independent class setup problem and is NP-complete.

SINGLE MACHINE SCHEDULING WITH DUE DATES AND CLASS SETUPS

ABSTRACT Standard scheduling problems bear the assumption that each job has a constant setup time and thus this setup time can be included in the processing time of a job. However, many realistic production systems suggest otherwise. This paper considers a single machine scheduling problem where jobs are classified into classes and a setup is required between jobs of different classes, but not between jobs of the same class. This class setup time depends only on the current job, not on the previous job. The performance measure is the maximum lateness of jobs. Several properties of the optimal solution and a lower bound of the optimal lateness are developed to construct a branch-and-bound algorithm in the search of an optimal schedule and the effectiveness of the proposed algorithm is empirically evaluated.

On First-Come First-Served Versus Random Service Discipline in Multiclass Closed Queueing Networks

We consider multiclass closed queueing networks. For these networks, a lot of work has been devoted to characterizing and weakening the conditions under which a product-form solution is obtained for the steady-state distribution. From this work, it is known that, under certain conditions, all networks in which each of the stations has either the first-come first-served or the random service discipline lead to the same (product-form expressions for the) steady-state probabilities of the (aggregated) states that for each station and each job class denote the number of jobs in service and the number of jobs in the queue. As a consequence, all these situations also lead to the same throughputs for the different job classes. One of the conditions under which these equivalence results hold states that at each station all job classes must have the same exponential service time distribution. In this paper, it is shown that these equivalence results can be extended to the case with different exponential service times for jobs of different classes, if the network consists of only one single-server or multiserver station. This extension can be made despite of the fact that the network is not a product-form network anymore in that case. The proof is based on the reversibility of the Markov process that is obtained under the random service discipline. By means of a counterexample, it is shown that the extension cannot be made for closed network with two or more stations.

Asymptotic optimality of statistical multiplexing in pipelined processing

The throughput of pipelined processing ofheterogeneous multitasked jobs is computed and optimized in this study. There areK job classes. Each job hasM tasks which have to be processed in a given order (same for all tasks) on a pipeline ofM processors. Tasks have random processing times. The jobs of each class form a stationary and ergodic sequence (with respect to their task processing times). Classes are differentiated by distinct statistics and may not be jointly stationary or ergodic. Thus, the jobs are overall statistically heterogeneous. We are interested in the average execution time per job\(\bar \tau \), when the job populations of the various classes become very large (asymptotically). This is shown to depend on the order in which jobs enter the pipeline. Under the natural class-based ordering, where all jobs of the first class enter first, followed by those of the second, third, and so on, the quantity\(\bar \tau \) is computed, but is shownnot to attain its minimal value in general. On the contrary, appropriate statistical multiplexing of jobs of different classes on the pipeline is shown to minimize the average execution time per job on every sample path (with probability one). The procedure, calledbalanced statistical multiplexing, is constructed and the minimal\(\bar \tau \) is computed in terms of the average execution times of the job tasks.

Optimal scheduling in a machine with stochastic varying processing rate

Addresses the problem of allocating the capacity of a machine among jobs of different classes when the machine processing rate varies stochastically over time. The authors establish that the policy that always allocates the maximum available processing rate to the class having the maximum weight minimizes, pathwise, a weighted sum of the remaining service requirements of the different classes, at any point in time. This result is based on the application of elementary forward induction arguments and holds over the class of all policies (e.g., including randomized policies). As an easy corollary of this result the authors generalize work by Hirayama and Kijima (1992) on the optimality of the /spl mu/c-rule in a multiclass G/M/1 queueing system in which the server processing rate varies stochastically with time. To the best of the authors' knowledge, their proof is the first one in this context that only uses direct pathwise arguments. >

Scheduling two job classes on a single machine

The problem of scheduling n jobs on a single machine is considered, where the jobs are partitioned into two classes and a set-up time is necessary between jobs of different classes. Gupta proposes an O( n log n) algorithm which, he claims, minimizes the sum of completion times. However, we present a counter-example which shows it can fail to generate an optimal schedule. Nevertheless, this problem is solvable in O( n 2) time by a well-known dynamic programming algorithm. An O( n 3) dynamic programming algorithm to minimize the sum of weighted completion times is also derived.

Efficient approximation for models of multiprogramming with shared domains

Queueing network models of multiprogramming systems with memory constraints and multiple classes of jobs are important in representing large commercial computer systems. Typically, an exact analytical solution of such models is unavailable, and, given the size of their state space, the solution of models of this type is approached through simulation and/or approximation techniques. Recently, a computationally efficient iterative technique has been proposed by Brandwajn, Lazowska and Zahorjan for models of systems in which each job is subject to a separate memory constraint, i.e., has its own memory domain. In some important applications, it is not unusual, however, to have several jobs of different classes share a single memory “domain” (e.g., IBM's Information Management System). We present a simple approximate solution to the shared domain problem. The approach is inspired by the recently proposed technique which is complemented by a few approximations to preserve the conceptual simplicity and computational efficiency of this technique. The accuracy of the results is generally in fair agreement with simulation.

Complete parameterized families of job scheduling strategies

The concept of a family of scheduling strategies in which a few parameters may be varied to achieve different performance levels is introduced. The use of such families in satisfying performance requirements stated in terms of average response times for jobs of different classes is studied. A performance requirement is said to be achievable if, given the loading conditions on the system, there exists a scheduling strategy which satisfies it. A family of scheduling strategies is said to be complete if every achievable performance requirement can be satisfied by a strategy from the family. Sufficient conditions for a parameterized family to be complete are proven. Three parameterized families are discussed, one in detail. Completeness of the three families is demonstrated and simulation results illustrating some properties of implementation are presented.

The Problems of Tardiness and Saturation in a Multi-Class Queue with Sequence-Dependent Setups

Abstract This paper deals with scheduling jobs of different classes on a facility with sequence-dependent setups between classes. Tardiness measures are of primary concern, and changeovers must be limited in order to avoid saturation. A method is given for classifying scheduling rules in this environment. Procedures are developed for avoiding saturation. A digital simulation model has been used to investigate the tardiness performance of various rules, and experimental results are given.