On M/G/1 system under NT policies with breakdowns, startup and closedown

AbstractThis paper studies the vacation policies of an M/G/1 queueing system with server breakdowns, startup and closedown times, in which the length of the vacation period is controlled either by the number of arrivals during the vacation period, or by a timer. After all the customers are served in the queue exhaustively, the server is shutdown (deactivates) by a closedown time. At the end of the shutdown time, the server immediately takes a vacation and operates two different policies: (i) The server reactivates as soon as the number of arrivals in the queue reaches to a predetermined threshold N or the waiting time of the leading customer reaches T units; and (ii) The server reactivates as soon as the number of arrivals in the queue reaches to a predetermined threshold N or T time units have elapsed since the end of the closedown time. If the timer expires or the number of arrivals exceeds the threshold N, then the server reactivates and requires a startup time before providing the service until the system is empty. If some customers arrive during this closedown time, the service is immediately started without leaving for a vacation and without a startup time. We analyze the system characteristics for each scheme

Analysis of operating characteristics for the heterogeneous batch arrival queue with server startup and breakdowns

In this paper we consider a like-queue production system in which server startup and breakdowns are possible. The server is turned on (i.e. begins startup) when N units are accumulated in the system and off when the system is empty. We model this system by an M[x]/M/1 queue with server breakdowns and startup time under the N policy. The arrival rate varies according to the server's status: off, startup, busy, or breakdown. While the server is working, he is subject to breakdowns according to a Poisson process. When the server breaks down, he requires repair at a repair facility, where the repair time follows the negative exponential distribution. We study the steady-state behaviour of the system size distribution at stationary point of time as well as the queue size distribution at departure point of time and obtain some useful results. The total expected cost function per unit time is developed to determine the optimal operating policy at a minimum cost. This paper provides the minimum expected cost and the optimal operating policy based on assumed numerical values of the system parameters. Sensitivity analysis is also provided

Association between metabolic body composition status and vitamin D deficiency: A cross-sectional study

This study aimed to investigate the risk of vitamin D deficiency in a relatively healthy Asian population, with (i) metabolically healthy normal weight (MHNW) (homeostasis model assessment-insulin resistance [HOMA-IR] < 2. 5 without metabolic syndrome [MS], body mass index [BMI] < 25), (ii) metabolically healthy obesity (MHO) (HOMA-IR < 2.5, without MS, BMI ≥ 25), (iii) metabolically unhealthy normal weight (MUNW) (HOMA-IR ≥ 2.5, or with MS, BMI < 25), and (iv) metabolically unhealthy obesity (MUO) (HOMA-IR ≥ 2.5, or with MS, BMI ≥ 25) stratified by age and sex. This cross-sectional study involved 6,655 participants aged ≥ 18 years who underwent health checkups between 2013 and 2016 at the Chang Gung Memorial Hospital. Cardiometabolic and inflammatory markers including anthropometric variables, glycemic indices, lipid profiles, high-sensitivity C-reactive protein (hs-CRP), and serum 25-hydroxy vitamin D levels, were retrospectively investigated. Compared to the MHNW group, the MHO group showed a higher odds ratio (OR) [1.35, 95% confidence interval (CI) 1.05–1.73] for vitamin D deficiency in men aged < 50 years. By contrast, in men aged > 50 years, the risk of vitamin D deficiency was higher in the MUO group (OR 1.44, 95% CI 1.05–1.97). Among women aged < and ≥ 50 years, the MUO group demonstrated the highest risk for vitamin D deficiency, OR 2.33 vs. 1.54, respectively. Our study revealed that in women of all ages and men aged > 50 years, MUO is associated with vitamin D deficiency and elevated levels of metabolic biomarkers. Among men aged < 50 years, MHO had the highest OR for vitamin D deficiency

The optimal control in batch arrival queue with server vacations, startup and breakdowns

This paper studies the N policy M[x]/G/1 queue with server vacations; startup and breakdowns, where arrivals form a compound Poisson process and service times are generally distributed. The server is turned off and takes a vacation whenever the system is empty. If the number of customers waiting in the system at the instant of a vacation completion is less than N, the server will take another vacation. If the server returns from a vacation and finds at least N customers in the system, he is immediately turned on and requires a startup time before providing the service until the system is empty again. It is assumed that the server breaks down according to a Poisson process whose repair time has a general distribution. The system characteristics of such a model are analyzed and the total expected cost function per unit time is developed to determine the optimal threshold of N at a minimum cost

Notes on the Computation of Laplace-Stieljes Transform for Lognormal and Weibull

In this paper, we perform the computation of the Laplace-Stieljes transform of Lognormal and Weibull distribution, in which the upper limit of the definite integral from infinite to 1 by using proper transformation. Some advantages are found in this study

A Batch-Arrival Queue with Multiple Servers and Fuzzy Parameters: Parametric Programming Approach

This work constructs the membership functions of the system characteristics of a batch-arrival queuing system with multiple servers, in which the batch-arrival rate and customer service rate are all fuzzy numbers. The α-cut approach is used to transform a fuzzy queue into a family of conventional crisp queues in this context. By means of the membership functions of the system characteristics, a set of parametric nonlinear programs is developed to describe the family of crisp batch-arrival queues with multiple servers. A numerical example is solved successfully to illustrate the validity of the proposed approach. Because the system characteristics are expressed and governed by the membership functions, the fuzzy batch-arrival queues with multiple servers are represented more accurately and the analytic results are more useful for system designers and practitioners

CONTROLLING ARRIVALS FOR A MARKOVIAN QUEUEING SYSTEM WITH A SECOND OPTIONAL SERVICE

This paper considers the optimal management problem of a finite capacity M/M/1 queueing system with F-policy in which some customers may demand a second service in addition to the first essential service. The F-policy investigates the most common issue of controlling arrival to a queueing system and it is required a startup time before starting to allow customers entering into the system. This system has potential applications in the wireless communication networks, the transport service and production system. By applying the birth and death process, some important performance measures are derived. A cost model, developed to determine the optimal control F-policy at a minimum cost, and sensitivity analysis are also studied

Computation Approaches for Parameter Estimation of Weibull Distribution

This paper examines the estimation comparison of two methods for Weibull parameters, one is the maximum likelihood method and the other is the least squares method. A numerical simulation study is carried out to understand performance of the two methods. Based on sample root mean square errors, we make a comparison between the two computation approaches. We find that the least squares method significantly outperforms the maximum likelihood when the sample size is small

Optimal Control of the M/G/1 and G/M/1 Queueing Systems with a Removable Server

本文旨在研究含有一可移動服務者M/G/1與G/M/1排隊系統之最佳控制，其中控制決策為N-方策。所謂N-方策，它的含意是指在系統中，如果要求服務之顧客數目累積到N個時，則服務站立即被開啟並開始提供服務；而當系統中所有顧客都被服務完時，則服務站立即被關閉，等系統中的顧客數目累積到N個時再被開啟。 首先我們利用輔助變數技巧(supplementary variable technique)和機率生成函數技巧(probability generating function technique)，探討在N-方策下兩個具無限容量的M/G/1和G/M/1排隊系統，分別獲得這兩個排隊系統正確的穩定特性結果，諸如顧客數在系統中之機率分配、顧客數之期望數、顧客在隊伍中之等候時間分配、服務者忙碌與閒置週期的期望長度等。我們也利用極大熵原理(maximum entropy principle)對於這兩個排隊系統分別建立其近似之穩定特性結果，同時藉由某些在這兩個排隊系統已存在之正確結果及其相對的近似結果實施其數值比較分析。 其次，我們研究在N-方策下兩個具有限容量的M/G/1和G/M/1排隊系統，利用輔助變數技巧(supplementary variable technique)分別為這兩個排隊系統發展一有效率之遞迴方法來計算顧客數在系統中之機率分配，藉由電腦程式求得系統穩定特性數值結果，並分別對兩個排隊系統之不同服務(或到達)時間分配實施其穩定特性結果之數值比較分析。 最後，對以上四個N-方策排隊系統，我們分別定義其單位時間之穩態期望成本，藉由此成本決定控制參數N的最佳解N*，使成本函數達到最小。 關鍵字：比較分析，G/M/1排隊系統，M/G/1排隊系統，極大熵原理，可移動服務者，遞迴方法，輔助變數技巧。In this dissertation, we deal with the optimal control of a single removable server in M/G/1 and G/M/1 queueing systems operating under the N policy in which the server may be turned on at arrival epochs or off at service completion epochs. The server begins service only when the number of customers in the system reaches a certain number, say $N$ $(N \ge 1)$. The supplementary variable technique and the probability generating function technique are used to develop the exact steady-state results for the N policy M/G/1 and G/M/1 queueing systems with infinite capacity. Examples are presented to calculate the steady-state probability distribution of the number of customers in the N policy M/G/1 queueing system for three different service time distributions, including exponential, 2-stage Erlang and 2-state hyperexponential distributions. We provide two special cases in the N policy G/M/1 queueing system, such as the ordinary G/M/1 queueing system and the N policy M/M/1 queueing system. We use the maximum entropy principle to develop the approximate steady-state results for the N policy M/G/1 and G/M/1 queueing systems with infinite capacity. We perform comparative analysis between some exact results and the corresponding approximate results in the N policy M/G/1 queueing system for two different service time distributions, such as exponential and 3-stage Erlang distributions. We also provide comparative analysis between some exact results and the corresponding approximate results in the N policy G/M/1 queueing system for the exponential interarrival time distribution. We study the N policy M/G/1 and G/M/1 queueing systems with finite capacity $L$. We provide a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining service (or nterarrival) time, to establish the steady-state probability distributions of the number of customers in two finite queueing systems. To illustrate analytically for the two recursive methods, we present examples of different service time distributions, such as exponential, 3-stage Erlang and deterministic distributions, in the N policy M/G/1 queueing system and exponential interarrival time distribution in the N policy G/M/1 queueing system. We provide the numerical results of system characteristics for different service (or interarrival) time distributions in the N policy M/G/1 and G/M/1 queueing systems, including exponential, 2-stage hyperexponential, 4-stage Erlang and deterministic time distributions.CONTENTS Page ABSTRACT i CONTENTS iii Chapter 1 Introduction 1 1.1 Background 1 1.2 Theoretical analysis techniques 3 1.2.1 Supplementary variable technique 4 1.2.2 Maximum entropy technique 6 1.3 Literature review8 1.2.1 Infinite source queueing system 9 1.2.2 Finite source queueing system 10 1.4 Problem statement 11 1.5 Scope of dissertation 12 Chapter 2 The N Policy M/G/1 Queueing System with Infinite Capacity 14 2.1 Assumptions and notations 15 2.2 Development of the equations and solutions 16 2.2.1 Partial differential equations 17 2.2.2 Steady-state results 18 2.2.3 Probability generating function 20 2.2.4 Simple examples 21 2.3 System characteristics 24 2.3.1 Expected number of customers in the system 25 2.3.2 Distribution of the waiting time in the queue 26 2.3.3 Expected length of the idle period and the busy period 27 2.4 Maximum entropy solutions 28 2.4.1 The constraints of the N policy M/G/1 queueing system 29 2.4.2 Maximum entropy model with uniform prior 29 2.4.3 Maximum entropy model with like-geometric prior 30 2.4.4 Comparative analysis 31 Chapter 3 The N Policy M/G/1 Queueing System with Finite Capacity 36 3.1 Assumptions and notations 37 3.2 Development of the equations and solutions 38 3.2.1 Partial differential equations 39 3.2.2 Steady-state results 40 3.2.3 Recursive method 41 3.3 The solution algorithm 43 3.3.1 Simple examples 44 3.4 System characteristics 50 3.4.1 Expected number of customers in the system 51 3.4.2 Distribution of the waiting time in the queue 52 3.4.3 Expected length of the idle period and the busy period 53 3.5 Numerical results 54 Chapter 4 The N Policy G/M/1 Queueing System with Infinite Capacity 59 4.1 Assumptions and notations 60 4.2 Development of the equations and solutions 62 4.2.1 Partial differential equations 62 4.2.2 Steady-state results 63 4.2.3 Probability generating function 66 4.3 System characteristics 68 4.3.1 Expected number of customers in the system 68 4.3.2 Expected waiting time in the queue 69 4.3.3 Expected length of the idle period and the busy period 70 4.4 Special cases 71 4.4.1 The ordinary G/M/1 queueing system 71 4.4.2 The N policy M/M/1 queueing system 75 4.5 Maximum entropy solutions 77 4.5.1 The constraints of the N policy G/M/1 queueing system 77 4.5.2 Maximum entropy model with uniform prior 78 4.5.3 Maximum entropy model with like-geometric prior 79 4.5.4 Comparative analysis 81 Chapter 5 The N Policy G/M/1 Queueing System with Finite Capacity 86 5.1 Assumptions and notations 86 5.2 Development of the equations and solutions 88 5.2.1 Partial differential equations 89 5.2.2 Steady-state results 90 5.2.3 Recursive method 93 5.3 The solution algorithm 96 5.3.1 Simple example 97 5.4 System characteristics 100 5.4.1 Expected number of customers in the system 102 5.4.2 Distribution of the waiting time in the queue103 5.4.3 Expected length of the idle period and the busy period 104 5.5 Numerical results 105 Chapter 6 The Optimal operating Policy 112 6.1 The optimal N policy M/G/1 queueing system with infinite capacity 113 6.1.1 The total expected cost function 113 6.1.2 Determining the optimal N policy 113 6.2 The optimal N policy M/G/1 queueing system with finite capacity 114 6.2.1 The total expected cost function 114 6.2.2 Determining the optimal N policy 114 6.3 The optimal N policy G/M/1 queueing system with infinite capacity 115 6.3.1 The total expected cost function 115 6.3.2 Determining the optimal N policy 115 6.4 The optimal N policy G/M/1 queueing system with finite capacity 116 6.4.1 The total expected cost function 116 6.4.2 Determining the optimal N policy 116 Chapter 7 Conclusions and Recommendations 117 7.1 Conclusions 117 7.2 Recommendations for further researches 120 References 122 Appendix 127 Curriculum vitae 13