Processing Time Distributions Research Articles

Diffusion approximations for large-scale buffered systems

Resource pooling is becoming increasingly common in modern applications of stochastic systems, such as in computer systems, wireless networks, workforce management, call centers, and health care delivery. At the same time, these applications give rise to systems which continue to grow in size. For instance, a traditional web server farm only has a few servers, while cloud data centers have thousands of processors. These two trends pose significant practical restrictions on admission, routing, and scheduling decision rules or algorithms. Scalability and computability are becoming ever more important characteristics of decision rules, and consequently simple decision rules with good performance are of particular interest. An example is the so-called least connection rule implemented in many load balancers in computer clouds, which assigns a task to the server with the least number of active connections; cf. the join-the-shortestqueue routing policy. From a design point of view, the search for desirable algorithmic features often presents trade-offs between system performance, information/communication, and required computational effort. In this paper, we study the trade-off between performance and computational effort in a stylized model of a system with a central server and large number of parallel buffers. We focus on randomized versions of the longest-queue-first scheduling policy. In this scheduling algorithm, the server works on a task from the buffer with the longest queue length among several sampled buffers; it approximates the longestqueue-first scheduling policy, which can be computationally prohibitive. It is our aim to quantify the system performance as a function of the computational effort expended on sampling. In our model, each buffer is fed with an independent stream of tasks, which arrive according to a Poisson process. All n buffers are connected to a single centralized server. Under the randomized longest-queue-first policy, this server selects d(n) buffers uniformly at random (with replacement) and processes a task from the longest queue among the selected buffers; it idles for a random amount of time if all buffers in the sample are empty. Tasks have random processing time requirements. The total processing capacity scales linearly with n and the processing time distribution is independent of n. We work in an underloaded regime, with enough processing capacity to eventually serve all arriving tasks. Note that this scheduling algorithm is agnostic in the

116 歩留まりと作業時間のばらつきを考慮した作業者の工作機械への配置とスケジューリング(OS4 : 生産スケジューリング・生産管理(4))

A systematic method has been proposed, in previous paper, to allocate human operators to machine tools and to determine machining schedules considering statistical distributions of machining times. It was assumed that the processing times executed by the human operators are represented by random variables according to normal distributions, in the previous paper. However, normal distribution cannot express distributions of machining time enough e.g. case that probabilities of delay are larger than that of earliness. A new method is proposed, in this paper, to assign human operators to machine tools and to determine machining schedules considering statistical distributions of processing times under the assumption that the processing times executed by the human operators are represented by random variables according to beta distributions.

Coordination of manufacturing, remanufacturing and returns acceptance in hybrid manufacturing/remanufacturing systems

This paper deals with the coordination of manufacturing, remanufacturing and returns acceptance control in a hybrid production-inventory system. We use a queuing control framework, where manufacturing and remanufacturing are modelled by single servers with exponentially distributed processing times. Customer demand and returned products arrive in the system according to independent Poisson processes. A returned product can be either accepted or rejected. When accepted, a return is placed in a remanufacturable product inventory. Customer demand can be satisfied as well by new and remanufactured products. The following costs are included: stock keeping, backorder, manufacturing, remanufacturing, acceptance and rejection costs. We show that the optimal policy is characterized by two state-dependent base-stock thresholds for manufacturing and remanufacturing and one state-dependent return acceptance threshold. We also derive monotonicity results for these thresholds. Based on these theoretical results, we introduce several relevant heuristic control rules for manufacturing, remanufacturing and returns acceptance. In an extensive numerical study we compare these policies with the optimal policy and provide several insights.

Performance analysis of large-scale parallel-distributed processing with backup tasks for cloud computing

In cloud computing, a large-scale parallel-distributed processing service is provided where a huge task is split into a number of subtasks and those subtasks are processed on a cluster of machines called workers. In such a processing service, a worker which takes a long time for processing a subtask makes the response time long (the issue of stragglers). One of efficient methods to alleviate this issue is to execute the same subtask by another worker in preparation for the slow worker (backup tasks). In this paper, we consider the efficiency of backup tasks. We model the task-scheduling server as a single-server queue, in which the server consists of a number of workers. When a task enters the server, the task is split into subtasks, and each subtask is served by its own worker and an alternative distinct worker. In this processing, we explicitly derive task processing time distributions for the two cases that the subtask processing time of a worker obeys Weibull or Pareto distribution. We compare the mean response time and the total processing time under backup-task scheduling with those under normal scheduling. Numerical examples show that the efficiency of backup-task scheduling significantly depends on workers' processing time distribution.

The expected asymptotical ratio for preemptive stochastic online problem

This paper considers the preemptive stochastic online scheduling problem on m uniform parallel machines with the objective to minimize the total weighted completion times, where the processing times of jobs are independent random variables. The actual processing time of each job is only learned upon its completion. In online environment, the existence and all parameters of each job, including its weight, release date and the distribution of processing time are unknown until its release date. During the processing of each job, it could be preempted and resumed later without extra cost. For this problem, a scheduling decision has to be made each time as the information of the processing time of the job is released step by step. Building on the Gittins index, we devise the Semi-Preemptive Policy for Uniform Machine Problem (SP) policy. With the assumption that all weights and processing times are bounded, we prove the expected asymptotical ratio of SP is one, which means E[∑wjCjOPT] approaches to one as the number of jobs increases to infinity, where Cj denotes the completion time of job Jj and OPT the offline optimum value.

Designing unpaced production lines to optimize throughput and work-in-process inventory

This article considers the optimal design of unpaced assembly lines. Two key decisions in designing an unpaced assembly line are the allocation of work to the stations and the allocation of buffer storage space between the stations. To the best of the author's knowledge, this is the first article to jointly optimize both the allocation of workload and the allocation of buffer spaces simultaneously when the objective is to maximize the revenue from throughput minus the cost of work-in-process inventory. Exact solutions are provided for small lines (three or four stations) with a fixed kind of processing time distribution (exponential or Erlang). Ten observations are made about the characteristics of the allocation of workload and buffer spaces. Heuristics are suggested for designing lines with more stations or different processing time distributions. A simulation study is done to test the observations and heuristics for longer lines and different processing time distributions (lognormal). Significant savings can be achieved by jointly optimizing both the workload and the buffer space allocations.

Performance analysis of closed-loop assembly lines with general processing times and finite buffer spaces

This article analyzes flow lines with converging and diverging material flow, limited buffer sizes, generally distributed processing times, and a constant number of workpieces as a closed assembly or disassembly queueing network. A decomposition approach in which each subsystem is modeled as a G/G/1/K queueing system is used. The population constraint is enforced by requiring that the sum of the expected number of customers in the subsystems is equal to the total number of workpieces. The results of a simulation experiment indicate that the proposed approximation provides accurate results and that it performs better than other approaches.

A polynomial time heuristic for the two-machine flowshop scheduling problem with setup times and random processing times

The two-machine flowshop scheduling problem to minimize makespan is addressed. Jobs have random processing times which are bounded within certain intervals. The distributions of job processing times are not known. This problem has been addressed in the literature with the assumption that setup times are included in processing times or are zero. In this paper, we relax this assumption and treat setup times as separate from processing times. We propose a polynomial time heuristic algorithm. Both Johnson algorithm and Yoshida and Hitomi algorithm, both of which developed for the deterministic problem, are special cases of the proposed algorithm. The heuristic algorithm uses a weighted average of lower and upper bounds for processing times. For different weights, the results of the proposed algorithm are compared based on randomly generated data. The computational analysis has shown that the proposed algorithm, with equal weights given to the lower and upper bounds, performs considerably well with an overall average error of 0.36%. The analysis has also shown that the proposed algorithm can safely be used regardless of processing time distributions and the range between lower and upper bounds.

Characteristics of Flameless Combustion in 3D Highly Porous Reactors under Diesel Injection Conditions

The heat release process in a free volume combustion chamber and in porous reactors has been analyzed under Diesel engine-like conditions. The process has been investigated in a wide range of initial pressures and temperatures simulating engine conditions at the moment when fuel injection starts. The resulting pressure history in both porous reactors and in free volumes significantly depends on the initial pressure and temperature. At lower initial temperatures, the process in porous reactors is accelerated. Combustion in a porous reactor is characterized by heat accumulation in the solid phase of the porous structure and results in reduced pressure peaks and lowered combustion temperature. This depends on reactor heat capacity, pore density, specific surface area, pore structure, and heat transport properties. Characteristic modes of a heat release process in a two-dimensional field of initial pressure and temperature have been selected. There are three characteristic regions represented by a single- and multistep oxidation process (with two or three slopes in the reaction curve) and characteristic delay time distribution has been selected in five characteristic ranges. There is a clear qualitative similarity of characteristic modes of the heat release process in a free volume and in porous reactors. A quantitative influence of porous reactor features (heat capacity, pore density, pore structure, specific surface area, and fuel distribution in the reactor volume) has been clearly indicated.

S142012 加工時間の確率分布を考慮した工作機械への作業者配置とスケジューリング([S142]生産システムの新展開(応用・実践))

S142012 加工時間の確率分布を考慮した工作機械への作業者配置とスケジューリング([S142]生産システムの新展開(応用・実践))

Computation of the steady state distribution for multi-server retrial queues with phase type service process

We consider a multi-server retrial queueing system with the Batch Markovian Arrival Process and phase type service time distribution. Such a general queueing system suits for modeling and decision making in many real life objects including modern wireless communication networks. Behavior of such a system is described by the level dependent multi-dimensional Markov chain. Blocks of the generator of this chain, which is the block structured matrix of infinite size, can have large size if the number of servers is large and distribution of service time is not exponential. Due to this fact, the existing in literature algorithms allow to compute key performance measures of such a system only for a small number of servers. Here we describe the algorithm that allows to compute the stationary distribution of the system for larger number of servers and numerically illustrate its advantage. Importance of taking into account correlation in the arrival process is numerically demonstrated.

Stochastic single machine scheduling with random common due date

This paper studies the single machine scheduling problem for minimising the expected total weighted deviations of completion times from random common due date. Jobs have exponentially distributed processing times and the common due date is a generalised Erlang distribution. The optimal schedules are shown to be ∧-shaped. Moreover, we give the optimal schedules when the machine is subject to stochastic breakdowns with independent and exponentially distributed uptimes and downtimes.

Process time distribution of driven polymer transport

We discuss the temporal distribution of dynamic processes in driven polymer transport inherent to flexible chains due to stochastic tension propagation. The stochasticity originates from the disordered initial configuration of an equilibrium polymer coil, which results in random paths for tension propagation. We consider the process time for when translocation occurs across a fixed pore and when stretching occurs by pulling the chain end. A scaling argument for the mean and standard deviation of the process time is provided using the two-phase picture for stochastic propagation. The two cases are found to differ remarkably. The process time distribution of the translocation exhibits substantial spreading even in the long-chain limit, unlike that found for the dynamics of polymer stretching. In addition, the process time distribution in the driven translocation is shown to have a characteristic asymmetric shape.

Time-varying priority queuing models for human dynamics

Queuing models provide insight into the temporal inhomogeneity of human dynamics, characterized by the broad distribution of waiting times of individuals performing tasks. We theoretically study the queuing model of an agent trying to execute a task of interest, the priority of which may vary with time due to the agent's "state of mind." However, its execution is disrupted by other tasks of random priorities. By considering the priority of the task of interest either decreasing or increasing algebraically in time, we analytically obtain and numerically confirm the bimodal and unimodal waiting time distributions with power-law decaying tails, respectively. These results are also compared to the updating time distribution of papers in arXiv.org and the processing time distribution of papers in Physical Review journals. Our analysis helps to understand human task execution in a more realistic scenario.

Generalized survivability analysis of systems with propagated failures

The paper presents a generalized model for quantifying and evaluating the survivability of systems and the services provided by the systems. For this purpose, we consider a multi-server system with infinite buffer, Markovian Arrival Process (MAP) and phase type (PH) service time distribution. The system is subject to the so-called propagated breakdowns. Accordingly, breakdowns arrive in bunches (we call them attacks) according to the MAP. Attacks consist of a random number of failures of different types defined by the required repair time. The process of arrival of different types of server failures within an attack is governed by phase type Markov process. The repair of a server takes an exponentially distributed time with intensity depending on the type of the occurred failure. We analyze the survivability of the system in terms of average time required for its complete recovery after an attack completion conditional no new attack arrives. In this paper, we consider two forms of recovery following the end of failure arrivals based on (1) the length of queue or (2) the number of broken servers reaching a preassigned level. To provide analysis of survivability, we first describe dynamics of the system by the multi-dimensional continuous time Markov chain. Then, we analyze survivability by means of matrix extension of so called method of collective marks. Numerical illustrations are also presented.

Development of Scheduling System Considering Statistical Distributions of Processing Times for Human Operators

Development of Scheduling System Considering Statistical Distributions of Processing Times for Human Operators

A Logistic Model for Solution Space of Job Shop Scheduling Problem

Lots of researchers have been attracted to the mystery of the solution space of the scheduling problem. In this paper, we introduce the Logistic Model, a core theory in population ecology, into job shop scheduling problem, surprised to find that the cumulative distribution of makespan has a perfect match with the S-curve. And what’ more, this neglected distribution property can truly and stably reflect the structure of the Fitness Landscape, whether it is a static randomly generated space, or a dynamic algorithm search space. Finally, we use the inflexion point of the S-curve as the characteristic number to study the difficulty level of the Job Shop Scheduling Problem, the processing path, the distribution of processing time, and the shape of the problem respectively. Experimental results show that compared with the previous methods, the Logistic Model can well reflect the intrinsic properties, and nicely distinguish the difficulty level of the problem. So, this is a new simple and accurate way to describe the characteristics of the problem, and to guide the design and evaluation of algorithms.

Asymptotical optimality of WSEPT for stochastic online scheduling on uniform machines

We study the stochastic online scheduling on m uniform machines with the objective to minimize the expected value of total weighted completion times of a set of jobs that arrive over time. For each job, the processing time is a random variable, and the distribution of processing time is unknown in advance. The actual processing time could be known only when the job is completed. For the problem, we propose a policy which is proved to be asymptotically optimal when the processing times and weights are uniformly bounded, i.e. the relative error of the solution achieved by our policy approaches zero as the number of jobs increases to infinity.

Analysis of passenger queues at airport terminals

This paper presents an approach for analyzing passenger queueing processes at several stages of the service chain in airport terminals with a focus on passenger check-in systems. The queueing process at the check-in is driven by the arrival process of the passengers and the service process, including the desired technical and personnel resources. The key features of these systems are the time-dependent process of arrivals, generally distributed processing times, and systems with multiple servers. Time dependent service quality measures are analyzed, which are related to managerial decisions about the required number of check-in operators. Different analytical methods are reviewed to derive service quality measures for such dynamic and stochastic systems. Due to the dynamics in the demand and in the provided capacity, the use of standard queueing models is rather limited. A stationary backlog carryover (SBC) approach is developed to approximate the relevant performance measures. Numerical experiments demonstrate the reliability of these different approaches. Based on economies of scale, positive effects for common check-in operations can be shown.

Predicting Cycle Time Distributions for Integrated Processing Workstations: An Aggregate Modeling Approach

To predict cycle time distributions of integrated processing workstations, detailed simulation models are almost exclusively used; these models require considerable development and maintenance effort. As an alternative, we propose an aggregate model that is a lumped-parameter representation of the workstation. The aggregate model is a single server with a work-in-process dependent aggregate process time distribution and overtaking distribution. The lumped parameters are determined directly from arrival and departure events measured at the workstation. An extensive simulation study and an industry case demonstrate that the aggregate model can accurately predict the cycle time distribution of integrated processing workstations in semiconductor manufacturing.